An exploration of mathematics students’ distinguishing between function and arbitrary relation.

ISSN 1463-6840 Proceedings of the British Society for Research into Learning Mathematics Volume 29 Number 3 Proceedings of the Day Conference held at The Loughborough University, Saturday 14th November 2009 These proceedings consist of short research reports which were written for the BSRLM day conference on 14 November 2009. The aim of the proceedings is to communicate to the research community the collective research represented at BSRLM conferences, as quickly as possible. We hope that members will use the proceedings to give feedback to the authors and that through discussion and debate we will develop an energetic and critical research community. We particularly welcome presentations and papers from new researchers. Published by the British Society for Research into Learning Mathematics. Individual papers © contributing authors 2009 Other materials © BSRLM 2009 All rights reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage retrieval system, without prior permission in writing from the publishers. Editor: M. Joubert, Graduate School of Education, University of Bristol, BS8 1JA ISSN 1463-6840 Informal Proceedings of the British Society for Research into Learning Mathematics (BSRLM) Volume 29 Number 3, November 2009 Proceedings of the Day Conference held at Loughborough University on 14 th November 2009 Interpretations of, and orientations to, “understanding mathematics in depth”: students in MEC programmes across institutions 1   Jill Adler1, Sarmin Hossain1, Mary Stevenson2, Barry Grantham2, John Clarke3, Rosa Archer4.   1 King’s College London, 2Liverpool Hope University, 3University of East London, 4St Marys College   Twickenham Symbolic addition tasks, the approximate number system and dyscalculia Nina Attridgea, Camilla Gilmorea and Matthew Inglisb 7     a Learning Sciences Research Institute, University of Nottingham, UK bMathematics Education Centre, Loughborough University, UK   The T-shirt task: Using a mathematical task as a means to get insights into the nature of the collaboration between in-service teachers and researchers 13   Claire Vaugelade Berg   University of Agder, Kristiansand, Norway   Motivating Years 12 and 13 study of Mathematics: researching pathways in Year 11 19   Rod Bond, David Green and Barbara Jaworski   Mathematics Education Centre – Loughborough University   Computer Based Revision 25   Edmund Furse   Swansea Metropolitan University   Children's Difficulties with Mathematical Word Problems. 31   Sara Gooding   University of Cambridge, UK   Some initial findings from a study of children’s understanding of the Order of Operations 37   Carrie Headlam and Ted Graham   University of Plymouth   The role of attention in the learning of formal algebraic notation: the case of a mixed ability Year 5 using the software Grid Algebra 43   Dave Hewitt   School of Education, University of Birmingham   Lower secondary school students’ attitudes to mathematics: Evidence from a large-scale survey in England 49   Jeremy Hodgena*, Dietmar Küchemanna, Margaret Browna & Robert Coeb   a   King’s College London, bUniversity of Durham Simon Says: Direction in Dialogue 55   Jenni Ingram, Mary Briggs and Peter Johnston-Wilder   University of Warwick   The relationship between number knowledge and strategy use: what we can learn from the priming paradigm 61   Tim Jay   Graduate School of Education, University of Bristol   Aspects of a teacher’s mathematical knowledge in a lesson on fractions 67   Bodil Kleve   Oslo University College   Post-16 maths and university courses: numbers and subject interpretation 73   Peter Osmon   Department of Education and Professional Studies, King’s College London   The role of proof validation in students' mathematical learning 79   Kirsten Pfeiffer,   School of Mathematics, Statistics and Applied Mathematics, NUI Galway   An exploration of mathematics students’ distinguishing between function and arbitrary relation. 85   Panagiotis Spyrou, Andonis Zagorianakos   Department of Mathematics, University of Athens, Greece   Identifying and developing the mathematical apprehensions of beginning primary school teachers 91   Fay Turner   Faculty of Education, University of Cambridge   What Might We Learn From the Prodigals? Exploring the Decisions and Experiences of Adults Returning to Mathematics 97     Robert Ward-Penny Institute of Education, University of Warwick Design Decisions: A Microworld for Mathematical Generalisation 103 Eirini Geraniou, Manolis Mavrikis, Celia Hoyles, Richard Noss Institute of Education, London Knowledge Lab Do students’ beliefs relating to the teaching of primary mathematics match their practices in school? Caroline Rickard Initial Teacher Education, University of Chichester, UK 109 BSRLM Geometry working group: tasks that support the development of geometric reasoning at KS3 115   Sue Forsythe; Keith Jones   University of Leicester; University of Southampton   Working group on trigonometry: meeting 4 121   Notes by Anne Watson   Department of Education, University of Oxford   Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Interpretations of, and orientations to, “understanding mathematics in depth”: students in MEC programmes across institutions Jill Adler1, Sarmin Hossain1, Mary Stevenson2, Barry Grantham2, John Clarke3, Rosa Archer4. 1 King’s College London, 2Liverpool Hope University, 3University of East London, 4St Marys College Twickenham In this paper we present initial findings from our study of interpretations and orientations to ‘understanding mathematics in depth’ among students in selected Mathematics Enhancement Courses (MEC) in the UK. The MEC is a 26-week pre-Initial Teacher Education (ITE) ‘mathematics subject knowledge for teaching’ course designed for, and undertaken by, graduates wishing to teach mathematics at secondary level, but do not have a Mathematics degree. It is completed before commencing with a PGCE. A common theme running through the MEC documentation is the importance of ‘understanding mathematics in depth’. We are interested in what and how MEC students interpret and orient themselves towards ‘understanding mathematics in depth’. In designing and conducting our empirical work we have drawn upon a related project in South Africa, which is exploring ‘mathematics for teaching’, specifically what and how mathematics and teaching are co-constituted in mathematics teacher education programmes. The MEC is an interesting empirical context for such study, as it is a mathematics course, or set of courses, specifically designed for future teachers. We have collected data through guided, semi-structured interviews with 18 students and 4 lecturing staff at three different institutions. The interpretations and orientations of MEC students towards mathematics and the notion of ‘understanding mathematics in depth’, we contend, provide additional insight into the developing notion of mathematical knowledge in and for teaching. Keywords: Understanding mathematics in depth, Mathematics Enhancement Course, Mathematics initial teacher education Introduction This study conducted in the UK extends from the QUANTUM project which is currently on-going in South Africa. Our focus here is on ‘understanding mathematics in depth’ as interpreted within the Mathematics Enhancement Course, (MEC). The MEC provides an alternative route into mathematics teacher education. It has been designed for graduates who do not have a mathematics degree but wish to teach mathematics at secondary level. It is a is a 26-week pre-Initial Teacher Education (ITE) ‘mathematics subject knowledge for teaching’ course which is completed before commencing with a PGCE. The programme has been running across a number of institutions in the UK for the past four years. The motivation in the UK, has been to encourage and attract more graduates into retraining as a secondary mathematics teacher. Graduates entering these programmes are required to have an A-level pass in mathematics, or some indication of post secondary study with mathematics. This seems to vary across institutions. Overall, students in these courses are moving from only some post school mathematics, to preparation for being a secondary mathematics From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 1 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 teacher. The MEC programmes are thus focused both on deepening and extending mathematical knowledge in ways appropriate to the profession of teaching. The empirical field of QUANTUM in South Africa (SA) has been focused to date on upgrading programmes for teachers whose qualifications were limited by apartheid teacher education policy and practice. Orientations in South Africa similarly contain intentions to deepen teachers’ subject knowledge in ways that are appropriate and useful in teaching. Despite differences between the UK and SA, programmes share the phenomenon of providing mathematical education specifically geared to the profession of teaching. The UK can be seen as another context where ideas and experience about content knowledge for teaching mathematics are being developed, particularly within the MEC. While the comparative advantage offered by looking across these two contexts will illuminate the field in interesting ways, the study in the UK will be of interest in the UK context itself, and more directly of benefit to the shared understanding that we hope develops across MEC course participants through their activity in the study. Of particular interest to us is the expressed commitment in MEC course materials to “a deep understanding of fundamental mathematics”. We are interested in what and how MEC students interpret and orient themselves towards ‘understanding mathematics in depth’. Our empirical work has been conducted across three UK institutions: a MEC tutor and six students from each of the institutions have been interviewed. We report here on our findings from the student interviews only. Data has been gathered related to: MEC students’ motivations for and concerns in pursuing a teaching career in mathematics and joining the MEC; the structure of the MEC and its activities; students’ orientations to learning on the MEC and the meanings (interpretations) they attach to ‘understanding mathematics in depth'. We are in the early stages of analyzing the data and in this paper we present and discuss emerging trends that lead us to suggest that MEC students’ ‘understanding of mathematics in depth’ is a discourse of mathematics interwoven with discourses of teaching and learning. Background In official and institutionally specific documentation for the MEC, there is a common theme that mathematics teachers need to know and understand ‘mathematics in depth’. This is a particular description of the specific ways in which teachers need to know and use mathematics in order to teach well. Within the field of research on mathematics teacher education (see Sullivan & Wood 2008; Even & Ball 2009) over the last 30 years there has been focus and a growing interest in content knowledge for teaching. There is increasing evidence of a positive relationship between student learning gains and what is being referred to as teacher’s mathematical knowledge for teaching (Hill, Rowan & Ball 2005) and that the nature of teachers’ mathematical knowledge and its use in practice matters for effective teaching. Ball, Thames and Phelps (2008) for example, have examined what teachers know and are able to use in practice and through this, elaborated and extended Shulman’s (1986) categories of content knowledge for teaching that included: subject matter knowledge, pedagogical content knowledge and curriculum knowledge. They distinguish common, specialized, and horizon content knowledge, as forms of subject matter knowledge; and knowledge of mathematics and students, mathematics and teaching, and mathematics and curriculum, as forms of pedagogical content knowledge. Rowland et al. (2005) also built on Shulman (1986) and developed a grounded model of mathematical knowledge used in teaching – the Knowledge Quartet. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 2 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 These and other studies related to mathematical knowledge for teaching (e.g. Neubrand 2008) do not cohere into a unified frame. However, they consistently reinforce Shulman’s insight that knowing mathematics for oneself is not synonymous with enabling others to learn mathematics. An expansive knowing of mathematics is illuminated in the MEC through its intention that prospective teachers need to understand mathematics ‘in depth’ and hence provides an interesting context in which to investigate what and how ‘understanding mathematics in depth’ is interpreted by MEC students who are close to completing their MEC. Research Method Three institutions named A, B, C, for the purpose of anonymity, took part in this study. The sample included 6 MEC students from each institution. The selection of students was purposive, and guided by a set of criteria so that the sample included students with: mathematical and non-mathematical backgrounds; different cultural and educational backgrounds; and ranging participation and performance in the course. A total of 18 semistructured interviews were conducted, guided by an interview agenda lasting approximately an hour. The interviews were recorded and transcribed. Each interview was conducted by a researcher from Kings College London and one of the MEC tutors in an institution which was not their own. This collaborative approach is beneficial. . Firstly, as insiders, the MEC tutors are well placed to probe students’ interpretations by drawing on aspects/activities of the MEC programme. Secondly, the tutors are all ‘new’ researchers and are enjoying participation in a collaborative research community. Sample Description- Within the study sample 14 students were educated in the UK and 4 educated abroad (2 students from Nigeria, 1 from Cyprus, 1 from Pakistan). Furthermore the point at which the students joined the MEC in their lives varied: 3 students joined straight after finishing their degrees at University. 4 students had had a short career and 11 students had had a long career before entry into the programme. Out of the 18 students, only 5 students had some form of teaching career or teaching experience before joining the MEC. In terms of educational qualifications 16 students had bachelor’s degrees, 1 qualified in a PG Programme and 1 qualified in Access to Primary School Teaching. The subjects studied by the students at degree level ranged from: Educational Studies, Business, Computer Science, Engineering, to Sports Science. In regard to mathematical background/qualifications: 12 students had a Level 3 Mathematics qualification (i.e. A Level); 3 students had an equivalent of a Level 3 Mathematics qualification and, interestingly, 4 students had a Level 2 Mathematics qualification (i.e. GCSE or its equivalent), with some mathematics related study or experience in their educational and work histories. The Interviews- Students were probed on: a). How and in what ways their learning of mathematics in the MEC has been different to their learning of mathematics at school and/or at University b). What and how they interpreted “understanding mathematics in depth”? c). How they rated five different statements related to “understanding mathematics in depth”. These statements had been elicited from earlier interviews with MEC tutors where the MEC tutors had defined in their own terms what it was to “understand mathematics in depth”. The MEC students were asked to put these five statements in the order of importance and then probed on their specific ordering. The statements included: Understanding mathematics in depth means being able to: 1). Justify your mathematical thinking. 2). Explain and/or communicate mathematical ideas and thinking to others. 3). Understand why and how these procedures work. 4). Make the connections between concepts and between procedures. 5). Identify structure and generality. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 3 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Data Analysis- We are in the initial processes of the data analysis, and will be concerned with 1). MEC students’ backgrounds, concerns and motivations for pursuing the MEC. 2). Their preferences within MEC activities. 3). Their orientations to mathematics in the MEC and their learning and interpretations of understanding of mathematics in depth. As a mathematics teaching focused programme, we anticipate interpretations and hence discourses (by which we mean representations of social practice, Van Leeuwen, (2008) ) of mathematics, of teaching and of learning to be present in the data. We are also interested in whether and how the students affiliate or dissociate themselves in their talk about mathematics, teaching and learning. We have developed an initial coding based on these theoretically informed assumptions, and our coding has been extended through our interaction with the data. For example, we found that students often associate mathematics and teaching elements when making reference to their Self and their Environment. For example, “I am more mathsy now” and “I am more confident to teach” in these instances for example the text would be coded as the “Self and mathematics”, and the “Self and teaching” respectively. These instances often show how the students are foregrounding themselves either in mathematics, in teaching or in both mathematics and teaching as they journey through on the MEC programme. So far the data has been analysed thematically. Our next step is to refine our analysis by conducting a critical discourse analysis, attending more closely to students’ positive and negative descriptions (e.g. what mathematics in depth IS and what it is NOT); and to whether and how they associate or distance themselves in what they say (e.g. I think this means vs. they say it means). Preliminary Findings As noted, to date we have focused our analysis on students’ spontaneous responses to what “understanding mathematics in depth” means for . As anticipated, there are mathematical, teaching and learning discourses that thread through the transcripts. Below is an example of some of the mathematical discourses that were prevalent across the data. Em, what it means to me is understanding, you know, the different concepts, how they originated, you know, what the idea was, therefore, a concept came about. How it can be applied is a useful lesson in everyday life, and how it’s interconnected with other aspects of mathematics. So that’s how I feel. We will have a deeper understanding of something. And, er, it’s like an open box – we don’t just look at the problem and say, ‘Okay, this is the calculus,’ ... No, you can have, you can try different methods, you can be flexible, you... and see what applies. So if you have a deeper understanding of it, you’re not going to be scared of anything because you know there is a solution, or you can attain to solution. – – – Here understanding Mathematics in depth includes Discourses of knowing the history of mathematical concepts – Discourse of mathematics as connected knowledge Other mathematically related discourses in the data were: Discourses of proof, knowing basic proofs or first principles Discourses of understanding formulae and their parts; when these are used Also reflected in the above quote is a discourse of learning, of acquiring a disposition towards mathematics as something you can do; knowing mathematics in depth means being able to be flexible. Across the data there were learning related discourses, where students talked about understanding mathematics in depth in terms of relearning mathematics, or learning new maths in particular ways. For example, From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 4 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 [Understanding mathematics in depth] means that whenever it comes to understanding it ... you’re not blinkered by what you’re relaying to someone else. ... you’ve actually looked at something, tried to look at it in depth... you may have been led down some paths that come to dead ends, you may have gone down some further paths that leads you to something else. Er, I think through doing that you're more able to build analogies within your own mind, so therefore if you’re trying to explain something to somebody. Because you’ve gone a bit beyond or possibly a bit more beyond what’s expected of you, or what you think should be expected of a student, you actually look at it from different perspectives. So I think whenever it comes to actually passing that knowledge on, you actually get an enthusiasm for it, and I think if you can actually gain the enthusiasm for what you’re looking at, there is a point beyond which it loses an effect, but if you get it just right, I think you can actually – I wouldn’t, I’m not going to say inspired because I think that’s an utterly wrong thing to say, but I think you can actually enthuse someone... And similarly, discourses of teaching thread through discussion of learning – indicating that students interpret understanding mathematics in depth as needed for teaching. For example: But the fact that, you know, you should never give a... a kind of an equation and there’s... there’s always like a practical example or an example in layman’s terms that can help somebody understand it rather than, em, put it into a lot of letters or numbers or something straightaway, because, when you... when you... when you... you often put, em... or express it in terms of, em, letters, for example. I’m... I’m... I’m one of these people that if I just see a lot of letters, it’s very intimidating, especially for children, like for somebody to see that, it’s very intimidating, which can even... even stop them starting the problem at all. So put that into a numerical example and then, for example, linking for patterns, for example, to start off with, and then going on to try and express that in terms of first principles or, em, or algebra...... it’s a way... well, it works for me, I think, em, and I think... I think it works well with children. Here and across the data is a representation that understanding mathematics in depth is enabling others to know mathematics. Within this are – Discourses of dealing with ‘difference’ (learners are diverse) – Discourses of making mathematics ‘practical’ Discussion/Conclusion In summary, our initial analysis of students’ orientations to ‘understanding mathematics in depth’ in these interviews reflects an amalgam of mathematical, teaching and learning discourses. Understanding mathematics in depth meant: knowing that and knowing why (knowing basic proofs or being able to work from first principles; understanding formulae, their parts and when these are used; knowing the history of mathematical concepts; and being able to connect different aspects of mathematics and its applications. These are resonant with Shulman’s elaboration of the Subject Matter Knowledge (SMK) that forms part of a teacher’s professional knowledge base (Shulman, 1986). Of course, these were not uniformly described with apparent different emphases a function of different schooling histories (e.g. some students were schooled in very different educational cultures outside the UK), different mathematical histories and different experiences in the MEC course. Across the interviews, however, was a relatively strong affiliation to teaching requiring depth understanding of mathematics, appreciation of opportunities to engage with mathematics in extended ways in the MEC, and particular appreciation of activities in the MEC that closely aligned with teaching (e.g. peer teaching activities). Many of the students pointed to the social relations in the MEC and the care taken by lecturers in their learning of mathematics. Some From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 5 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 expressed directly that they believed the MEC would give them a ‘leg-up’ in the PGCE, as they had had opportunity to revisit school mathematics and relearn it ‘in depth’. The next question of course, is how their appreciation for understanding mathematics in depth recontextualises into the actual teaching experience and practices and this is what we hope to be pursuing next. References Adler, J., D. Ball, et al. 2005. Reflections on an emerging field: Researching mathematics teacher education. Educational Studies in Mathematics 60(2): 359-381. Adler, J. and Davis, Z. 2006. Opening another black box: Researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education 37(4): 270-296. Ball, D.L., Thames, M.H., and Phelps, G. 2008. Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5): 389-407. Even, R., & Ball, D. L. 2009. The Professional Education and Development of Teachers of Mathematics. The 15th ICMI Study Springer. Hill, H. C., B. Rowan, and Ball, D.L. 2005. Effects of Teachers' Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal 42(2): 371-406. Neubrand, M. 2008 Knowledge of teachers—Knowledge of students: Conceptualisations and outcomes of a Mathematics Teacher Education Study in Germany. Proceedings of the Symposium on the 100th Anniversary of ICMI. Rowland, T., P. Huckstep, et al. 2005. Elementary teachers' mathematics subject knowledge: the knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education 8(3): 255-281 Shulman, L. S. 1986. Those who understand knowledge growth in teaching. Educational Researcher 15(2): 4-14. Sullivan, P., and Wood, T. (Eds.). 2008. The Handbook of Mathematics Teacher Education. Knowledge and Beliefs in Mathematics Teaching and Teaching Development. (Vol. 1). Rotterdam: Sense Publishers. Van Leeuwen, T. (2008) Discourse and practice: New tools for critical discourse analysis. Oxford. Oxford University Press. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 6 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Symbolic addition tasks, the approximate number system and dyscalculia Nina Attridgea, Camilla Gilmorea and Matthew Inglisb a Learning Sciences Research Institute, University of Nottingham, UK bMathematics Education Centre, Loughborough University, UK Several recent theorists have proposed that dyscalculia is the consequence of a disconnect between the so-called ‘approximate number system’ and formal symbolic mathematics. Such theories propose that symbolic exact mathematics is built out of approximate representations of quantity. Here we investigate this proposal by testing whether non-dyscalculic adults appear to use their approximate number systems when tackling symbolic tasks. We find a strikingly similar pattern of responses on two approximate addition tasks, one where participants saw numerosities represented as dots and one where the numerosities were represented with Arabic symbols. These findings are consistent with the view that non-dyscalculic adults do indeed use the approximate number system when dealing with symbolic mathematics. Keywords: approximate number system, arithmetic, dyscalculia, number sense. There is an increasing body of evidence that humans have an inbuilt ‘number sense’ – or approximate number system (ANS) – which supports approximate numerical operations (e.g., Cordes, Gelman, Gallistel & Whalen 2005, Dehaene 1992, 1997). The ANS, which is present in infants, children and adults, involves approximate, abstract representations of number that support both the comparison and manipulation of numerosities (Barth, Kanwisher, & Spelke 2003, Barth et al. 2006; Pica, Lemer, Izard, & Dehaene 2004). When adults and children compare or add symbolic numerals, these approximate representations seem to be activated (Dehaene 1997, Gilmore, McCarthy, & Spelke 2007, Moyer & Landauer 1967). It is even suggested that a disconnect between the ANS and formal symbolic mathematics is the root cause for so-called mathematics disorder or dyscalculia. The Diagnostic and Statistical Manual of Mental Disorders (American Psychiatric Association 2000) describes mathematics disorder as a condition which causes a person to have substantially lower scores on mathematics tests than would be expected given their age, intelligence and educational background. The Department for Education and Skills (2001), using the term dyscalculia, offered a similar characterisation, emphasising how difficult it is for dyscalculic students to acquire “simple number concepts”. Rouselle & Noël (2008) have suggested that dyscalculia is the result of a disconnect between approximate numerical representations held in the ANS and the symbols used in formal mathematics. They found that children classified as dyscalculic could perform well on non-symbolic comparison tasks (i.e. tasks of the form “are there more blue dots or red dots?”), but could not successfully tackle the equivalent tasks represented symbolically. The suggestion that dyscalculia is the result of a disconnect between the ANS and symbolic mathematics is consistent with other recent claims that the ANS provides the basis of all exact mathematics (e.g. Gilmore, McCarthy & Spelke 2007). But if these accounts were correct, we would expect that non-dyscalculic adults, who are fluent in simple arithmetic tasks, would somehow harness the ANS when tackling approximate symbolic mathematics tasks. Our goal in this paper is to explore this account of dyscalculia by examining the behaviour of non-dyscalculic students during both symbolic and non-symbolic addition tasks. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 7 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Experiment 1 The primary goal of Experiment 1 was to provide a baseline for the ANS in the context of non-symbolic addition tasks. Participants saw three numerosities, n1, n2, n3, represented as coloured dots. Their task was to determine which was greater out of n1+n2 and n3. Previous research has found that tasks during which the ANS is used to compare numerosities tend to show a ratio-effect. That is to say that, because representations within the ANS are only approximate (internal representations of a numerosity n are assumed to be distributed normally around n), comparisons of numerosities which have a ratio close to one lead to more errors than those of numerosities with a ratio far from one (e.g., Barth et al. 2006, Gilmore et al. 2007). We would expect to find such an effect in the current task. Method Twelve staff and students (seven male) from the University of Nottingham (aged 23-36, M=29) received £4 for participating in the study. Participants were tested individually. Displays were presented on a 17'' Philips 170B LCD placed at eye level, and were viewed from approximately 60cm away. The stimuli consisted of three dot arrays. The two addend arrays were blue dots against a white background and the comparison set array was red dots against a white background. To prevent participants using strategies based on continuous quantities correlated with number (dot size, luminance, total enclosure area), the stimuli were created following the method of Pica et al. (2004). For each problem two sets of stimuli were created: one in which the dot size and total enclosure area decreased with numerosity, and one in which the dot size and total enclosure area increased with numerosity. Fifty problems were used for each approximate ratio. Sum totals for number to sum ratios less than 1 were the integers from 21 to 70, sum totals for number to sum ratios greater than 1 were the integers from 11 to 60. Comparison numbers were related to sum totals by approximate ratios 8:5, 7:5, 6:5, 5:6, 5:7, 5:8. The addends were randomly chosen in such a way that neither was larger than the comparison number, or less than 5 (for those problems with a sum total less than 30) or 10 (for all other problems). Each set of dots was displayed for 300msec, in the order shown in Figure 1. Figure 1: A non-symbolic addition task. Each problem appeared twice: once with each stimuli set (according to Pica et al.’s method (2004)). Responses (and response times) were recorded via coloured blue (leftmost) and red (rightmost) buttons on a five-button response box. In summary, the experiment followed a 3 (approximate ratio: 8:5, 7:5, 6:5) × 2 (ratiodirection: number-larger, sum-larger) design. This yielded a total of 600 trials for each participant. The experiment was preceded by a practice block of 10 trials. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 8 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Results Mean accuracy rates for each ratio and ratio-direction are presented in Figure 2. These were analyzed using a 3 (approximate ratio: 5:6, 5:7, 5:8) ×2 (ratio-direction: number-larger, sumlarger) repeated-measures analysis of variance (ANOVA). There were main effects of ratio, F(2,10)=110.72, p<.001, and ratio-direction, F(1,11)=6.09, p=.031. Accuracy was higher for problems with the number larger, M=79%, SD=7.6%, than the sum larger, M=64%, SD=15.9%. As predicted, we found the characteristic ratio-effect, as there was a significant linear trend of ratio, F(1,11) = 188.6, p<.001. Figure 2: Accuracy rates for sum-larger and number-larger problems in Experiment 1, by ratio. Error bars show ±1 SE of the mean. Discussion The results from Experiment 1 clearly showed that participants were able to accurately tackle non-symbolic addition tasks at accuracies well above chance. Furthermore we found the so-called ratio-effect: accuracies on problems of ratio 5:6 were lower than those of ratio 5:8. Surprisingly, however, we also found an unpredicted main effect of ratio-direction: number-larger problems had higher accuracy rates than sum-larger problems. Note however, that like earlier researchers (Barth et al. 2006) we did not counterbalance the order in which the numerosities were presented (the sum was always presented first, followed by the comparison number). Consequently, it is possible that this effect might be a consequence of a confound in the experimental design. Experiment 2 Our goal in Experiment 2 was to ask non-dyscalculic participants to tackle problems similar to those used in Experiment 1, but where the numerosities were represented symbolically. If, as we hypothesised, the ANS is used in such tasks we would expect to see similar patterns of results as in Experiment 1. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 9 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Method Twelve staff and students (five male) from the University of Nottingham (aged 19-37, M=24) received £4 for participating in the study. Participants were tested individually. Displays were presented on a 17'' Philips 170B LCD placed at eye level, and were viewed from approximately 60cm away. Stimuli consisted of two symbolic items, a sum and a comparison number, as shown in Figure 3. Figure 2: A symbolic addition task. Numerosities were from the same range as those used in Experiment 1. A total of 25 problems were used for each approximate ratio, with the task being to determine as rapidly as possible which item (left or right) was numerically larger. Again six comparison ratios were used, 8:5, 7:5, 6:5, 5:6, 5:7 and 5:8. Each problem appeared four times (once as each of a+b vs. c, b+a vs c, c vs a+b and c vs b+a). Both responses and response times were again recorded via the leftmost (left-larger) and rightmost (right-larger) buttons on a five-button response box. In summary, the experiment followed a 3 (approximate ratio: 8:5, 7:5, 6:5) × 2 (ratiodirection: number-larger, sum-larger) design. This yielded a total of 600 trials for each participant. The experiment was preceded by a practice block of 10 trials. Results Mean accuracy rates for each ratio and ratio-direction are presented in Figure 4. These were analyzed using a 3 (approximate ratio: 5:6, 5:7, 5:8) ×2 (ratio-direction: number-larger, sumlarger) repeated-measures ANOVA. There were main effects of ratio, F(2,22)=22.08, p<.001, and ratio-direction F(1,11)=6.17, p=.030. Accuracy was higher for problems with the number larger, M=97%, SD=2.6%, than the sum larger, M=93%, SD=5.2%. There was a significant linear trend of ratio, F(1,11) = 6.17, p=.030. Discussion The data from the symbolic tasks we used in Experiment 2 followed essentially the same pattern as that from the non-symbolic tasks we used in Experiment 1. We found the ratio-effect characteristic of the ANS, as well as the unexpected main effect of ratio-direction (this latter finding casts doubt upon the suggestion that it was the result of insufficient counterbalancing, as all stimuli were presented simultaneously in Experiment 2). This pattern of results is exactly what we would expect if the ANS was being used by participants in Experiment 2 to help them tackle the symbolic addition and comparison tasks we used. However, unlike with the non-symbolic tasks used in Experiment 1, participants had various different strategies available to them when tackling the symbolic tasks in Experiment 2. Might it be that adopting non-ANS strategies would result in this pattern of results? To test for this possibility we conducted several additional analyses. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 10 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Figure 4: Accuracy rates for sum-larger and number-larger problems in Experiment 2, by ratio. Error bars show ±1 SE of the mean. Space constraints prevent a detailed presentation of the statistical tests we used to rule out the possibility that participants were using alternative strategies. Our method in each case however, was to compare mean accuracy rates and response times on the subset of the problems where the alternative strategy would have worked with those problems where it would not have. If participants had consistently been using the given alternative strategy we would have expected to see a difference in the means associated with these two subsets of problems. In those cases where such a difference was also predicted by the ANS-based account we instead looked to see whether performance was above chance on those problems where the alternative strategy would not have worked. No evidence was found that any of the following strategies were being used by participants: • Rounding Down. A participant using this strategy would have added the two leftmost digits of each of the addends and compared to the leftmost digit of the comparison number. • Comparison Number Range. Here a participant would have based their answer on the size of the comparison number alone. If it was higher than the median comparison number used across the experiment it would have been selected. • Addend Range. Alternatively, participants could have based their answer on the size of the largest addend. If it was above the median it would have been selected. • Addend-Comparison Difference. Participants may simply have compared the size of the largest addend with the comparison number, and assumed that the sum is largest when that difference was low. General Discussion The ANS appears to be an inbuilt cognitive system that supports rapid – but approximate – numerical calculations. The evolutionary benefit of such a system is clear, for example one can easily imagine how a predator might find it advantageous to be able to rapidly decide which of two groups of prey is the larger. But is the ability to connect the ANS to formal symbolism a prerequisite for higher level mathematics? Here we explored the suggestion that dyscalculia is the result of a disconnect between the ANS and symbolic From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 11 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 mathematics. If this were the case we would predict that participants tackling tasks where numerosities are represented non-symbolically would show a similar pattern of behaviour to those who tackle the equivalent tasks where the numerosities are represented with Arabic numerals. This is exactly what we found. The ANS-symbolism disconnect proposal is a strong hypothesis, and further work is needed to investigate its potential. If it is correct, however, then it seems that an important early goal for mathematics education is to form and strengthen connections between various different representations of numerosity and the internal representations of the ANS (e.g. Wilson et al. 2008). References American Psychiatric Association 2000. Diagnostic and statistical manual of mental disorders (4th ed, text revision). Washington DC: Author. Barth, H., N. Kanwisher and E. S. Spelke. 2003. The construction of large number representations in adults. Cognition 86: 201-221. Barth, H., K. La Mont, J. Lipton, S. Dehaene, N. Kanwisher and E. S. Spelke. 2006. Nonsymbolic arithmetic in adults and young children. Cognition 98: 199-222. Butterworth, B. 2005. Developmental dyscalculia. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 455-468). New York: Psychology Press. Cordes, S., R. Gelman, C. R. Gallistel and J. Whalen. 2005. Abstract number and arithmetic in preschool children. Proceedings of the National Academy of Sciences 102: 1411614121. Dehaene, S. 1992. Varieties of numerical abilities. Cognition 44: 1-42. Dehaene, S. 1997. The number sense. Oxford: Oxford University Press. Department for Education and Skills 2001. Guidance to support pupils with dyslexia and dyscalculia (No. DfES 0512/2001). London: DfES. Gilmore, C. K., S. McCarthy and E. S. Spelke. 2007. Symbolic arithmetic knowledge without instruction. Nature 447: 589-591. Moyer, R. S., and T. K. Landauer. 1967. Time required for judgements of numerical inequality. Nature 215: 1519-1520. Pica, P., C. Lemer, V. Izard, and S. Dehaene. 2004. Exact and approximate arithmetic in an Amazonian indigene group. Science 306: 499-503. Rousselle, L., and M. P. Noël. 2007. Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs. non-symbolic number magnitude processing. Cognition 102: 361–395. Wilson, A. J., S. Dehaene, P. Pinel, S. K. Revkin, L. Cohen and D. Cohen. 2006. Principles underlying the design of “The Number Race”, an adaptive computer game for mediation of dyscalculia. Behavioral and Brain Functions 2: #20. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 12 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The T-shirt task: Using a mathematical task as a means to get insights into the nature of the collaboration between in-service teachers and researchers Claire Vaugelade Berg University of Agder, Kristiansand, Norway By following a mathematical task, from its design by researchers to its implementation by a teacher, it is possible to get some insights into the collaboration between researchers and teachers. Activity Theory is used as a theoretical approach in this research. Keywords: designing and implementing tasks, collaboration between researchers and teachers, activity theory. The research setting The research reported here examines the way a mathematical task is transformed and adapted, from its design among researchers to its implementation by a teacher in lower secondary school level. This research is situated within an ongoing research project at the University of Agder (UiA) called TBM (Teaching Better Mathematics). The name of the project reflects two goals: the first one concerns Teaching (Better Mathematics) and aims at developing better understanding of, and competency in mathematics for pupils in schools. The second one concerns (Teaching Better) Mathematics and aims at exploring better teaching approaches in order to achieve the first aim. The research involves collaboration between researchers from university and teachers working at different levels, from kindergarten up to upper secondary school. Within the project, we collaborate with 4 kindergarten, 6 primary schools and lower secondary schools, and 3 upper secondary schools. The collaboration with the teachers is organised around workshops which happen approximately once a month and consist of a plenary presentation of a theme within mathematics and some group work where the teachers have the opportunity to work collaboratively with colleagues at the same school level. In addition, within each school, a group of two or three teachers (called the TBM group) is responsible for the continuity between the work done at the university during the workshops and the teaching of mathematics in their respective schools. An important feature of this research is the recognition of the researchers and the teachers as working together as co-learners and getting the opportunity to develop a better understanding of each others “world and its connections to institutions and schooling” (Wagner 1997, 16). The centrality of inquiry The idea of inquiry plays a crucial role in the TBM project, as we consider that through the project we, as researchers, are able to address inquiry at three different levels (Jaworski 2006): at a first level, as pupils engage with a task and inquire into the mathematics, at a second level, as teachers engage with inquiring into the developmental process of planning for their teaching, and at a third level, as researchers engage with inquiring into the research process of systematically exploring the developmental process and practices as presented at the two first levels. Thereby, all participants of the TBM project engage with inquiring into how to improve mathematics learning and teaching in classrooms. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 13 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Furthermore, we consider that inquiry could form a basis for the teachers’ teaching practice in mathematics. Activity Theory as a means to characterize development In our project we use Activity Theory as a means to describe and characterise both the researchers’ and the teachers’ development. Within Activity Theory, the idea of “activity” endorses a precise meaning: human activity is understood as directed by a motive and is firmly rooted in actions and goals. Furthermore, these actions are carried out through operations and conditions (Engeström 1999). Thereby, we see our motive within the TBM project as being to engage, collaboratively with teachers, in inquiry about teaching and learning of mathematics in order to improve pupils’ achievement in mathematics. Our actions consist of organising workshops and designing suitable mathematical tasks, while the notions of operations and conditions refer to searching and collecting ideas and adapting these to the workshop environment and classroom situation. In order to deepen the ideas of actions and operations, I propose to introduce the theoretical constructs of “didactical aim” and “pedagogical means” (Berg 2009). Didactical aim refers to the choice of a particular area or subject-matter as for example symmetry, algebra or proportionality, while pedagogical means refers to a task which is chosen and used in order to address the chosen didactical aim. Here I consider that the construct of “didactical aim” as a useful theoretical tool enabling me to pull out, articulate, and make visible central issues in relation to both the collaboration with teachers, and more specifically, the design and preparation of workshops, and the teachers’ planning and preparation of their own teaching. Thereby, “didactical aim” relates both to the researchers’ actions (preparation of workshops) as it emphasises which mathematical goal the researchers plan to address during the workshops, and to the teachers’ actions (preparation of teaching) as it emphasises which mathematical goal the teachers plan to address during their teaching. Likewise, “pedagogical means” serves as a theoretical tool describing the result of the processes of choosing, transforming and adapting a particular task to a specific social setting in order to address a chosen didactical aim. Thereby, “pedagogical means” refers both to the researchers’ operations in the sense of assembling a set of tasks and adapting these to the group session during the workshops, and to the teachers’ operations as they prepare and adapt a particular mathematical task to their own classes. Therefore I consider that “by presenting a particular task within a specific social setting, a didactician creates a mathematical environment whose characteristics depends both on the mathematical task and on the setting” (Berg 2009, 103). In this article, I present and compare two mathematical environments: the first one relates to a specific task prepared by the researchers in order to engage collaboratively with teachers during a particular workshop, the second one relates to how a teacher implement this particular task in his teaching. Focusing on a specific task In my current research, I follow a particular task, the T-shirt task, from its design during a meeting among researchers at UiA (TBM meeting, 26.11.08), its presentation to the teachers during a workshop (03.12.08), and to its implementation in a primary school (11.12.08) and in a lower secondary school (05.05.09). The rationale for choosing this particularly task is the following: in December 08, a teacher from primary school contacted me and invited me to follow her teaching as she wanted to implement a task from a previous workshop in her teaching. Similarly, in May 09, another teacher (Per) from lower secondary school contacted me as he planned to implement a task from a previous workshop in his teaching. The fact that these teachers were referring to the same mathematical task (the T- From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 14 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 shirt task) encouraged me to focus my research on the processes behind the design and implementation of this specific task. Because of space limitation, I focus on the results of the analysis of the mathematical environment created by Per. The TBM meeting The T-shirt task was elaborated during the TBM meeting on November 26th . During that meeting, the discussion among the group of researchers was focusing on how to address “communication in mathematics classroom”. This theme had been chosen in advance by the TBM group of the different schools. While exploring different possibilities for addressing this subject, we decided to contextualise the discussion and to address “communication in mathematics classroom” through engaging with a particular task. We chose the T-shirt task as we agreed that this specific task offered a rich approach to communication. As one of the participants emphasised: … in communicating mathematics, questions are a far more effective way of communicating than telling. In order to make sense of mathematical knowledge, pupils need to take the responsibility for exploring which means questioning the teacher, questioning others. The fundamental aspect about communication is questioning. (TBM meeting, 26.11.08, translated from Norwegian by the author) Thereby, our group decided to focus on the ability to ask “good” questions, in the sense of engaging with task by inquiring into the mathematics (Jaworski 2006). The T-shirt task was elaborated in the following way: the context is a phone call where one person has to explain to another one the motive of a logo to be reproduced on a T-shirt (see Figure 1). Figure 1: The T-shirt task presented within a grid system behind the logo The workshop (03.12.08) Usually the workshops are organised according to the following pattern: first one of the didacticians from the university gives a plenary presentation on a chosen theme. Second, a group session is organised where all participants are divided into groups according to the level at which they teach. Finally, all participants gather together in order to exchange experiences from the group sessions. The title of the workshop on December 3rd was “To ask good questions in mathematics”, and it was one of the didactician from the university who had the responsibility to introduce to the participants of the workshop (teachers from different school levels, from kindergarten to upper secondary school) ideas related to questioning and From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 15 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 communicating in mathematics and to link these to the idea of “inquiry”. During the group session, all teachers engaged with the T-shirt task and discussed different approaches to it with emphasis on the kind of questions which are relevant to ask in order to reproduce the logo in an accurate way. In addition, they discussed the possibility of integrating, changing and adapting this task to their respective classes. Summary of the design process Looking at the design process of the T-shirt task, the discussion during the TBM meeting concerned “Communication in mathematics” and we decided to emphasise questioning and its relation to inquiry. Furthermore, we chose to contextualise the discussion through presenting a mathematical task and the T-shirt task was selected as a relevant example. Using the theoretical constructs from Activity Theory, I understand the idea of “actions” as referring both to the researchers’ preparation of workshops, and to the teachers’ preparation of their own teaching. These actions are linked to the didactical aims chosen by the researchers and by the teachers. Furthermore, I consider the construct of “operations” as referring both to how the researchers assemble a set of tasks and adapt these to group sessions during the workshops, and to how the teachers prepare and adapt tasks to their own classes. Likewise, I consider that these operations offer a description both of the researchers’ and of the teachers’ pedagogical means. Implementation of the T-shirt task in lower secondary school Before observing how the teacher, from lower secondary school, implemented the Tshirt task in his class, I had the opportunity to interview him and, thereby, to make visible his aims for the teaching period. Likewise, I conducted an interview right after the teaching period in order to summarise and evaluate it with the teacher. In this article I focus on the interview before the teaching period. Visiting a lower secondary school: in class with Per On May 5th, I had the opportunity to visit Per and to observe his teaching in grade 8. During the interview he explained the rationale for implementing the T-shirt task in his class and he emphasised particularly on the following aspects: From the curriculum, there are first of all two aspects which I would like to have as goals for my teaching, it is the use of coordinate system, and the second is the introduction of functions… And you can say, what I want to emphasise is communication, I would like the pupils to have an understanding of how one communicates in mathematics. From his utterance, it seems that Per decided to implement the T-shirt task in his class since he could recognise the possibility to address two aspects from the curriculum through this task. The first one refers to the use of coordinate system, the second one relates to the introduction of functions. In addition, Per emphasised communication in mathematics. It was the first aspect (use of coordinate system) which was emphasised during the teaching period I observed (right after the interview). Per organised his teaching by dividing the lesson into two parts. During the first part, he asked one pupil to be responsible for explaining the logo of the T-shirt task to another pupil who was sitting behind a blackboard and could not see the logo. Here Per had prepared in advance a slightly different representation of the logo: the logo of the T-shirt was drawn without a grid system behind it and he presented this version to the pupil. From the classroom observation, it is possible to follow how the pupil struggled to From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 16 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 explain the respective positions of the circle and triangles, as drawn on the logo. During the second part of the lesson Per asked another pupil to be responsible for explaining the logo of the T-shirt task, this time with the grid system (see Figure 1). Here it is possible to follow how the pupil used the grid as a coordinate system and could refer to the circle and the triangles by indicating the coordinates of particular points. During both exchanges (description with and without a grid system) the rest of the pupils could follow how the communication of the logo was influenced by having the opportunity or not to use a coordinate system. My interpretation of Per’s way of implementing the T-shirt task is the following: as the T-shirt task was introduced during the workshop of December 3rd, Per was able to identify the possibility to address two aspects from the curriculum, the use of coordinate system and the introduction of functions. I understand these as didactical aims which Per plan to address during his teaching, or using the theoretical tools from Activity Theory, these didactical aims are the “goals” for his “actions”. Thereby, during the lesson I observed, Per was addressing the use of coordinate system as one of his didactical aims. At the same time, as he implemented the T-shirt task by contrasting the presentation of the logo with and without the grid system behind it, he was in a position of emphasising communication in mathematics. Thereby, my understanding of Per’s teaching is that his didactical aim for that lesson was to address the use of coordinate system and he modified and adapted the T-shirt task to his class in order to achieve his didactical aim. Here Per’s “operations” were to produce the modified version of the T-shirt task (presentation with and without the grid system behind the logo) and therefore this new task acted as a pedagogical means which was chosen and used in order to address the chosen didactical aim. In addition, this comparison allowed Per to emphasise on communication in mathematics. Comparing the researchers’ and Per’s didactical aims and pedagogical means Looking back to the researchers’ elaboration and preparation of the workshop comparing to the interview with Per before his teaching, it is possible to observe an inversion between the didactical aims and pedagogical means: Researchers Per Didactical aim Communication Coordinate system Pedagogical means T-shirt task (coordinate system) T-shirt task (communication) Table 1: Comparing the researchers’ and Per’s didactical aims and pedagogical means A possible explanation for the observed inversion consists of taking into consideration and recognising the fact that the teacher and the researchers belong to two different activity systems. Furthermore, this recognition begs the following question: how can this inversion be understood using the theoretical constructs available within Activity Theory? As explained earlier, I conceptualised our research group at UiA in terms of activity system where our activity is motivated by a desire to engage, collaboratively with teachers, in inquiry about teaching and learning of mathematics as a means to improve pupils’ achievement in mathematics. The goals of our actions, consisting of workshops and school visits, were to initiate and make possible the collaboration with teachers. In addition, according to Engeström (1999), an activity system is defined by the “rules”, the “community”, and the “division of labour” followed by the activity system. Concerning our research group, the community of researchers consists of 5-6 persons working within the TBM project, and the From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 17 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 division of labour is made visible as we prepare the different workshops and are responsible for school visits. Concerning the dimension called “rules”, one aspect of it is visible in our community as our group takes into consideration the wishes emerging from the teachers’ TBM group at each school in relation to the choice of didactical aims. The teachers usually suggest several themes to us and from these ideas we discuss how to organise the different workshops in a coherent way. Thereby, within our activity system, one of the rules is to recognise and elaborate on the different themes proposed by the teachers. Communication in mathematics is one example of the teachers’ suggestions. Looking at the teachers’ activity system at their respective schools, I understand the motive for their activity as creating opportunities for engagement with mathematics, and offering critical guidance for what mathematics achievement means. Concerning the goals of their actions (teaching), I see these as being the organisation of pupils’ participation into mathematics classrooms. At the teachers’ schools, the community consists of all teachers and colleagues working within the administration, and the division of labour is clearly decided by the head teacher within each school. Concerning the dimension “rules”, one of the constraints for the teachers is to follow the curriculum. It seems that it is this rule which became visible through Per’s utterance. Thereby, I consider that by following a specific mathematical task from its design by researchers to its implementation in Per’s class, the research reported here enables me to compare and to observe an inversion between the researchers’ and Per’s didactical aims and pedagogical means. A possible explanation for this inversion consists of recognising the researchers’ and the teachers’ communities as belonging to two different activity systems each of them having different rules. I argue that this recognition helps us, as researchers, to get deeper understanding of each others’ world (Wagner 1997). References Berg, C. V. 2009. Developing algebraic thinking in a community of inquiry: Collaboration between three techers and a didactician. Doctoral Dissertation at the University of Agder. Kristiansand, Norway: University of Agder. Engeström, Y. 1999. Activity theory and Individual social transformation. In Perspectives on activity theory, ed.Y. Engeström, R. Miettinen and R-L Punamäki, 19-38. Cambridge: Cambridge University Press. Jaworski, B. 2006. Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187-211 Wagner, J. 1997. The unavoidable intervention of educational research: A framework for reconsidering researcher-practitioner cooperation. Educational Researcher, 26(7), 1322 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 18 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Motivating Years 12 and 13 study of Mathematics: researching pathways in Year 11 Rod Bond, David Green and Barbara Jaworski Mathematics Education Centre – Loughborough University We report on a collaboration, between 4 teachers in 4 schools and a university team of 3, over a period of 21 months, to enthuse Year 11 students (taken from the top 25% ability range) about mathematics and encourage their further study of mathematics in Years 12 and 13. Each school used a different pathway to achieve these goals: this involved acceleration, enrichment, the Free Standing Mathematics Qualification or an early start to A level. The research was developmental in both studying the practices and processes involved while contributing to teachers’ continuing professional development in mathematics. The project Funded by the NCETM1, this project involved a 21 month investigation into how teachers can motivate and enthuse able mathematics students (taken from the top 25% ability range as judged by tests and examinations) by developing pedagogy in different approaches at KS4. It was motivated by a desire to encourage more young people to enjoy mathematics and take it further in their studies. From an NCETM perspective, the professional development of the teachers concerned was paramount. To achieve these various aims, the project took a developmental research approach which involved 4 teachers in 4 schools and 3 university academics. Project activity included: For the teachers • Creating a mathematical pathway in Year 11, designing and delivering the related course, and studying its progress and outcomes; • Participating in collaborative activity through meetings at the university and visits to each others’ schools; • Reflecting on, evaluating and reporting outcomes. For the academics • Working with the teachers to encourage and study the creation and implementation of a pathway; • Collecting data to chart progress and evaluate outcomes; • Conducting analyses and reporting on outcomes. Methodology The project was conceived by Bond as a result of contacts with headteachers who wanted to improve exam performance and Year 12 uptake, and discussions with Heads of Mathematics faced with developing strategies for students who had taken GCSE (General Certificate of Secondary Education) in mathematics at the end of Year 10. It was clear that schools used a range of approaches and questions arose as to ways in which such approaches contributed to achieving project aims. Four schools agreed to participate in the project, each with a lead teacher and a different chosen pathway, as follows. A. Entry for GCSE at the end of Year 10 followed by A/S Mathematics in Year 11 1 National Centre for Excellence in the Teaching of Mathematics. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 19 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 B. Entry for GCSE at the end of Year 11 and the Free Standing Mathematics Qualification (FSMQ) at the end of Year 11 C. Entry for GCSE at the end of Year 10 followed by the FSMQ in Year 11 D. Entry for GCSE at the end of Year 11 with no additional qualification offered, but with the course enhanced by practical work and ICT The project provided, for each school, two hours per week off timetable for the lead teacher, with release time for regular meetings as a group at the university to discuss progress and share ideas, and £600 to buy resources of their choice. It provided also the support of a Project Officer and staff at the university and the guidance of an advisory group which would monitor the direction of the project. The project used a developmental research methodology such that research activity contributed to promotion of development (Jaworski, 2008). Research was designed to explore the nature and outcomes of each particular pathway, and the associated development of the lead teacher. Each lead teacher organized the activity of the project within their school, liaising with their mathematics department as appropriate and teaching their own class of students throughout the year. In two schools, two teachers were involved in the teaching. Teachers were encouraged to keep a record of progress according to their aims in the project. The Project Director (Bond) and the Project Officer (Green) liaised closely with schools to support the initiation of the project and to collect relevant data (see below). Project meetings of the four lead teachers and the university team were organised to take place at the university 9 times during the life of the project. Teachers were encouraged to reflect on activity and progress and to discuss issues and concerns; the university team encouraged reflection and asked probing questions. The team asked teachers to state research questions for their own pathway and discussion in the meetings allowed these questions to be refined as the academic year progressed. Towards the end of the project, theoretical perspectives were discussed to enable teachers to consider the theories motivating their teaching and its development. Data, both quantitative and qualitative, were collected throughout the project and analysed by the university team. Quantitative data included school data, three student questionnaires, three teacher questionnaires and public examination results. Data were also collected from a fifth school to act as a ‘control’. Qualitative data included recordings of the meetings (summarised), the developing research questions, two interviews with each of the four teachers (first interviews transcribed), interviews with students in two schools, teachers’ written reports , and a diary (one teacher only). Quantitative data were first coded by hand and entered into Microsoft Excel spreadsheets for checking. Data from the second questionnaires, for both students and teachers, were analysed and graphs produced using Excel. All other data were transferred to SPSS for analysis and production of tables and graphs. Due to the small and highly specific nature of the sample of schools, caution has been used in interpretation of these data. Results which may appear significant (e.g. using a multi-dimensional chi-squared test on crosstabulated data) might well not generalise to a wider population or other circumstances. Our hope is that teachers in other schools where circumstances seem similar to those in the project schools will feel able to draw their own tentative conclusions and then to conduct research to verify them. Of the extensive qualitative data, the first set of interviews with teachers (June 2009, transcribed) and the teachers’ written reports have so far provided the main data for analysis, supported by the other forms of data. Analysis has involved a cyclic process of reading, rereading and categorisation of the data. This process is still ongoing and what we present here are tentative initial categorisations. A detailed report has been written for the NCETM, From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 20 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 including detail of data and analyses and the full teacher reports. A copy can be obtained by contacting one of the authors2. Findings from the project Quantitative analysis School data, examination results and progression Data collected from the four schools and the control school included information on students numbers, gender, SATS and examination results. These data varied according to the nature of the top set – for example in School A, where the top set was 1 out of 10, various results and indices were not surprisingly higher than in the other three schools which had two or three parallel four-class streams each with a top set (i.e. 1 out of 4). Also, the time allocated to mathematics varied considerably between schools with School A having 5 hours per week and School D only 2 hours and 20 minutes. The gender balance was quite even in all classes except in School A (59% male) and School D (66% male). With regard to progression rates – i.e. students transferring into Year 12 – complete data were hard to obtain due to transfers between schools, students dropping out of Y12 mathematics courses at various points and even students who leave school, only to return later. In almost all classes a decline is indicated in uptake expectations between October 2008 and April 2009, and a further decline in the actual uptake, after drop-outs. Results from the control school suggest that this decline is a common feature to which the project made little difference. Student questionnaires (initial and final) These two questionnaires were identical, including: 12 questions on a 5 point scale to measure perceptions of a) confidence, b) teacher supportiveness, and c) usefulness of mathematics for themselves; and 10 questions on a 5 point scale to measure d) enjoyment and e) usefulness of mathematics for society; each index was tested for reliability. Data and boxplots can be found in our NCETM report. There was no appreciable change in confidence levels (which were quite high) between October 2008 and April 2009 although an increase in lower end outliers suggested that some students were feeling examination pressure or starting to feel that mathematics was not for them. Enjoyment levels remained quite high overall. Project classes had an increased spread leading to a higher median than the control. Student views on teacher supportiveness, which were largely positive, increased for classes A and D. Perceptions of control classes were lower. Students’ views on the usefulness of mathematics for themselves and for society remained at a high level for both project schools and control. Students were largely graded at SATS levels 7 and 8. An analysis was done for each of the 5 indices against the two SATS levels. SATS-8 students demonstrated mainly high confidence levels and positive enjoyment levels, whereas there was a wider range of confidence for SATS-7 students whose enjoyment levels were generally lower though still mainly positive. Levels of perceived teacher supportiveness were generally high for SATS-8, and lower, although still positive, for the SATS-7 students. The Spearman’s rho correlation of confidence with teacher supportiveness was much higher for SATS-7 than for SATS-8, suggesting the crucial role that (perceived) teacher attitude can play for the SATS-7 students. 2 Corresponding author: R.M.Bond@lboro.ac.uk From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 21 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Project teachers’ predictions of their students’ perceptions Half way through the 2008/9 academic year, a second, shorter questionnaire was completed by students with questions 1 to 5 on 3 levels (a lot, a bit, not at all) and question 8 on 2 levels (Yes/No). The main statements for their consideration were: 1. I am enjoying Year 11 mathematics 2. The course is demanding in terms of workload 3. I am being stretched mathematically 4. My understanding of mathematics has improved in Year 11 5. The course has inspired me to carry on studying mathematics next year 8 Could the teaching be improved? At the same time, their teachers were asked to predict the response profiles they expected from their classes. Findings and predictions on enjoyment matched fairly well, with teachers only slightly overestimating. However, teachers greatly overestimated their students’ workload. We noted interestingly that class B (taking GCSE and FSMQ simultaneously) recorded a high level of enjoyment and also the highest incidence of the workload being very demanding. Teachers substantially overestimated the intellectual demand level (“I am being stretched mathematically”), although almost all students were finding the work challenging to some degree. Almost all students reported some improvement in understanding, and teachers’ estimates were close to students’ recordings. Teachers slightly overestimated the numbers of students who said they intended to continue to Year 12, although the overestimate was most marked on just one school (C) which ultimately had a lower uptake than the other schools. Regarding Question 8, remarkably, the prediction of Teacher A, that 100% of students would say that teaching could not be improved was exactly correct. Teachers B and C rather overemphasised the number who would say “yes”. Teacher D estimated his students’ responses very accurately. Teaching styles At the beginning and end of the academic year, teachers were asked to complete a questionnaire designed to assess the strengths of three factors comprising their teaching styles: transmission, connection and discovery (Askew et al, 1997). Briefly, transmission teaching views mathematics as a body of knowledge and skills to be passed on from teacher to student; connectionist teaching views mathematics as an interconnected body of ideas and reasoning processes which the teacher and student construct together, and discovery teaching views mathematics as a personal construction of the student. Responses indicated that Teacher A moved from being just within the connectionist zone, to being substantially within this zone; teachers B and C stayed in a very similar positions within their zone (B in the transmission zone and C in the discovery zone) and D moved from being substantially within the transmission zone to the boundary between transmission and discovery. Thus the project seemed to have the most marked effect on Teacher D. Qualitative analysis While the quantitative data pointed mainly towards students’ perceptions and achievements, and their teachers’ associated expectations, the qualitative data pointed mainly towards teachers’ perceptions of their teaching and its development through the year. We present here the broad picture of findings so far. Teachers’ goals for the project Teachers were asked to say something about their goals for the project (or the goals of their school). Teacher A, who had taught his class (top set of 10 sets) since Year 9 and whose class From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 22 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 had already taken GCSE in Year 10, wanted to see if taking A/S level in Year 11 was “doable”. Could he take a class of 30 students through half an A level in Year 11? Teacher B, who was new to the school and who had had the project thrust onto her in her first year in the school, presented a school perspective. They wanted to “do something extra” with their “more able” students (2 parallel top sets), “give them more experience with maths”. They wanted “to push them a bit further than GCSE” (offering also the FSMQ). Teacher C, whose students had taken GCSE at the end of Year 10 (some were trying to raise their GCSE grades) said that he was looking for a course that would “stretch our more able pupils”, something that would take them a bit further, give them insight into what A level maths would be like”. Thus the school offered the FSMQ to parallel top sets alongside GCSE retakes. Teacher D also wanted to “stretch our more able students” but without fast-tracking. The students were taking GCSE in Year 11. He wanted to introduce them to mathematics that would give them a taste of A level, and would reinforce and be complementary to GCSE, but without offering a separate qualification. Thus the goals expressed by the teachers fitted well with the stated goals of the project. The pathways were different; two schools having already taken Year 10 classes through GCSE were looking for a suitable course for Year 11; two schools were in the process of preparing students for GCSE at the end of Year 11. One school in each category decided to take on the FSMQ course, one (B) alongside GCSE in Year 11, and one (C) in parallel, for some students, with some retaking GCSE and seeking higher grades. We now report on particular issues or outcomes. DO-ability – what works Teacher A asked whether the plan to take 30 Year 11 students through half an A level in one year was “do-able”. His research showed that it was do-able, and outcomes from the process indicated considerable success as detailed in his diary. The idea of ‘do-ability’ seemed to permeate the rhetoric of the school research for all four teachers. “What works” was a common focus. Teachers initially all focused on what they would do, or what they had done, and the extent to which it worked relative to their goals in the project and the context of their school. What works included: planning for the classroom, types of activities and tasks, how students respond, what issues arise and what all of this looks like in practice. In the early stages, not much was said about what teachers learned or could learn from the project. The focus for all of them was on what students would or could gain from the planned activity, and on outcomes in terms of examination results and achievement of student targets. Preparing for teaching Teachers put a lot of time into their planning of activities and resources for their students. This involved thinking hard about what would be interesting and motivating for students. Teacher B, who was new to her school and to teaching A level mathematics, spent considerable time working on mathematics herself. Teacher A prepared songs and quiz-based tasks related to curriculum areas. Teacher D prepared computer-based activities such as spreadsheets for numerical differentiation and activities with graphical calculators. Teacher C engaged in collaborative planning with a colleague to find “different ways of doing the topics”. All produced video-recorded examples of innovative practice for sharing in project meetings Valuable to work with colleagues (in school and in the project) Teachers emphasised the importance for them of sharing their thinking, planning and reflecting with colleagues, both in terms of sharing ideas and gaining ideas from others, and From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 23 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 in terms of gaining support where there were issues and problems. Teachers B and C valued opportunities to share ideas with and/or gain support from colleagues in their own school. All valued the opportunity to share ideas and issues with one other. Schools A, B and C made choices at department and school level, whereas in School D it was largely the choice of the one teacher, albeit supported by his Principal. Disappointingly, for Teacher D, his department showed little interest. Time pressure Teachers B, C and D emphasized pressure of time on what they were able to achieve. The time allocation for mathematics in School A was more generous than the other schools, with School D the least generous. While we are aware that teachers generally experience time pressure and that there are competing demands on allocation of time to subjects within a school, it was clear that achieving the goals of this project was considerably circumscribed by time factors and associated pressures. Teachers’ learning through reflection Project meetings put emphasis on what the teachers were learning from their activity and its outcomes. For all of the teachers this seemed to require a refocusing of their attention and a use of different language to describe what they were experiencing. Reflecting on their experience and offering some analysis of it in terms of their own learning in a project meeting required a more personal introspection. However, the supportive nature of project meetings encouraged the teachers to share personal issues and concerns. All expressed growth of understanding of, and confidence in, the new activity. The teachers commented overtly on the value of project time in school in which to reflect, and the nature and outcomes of this reflection: e.g., “I have certainly done more reflecting on what has happened.… You start thinking about what would you do differently next time. I have also been trying to notice when something has been successful and then try to come up with something similar next time … ways of teaching … which would work in a similar way, so yes I have developed that way”. In conclusion Perceptions of pathway success were strongly related to school factors such as time devoted to mathematics and the degree of support for the project within a school. Schools A, B and D indicated their intention to continue the same pathway in future years. School C saw problems between students retaking GCSE and coping with the demands of the FSMQ. Early entry for GCSE is being discontinued. Further research can usefully explore whether these findings accord with practices more widely. References Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective Teachers of Numeracy. London: King’s College, London. Jaworski, B. (2006). Developmental research in mathematics teaching and learning: Developing learning communities based on inquiry and design. In P. Liljedahl (Ed.), Proceedings of the 2006 annual meeting of the Canadian Mathematics Education Study Group. Calgary, Canada: University of Calgary. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 24 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Computer Based Revision Edmund Furse Swansea Metropolitan University A computer system on the web known as xplus12 has been developed which supports KS3 revision in Number and Algebra. This has been evaluated with a year 8 top-set class on two trials in Number with a little Algebra. Since all the data is stored in a database including timings and all student attempts at answers, it is possible to identify a number of behaviour patterns among the pupils. These include the identification of groups such as "rubbishers" and "rushers" who essentially abuse the system. Although an effect size of 0.9 was found in the first trial it was not statistically significant due to the low numbers of pupils completing the test, and no improvement was found in the second trial. Misconceptions were also identified by the system. A number of suggestions are discussed for improvements of the system including improved examples with animation and explaining answers. Also included are techniques to handle rubbishers and rushers. Keywords: web, revision, computer based learning Introduction The use of computer systems to assist in the teaching of mathematics is growing worldwide, especially in the USA. Anderson's Cognitive Tutor (Algebra) from Carnegie Melon University is installed in over 2000 schools. Computer remedial teaching of algebra has been a growing market in American Universities for some time, and in the last few years this is now filtering down into schools. It is convenient to use computers for remedial teaching since the pupils can do it at their own pace and institutions may use cheaper staff. Here we report on an evaluation of the xplus12 revision system that runs on the web (www.xplus12.com). The system has been developed over a number of years and uses AI techniques both in the automatic synthesis of worksheets, and in sophisticated answer checking. The development system is built in LISP and Java, and the delivery system runs on all computer platforms that support Java. The website has sections on worksheets, starters, games and revision and has over 40 worksheets. A unique property of the system is automatic transition from type-in questions into multiple-choice questions after a suitable number of wrong attempts. Most systems available only support multiple-choice questions but these are too reliant on recognition memory. But the danger of type-in questions is that a pupil can get stuck on a question and not be able to progress. Mathematics is an ideal subject for a computer based learning system since the computer is able to correctly check if a pupil's answer is correct. This can even be done for algebraic answers by use of suitable AI techniques. Most systems fudge this and cannot handle all the possible algebraic answers. This study finds an impact of the revision software in xplus12 but it is not statistically significant. Too many pupils had to be excluded from the trials due to non-completion of the test or writing rubbish. The paper discusses in particular two important groups of students: "rubbishers" and "rushers". The discussion section looks to a number of improvements of the system, which should make learning more effective. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 25 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Literature Review Several studies have been done on the effectiveness of the use of computers in the teaching and learning of mathematics. The results appear to be mixed, with some finding no effect and others a small positive effect. Tienken and Wilson (2007) have a review of the impact of CAI on mathematics achievement. They found an effect size of 0.12, which was statistically significant p < 0.05. Whereas, The Electronic Education Report (2007) reports on a large study of seven maths computer packages and found none had a statistically significant improvement on performance. Zhu and Polianskaia (2007) provide a comparison between traditional lecture and computer-mediated instruction and find no statistically significant difference over a ten-year period. Paper and pencil was more effective. Clearly computer systems differ widely in many aspects and this has impacts on their effectiveness. One of the highest effect sizes (of 1) is reported by Anderson (1995) for his Cognitive Tutor in Geometry, but surprisingly there is no difference in performance with his Algebra Tutor between controls and students using the system. Anderson attributes this to poor transfer from the computer system to paper and pencil solutions. One of the problems of some of the positive results studies is that the students were self-selected. This is particularly true for revision systems, like SAM Learning. It is also an issue that is recognised in the literature, for example Biesenger & Crippen (2008). However, several systems have been used with whole classes that then encounter problems of nonengagement by some students. Egan, Jefferies and Johal (2006) introduce the classification of lurkers, workers and shirkers in an online teaching system. There have been a number of studies of guessing including Beal et al (2008). Several systems, including Aleven et al (2006) attempt to get the computer system to identify the students abusing the system and get it to react appropriately; for example students who click through hints at speed are told to slow down. Abusing the computer system is part of a more general problem of disaffected pupils and those who avoid engaging with lessons (Dowson & McInerney 2001). Most studies of the effectiveness of computer systems compare with a control group that uses paper and pencil. But paper and pencil questions do not always easily translate into a computer form. This is especially a problem with assessment and is one of the reasons for the prevalence of multiple-choice type questions. There is also the problem of the assessment of partially correct answers, since this is usually dealt with in an explicit way in paper-based assessments; but it is difficult to do on computers. Lindsay (1999) compared the use of a computer algebra system with paper and pencil techniques. Paper and pencil was more effective. Threlfall et al (2007) analysed the differences between paper versions and computer versions of KS2 and KS3 questions. Pupils sometimes do better with the computer versions because there is more opportunity to explore different answers even though the system did not give them feedback to tell them if it was correct or not. Ashton et al (2006) argue the importance of computer systems being able to evaluate partially correct answers to mathematics questions, but there are few if any that can do this. Computer systems may also teach a subject in a completely different way to how a student may do it in class using pencil and paper. For example, Anderson's Algebra Tutor is used in over 2000 American schools but uses box diagrams to explain algebraic simplification which does not naturally translate to a paper based method. The quality of feedback from a teacher or a computer system is an important factor in how well pupils learn. Many systems only give a correct/incorrect response, which may not be very effective to aid learning unless they can eventually discover the correct result. Some systems give the correct answer and this can be done in a three ways: (1) immediately when the pupil gets the question wrong; (2) after a fixed number of attempts; (3) via a help system which has run to the end of helpful advice and just tells the student the answer. A few From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 26 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 systems go further and give an explanation of why the answer is correct. There is also an issue of the timing of feedback. Some systems, including MyMaths, do not give feedback until all the questions have been answered. But Tallent-Runnels et al (2005) give a review on how to teach online and identify the importance of prompt feedback. Received wisdom in Tutoring Systems suggested that users should be given a large amount of user control in navigation of the system (e.g. Wenger (1987)). But more recent research suggests that weaker students need stronger control by the system in order to learn effectively (Kopcha and Sullivan 2008). Ketamo and Alajaaski (2008) found that in the use of a multiple-choice based system there was a lot of guessing and students did not have the skill to choose appropriate materials to study. Similarly, Mezirow (1995) found that students tended to choose topics to study that they were already familiar with rather than the ones they needed to work on. Computer System and Method The xplus12 revision system covers a subset of KS3 Number and Algebra by means of interactive worksheets. The pupil is given simple feedback to each question on typing a return after the answer in the field. They are normally given five attempts, after which a multiplechoice version of the question replaces the type-in question. When the pupil has completed the first worksheet and attempted all the questions, the computer determines which are the pupil's weakest sections. The pupil then does a short remedial worksheet for each topic they are weak on. In Number 3 these worksheets just contain 4 interactive questions, but in Number/Algebra 2 there was an example as well. Once the pupil has completed the remedial worksheets the pupil then goes on to the last worksheet that is very similar to the first one. It has the same sections and also 2 questions per section. But there is no feedback and it works as a test. Qualitative Study Results A questionnaire was given to the pupils and there was also a discussion with four pupils. The closed questions had a five point Likert scale response. The question "How easy do you find xplus12 to use?" had a median of 4 as did the question "How much did xplus12 help you to learn maths?". The results indicated that the pupils liked the xplus12 system, found it easy to use, and felt they learned mathematics from it. There were also open questions such as "What do you particularly like about xplus12?" and several pupils indicated they liked the type-in questions turning into multiple-choice questions. On the negative side they thought there was more need of colour, animation and more games. Quantitative Study Results First Trial 22 students did the revision exercises but only 14 students provided reliable data that can be used for analysis. Eight students had to be excluded since they did not complete the revision exercises and so it is not possible to compare pre and post test results. The pre test scores are based on the pupils' first attempt at the questions. They may go on to attempt each question several times, and since the system eventually ends up with a multiple-choice question they eventually get every question correct. The post-test scores are just based on the answer since they are only given one attempt with no feedback. The pre test mean was 12.9 and the posttest mean of 15.6 with a standard deviation of 3, which represents an effect size of 0.9. Unfortunately this difference is not statistically significant at the 0.05 level. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 27 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 One of the problems encountered was that some pupils would attempt to get to the multiple-choice version of a question as soon as possible by guessing or writing rubbish. Five pupils were identified using this tactic and they may be labelled with the name "rubbishers". Second Trial In order to try to decrease the number of pupils "rubbishing", the mechanism for deciding when to present the multiple-choice question was changed from a threshold of 5 attempts. Instead, whenever the pupil answered a question too fast (less than a second) and had the wrong answer, it was assumed they were guessing. The penalty for this was that the threshold number of attempts was increased by one. Unfortunately this change was a disaster with two pupils making 60 attempts at questions! Four pupils had at least 30 attempts at a question. We can operationalise this by identifying pupils who have at least two questions with 15 or more attempts as "rushers". An unfortunate consequence of the changed attempt threshold policy was that the rushers spent a great deal of time on the pre test and most did not complete the post-test. A few pupils also repeated the test thus invalidating the pre-scores. One of the advantages of a computer system of this type is that it is possible to store all the attempted answers a pupil makes to a question and to identify misconceptions. A particular case of this occurred in question one of the pre test in BODMAS: 3 - 12 + 4 = ? Eleven pupils gave the correct answer of -5 but ten students gave the answer of 13, and three -13. Clearly there is a misconception here and pupils are taking the smaller number from the larger one. Interestingly this misconception did not occur in the pre-test with the BODMAS questions: 2+3×5=? 2+7-3=? and since the post test question is easier, they are not matched questions. Discussion It is clear that there is a need for the computer system to be improved. Too many pupils are disaffected and not enough learning is taking place. I take the theoretical stance that if pupils want to learn, then it should be possible to design a computer system to help them to learn. It can be argued that pupils behave as rushers and rubbishers because they cannot do the questions however hard they try. There is evidence of rusher and rubbisher personality types from the data since these behaviours persist, even across trials. Furthermore, several pupils type rubbish immediately they start a question, indicating no effort to try to answer the question correctly. On the other hand, no pupil is 100% a rusher/rubbisher (RR), but RRs are likely to attempt questions they find easier. It appears that they have a low threshold of effort and may resort to RR if their first 3 attempts are wrong. It is also true that rushers are likely to exacerbate rubbishing since the computer system was postponing the multiple-choice question as a result of their behaviour. A simple solution to engaging and helping RRs is to provide interactive help. A later version of the system provides optional interactive help after three attempts. This is also the approach taken by Anderson’s Cognitive Tutor, although their help system is much more complex. It is also clear that the multiple-choice version should be triggered always after five attempts, and not penalise RRs by making it later. Rushers can be slowed down by giving them a pop up message that tells them to slow down and they have to click to continue. It is easy for the system to identify if the pupil is answering questions too fast and incorrectly. It is more difficult to check if the pupil is From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 28 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 writing rubbish but long answers of consecutive numbers are usually indicative. A strong rubbish checker would be safer than a weak one since a genuine pupil would take a dim view of being challenged at writing rubbish. It is also clear that not enough pupils are learning from their mistakes. It is not clear if the pupils are actually benefiting from the multiple choice questions apart from letting them get onto the next question rather than being permanently stuck. The system does not currently record the timing data of the multiple-choice attempts but this could be changed. Thus, again the pupils could be slowed down with a suitable dialog if they are clicking the options too fast. More importantly, the pupils need to have an explanation of the answer. It is not sufficient just to be told the correct answer, and rushers are likely to just want to get onto the next question rather than digest the answer. One simple solution is to leave the correct answer and its explanation up for 5 seconds before they can proceed. A more sophisticated approach is to use animated text. Also, it is probably desirable for the system to check if the pupils think they understand their mistakes. Animation has also been used to improve the examples. Direct observation in class suggests that many pupils rush straight into answering the questions in a worksheet without reading the examples. Animating the examples makes them more interesting and encourages them to read them. Admittedly a disaffected pupil may not look at the animation, but there is nothing else for them to do on the computer. The system now also asks them how well they understand the example before proceeding to the questions. In principle, their answer can inform future navigation. It is straightforward to incorporate misconception handling in the system if the author knows of suitable misconceptions. These can be included as alternative multiple choice questions and checked for also during the type-in stage. It is clear from the qualitative study that the colour and appearance of the system needs to be improved. This came out clearly from the qualitative study, but it is possible that deeper issues may be more important. They also wanted more games. Some might argue that provision of games having completed the revision might provide motivation to do the work. But there is a danger that this will encourage some pupils to rush their work to get to the games. It is better to provide some form of intrinsic motivation within the system. Conclusion The xplus12 revision system improves learning but the quantitative results are not statistically significant. The new improvements of animated examples, preventing rushing, misconception handling and explanations of answers have been implemented and should improve learning further. If more schools use the system then further research should prove the advantages of these enhancements. References Aleven V., B. McLaren , I. Roll, and K. Koedinger. 2006. 'Toward Meta-cognitive Tutoring: A Model of Help Seeking with a Cognitive Tutor. International Jnl. of Artificial Intelligence in Education; Jul2006, Vol. 16 Issue 2, 101-128 Anderson J.R., A. T. Corbett, K. R. Koedinger and R. Pelletier. 1995. Cognitive Tutors: Lessons Learned. Journal of the Learning Sciences; 1995, Vol. 4 Issue 2, p167, 41p Ashton, H.S., C. E. Beevers, A. A. Korabinski, and M. A. Youngson. 2006. Incorporating partial credit in computer-aided assessment of Mathematics in secondary education. British Journal of Educational Technology, Vol 37, No 1, 2006, 93–119. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 29 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Beal, C. R., L. Qu, and H. Lee. 2008. Mathematics motivation and achievement as predictors of high school students' guessing and help-seeking with instructional software. Journal of Computer Assisted Learning, Dec2008, Vol. 24 Issue 6, p507-514 Biesenger K. & K. Crippen. 2008. The Impact of an Online Remediation Site on Performance Related to High School Mathematics Proficiency. Journal of Computers in Mathematics and Science Teaching (2008)27(1), 5-17 Dowson, M., & D. M. McInerney. 2001. Psychological Parameters of Students' Social and Work Avoidance Goals: A Qualitative Investigation. Journal of Educational Psychology, 93, 35-42. Egan C., A. Jefferies and J. Johal. 2006. Providing Fine-grained Feedback Within an On-line Learning System – Identifying the Workers from the Lurkers and the Shirkers. The Electronic Journal of e-Learning Volume 4 Issue 1, 15-24. Electronic Education Report. 2007. Study Reports Software Did Not Raise Test Scores in Reading and Math, Simba Information, Stanford, April 20 2007, 1-3. Kopcha T.J. and H. Sullivan. 2008. Learner preferences and prior knowledge. In Learner-controlled computer-based instruction. Education Tech Research Dev (2008) 56:265–286. Mezirow, J. 1995. Fostering critical reflection in adulthood : A guide to trans-formative and emancipatory learning. Jossey-Bass. Nguyen, D., J. Yi-Chuan, and D. Allen. 2006. The Impact of Web-Based Assessment and Practice on Students' Mathematics Learning Attitudes. Jnl. of Computers in Mathematics & Science Teaching; 2006, Vol. 25 Issue 3, 251-279 Tallent-Runnels M.K., S. Cooper, W. Y. Lan , J. A. Thomas, and C. Busby. 2005. How to Teach Online: What the Research Says. Distance Learning Vol 2. Issue 1 Thelfall J., P. Pool, M. Homer and B. Swinnerton. 2007. Implicit aspects of paper and pencil mathematics assessment that come to light through the use of the computer. Educational Studies in Mathematics (2007) 66:335–348. Tienken C.H. and M. J. Wilson. 2007. The Impact of Computer Assisted Instruction on Seventh-Grade Students Mathematics Achievement. Planning and Changing, Vol. 38, No. 3 & 4, 2007, 181-190. Wenger, E. 1987. Artificial Intelligence and Tutoring Systems. Morgan Kaufmann. Zhu, Q and G. Polianskaia. 2007. A Comparison of Traditional Lecture and Computermediated Instruction in Developmental Mathematics, Research and Teaching in Developmental Education. Fall2007, Vol. 24 Issue 1, 63-82 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 30 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Children's Difficulties with Mathematical Word Problems. Sara Gooding University of Cambridge, UK This article reports a study of the difficulties that primary school children experience whilst tackling school mathematical word-problems. A case study of four Year 5 children was conducted; this involved interviews which probed the children’s views of their own difficulties and discussions with the children as they tackled word problems. The data were qualitatively analysed using a thematic analysis approach based on categories of difficulty identified from existing literature. Examples of transcripts and responses which show the children experiencing difficulties are included, as well as the children's opinions on their difficulties. My interpretation of these findings, including proposed subcategories of difficulty, is also given. The report concludes with suggestions of methods – subject to further research – that teachers may use to help children overcome their difficulties with school mathematical word problems. Background Children’s poor performance with mathematical word problems is a trend that I became aware of very early on in my teaching career and one that an interest has been taken in by many who are involved in Mathematics education. By looking at the existing literature on children’s difficulties with mathematical word problems, I was able to gain a more detailed insight into the causes of children’s difficulties. Using the evidence from existing research, I formulated five categories of difficulties that children may experience whilst tackling mathematical word problems. These categories are presented below. Reading and Understanding the Language Used Within a Word Problem Difficulties in this category involve children not being able to decode the words used in a word problem, not comprehending a sentence, not understanding specific vocabulary and not having confidence or the ability to concentrate when reading. (Ballew and Cunningham 1982: Shuard and Rothery 1984: Cummins et al 1988: Bernardo 1999). Recognising and Imagining the Context in Which a Word Problem is Set These difficulties arise when children cannot imagine the context in which a word problem is set or their approach is altered by the context in which the word problem is given. (Caldwell and Goldin 1979: Nunes 1993). Forming a Number Sentence to Represent the Mathematics Involved in the Word Problem Children appear to find it harder to form a number sentence for some word problems structures than others. These difficulties can result in children not being able to select a calculation to perform or selecting an incorrect calculation. (Carey 1991: English 1998). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 31 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Carrying Out the Mathematical Calculation Difficulties can occur here with children’s selection of, and aptitude with calculation strategies (for example formal algorithms, pencil and paper methods and calculators). The context in which a word problem is given and the size of numbers involved can impact on children’s choice of a calculation strategy. (Verschaffel, De Corte and Vierstraete 1999: Nunes 1993: Anghileri 2001). Interpreting the Answer in the Context of the Question Children have been shown to not consider real-life factors and constraints when giving an answer to word problems which can result in giving an answer that is impossible in the context and therefore incorrect. (Verschaffel, De Corte and Lasure 1994; Wyndham and Säljö 1997; Cooper and Dunne 2000). Method Aims The aims of the study were to establish whether difficulties within the identified categories occur in English Primary Schools and, if they do, to find examples of children experiencing difficulties within the categories. I hoped that examples of children experiencing the range of difficulties may provide a resource for increasing teachers’ awareness of the difficulties. Data Collection Four children were selected to take part in the study. These children were from the Year 5 class that I taught; hence, I knew them well. They were selected on the criteria that they were willing and able to discuss the mathematics that they were doing and were working at a range of attainment levels in Mathematics lessons. The first element of the data collection involved interviews on the children’s views of difficulties they had experienced with mathematical word problems. The second element involved the children working individually through sets of equivalent word problems and discussing their processes and difficulties with me. There were five sets of equivalent word problems that each child attempted. Each set was given in a different condition, with a different form of help given in each. Each form of help corresponded to one of the previously identified category of difficulty. For example I read the word problem to the child, explained any vocabulary and simplified sentences in condition one to correspond to the first category. I offered forms of help in the belief that if I gave a specific type of help and children then solved a problem, I could identify where the original difficulty lay and be aware of which kinds of help allow children to overcome certain types of difficulty. Data analysis The first stage of data analysis involved analysing and coding interview transcripts and recordings. Excerpts were coded under a category of difficulty if they showed opinions on that difficulty, a child experiencing that difficulty, or a child competently completing a process, therefore not having that difficulty. Any un-coded data were then checked for a need for new categories or reported as ‘Other Findings’. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 32 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The second stage of data analysis involved analysing and coding all incorrect or no responses to word problems. Transcripts and children’s jottings or workings were used to code as to which difficulty prevented a correct answer being given. Using the coded data, I was able to create subcategories within some of the five main previously identified categories of difficulty. Finally, I selected illustrative examples of children experiencing difficulties or giving opinions on difficulties from each category and subcategory. Examples were picked using the criteria of being typical and not extreme. The Results and Discussion • • • • • • • The finalised categories and subcategories of difficulties formed are: Reading and Comprehension o Decoding the Words in a Word-Problem o Understanding the Meaning of the Words and Sentences Reading All of the Information Distracting Information Imagining the Context Writing a Number Sentence Carrying Out the Calculation o Lack of Accurate Methods for Calculating o Making a Mistake When Calculating Interpreting the Answer in the Context of the Question o Giving an Answer that is Possible or Likely o Transferring an Answer into the Required Format I have selected examples of children experiencing the above difficulties, or opinions on difficulties from the categories ‘Reading and Comprehension’ and ‘Interpreting the Answer in the Context of the Question’ to present below. Reading All of the Information The following example shows Liam giving an incorrect response to a word problem because he has not read or comprehended all of the text in the question. Figure 1, a word problem given to Liam and the transcript of the conversation that followed. Liam: This is a tricky one. I’m gonna have to say it’s the big pack there. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 33 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Researcher: Why is that then? Liam: Well the big pack is 80p, now half price so 40p. But these packs cost 30 plus 30 plus 30 which is … 90 so yeah 40p. Researcher: How much did you say they would be [small packs]? Liam: 90p and you could buy 2 big packs for that! I acknowledge the possibility that Liam may not have understood the term ‘3 packets for the price of 2’ but his confidence in his final answer leads me to believe that he has simply not read or disregarded the information in the second star, leading him to get the correct answer, but for the wrong reasons. His mistakes here may be related to how the word problem is arranged on the page as all of the required information for the larger packet of pencils is in the star, but the information for the smaller packets is not. Distracting Information Two comments related to this category are shown below: Liam: I like the little flashy. Liam: Hah Patrick! Either you watch SpongeBob or you have a kid that watches SpongeBob. Liam’s first comment is related to the illustrations in the word problem in Figure 1 and his second to a word problem featuring a character called Patrick. Although Liam’s observations and comments may not lead to incorrect answers, they show that his attention may not be focused on the mathematics required to answer the problem and therefore these distractions may cause him difficulties. Giving an Answer that is Possible or Likely Rachel was given a calculator to use to answer a word problem about the number of children going on a school trip and gave the answer of ‘8.333333 children’. Rachel: Researcher: 8 .333333 so 8.3 dot [recurring]. Is that the number of children? Rachel: Yeah. Here, although Rachel had carried out an appropriate calculation, she has not given a correct answer and does not appear to realise, or consider it important that it is impossible to have a third of a child on a school trip. Another example of a child experiencing difficulties within this category is below: Liam: I can tell you this drink’s gonna cost loads more than a Mars bar. It’s 35 for the Mars bars so take away 35 which would be …that would be 85. It’s £85. £85 for a fizzy drink. Researcher: That sounds a lot doesn’t it? Liam: Mmmm [in agreement] I’m not going to this shop. Researcher: Do you think that this sounds like a realistic shop then? Liam: Yes cos I’ve seen things like my Pokemon cards and they cost £3.99 for one pack. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 34 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Researcher: So you think that there could be a shop that sells a fizzy drink for £85? Liam: Yes. Figure 2, a word problem given to Liam and the transcript of the conversation that followed. Liam’s difficulties are caused by problems transferring between pounds and pence, but his perception that £85 is a possible answer, even after attention was drawn to it, mean that he could not identify his error and went on to give an incorrect answer. This example is closely linked to the subcategory of ‘Transferring an Answer into the Required Format’ due to the difficulties he had with dealing with money. Conversely to this example, when answering an equivalent word problem to the one shown in Figure 2, Liam experienced similar difficulties with knowing when values were in pounds or pence, but was able to use his judgment of what is realistic to identify his error and go on to give a correct answer. Transferring an Answer into the Required Format The following excerpt shows Fiona having difficulties transferring her answer from a decimal number into money: Fiona: Researcher: Fiona: Researcher: Fiona: He has to pay 15 for four. But it’s half price isn’t it so how much is that? Uuuuh 7.5. OK. What’s that in money? £7. 05. Although Fiona was able to carry out the calculation of ‘15 ÷ 2 =’ correctly, she had difficulties when trying to write that value in the standard format for money. Other Findings Other findings were also identified. These involved a child having difficulties because he was not using jottings. When prompted to write numbers down as he was calculating mentally, he was able to carry out a calculation more effectively. Manipulatives were also shown to help a child to find a correct answer to a word problem after previously not being able to for equivalent word problems. This shows that manipulatives may be a useful tool for helping children to answer word problems marginally beyond their current grasp. Children also made comments which suggested that they were able to identify the equivalence between word problems. This ability to recognise equivalence could imply that showing children how to correctly find an answer to a word problem may help those children to also solve equivalent word problems. Recommendations As a result of examining my findings alongside existing literature, I have compiled a list of strategies that teachers and researchers could trial to help children to overcome difficulties with mathematical word-problems: • Encourage children to read the word-problems thoroughly; • Teach children which kinds of information may be important; • Ensure that children practise solving word-problems to allow them to be able to recognise the structure of word-problems and therefore know when to use each calculation; From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 35 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 • • • • • Consider giving children manipulatives to support the solving of word-problems currently beyond the scope of their ability; Encourage children to write down their workings so that they do not become unnecessarily confused; Encourage children to check if their answer satisfies the criteria of a question. For example if it is in the correct format; Teach children to calculate with monetary values; Encourage children to check if an answer is possible in the context of the question. References Anghileri, J. 2001. Development of Division Strategies for Year 5 Pupils in Ten English Schools. British Educational Research Journal, 27(1), 85-103. Ballew ,H., and Cunningham, J. 1982. Diagnosing Strengths and Weaknesses of Sixth-Grade Students in Solving Word Problems. Journal for Research in Mathematics Education, 13(3), 202-210. Bernardo, A. 1999. Overcoming Obstacles to Understanding and Solving Word Problems in Mathematics. Educational Psychology, 19(2), 149-163. Caldwell, J., and Goldin, G. 1979. Variables Affecting Word Problem Difficulty in Elementary School Mathematics. Journal for Research in Mathematics Education, 10(5), 323-336. Carey, D. 1991. Number Sentences: Linking Addition and Subtraction Word Problems and Symbols. Journal for Research in Mathematics Education, 22(4), 266-280. Cooper, B., and Dunne, M. 2000. Assessing Children’s Mathematical Knowledge: Social Class, Sex and Problem-solving. Buckingham: Open University Press. Cummins, D., Kintsch, W., Reusser, K. and Weimer, R. 1988. The Role of Understanding in Solving Word Problems. Cognitive Psychology, 20, 405-438. English, L. (1998). Children’s Problem Posing within Formal and Informal Contexts. Journal for research in Mathematics Education, 29(1), 83-106. Nunes, T., Schliemann, A.D., and Carraher, D.W. 1993. Mathematics in the Streets and in Schools. Cambridge: Cambridge University Press. Shuard, H., and Rothery, A. 1984. Children Reading Mathematics. London: John Murray (Publishers) Ltd. Verschaffel, L., De Corte, E and Lasure, S. 1994. Realistic Considerations in Mathematical Modeling of School Arithmetic Word Problems. Learning and Instruction, 4(4), 273294. Verschaffel, L., De Corte, E. and Vierstraete, H. 1999. Upper Elementary School Pupils’ Difficulties in Modelling and Solving Nonstandard Additive Word Problems Involving Ordinal Numbers. Journal for Research In Mathematics Education, 30 (3), 265-285. Wyndhamn, J., and Säljö, R. 1997 Word Problems and Mathematical Reasoning: A Study of Children’s Mastery of Reference and Meaning in Textual Realities. Learning and Instruction, 7(4), 361-382. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 36 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Some initial findings from a study of children’s understanding of the Order of Operations Carrie Headlam and Ted Graham University of Plymouth This paper presents some of the initial findings of a study into the strategies used by children to solve arithmetic and algebraic problems requiring the appropriate use of the order of arithmetic operations. The research has utilised graphics calculators which have been programmed with Key Recorder Software as a data collection tool. This has enabled the researchers to analyse the children’s approaches to some of the questions posed by observing their calculator keystrokes. Interviews with both teachers and pupils will be used to link the pupils’ strategies with the teaching methods used, and an initial analysis of observed misconceptions has been carried out. Initially this study has involved children in the UK and in Japan, where teaching methods differ substantially. Introduction The principle of the Order of Operations is a cornerstone of the understanding of arithmetic. It is necessary in order to correctly perform arithmetic calculations and it is also an essential prerequisite to the beginnings of the understanding of algebraic structure and the ability to understand and apply the principles of algebraic convention correctly. More fundamentally, it could be argued that an acknowledgement of the need for a convention in arithmetic is an important step in the development of an appreciation of mathematical convention in other areas of mathematics and indeed to the sense of learning the language of mathematics, where standard rules are necessary in order to assist in clear communication. In considering the ways in which algebra is developed in different countries, the relationship between algebra and arithmetic is usually characterised by the definition of algebra as generalised arithmetic. Thus the view of ‘arithmetic then algebra’ dominates school curricula in most countries. The reason for this, according to Lins and Kaput (2004) can be found in the strong dominance of Piagetian constructivism. As algebra would require formal thinking, while arithmetic would not, and as formal thinking would correspond to a later developmental stage, algebra should come later than arithmetic. (page 50) This is seen in the work of Kuchemann in Hart (ed) (1981) for the Concepts in Secondary Mathematics and Science (CSMS) project who combined the view of algebra as generalised arithmetic with the Piagetian developmental view. Lins and Kaput (2004) argue that the most visible result of Kuchemann’s work is a reported link between different uses of letters in ‘generalised arithmetic’ and Piaget’s levels of intellectual development. (page 50) Hewitt (2003) considered students’ reading of formal algebraic notation and he observed that many errors made by students could be accounted for by the strict left-to-right reading of formally written arithmetic statements. He also considered how students read word statements, acknowledging that expressing non left-to-right order in written words can be problematic as well since words do not possess a set of notational conventions, such as brackets (page 34) In the National Strategies Secondary Mathematics Exemplification (DCSF, 2008) the learning objective relating to this states that pupils should be taught to: Use the order of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 37 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 operations, including brackets. (page 86) One example of a learning outcome is that a year 7 student should be able to perform the calculation either mentally or using jottings. This objective is also linked to calculator methods, with the expectation that a pupil should be taught to carry out more complex calculations using the facilities on a calculator (page 108) and with the order of algebraic operations, where pupils are expected to understand that algebraic operations follow the same conventions and order as arithmetic operations. (page 114) The exemplification makes it clear that pupils are expected to be able to use a scientific calculator efficiently when evaluating more complex mixed operations. The methods for teaching this topic can vary, but one common theme in some countries is to use a mnemonic to aid the memorisation of the order of operations. In the UK this is commonly BIDMAS or BODMAS: Brackets Index Division Multiplication Addition Subtraction Or Brackets Of (Order) Division Multiplication Addition Subtraction In the USA the mnemonic PEMDAS is commonly used: Parentheses Exponents Multiplication Division Addition Subtraction Clearly this may have its uses in remembering the “rule” once the concept has been understood, but it is the clear understanding of the underlying principle and conventions that enable it to be put into practice. This includes the understanding of index notation and the recognition of a fraction for division. Thus it is far from merely being a case of learning a mnemonic; a sound understanding of mathematical notation and structure is required in order to carry out a calculation of the type given in the National Strategies Mathematics Exemplification. It is this deep understanding that lays the foundations for an understanding of algebraic structure. It is interesting to note that the use of mnemonics is not referred to at all in the National Strategies documentation, and yet many text books and other resources used in the UK and in the USA encourage it. In contrast, from conversations with Japanese teachers it would seem that mnemonics are never used in Japan. Indeed in the Japan National Mathematics Program (2000) the order of operations is not specifically referred to at all. The teaching methods are very didactic with a large emphasis on whole-class teaching and repetition of questions, focusing on algebraic structure. The study: Context and Methods The primary aim of this study is to examine the ways that pupils perform calculations which require the correct use of the order of operations and to study the misconceptions that may arise. One tool that will be utilised in order to carry out the research will be a piece of software that was developed as a research tool by Texas Instruments in conjunction with the University of Plymouth. This software is called the Key Recorder and can be loaded onto the more recent models of the TI graphics calculator. It has been used as a data collection tool in a small number of research projects (For example: Graham, Headlam, Honey, Sharp and Smith, (2003), Berry, Graham and Smith (2003), Smith (2003) Berry, Graham and Smith (2005), Berry, Graham and Smith (2006), Sheryn (2005), Sheryn (2006a), Sheryn (2006b) Graham, Headlam., Sharp and Watson (2007) ) This study involves classes of children who have been taught about the Order of Operations and who would therefore be expected to be able to perform calculations based From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 38 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 upon these principles. These children are in the age range 12 – 14 (Years 8 and 9 in UK schools). The children complete worksheets of questions involving a variety of calculations, some with and some without a calculator. For the calculator-based questions the children are provided with a TI-84 graphics calculator which has the Key Record software running. When the children’s work has been initially analysed, some of the children are then interviewed in order to follow up and pursue questions that arise. Initial Findings A pilot study was carried out in the UK and in Japan. In each country one class of students was involved. In the UK this was a class of 20 middle ability students in year 8 (age 13). The Japanese class consisted of 33 mixed ability students aged 14. Both classes had been taught the principles of the order of operations as part of their scheme of work, and had also been taught simple algebraic conventions, including substitution of letters for numbers in algebraic expressions. In the pilot study the graphics calculators were not used; the children were given one worksheet to complete without using a calculator. As a result of this study the worksheets were adapted and a second worksheet produced. The second worksheet contained questions which were identical in structure to those in the first worksheet but involving decimal numbers which would encourage the use of a calculator. The children would be given a graphics calculator with the Key Record Software running which they were asked to use when completing the second worksheet. The main study has now been carried out in a further two classes in UK schools, both middle ability year 8 classes. The children completed both worksheets, and afterwards their worksheets was analysed alongside the Key Record data. Some children were then interviewed and the teachers were also interviewed. From the pilot study it was interesting to investigate the questions that the Japanese children got wrong, and to analyse their ways of working. There was a general tendency to treat all the questions as algebraic, even though they were mainly numerical. The calculation that was answered incorrectly by most Japanese pupils was question 10: the calculation (1 + 2)2 was in many cases calculated by expanding the brackets first: Figure 1 Examples of three Japanese pupils’ work on question 10 and it was observed that the incorrect answers were more likely to be due to careless errors rather than revealing misconceptions. When calculated in this way, the need to remember a rule such as BIDMAS becomes unnecessary, although there is still a need to know that indices are evaluated first in the numerator. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 39 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 This question was approached differently by the children in the UK. One girl’s attempts at questions 9 and 10 of the non- calculator worksheet are shown in figure 2: Figure 2 One UK pupil’s work on questions 9 and 10 on non-calculator worksheet In question 10, although she correctly evaluated the denominator, involving brackets, she did not evaluate the index first in the numerator. When interviewed, she was asked what she was thinking about when doing the worksheet, she immediately answered “I was thinking of BIDMAS” but when asked what this stood for she hesitated and then answered Brackets, individual, divided, multiply, addition and subtraction” When prompted about what the “I” stood for she did not know, and even when asked about the word “Index” she was not sure what this meant, although when she was shown the number she immediately said “ oh – to the power of 2?” which revealed that she understood what a power was, but had not related this to the word “index” and therefore could make no sense of the I in BIDMAS. The same misconception is also revealed in her answer to question 9. Her attempts at the corresponding questions on the calculator paper are shown in figure 3: Figure 3 The same pupil’s work for questions 9 and 10 of the with-calculator worksheet Analysing her keystrokes revealed that she used her calculator efficiently with a good grasp of the need to evaluate each of the numerator and denominator first before dividing. Her incomplete understanding of the BIDMAS rule was overcome by using the calculator efficiently. For question 9 the pupil correctly evaluated the numerator on her calculator, and then evaluated the denominator: Once happy with the denominator she proceeded to finish the calculation: It would seem that she wanted to check that the answer from the first line was the same as evaluating the power first, then adding, thus indicating that she had an idea of the correct order, even though she had got the equivalent non-calculator question wrong. When she was using her calculator she was able to investigate the effect of calculating the power first and successfully confirm that this was the correct way to carry out the calculation. In From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 40 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 question 10 the pupil now seemed satisfied that the calculator would produce the correct result for both numerator and denominator and worked efficiently to produce the correct answer: Conclusions In this first part of the study it would seem that there are substantial differences between Japanese and British childrens’ ability to carry out arithmetic calculations and the approaches used. The Japanese pupils relied upon algebraic approaches which they generally employed correctly, but this sometimes caused them to make the calculation unnecessarily complicated and they made algebraic mistakes. The children in the UK relied heavily upon remembering BIDMAS which worked well if they remembered it correctly but broke down if they did not fully understand what all the letters stood for. However the use of a calculator enabled the pupils to experiment and discover the conventions, which reflects the teaching approaches used. References Berry, J., T. Graham and A.Smith. 2006. Observing Student Working Styles When Using Graphic Calculators to Solve Mathematics Problems International Journal for Technology in Mathematics Education 37 no.3 291-308 DCSF 2008. The National Strategies: Secondary Mathematics Exemplification onlinehttp://nationalstrategies.standards.dcsf.gov.uk/strands/881/66/110129 Graham, T., C. Headlam, S. Honey, J. Sharp and A.Smith. 2003. The Use of Graphics Calculators by Students in an Examination: What do they really do? International Journal of Mathematical Education in Science and Technology 34 no.3 319 – 344 Graham.T., C. Headlam, J. Sharp, and B. Watson. 2007. An investigation into whether student use of graphics calculators matches their teacher’s expectations International Journal of Mathematical Education in Science and Technology 9 no.2 179 – 196 Hart, K.M. (Ed). 1981. Children’s Understanding of Mathematics: 11-16 Murray Hewitt, D. 2003. Some issues regarding formal algebraic notation in Pope,S. (Ed) Proceedings of the British Society for Research into Learning Mathematics’ 23 no.1 Japan Society of Mathematical Education. 2000. Mathematics Program in Japan Lins, R and J. Kaput. 2004. The Early Development of Algebraic Reasoning: The Current State of the Field New ICMI Studies Series 8 47 – 70 Sheryn, S.L. 2005. Getting an Insight into How Students Use their Graphical Calculators Proceedings of the British Society for Research into Learning Mathematics 25 no.2 103 – 108 Sheryn , S.L. 2006a. What do Students Do with Personal Technology and How Do We Know? How one student uses her graphical calculator International Journal of Technology in Mathematics Education 13 no.3 151 – 158 Sheryn, S.L. 2006b. Investigating the Appropriation of Graphical Calculators by Mathematics Students Ed.D. Thesis University of Leeds School of Education From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 41 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 42 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The role of attention in the learning of formal algebraic notation: the case of a mixed ability Year 5 using the software Grid Algebra Dave Hewitt School of Education, University of Birmingham The learning of formal algebraic notation is seen as a challenge for many students (Van Amerom, 2003). The act of symbolising is not so much a problem. Hughes (1990) showed that very young children are able to symbolise in order to record how many items were placed into a tin. The problem is more concerned with interpreting and using someone else’s notation, in this case the socially agreed convention of formal algebraic notation. In activities where students are asked to find rules for pictorial patterns, they can often find rules but expressing those rules in formal notation is seen as difficult. An indication of this is seen in the levels of “patterning abilities” used by Ma (2008) and based upon Orton and Orton (1999) where the highest level of this scale is students’ ability to express their rule in formal notation. The difference between finding rules and expressing those rules in formal notation highlights a difference I see between these two aspects of mathematics. Spotting patterns and finding rules is algebraic in nature whereas how to express those rules is a matter of language and notation. This is an example of the arbitrary and necessary divide (Hewitt, 1999) where I call those things which are socially agreed, such as names and conventions, arbitrary as they can appear to feel so for a learner and are a matter of choice. Those things which are necessary concern properties and relationships and are not a matter of choice. For students to learn the arbitrary they need to be informed of these socially agreed names and conventions, whereas students can come to know the necessary through their own mathematical activity. This implies there are different pedagogic challenges for a teacher between the arbitrary and the necessary. The learning of algebraic notation is essentially about accepting and adopting a socially agreed convention and thus lies in the realm of the arbitrary. The finding out of rules is a different matter and lies in the realm of the necessary. Thus, the teaching of notation is pedagogically a different challenge to the teaching of algebra and requires me providing notation one way or another and helping students to accept, rather than question, notation and adopt it within their work. The Software Figure 1: the first two rows of the grid The vehicle I used to provide notation is the software Grid Algebra3 which is based upon a multiplication grid (see Figure 1 where just the first two rows are shown). Movements between the numbers can be made through dragging a number from one cell to another cell. A move to the right would involve addition, to the left subtraction, down involves multiplication 3 Available from the Association of Teachers of Mathematics From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 43 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 and up division. These movements can be carried out with the software and result in formal notation as indicated in Figure 2. Figure 2: some movements on the grid As movements were made students began to see the formal expressions as historical artefacts, representing journeys which had been made on the grid. At times the notation was the only evidence available for journeys which had taken place and thus there was a need for students to use this notation in order to engage with tasks such as trying to re-create the journey made. Brown and Coles (1999) have talked about setting up activities so that there is a need to use algebra and here activities were set up where there was a need to use the notation. Indeed the formal notation was subordinate (Hewitt, 1996) to some of the tasks as the notation was not a requirement in students’ understanding the task yet the students were forced to go through the notation in order to carry out those tasks. Such activities involve students meaningfully practising interpretation of the formal notation even if that notation is relatively new to them. The study The study took place in an inner city multi-cultural primary school which has a greater percentage of children with free school meals than the national average and whose KS2 results are below the national average. There were 21 students in the group taught and these were the total number from two Year 5 classes who returned parental permission slips and who were personally happy to be part of the study. They were a mixed ability group ranging from teacher assessed National Curriculum levels two to five. The class was taught by myself on three consecutive days for one hour on the first day and one and half hours on the other two days, giving a total of four hours. Some of this time was spent with them working on pen and paper tasks and some of this time included two half hour sessions with them working individually or in pairs in a computer room. The rest of the time the class worked with a whole-class focus on the Interactive Whiteboard. Except for the pen and paper tasks, all the time was spent working with the Grid Algebra software, and even the pen and paper tasks were largely based upon the software, some of them being printout of sheets accompanying the software. Their class teachers reported that none of them had been taught any formal algebra, including use of a letter, or used formal algebraic notation. Some key aspects of the way in which I worked with the students included a general absence of anything being explained by myself. Instead there was use of questioning and an expectation that students would notice and abstract rules concerning how notation was written along with the mathematical processes required to solve linear equations. There were a number of stages in the foci of the teaching sessions, which were: students meeting notation for multiplication (brackets) and division (division line) and how addition and subtraction fit in with those in larger expressions (placing of an addition/subtraction sign following a division); order of operations within an expression; introduction of letters; substitution; multiplying out brackets; inverse operations and solving equations. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 44 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Several of the tasks involved students re-creating journeys given final expressions, or finding where a journey began given only the final expression. These involved students having to work through the notation as it was the only information provided and in this way the notation was subordinate to the task in that reading the notation, and order within the notation, was a necessary aspect of carrying out the task. This work I am not reporting here but I will comment that all students at the end of the lessons were writing their work in correct algebraic notation. What I do report here is a small window of time when the students first met and began to work on writing expressions involving division. Attention Attention has been discussed as part of the teaching and learning dynamic in different ways. Mason (1989) has talked of a shift of students’ attention which is indicative of a learning process whilst Ainley and Luntley (2007) have discussed the “attentional” skills which experienced teachers exhibit. Wilson (2009) has discussed the relationship between teacher and student in relation to attention. She talks of the notion of alignment between teachers’ practices and students’ focus of attention. My interest in attention concerns pedagogic decisions about where a teacher might wish students’ attention to be placed at particular moments in a lesson. This will involve deliberate teaching acts which attempt to direct students’ attention onto particular aspects of what is being discussed. My interest is in catching such teaching moments but also following through and examining evidence of where students’ attention might actually be placed over time following such teaching acts. Methodology Lessons were video recorded and written work was collected. The later viewing of these I describe through the metaphor of glasses. The viewing of anything is never carried out neutrally. In viewing something I bring with me my experience and the particular interests which are currently present for me. In this case my viewing glasses contained lenses which represented my particular interest in the placement of attention. Thus the viewing of video and students’ work was carried out through these lenses. The act of noticing (Mason, 2002) particular incidents or aspects of writing is indicative of links made between those artefacts and my particular interest. What follows is a description of some of those artefacts along with my accounting for their significance to me in relation to my interest in the placement of attention. In particular, this paper will focus the notation associated with the division line. The notation for division: attention on particularities of expressions The notation of addition and subtraction was familiar to the students and so the focus up to this point in time was on seeing an expression as an object rather than a process to be carried out. For example, seeing ‘2+1’ as how it is rather than wanting to carry out that addition and say three. However, the notation of division was new to these students and attention was now focused onto particular aspects of the expression which were novel for the students. Having made a downwards movement to create a multiplication I asked them what was the opposite of doing this. They responded almost in unison with “divide” and after me asking “…by what?” they said “Divide by two”. This continued as follows: DH: Divide by two? OK. So when I go from 8 to 4 I’ve got to do eight divided by two. Is that right? Students: Yes. DH: OK, and here it is [movement made resulting in ,8-2. ] From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 45 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Students: Oh... [A few seconds of quiet]... Fraction [several voices] / It’s a fraction but it stands for divide [one voice]. DH: So, say this to me [pointing at the 8]. Students: Eight… [DH points to the divide line] … divide/over/add [DH points to the 2] … two. DH: OK. Just say divide. OK? Ready? [Points to the 8 then the divide line and then 2]. Students: [In time with the pointing] Eight... divided by... two. DH: OK? Eight divided by two. For many students this notation appeared to be a surprise. There were also several voices saying that ,8-2. was a fraction and I would conjecture that this form of notation had previously only been used under the topic of fractions, whereas the notation 8÷2 had been used for division. Here attention was brought to the individual parts of ,8-2. in order to learn to read how this expression was going to be said within these lessons, as a division rather than as a fraction. This required taking something which the students viewed as an object, a fraction, and seeing it as a process, a division. This was the opposite shift of attention which was involved earlier when I wanted to help students see an expression as an object rather than a process. Following the above sequence, students were asked to write down what it would look like if I dragged the ‘6’ in row two up to the cell above, then (having already hidden the ,8-2. ) when the ‘8’ would look like dragged up (so a repeat of what we had done above) and then, having dragged the ‘8’ in row two to the left to produce ‘8-4’, what it would look like if ‘8-4’ was dragged up to the cell above. For each of these, students wrote down what they thought it would look like and then the action was carried out on the grid to reveal what it did actually look like. Out of the 18 booklets which were finally collected in on the last day, 12 of them wrote down ,6-3. , two of them wrote ,3-6. , another wrote 6÷3 and the remainder wrote ,6-2. . My conjecture is that the attention of many of the students was on the grid and the start and finish cells of this little journey. It started with the number ‘6’ and the cell which it would end on had ‘3’ in it. What was not visually present on the grid was the fact that the operation was division by two. Instead many students have used numbers which were visible. In particular the ‘3’ is visually above the ‘6’ and this might be a factor why two of them wrote ,3-6. . What was adopted by all but one student was the use of a horizontal line for division. The challenge of ‘8-4’ divided by two was new to them as they had only seen single numbers being divided. This was successfully written by 11 of the 18 students. However, since the finishing cell of this movement contained the number ‘2’ it was not clear whether these students wrote dividing by two because they were aware of the mathematical operation or whether the ‘2’ came from them attending to the final cell. So, in retrospect, this was not a good example to have chosen. Four of the students who wrote something different seemed to carry out the calculation of 8-4 and so wrote either ,4-2. or ,2-4. . Two other students wrote: This appeared to indicate the two stage nature of what was happening. The first operation of subtraction was carried out and the second operation of division shown with that answer. This reminded me of the common misuse of the equals sign when someone writes 8−4=4÷2=2, only this time without the final division calculation taking place. One other person wrote: From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 46 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The fact that previous examples had a division line underneath a single number might have been generalised to this example. Since notation is arbitrary, there would be no way of someone knowing how the case for single numbers is generalised to cases for expressions, and so this seemed a perfectly reasonable way to generalise the use of the division line. Division not only brings with it the issue of the division line but also adds a notational issue when it is followed by addition or subtraction. Shortly after the sequence above I said I would drag ,8−4+6-2. one cell to the left (where this would produce ,8−4+6-2.−1 ). Of the 13 who attempted this, six wrote down the conventionally correct answer with five writing ,8−4+6−1-2. . This is again a novel situation for them and so how would they be expected to know what is the conventional way of writing this? When I moved the expression across to produce the result there were a number of students who said “Yessss!” followed by a small gap of a couple of seconds followed by several voices saying “Uh?” with one student saying “Why’d the one have to go there?” I did not respond to this and I am not convinced the question was actually directed at me. It appeared more an expression of surprise. I did, however, want to focus attention, in a factual way rather than an explanatory way, as to this feature of where the subtraction sign was written. In doing so I offered a visual and aural image; I pointed the pen at the left hand end of the division sign and moved to the right along the line saying “Just notice, zzzzzippp,...” and then continued along this line taking my pen off briefly before putting it back down again along the subtraction line and continued by saying “... ping!” This aural image was heard being repeated by a number of the students as I then went about clearing the grid and beginning to start the next challenge. None of the students asked again for any explanation of why the notation was how it was. In the expressions the students wrote down for the above, there was only one which involved a calculation: ,8−4+6−1=4-2. . The ‘=4’ is where we finished up after this movement and so this equation could be considered mathematically correct and just not written conventionally. This whole sequence of the lesson from when the first division was carried out lasted under seven minutes and in this time there was a significant shift towards writing expressions conventionally; the division line was adopted and there were nearly no calculations being carried out, indicating a shift back from process to object. Final remarks Coming to accept and adopt formal notation involved a development where attention was shifting from process to object, and occasionally vice versa. At times attention appeared to be on what was visible within the grid rather than the operations which were visually implicit even if verbally explicit. Pedagogic techniques were occasionally used to draw attention onto particular detail and at other times to ignore detail and treat an expression as an object upon which further operations were being carried out. I note that there was a general sense of acceptance of the notation, rather than questioning why it appeared how it did. Even when this did happen, with the initial creating of a division and then with a subtraction following a division, any comments did not appear to be directed towards me. I conjecture that this was because I was not doing the writing of those expressions. Instead the computer created them and students are quite used to accepting all sorts of arbitrary conventions which form part of interactions with technology. References Ainley, J. and Luntley, M. (2007), 'The role of attention in expert classroom practice', Journal of Mathematics Teacher Education, 10(1), pp. 3-22. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 47 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Brown, L. and Coles, A. (1999), 'Needing to use algebra - a case study', in O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, Haifa, Israel Institute of Technology, pp. 153-160. Hewitt, D. (1996), 'Mathematical fluency: the nature of practice and the role of subordination', For the learning of mathematics, 16(2), pp. 28-35. Hewitt, D. (1999), 'Arbitrary and Necessary: Part 1 a Way of Viewing the Mathematics Curriculum', For the Learning of Mathematics, 19(3), pp. 2-9. Hughes, M. (1990) Children and Number. Difficulties in Learning Mathematics, Oxford: Basil Blackwell. Ma, H.-L. (2008), 'The algebraic thinking of 5th and 6th graders to solve linear patterns with pictorial contents ', Research and Development in Science Education Quarterly, 50, pp. 35-52. (In Chinese). Mason, J. (1989), 'Mathematical abstraction as the result of a delicate shift of attention', For the learning of mathematics, 9(2), pp. 2-8. Mason, J. (2002) Researching your own practice: the Discipline of Noticing, London: RoutledgeFalmer. Orton, A. and Orton, J. (1999), 'Pattern and the approach to algebra', in A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics, London: Cassell, pp. 104-120. Van Amerom, B. A. (2003), 'Focusing on Informal Strategies When Linking Arithmetic to Early Algebra', Educational Studies in Mathematics, 54(1), pp. 63-75. Wilson, K. (2009), 'Alignment between teachers’ practices and pupils’ attention in a spreadsheet environment', in M. Tzekaki, M. Kaldrimidou and H. Sakonidis (Eds), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, Thessaloniki, Greece, 353-360. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 48 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Lower secondary school students’ attitudes to mathematics: Evidence from a large-scale survey in England Jeremy Hodgena*, Dietmar Küchemanna, Margaret Browna & Robert Coeb a King’s College London, bUniversity of Durham In this paper we present some preliminary data from the ESRC funded ICCAMS project about current student attitudes to mathematics at Key Stage 3 in England. We compare attitudes by sex and by attainment. Whilst the data largely confirms existing findings, an unexpected result was that a very high proportion of students responded that, for mathematical success, effort was more important than ability. We also present some interview data concerning student attitudes. Keywords: Attitudes, attainment, gender Background Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) is a 4-year research project funded by the Economic and Social Research Council in the UK (Hodgen et al. 2008, 2009). In this paper, we report and discuss early findings of the study regarding students’ attitudes drawing on both survey and interview data. Methods and theoretical framework Phase 1 of the ICCAMS project consists of a large-scale survey of 11-14 years olds’ understandings of algebra and multiplicative reasoning in England using three tests of mathematical understanding and an attitudes questionnaire. The three mathematics tests, covering algebra, decimals and ratio, were originally used in the late-1970s as part of the Concepts in Secondary Mathematics and Science (CSMS) study. (See Hart 1981, for a discussion of the test development.) The attitudes test is adapted from previous work (Boaler, Wiliam, and Brown 2000). In Phase 2 of the study we are conducting a collaborative research study with eight teachers extending the investigation to classroom / group settings and examining how assessment can be used to improve attainment and attitudes. Participants In June and July 2008, tests were administered to a sample of approximately 3000 students across Key Stage 3 (KS3) from 10 schools and approximately 90 classes. In England, KS3 refers to the first three years of secondary school: Years 7 (ages 11-12), Year 8 (ages 12-13) and Year 9 (ages 13-14). Since the survey was conducted at the end of the school year, the vast majority of these students were at the older end of these age ranges: 12, 13 and 14 years old, respectively. We report here on the attitudes of a sub-sample of 1422 students for whom we have linked data on attitudes and attainment on the Ratio test, enabling us to report on the relationship between students’ attitudes and attainment. The sub-sample consisted of 494 Year 7 students, 524 Year 8 students and 394 Year 9 students. Of the total, 748 were boys and 674 were girls. The Ratio test reports students’ mathematical attainment using a hierarchy of levels from Level 0 up to Level 4 (Hart 1981; Brown, Küchemann, and Hodgen Forthcoming). In the sub-sample, the attainment of boys was slightly higher than girls (see Table 1). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 49 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Boys Girls Ratio Level 0 1 2 3 4 0 1 2 3 4 Year 7 [12] 24% 49% 16% 8% 4% 23% 56% 13% 6% 2% Year 8 [13] 16% 49% 14% 16% 6% 20% 52% 15% 9% 5% Year 9 [14] 10% 41% 15% 19% 15% 9% 48% 19% 16% 9% Table 1: Comparison of the attainment of boys and girls on the Ratio test by Year group [age] for the subsample. We note that these early results should be treated with caution. In particular, we note that this sub-sample of students appears to be slightly higher attaining than the general KS3 population in England. A further sample of approximately 3000 students took part on the survey in Summer 2009 and these results are currently being analysed. When this process is complete, the sample will be representative of schools and students in England. The full sample was randomised and drawn from MidYIS, the Middle Years Information System. MidYIS is a value added reporting system provided by Durham University, which is widely used across England (Tymms and Coe 2003). Research background: Attitudes, attainment and participation In England, in common with many other countries, too few students choose to continue studying mathematics once it ceases to be compulsory. There is considerable research in England addressing reasons for non-participation in mathematics - students stop studying mathematics because they experience it as difficult, abstract, boring and irrelevant (Osborne et al. 1997). The most recent findings relating to 16 year-olds (Matthews and Pepper 2007; Brown, Brown, and Bibby 2008) suggest that students’ attainment and attitudes are strongly inter-related. A major factor is that even relatively successful students perceive that they have failed at the subject and lack confidence in their ability to cope with it at more advanced levels, especially in comparison to the perceived ‘clever core’ of fellow-students. When pressed about the reasons for their feelings of failure, students suggest that they do not understand parts of what they have been taught and point to the predominance of routine and formal work on algebra and multiplicative reasoning (Nardi and Steward 2003). These perceptions of failure appear to be strongly linked to ideologies of ‘ability stereotyping’ (Ruthven 1987) and ability grouping (Boaler, Wiliam, and Brown 2000). Girls’ attitudes to mathematics tend to be more negative than boys. Boaler and Greeno (2000) link these more negative attitudes to mathematical teaching practices that do not emphasise understanding. However, the nature of the relationship between attitudes and attainment is poorly understood. In common with other highly and relatively highly attaining countries in TIMSS 2007, English students’ attitudes fell in comparison to TIMSS 1999, the last comparable data (Sturman et al. 2008). The fall in England at Grade 9, 25 percentage points, was greater than for other comparable countries, despite an increase in attainment elative to previous TIMSS surveys. One puzzling result is that, although within countries higher attainment is associated with more positive attitudes, the between country effect is in the opposite direction – countries with higher mathematical attainment tend to have more negative attitudes (Askew et al. 2010). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 50 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Early analysis of survey data: Student attitudes to mathematics This early analysis of the survey largely confirms existing findings. For example, students’ attitudes dropped as they got older. 63% of 12 year olds responded that they enjoyed mathematics lessons, but this had fallen to 54% of 14 year olds. (See Table 2.) However, the drop was greater for girls than for boys. Although a similar proportion of 12 year-old boys and girls said that they enjoyed mathematics lessons (64% and 62%, respectively), by age 14 the proportions were 59% of boys compared to 50% of girls. This reflects the TIMSS 2007 finding of greater levels of self-confidence amongst Grade 9 boys than girls in England (Sturman et al. 2008). Similarly, although boys’ perceptions of their own ability were largely stable and positive across the age range, girls’ perceptions of their ability had dropped to about half the sub-sample by age 14. (See Table 3.) Year Group [Age] Year 7 [12] Year 8 [13] Year 9 [14] Boys 64% 61% 59% Girls 62% 61% 50% Total 63% 62% 54% Table 2: Positive responses to ‘Do you enjoy maths lessons?’ by age and gender. Boys Girls Year 7 [12] 78% 67% Year 8 [13] 80% 66% Year 9 [14] 81% 52% Table 3: Positive responses to ‘Do you think you are good at maths?’ by age and gender. Across the age range, more boys than girls thought that they would study mathematics after GCSE and for both boys and girls this dropped. However the drop was greater for girls with only a quarter of the sub-sample of 14 year olds saying that they would continue post-16. One potentially positive finding is the relatively high proportion of students of both sexes who were undecided. (See Table 4.) Boys Girls Year [Age] Year 7 [12] Year 8 [13] Year 9 [14] Year 7 [12] Year 8 [13] Year 9 [14] Yes 38% 34% 35% 30% 28% 26% No 15% 15% 22% 16% 19% 32% Don't know 46% 51% 44% 54% 53% 43% Table 4: Responses to ‘Do you think you will continue to study maths after GCSE?’ by age and gender. One surprising result was that a very high proportion of students responded that working hard was more important for success in mathematics than natural ability: 89% of 12 year olds and 85% of 14 year olds, with the drop being almost wholly due to a change in boys attitudes. (See Table 5.) This result appears to contradict previous findings. Year Group [Age] Year 7 [12] Year 8 [13] Year 9 [14] Boys 88% 81% 82% Girls 89% 89% 87% Total 89% 85% 85% From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 51 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Table 5: ‘Working hard’ responses to ‘Which do you think is more important for success in maths? Working hard or Being naturally clever’ by age and gender. The relationship between attitudes and attainment We now turn to examine the relationship between attainment and attitudes. Unsurprisingly, a greater proportion of the highest attaining students said that they enjoyed maths and, in contrast to other students, this proportion did not fall for older students. (See Table 6.) ~ Bottom 75% ~ Top 25% ~ Top 10% Year 7 62% 64% 71% Year 9 52% 59% 69% Table 6: Comparison of positive responses to ‘Do you enjoy maths lessons?’ between Year 7 [age 12] and Year 9 [age 14] by attainment. Rough attainment proportions calculated as follows: for bottom 75% by aggregating Levels 0 and 1 for Year 7 [75%] and Levels 0, 1 and 2 for Year 9 [71%]; for Top 25% by aggregating Levels 2, 3 and 4 for Year 7 [25%] and Levels 3 and 4 for Year 9 [29%]; for Top 10% by aggregating Levels 3 and 4 for Year 7 [10%] and taking Level 4 for Year 9 [12%]. Again, unsurprisingly, a greater proportion of high attaining students were intending to continue to study mathematics after GCSE. However, the proportion of the highest attaining 10% intending to continue studying post-16 was only 57%. It is worth noting that 24% of the bottom 75% said that they intended to continue with mathematics post-16. This figure is relatively high in comparison to the limited options in English education for this group of students post-16. (See Table 7.) ~ Bottom 75% ~ Top 25% ~ Top 10% Year 7 32% 43% 59% Year 9 24% 46% 57% Table 7: Comparison of students intending to continue to study mathematics post-GCSE at Year 7 [age 12] and Year 9 [age 14] and by attainment. See Table 6 for how rough attainment proportions calculated. It is also noteworthy that the proportion of students who felt that the working hard was more important than natural ability was high at all attainment levels. (See Table 8.) Year Group [Age] Year 7 [12] Year 8 [13] Year 9 [14] 0 89% 81% 86% Attainment (as measured by Ratio Level) 1 2 3 89% 89% 91% 87% 87% 82% 86% 86% 81% 4 86% 82% 85% Table #: Proportions of ‘working hard’ responses to ‘Which do you think is more important for success in maths? Working hard or Being naturally clever’ by age and attainment. Findings from interviews We have conducted several group interviews with students from Phase 2 schools. These interviews have followed a semi-structured format. Here, for reasons of space, we discuss just one interview. This interview is of particular interest because it sheds light on the issue of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 52 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 ‘working hard’ and success at mathematics referred to in the analysis of survey data above. The interview was with three Year 8 students from the second set of a relatively high attaining state comprehensive school. The students were typical in disliking algebra and describing it as “bad”. Similarly, when asked whether they themselves were “good at maths”, they referred to their test results and levels from the end of Year 7 (Hodgen and Marks 2009). The following is an extract about what it takes to be good at mathematics: Researcher: What’s it take to be good at maths? Student K: - dedication... Student M: -knowing your numbers... Student C: I think it just comes normal to you... with some people, like you might not be good at many subjects, but when it comes to maths you could just be brilliant... Researcher: And is that what those three were... Student C: I reckon one them was... J, when it came to him, because he... I don’t think he’s that great, like superb at all the others but when it comes to maths he’s superb, he gets every answer first, so... Researcher: So what has he got...? What’s it take to be good at maths? Student M: -study... Researcher: Do you think he worked hard, this boy...? Student K: ...um, I think yeh, he worked hard.. I think ... if he didn’t know what he had to do in class, if he didn’t understand it, then he would go home and like get hold of a maths book or something which would explain it better and also give him questions that he could do to make sure he actually knew what he was doing when he came back to class the next day, so he could understand it An interesting feature of this discussion was the way in which the student referred to J as “great” and “superb” but also as someone with “dedication” who “worked hard”. Conclusion As we have already noted, the analysis is at an early stage. The next stage of analysis will further investigate the relationship between students’ attitudes and attainment. References Askew, M., J. Hodgen, S. Hossain, and N. Bretscher. 2010. Values and variables: A review of mathematics education in countries with high mathematics attainment. London: The Nuffield Foundation. Boaler, J., and J. G. Greeno. 2000. Identity, agency and knowing in mathematics worlds. In Multiple perspectives on mathematics teaching and learning, edited by J. Boaler. Westport, CT: Ablex Publishing. Boaler, J., D. Wiliam, and M. Brown. 2000. Grouping - disaffection, polarisation and the construction of failure. British Educational Research Journal 26 (5):631-648. Brown, M., P. Brown, and T. Bibby. 2008. “I would rather die”: Attitudes of 16 year-olds towards their future participation in mathematics. Research in Mathematics Education 10 (1):3-18. Brown, M., D. E. Küchemann, and J. Hodgen. Forthcoming. The struggle to achieve multiplicative reasoning 11-14, The Seventh British Congress of Mathematics From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 53 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Education (BCME7). Paper submitted to The Seventh British Congress of Mathematics Education (BCME7), University of Manchester. Hart, K., ed. 1981. Children's understanding of mathematics: 11-16. London: John Murray. Hodgen, J., D. Küchemann, M. Brown, and R. Coe. 2008. Children’s understandings of algebra 30 years on. Proceedings of the British Society for Research into Learning Mathematics 28 (3):36-41. ———. 2009. Secondary students’ understanding of mathematics 30 years on. Paper read at British Educational Research Association (BERA) Annual Conference, at University of Manchester. Hodgen, J., and R. Marks. 2009. Mathematical ‘ability’ and identity: a socio-cultural perspective on assessment and selection. In Mathematical Relationships in Education: Identities and Participation, edited by L. Black, H. Mendick and Y. Solomon. London: Routledge. Matthews, A., and D. Pepper. 2007. Evaluation of participation in GCE mathematics: Final report. QCA/07/3388. London: Qualifications and Curriculum Authority. Nardi, E., and S. Steward. 2003. Is mathematics T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroom. British Educational Research Journal 29 (3):345-367. Osborne, J., P. Black, J. Boaler, M. Brown, R. Driver, and R. Murray. 1997. Attitudes to Science, Mathematics and Technology: A review of research. London: King's College, University of London. Ruthven, K. 1987. Ability stereotyping in mathematics. Educational Studies in Mathematics 18:243-253. Sturman, L., G. Ruddock, B. Burge, B. Styles, Y. Lin, and H. Vappula. 2008. England’s Achievement in TIMSS 2007 National Report for England. Slough: NFER. Tymms, P., and R. Coe. 2003. Celebration of the Success of Distributed Research with Schools: the CEM Centre, Durham. British Educational Research Journal 29 (5):639653. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 54 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Simon Says: Direction in Dialogue Jenni Ingram, Mary Briggs and Peter Johnston-Wilder University of Warwick There has been a steady increase in the quantity of mathematics education research focusing on language, discourse and interaction. A wide variety of theoretical frameworks and methodological approaches have been taken including discursive psychology, commognition, and discourse analysis. This paper explores the use of a conversation analysis approach to analyzing interactions in mathematics classrooms. In particular what this approach can tell us about the structure of interactions and the use of repair in the negotiation of mathematical meanings. Keywords: classroom discourse, conversation analysis, repair Introduction and Background Several authors have focused on language, discourse and communication in the mathematics education literature over the past twenty years. Some authors have explored the relationship between discourse and identity (Boaler, Wiliam and Zevenbergen 2000, Lee 2006), beliefs about mathematics and beliefs about teaching and learning. Others have focused on interactional strategies and the implications of these for the learning of mathematics (O'Connor and Michaels 1993). More recently, Sfard (2007) has proposed a theoretical framework which conceptualises learning mathematics as a transformation and extension of learner’s discourse. This “commognitive” framework treats a learner’s discourse as the object of learning and not just the means of learning, raising the importance of research on mathematics classroom discourse. There are many methodological approaches available in the research on discourse: discourse analysis including systemic functional linguistics, critical discourse analysis, discursive psychology and conversation analysis to name a few. This paper explores the conversation analytic approach as a means of explicating the complexity of interaction in the classroom setting. After outlining the conversation analysis methodology the notion of repair is explored in the context of two extracts taken from transcripts of whole-class interactions. Differences in the organisation of repair between the classroom contexts and everyday conversation are then examined and the implications these differences have on the learning of mathematics are discussed. Conversation analysis The origins of conversation analysis lie in the analysis of naturally occurring conversation but have been extended to include the analysis of institutional settings such as courtrooms and emergency help lines (see Drew and Heritage 1992 for more examples). McHoul (1990) used a conversation analysis approach in his study of geography classrooms and Seedhouse (1996) offers an in-depth analysis of second language classrooms. Conversation analysis (CA) as a methodology assumes that interaction is structurally organised and the goal of CA is the exposition of this structure from the perspective of the participants themselves (Levinson 1983). Consequently, claims about the existence of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 55 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 various structures of interactions need to be supported with evidence in the transcripts that the participants orient themselves to these structures. This is often referred to as the next-turnproof-procedure and serves as an attempt to offer empirical proof and prevent the imposition of the researcher’s preconceptions. Of particular importance to the analysis of classroom discourse is the CA dynamic view of context. The well-known initiation-response-feedback (IRF) triad from a discourse analytic approach is proposed as a common feature of classrooms but this takes a static view of the context of the classroom. The CA approach examines why the IRF triad shapes the classroom context. In particular, each occurrence of an IRF sequence can only be understood within the sequential context of the interactions as they are affected by previous utterances and it influences those utterances that follow. Each part of the triad has multiple roles in relation to what has occurred before and in what is to follow. A response, for example is constrained by the nature of the initiation it follows but also constrains the feedback that is yet to come. The CA approach therefore offers a tool for explicating the fluidity of classroom interaction, in particular how the context (mathematical, task-based, management) can shift during and after individual turns. A CA approach provides an in-depth analysis of a specific context but also limits the generalisability of any analysis. Talk in classrooms is usually goal oriented. The multidimensionality (Doyle 1990) of classrooms means there are multiple goals influencing interaction in the classroom. These include the pedagogic goals of the teacher and the wider goals associated with the behaviour, motivation and learning of the participants. CA examines interactions with reference to these goals, but the analysis seeks only to examine the interactional advantages and disadvantages and does not evaluate the effectiveness of these interactions from a pedagogical perspective. This could lead to a conflict between pedagogic goals and the interactional goals controlling the discourse. The structures of turn taking and repair are particularly relevant to the classroom contexts and it is the latter that is the focus for this paper. There are three features of a repair: the trouble source; the initiation; and the outcome. Trouble has a broader definition than that of errors and mistakes, including difficulties in understanding, hearing or in the structure of the interaction itself. The repair initiation and outcome are sequential and can be performed by the speaker in which the trouble occurred (self) or other participants (other). Self-initiated self-repair is the preferred type of repair, occurring most often in everyday conversation, whilst other-initiated other-repair is very rare (Schegloff, Jefferson and Sacks 1977). In CA, the preference for self-initiated self-repair is not that the participants like or want to do this, but the act is socially affiliative. Preferred actions are normally bald and direct, without hesitation or delay, whilst dispreferred actions usually include hesitation, mitigation, and delay. There is some evidence that the preference organisation of repair is different in the classroom context (McHoul 1990) with other-initiated repairs occurring more frequently. This is perhaps unsurprising in many classrooms as it is often the teacher who not only has the expertise to identify the trouble source but also the authority to initiate a repair. The data discussed in this paper is taken from the transcript of a lesson with 12-13 year old students, considered to be high achievers in relation to their peers, focusing on measures of central tendency, and is chosen to exemplify the features of repair outlined above. The students have been asked to calculate the missing value if the mean is 70 and the other values are 72, 43, 85 and 71. The extracts are taken from the whole-class interactions that occurred after the students had been given some time to work on the problem individually. Extract 1 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 56 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 1 Sam: I added them all up 2 Simon: you added them all up Sam has been nominated by the teacher (Simon) to explain what he did. His answer is incorrect and Simon initiates a repair in line 2. Immediately we can see the importance of knowing the interactions that preceded this interaction in order to identify Sam’s utterance as causing trouble. However, these two utterances do not themselves offer evidence that this is in fact an other-initiated repair. We need to look at the utterances that follow for that: Extract 2 3 Sam: yeah 4 Simon: so you did (0.9) what that (0.6) plus that (0.5) plus that, did you add that one on as well. 5 Sam: u:m: no 6 Simon: okay 7 Sam: and then (0.6)I: (1.5) divided that by five (0.7) to get the how much she needed (0.7) in the last (0.3) um: (.) test. 8 Simon: so you added up the four numbers, (0.8) you added up four numbers (0.3) and then you divided by five? (1.8) is that it? 9 Sam: yeah In line 3, Sam’s affirmative response indicates that Simon’s repetition of his answer in line 1 has been understood by Sam as a check that Simon has heard and understood Sam correctly. In lines 4 and 8, it becomes clear that Simon in fact meant his repeat as a repair initiation by repeating, expanding and recasting Sam’s initial response further and emphasising the words four and five in line 8, locating the source of the trouble. The repair itself is not performed in the entire episode but the dispreference for other-repair is clear from Simon’s final utterance: Extract 3 18 Simon: Sam added them up. okay (1.2) Sam added them up shhshh shh shh. Sam added them up, (0.7) they added up to two hundred and seventy one, that is a useful bit of information (1.0) bu:t that thing about dividing by five. that seemed to me, I don't know, a little bit nonsensical cause you've only got four From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 57 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 numbers, dividing by five I'm not sure. Phillip. Here Simon is now addressing the class as a whole. He starts by repeating Sam’s response and follows it with a positive evaluation. Then he pauses before locating the trouble again. This second half of the response is hesitant and mitigated with vague phrasing “that thing about” and hedged comments “seemed to me”, “I don’t know”, “I’m not sure”. Simon is avoiding making the repair, and in this example, the repair is not directly made until the next student speaks and offers the expected answer: Extract 4 19 Phillip: um you need, if yo-, you can find th- like all the numbers, the end mark, the end percentage means that there's like three hundred and fifty percent altogether divided by five it comes up to seventy. 20 Simon: right hold on a sec. (0.5) three hundred and fifty percent,(.) er I suppose, can you add percentage together and then get three hundred and fifty per[cent I suppose so ]ok (0.9) 21 Phillip: 22 Simon: [no what we ] so you're saying that if you've got five numbers (0.4) and you want to get a mean (.) of seventy In this example, Phillip is searching for the words he needs to explain what he did. In line 2, Simon initiates a repair focused around the trouble associated with whether you can have three hundred and fifty percent or not. There is no indication in the interaction as to whether this is Simon’s interpretation of Phillip’s trouble (in which case it is an other-initated repair) or a new trouble source which is Simon’s himself (in which case it is a self-initiated repair). Phillip’s interruption indicates that this was not the source of trouble for Phillip; Simon recasts Phillip’s answer in line 4, consequently performing the repair. Each of the two repairs in extract 4 is clearly different in nature. The first is a difficulty in communication revealed by Phillip’s word search, whilst the second is mathematical (can you have three hundred and fifty percent?). The latter appears to be repaired in the same turn as the trouble (line 2) yet in line 4, Simon’s recasting uses the word number instead of percent, indicating that a repair of meaning has not actually been performed. It is interesting to note that the trouble in the first extract was also mathematical and in both cases, the repair was to some extent not performed. In conclusion, a conversation analytic approach offers an effective tool for exploring and exposing the structure and complexity of mathematics classrooms. In particular, the preference organisation of repair explicates the roles adopted by the participants of teacher and students. The more frequent prevalence of other-initiated repair defines the teacher as expert and in a position of authority, similar to adult-child interactions in other contexts. The dispreference for other-initiated other-repair remains evident in the extracts discussed above. Future work will include the analysis of a broader data set to examine the structure of repair in the secondary mathematics classroom. This includes exploring the similarities and differences between the nature of the trouble source, the participants in the repair and the repair trajectory. The implications of each of these on the learning of mathematics are of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 58 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 particular interest. For example, the relationship between the dispreference of other-initiated other-repair and the role of errors and mistakes in the learning of mathematics. Bibliography Boaler, J., D. Wiliam, and R. Zevenbergen. 2000. “The construction of identity in secondary mathematics education.” Proceedings of the Second International Education and Society Conferenc, ed by J. Matos and M. Santos. Univeridade de Lisban. Doyle, W. 1990. “Classroom management techniques.” In Student Discipline Strategies: Research and Practice, by O. C. Moles. SUNY Press. Drew, P., and J. Heritage. 1992. Talk at Work: Interaction in Institutional Settings. Cambridge: Cambridge University Press. Lee, C. 2006. Language for Learning Mathematics: Assessment for Learning in Practice. Buckingham: Open University Press. Levinson, S. 1983. Pragmatics. Cambridge: Cambridge University Press. McHoul, A. W. 1990. “The organization of repair in classroom talk.” Language in Society 19: 349-377. O'Connor, M. C., and S. Michaels. 1993. “Aligning academic task and participation status through revoicing: Analysis of a classroom discourse strategy.” Anthropology and Education Quarterly 24, no. 4: 318-335. Schegloff, E. A., G. Jefferson, and H. Sacks. 1977. “The preference for self-correction in the organization of repair in conversation.” Language 53: 361-382. Seedhouse, P. 1996. Learning Talk: A Study of the Interactional Organisation of the L2 Classroom from a CA Institutional Discourse Perspective. Unpublished PhD Dissertation: University of York. Sfard, A. 2007. “When the Rules of Discourse Change but Nobody Tells You: Making Sense of Mathematics Learning From the Commognitive Standpoint.” The Journal of the Learning Sciences 16, 4: 565-613. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 59 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 60 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The relationship between number knowledge and strategy use: what we can learn from the priming paradigm Tim Jay Graduate School of Education, University of Bristol Priming methods involve showing a stimulus for a short amount of time (the prime), followed by a second stimulus (the target), which children are asked to perform some operation on. If there is a strong association between the prime and target for a particular child, then the operation on the target will be facilitated by the presence of the prime. This paper describes a project in which priming methods are used to add to our understanding of strategy development for simple addition problems. Children were asked to complete two activities; a priming trial designed to demonstrate priming effects for doubling, and a set of addition problems where participants were asked to explain how they arrived at their answers. Approximately half of the participants used counting strategies (count-on from first, count-on from smallest), while half used non-counting strategies (decomposition, tie or retrieval). Results indicate that a priming effect for doubling relationships but only for the group of children using non-counting strategies. This result could help to explain the relationship between the development of number knowledge and the development of new strategies. Introduction There is a well established understanding of the normal course of development of strategy-use when solving single-digit arithmetic problems (e.g. Fuson 1992). Children begin this course of development by counting both addends in an addition problem, often using concrete objects such as fingers to aid the process. The next strategy to appear is the count-one strategy, in which children start with one of the addends, then count on from there to find the answer. The 'min' strategy usually comes next, which involves children choosing to count on from the largest addend. These three strategies all involve children counting in order to arrive at an answer to a problem. At some point, children will begin to be able to solve some simple problems using retrieval – directly accessing answers to problems stored in memory. Although strategies generally appear in this order, children maintain a repertoire of several strategies any any given point during this development, and show a high degree of variability in their application of strategies to problems (Siegler 2007). Fewer studies have addressed the nature of strategies used to solve problems resulting in answers greater than 10. However, this aspect of the literature is growing due to the recent focus on children's adaptive expertise in selecting amongst strategies (Verschaffel, Torbeyns, De Smedt, Luwel, & Van Dooren 2007). When children are solving addition problems that bridge 10, there are heuristic strategies available that sit between the counting strategies and direct retrieval in terms of efficency. The problem '8 + 7', for example, might be solved by converting the problem to '7 + 7 + 1' – this is often referred to as the 'tie' strategy and is often used in the case of 'near tie' problems where the addends differ by 1. Alternatively, '8 + 7' might be solved by converting to '8 + 2 + 5', if a child knows their number-bonds to 10 (sometimes known as 'ten-friends') – this is often referred to as a 'decomposition' strategy. It is not clear what factors are involved either in stimulating the adoption of new strategies in From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 61 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 response to problems or in determining the selection of one strategy over others that are available in relation to a given problem. The relationship between number knowledge and strategy development Torbeyns, Verschaffel and Ghesqiere (2005) give a hypothetical example of a child who is able to accurately retrieve the answer to 6+6, but not 7+7 or 8+8. This child would be expected to be more likely to use the tie strategy when solving 6+7 (by transforming the problem to 6+6+1) than when solving 7+8 or 8+9. Torbeyns et al. showed that children in their study differed in the efficiency with which they carried out counting, decomposition and tie strategies, but that children at a range of different ability levels all showed similar levels of adaptivity, generally choosing the strategy that would generate a correct answer most quickly in response to a particular problem. Torbeyns et al. only analysed data from children who were already using either the decomposition or tie strategy – their aim was to study variation in adaptivity related to differences in achievement in mathematics, not to investigate the necessary conditions for the development of these strategies. Very relevant to the current discussion is the existence of a “tie effect” (LeFevre, Shanahan, & DeStefano 2004), whereby the problem-size effect (the fact that arithmetic problems with larger answers tend to be answered more slowly than those with smaller answers) can generally not be observed for tie problems (where both addends in an addition problem are the same). LeFevre et al. showed that the tie effect is not due to facilitation of encoding (the fact that the same number appears twice means it is more quickly encoded the second time), but is due to calculation and memory access. It seems reasonable to argue, as do Torbeyn, Verschaffel and Ghesquiere (2005), that good knowledge of doubling relationships (pairings between 6 and 12, 7 and 14 and so on) is required in order for children to begin using the tie strategy. However, this paper aims to go a step further and make the claim that implicit knowledge of doubles is a prerequisite for use of the tie strategy. Using priming to investigate number knowledge The first study of number knowledge that employed priming as a method was that of den Heyer and Briand (1986), in which a priming distance effect (PDE) was observed. The PDE is the phenomenon that a reponse to a target stimulus is facilitated by the presentation of a prime that is similar in magnitude to the target. For example, in the lexical decision task used in den Heyer and Briand's study, participants were quicker to respond to 'five' after the prime 'four' than after the prime 'three'. The PDE has been shown to be equally strong in both directions – so the prime '4' facilitates processing of a target '5' as well as it does '3' – and has also been demonstrated in different modes, whereby the prime 'six' facilitates processing of 'seven' or '7', for example (Reynvoet, Brysbaert, & Fias 2002). There is some debate regarding the mechanism underlying the PDE. Some researchers have explained the effect in terms of operations on a 'sub-symbolic' number-line, used in order to compare number information in terms of magnitude. However, there is evidence to suggest that a connectionist approach might generate a more satisfactory explanation. There is evidence, for example, that as well as proximity on the number line, other relationship amongst numbers can give rise to priming effects. Garcia-Orza, Damas-Lopez, Matas and Rodriguez (2009) show that, for adult participants, the prime '2x3' facilitates processing of target '6', using a masked prime protocol. This suggests that, rather than sub-symbolic processing, these priming effects reflect symbolic processing within something like Collins and Loftus' (1975) semantic activation network. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 62 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Method Participants 57 children, from two primary schools, took part in this experiment. They were aged between 7 years, 2 months and 9 years, 11 months. In each school, the Mathematics Coordinator was asked to select those children who were able to reliably solve single-digit addition problems, but did not yet consistently use a retrieval strategy. All of the children who participated in the study had experienced classroom instruction in the use of a range of strategies for solving addition problems, including counting strategies, decomposition and tie. Instruments and measures Two tasks were prepared, using the DirectRT psychology experiment software package. Stimuli were presented to participants in the centre of a 17 inch monitor, using a 48 point font. A microphone was used in order to measure verbalisation latency. Addition problem task For this task, a set of addition problems was created. All single-digit addition problems with two addends, where the two addends were different and the answer was greater than 10, were included. Participants were asked to respond with an answer to each problem. Following an answer, participants were prompted with the question, “How did you solve the problem x + y?” Problems were presented to participants at random, without replacement. Strategies were coded as being either “count-one”, “min”, “decomposition”, “tie”, “retrieval” or “other – including don't know”. Three practice problems were given before starting the main set of problems. The practice problems were “7 + 3”, “6 + 2” and “4 + 4”. Priming task 60 prime-target pairs were created. Of these, 15 pairs related to the present study. Primes used were “5”, “6”, “7”, “8” and “9”. The target stimulus in each pair was either the exact double of a prime, or the double +/- 1. The remaining prime-target pairs presented to participants were included in order to ensure that participants could not predict that the purpose of the study was to assess knowledge of doubles, and were constructed as if intended to address participants' knowledge of number bonds and proximity on the number line. Primetarget pairs were presented at random, without replacement. Timings were as follows: Fixation “*”: 1000ms; Prime stimulus: 200ms; Fixation “*”: 500ms; Target stimulus. Participants were instructed to say into the microphone the second number in each pair (the target), as quickly as possible. The reaction time recorded for each trial was the time it took for the participant to begin reading the target number, following its presentation. Design The experiment employed a mixed design, with two independent variables. The first independent variable was the relationship between prime and target in a prime-target pair. There were 3 conditions of this variable; the target was either double the prime minus one, double the prime exactly, or double the prime plus one. The second independent variable was whether or not participants claimed to use the tie strategy at least once whilst completing the addition problem task. The dependent variable was the time that it took to read aloud the target stimulus. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 63 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The hypothesis was that reaction time would be least when the target was the exact double of the prime and that this effect would be observed only for those participants who used the tie strategy at least once during the addition problem task. Procedure Participants completed the two activities individually, in a quiet room outside of their usual classroom. A laptop computer was used to generate stimuli and record reaction times via a microphone. The experimenter watched the laptop screen during each trial, whilst participants watched a second monitor, synchronised with the laptop. Half of the participants completed the addition problem task first, followed by the priming task, while half completed the two tasks in the reverse order. The researcher introduced the task, and gave the participant an opportunity to ask questions. Three practice trials were completed, followed by a further opportunity to ask questions. The block of experimental trials for the task were then completed. Results and Discussion The addition problem task was used in order to divide participants into two groups. 29 participants reported using the tie strategy on at least one occasion during this task, while 28 participants did not. A 3 x 2 mixed ANOVA was carried out. The independent variables were prime-target pair (repeated measures: target = 2 x prime – 1; target = 2 x prime; target = 2 x prime + 1) and whether children used the tie strategy at least once during the addition problem solving activity (independent groups). The dependent variable was the median time it took for a participant to begin reading the target stimulus. There was a significant main effect of prime-target relationship (F2, 84=4.867, p=0.01). This indicates that participants were significantly quicker to read a target that was exactly double the preceding prime than a target that was 1 greater or 1 less than the exact double of the preceding prime. There was also a significant interaction between prime-target pair and strategy use (F2, 84=6.879, p=0.002). As can be seen in Figure 1, the effect of variation in prime-target pair on RT is accounted for entirely by the group of children using the tie strategy. There was no significant main effect of strategy use on reaction time. Participants who used the tie strategy to solve at least one of the set of addition problems were quicker to read a target that was the exact double of the prime than a target that was the exact double of the prime plus or minus 1. The participants in the study had no way of predicting the relationship(s) under investigation. This means that on perceiving the prime stimulus, the double of the prime (amongst other cognitive resources including numbers and concepts) was automatically activated. When the target stimulus was the exact double of the prime, participants' reading of the target was facilitated due to that number already having been activated. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 64 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Figure 1: Graph to show effect of prime-target pair type on RT, by strategy use Thus this study clearly demonstrates the fact that children using the tie strategy have implicit knowledge of the relationship between numbers and their doubles (at least for numbers between 5 and 9). This result contributes substantially to the literature on both children's arithmetic strategy development and its relation to the literature on the nature of children's representation of number. The findings do not directly identify a causal relationship between the development of knowledge of relationships between numbers and their doubles and the development of the tie strategy. However, of the two possible interpretations (either the implicit knowledge of doubles is a necessary condition for the development of the tie strategy, or children's knowledge of the tie strategy encourages the rapid development of knowledge of doubles) it is intuitively most likely that children must develop a knowledge of the relationships between numbers and their doubles before they can add the tie strategy to their repertoire. This fits fits well within a resource activation framework (Hammer, Elby, Scherr, & Redish 2005). Within this framework, cognitive resources are activated in response to a problem situation. These resources are used in the assembly of ad hoc theory in order to generate a solution. Further work must be done in order to fully understand the relationship between the development of implicit knowledge and the development of strategies, but some important implications should be considered at this stage. Most importantly, the study calls into question the claim that teachers should be helping children develop ways to select amongst available strategies for solving problems. Torbeyns, Verschaffel and Ghesquiere (2005) found there was no difference in levels of adaptivity (the ability to select the most efficient strategy from a repertoire of available strategies for a given problem) between children across a range of mathematical ability. The present study shows that children do not use the tie strategy if they do not have implicit knowledge of relationships between numbers and their doubles. Taken together, the evidence from these studies shows that the development of new strategies, and the development of the adaptivity necessary to select amongst strategies, occur as a result of the development of associated cognitive resources such as the knowledge of particular types of relations amongst numbers. These results help to provide an explanation for some effects observed in previous work. For example, Siegler and Stern (1998) observed that a majority of children in a study From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 65 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 were using a new strategy for solving a number of problems before they were aware of using it. Within the resource activation framework, the learner's use of a particular procedure (resulting from the automatic activation of relevant cognitive resources) and the learner's representation of that procedure are quite different things. The representation of a particular strategy will always follow that strategy's first use (whether it follows immediately or at some later point). Conclusion This study represents an important step in our growing understanding of children's development of mathematical thinking. Its main contribution consists in the argument that implicit knowledge of relationships between numbers and their doubles is a necessary prerequisite for the development of the tie strategy. References Alibali, M. W. 1999. How children change their minds: strategy change can be gradual or abrupt. Developmental Psychology, 35: 127-45. Collins, A. M., and E.F. Loftus. 1975. A spreading-activation theory of semantic processing. Psychological review 82: 407–428. Fuson, K. 1992. Research on Whole Number Addition and Subtraction. In Handbook of Research on Mathematics Teaching and Learning, ed. D. A. Grouws, 243-275. New York: Macmillan. Garcia-Orza, J., J. Damas-López, A. Matas, and J. M. Rodriguez. 2009. " 2 x 3" primes naming" 6": Evidence from masked priming. Attention, perception & psychophysics 71: 471-80. Hammer, D., A. Elby, R. E. Scherr, and E. F. Redish. 2005. Resources, framing, and transfer. In Transfer of learning from a modern multidisciplinary perspective ed. J. P. Mestre, 89–120. Information Age Publishing Inc. den Heyer, K., and K. Briand. 1986. Priming single digit numbers: Automatic spreading activation dissipates as a function of semantic distance. The American Journal of Psychology 99: 315–340. LeFevre, J. A., T. Shanahan and D. DeStefano. 2004. The tie effect in simple arithmetic: An access-based account. Memory and Cognition 32: 1019–1031. Reynvoet, B., M. Brysbaert, and W. Fias. 2002. Semantic priming in number naming. The Quarterly Journal of Experimental Psychology Section A 55: 1127–1139. Siegler, R. S. 2007. Cognitive variability. Developmental Science 10: 104-109. Siegler, R. S., and E. Stern. 1998. Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology-General 127: 377-397. Torbeyns, J., L. Verschaffel, and P. Ghesquiere. (2005). Simple addition strategies in a firstgrade class with multiple strategy instruction. Cognition and Instruction 23: 1-21. Verschaffel, L., J. Torbeyns, B. De Smedt, K. Luwel, and W. Van Dooren. 2007. Strategy flexibility in children with low achievement in mathematics. Educational & Child Psychology 24: 16-27. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 66 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Aspects of a teacher’s mathematical knowledge in a lesson on fractions Bodil Kleve Oslo University College This paper is about a mathematics teacher, and how aspects of his mathematical knowledge surfaced in a 5th grade (11 years old) fractions lesson in Norway. The teacher’s responses to pupils’ (unexpected) comments and questions, ‘contingent moments’, are discussed. Difficulties in dealing with improper fractions, which were mirrored in the pupils’ inputs in the lesson, are discussed. Considerations are made whether the problems the pupils expressed can be traced back to aspects of the teacher’s mathematical knowledge. Keywords: Improper fractions, teacher knowledge, contingent moments Background and introduction What knowledge is required for the teaching of mathematics has been widely discussed within mathematics educational research, both with regard to what comprises the knowledge, and how this mathematical knowledge is made accessible to others. Through classroom observations and focus-group meetings with four mathematics teachers in 5th grade elementary school in Norway I have been studying how teachers drew on their knowledge in mathematics and mathematical didactics in their teaching. Lessons were videotaped and I have used the Knowledge Quartet developed by Rowland, Huckstep and Thwaites (2005) as an analytical framework to study how a teacher’s (Hans’) mathematical knowledge surfaced in the lesson. How examples and illustrations of improper fractions influence the pupils’ conceptions and difficulties are discussed. Before presenting the lesson with the teacher Hans, I will report some research about mathematical knowledge for teaching which I have used in my study, and also briefly report research about fractions which suggests some factors explaining why pupils’ concepts of fractions only become partly developed. Mathematical knowledge for teaching By questioning how teachers’ use their knowledge in the subject they teach and where teachers’ explanations come from, Shulman (1986) brought didactics into mathematics educational research. Shulman suggested distinguishing among three categories of content knowledge: Subject Matter Content Knowledge, Pedagogical Content Knowledge and Curricular Knowledge. Subject Matter Content Knowledge (SMK) refers to the knowledge the teacher has in mind, both substantive and syntactic. Pedagogical Content Knowledge (PCK) goes beyond knowledge of the subject and refers to content knowledge for teaching, “It is the capacity of a teacher to transform the content knowledge he or she possesses into forms that are pedagogical powerful” (Shulman, 1987, p.15). Curricular Knowledge is both lateral and vertical. Lateral curriculum knowledge is how the teacher is able to relate the content and issues discussed in his/her subject to that being discussed in other subjects. Vertical curriculum knowledge is about what has been taught in earlier lessons (and years) within a subject as well as what is relevant to be taught in the next lessons. Rowland et al (2005) based their work on Shulman’s categories of knowledge. Through a grounded approach to data from video studies, the Knowledge Quartet (KQ) was From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 67 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 identified. In the KQ the classification of the situations in which mathematical knowledge surfaces in teaching is of importance (Rowland and Turner, 2008). The Knowledge Quartet has four broad dimensions; Foundation, Transformation, Connection and Contingency. Foundation is the mathematical knowledge the teacher has gained through his/her own education, it is knowledge possessed and which can inform pedagogical choices and strategies. It is the reservoir of pedagogical content knowledge you draw from in planning and carrying out a lesson and thus informs pedagogical choices and strategies. Transformation focuses on the teacher’s capacity to transform his or her foundational knowledge into forms which can help someone else to learn it. It is about examples and representations the teacher chooses to use. The third category, Connection, binds together distinct parts of the mathematics and concerns the coherence in the teacher’s planning of lessons and teaching over time and also coherence across single lessons. Contingency is the category which concerns situations in mathematics classrooms that are impossible for the teacher to plan for; the teacher’s ability to deviate from what s/he had planned and the teacher’s readiness to respond to pupils’ ideas are important classroom events within this category. Fractions As with decimals and percentages fractions occur with different meanings. These meanings can also be seen in everyday life. A fraction can be a part of a whole, a place on the number line, an answer to a division calculation or a way of comparing two sets or measures (part group). Novillis (1976) studied the hierarchical development of various aspects of fractions among American children. She found that the part-whole and part-group models were significantly easier for the children to understand than the number line. Her study referred to work with fractions not bigger than one. As opposed to the part whole or part group model, the number line does not incorporate that a fraction can be thought of as a concrete object. But according to Dickson, Brown and Gibson (1984), a number line makes improper fractions appear more natural. They claimed that “the representation of fractions as sub areas of a unit area does not lend itself very well to the representation of improper fractions” and that “The acceptance of the definition of a fraction as meaning ‘part of a whole’ is inconsistent with the very existence of such improper fractions” (p. 279). According to Anghileri (2000) much of the focus when working with fractions in school is identification of fractions as part of a whole. She claimed that success in working with fractions depends on the ability to see the fraction both as representing a number and a ratio which reflects the procedure for finding the number. She wrote: Research suggests that an approach to fractions which identifies each as numbers to be located on a number line, without emphasizing the way of partitioning a whole, will help to establish the equivalence with decimals and percentages (p.115) She thus warned against emphasizing fractions as parts of a whole in schools. This is in accordance with Askew (2000) who claimed that if one focuses on fractions as part of a whole so that becomes a social convention, possibilities for a obtaining a well developed fraction concept are limited. Hans Based on an analysis of a lesson with Hans using the KQ as an analytical framework, I will discuss how aspects of Hans’ mathematical knowledge became visible in this lesson, and if difficulties in dealing with fractions bigger than one whole can be traced back to aspects of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 68 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 the teaching. I will present an account of the lesson before I analyze it in terms of the aspects of the KQ, followed by a discussion. The lesson objective was written on the smart board when the lesson started: ‘To be able to calculate with fractions which are bigger than one whole’. Hans asked the class what the objective involved and a pupil suggested 4/3 or 1 1/3. The first task was 6/8+5/8 and it was illustrated by two rectangles on the smart board. A pupil was taken to the board. He worked it out and used the smart board to illustrate how one rectangle was filled up to a whole and three remained in the other. The next task was illustrated by circles divided into eight pieces. There were two circles on each side of an equal sign. Five pieces in the first and four in the other on each side were shaded. To Hans’ question how much it was all together Jens first suggested 11/9. When the teacher did not confirm, Jens suggested 10/9. Hans then asked how much was shaded in the first circle, and Jens said 5/9. ‘Are there nine?’ Hans then asked, and he let a pupil in class answer 8. What about the next circle? Hilde suggested ½. Hans confirmed but converted the half into 4/8 ‘for the sake of clarity’ he said. The answer 9/8 was agreed upon, and a pupil came to the board converting it into 1 1/8. A girl, Petra, then demonstrated uncertainty (misconception) asking: ‘How is it possible to take nine eighths? When there are eight bits and then take one more? How is that possible?’ Pupils in class explained. For the next two examples, Hans chose not to use illustrations; ‘let us try without’, he said. The tasks to be done were: 3/5+3/5=, and 7/10+5/10=. For the last one a pupil suggested converting 12/10 into 1 1/10. After having clarified that it became 1 2/10, Hans wanted to go on to the next task, when Mads had his hand up suggesting converting 1 2/10 into 1 1/5 ‘like the first one’, he said. The last task was to take away ¼ from 2. Hans had drawn 2 circles on the smart board, and he wrote 2-1/4 under the circles. The circles were divided into four pieces which all were shaded. Espen was taken to the board to work it out. He said: ‘it is two minus one fourth’ and he erased two ¼ pieces of the second circle. Mads then shouted: ‘Now you are erasing one half’, followed by: ‘not one fourth of two’. Hans asked Espen if he was sure and how much he actually should have taken away and Espen answered: ‘If it is two so half of that’ (pointing to one of the circles). Hans emphasized ‘what is the whole’. Then Espen realized that he had not done what the teacher had expected. He shaded one of the quarters he had erased and wrote 1 ¾ =7/8. Then Hans asked him to explain what he had been thinking. After this episode Ella went to the board and pointed to the digit 1 in 1 ¾ and asked: ‘what does the one actually do there?’ Hans responded to her question by taking it all from the beginning, asking questions for Ella to answer throughout his review. He started by asking what the whole was and how many pieces they originally had. Four more times during Hans’ review, Ella repeated her question. When he had finished and Ella expressed not understanding, Hans erased the board and tried again. This time he just focused on 4/4 being one whole. Then Ella said she understood. Foundation What subject knowledge (SMK) did Hans have on which he could draw in this lesson? In the textbook fractions were illustrated as part of rectangles, as part of circles and on number line. Hans used rectangles and circles, but not the number line. He demonstrated that he knew how to make fraction tasks adding up to more than one. Hans ‘converted’ ½ into 4/8 ‘for the sake of clarity’. He foresaw that writing ½ could cause confusion and wanted to avoid the issue of common denominator at this stage. He also inserted an equal sign when that was missing in Espen’s work with the subtraction task on the board. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 69 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Transformation How was Hans’ foundational knowledge set out in practice? The lesson seemed to be well planned. He had prepared examples and illustrations in form of rectangles and circles. The smart board was used to illustrate how it was possible to fill up one whole and then ‘see’ what fraction was left. There was a dialogic approach, Hans invited the pupils to participate and he let them come to the board to work out the exercises he had chosen. He also let other pupils explain when errors and misconceptions surfaced. Thus the pupils took actively part in the lesson. Hans’ choice of examples and illustrations mirror a view on fractions as part of a whole. He did not use the number line. Neither did he use examples which mirrored fractions as part of a group or proportions. In the first example Hans used the expression ‘eleven out of eight’ which may have caused the question from a pupil about how it was possible to take nine eighths, when there were eight bits and then take one more was grounded in that expression. All exercises included converting an improper fraction into a mixed number. In all but the first example (which was taken from the text book), the fraction part of the mixed number was a unit fraction. This suggests why a pupil converted 12/10 into 1 1/10 which again reveals that she had an undeveloped conception about the link between an improper fraction and a mixed number. The reason why the tasks Hans made all had a unit fraction as the one remained is not clear. One suggestion is that Hans only thought about making the answer bigger than one whole, and one bigger was sufficient. Another suggestion is that he wanted to avoid abbreviations since that was not the goal of the lesson. With regard to the illustration of 2-1/4, it seemed that it was the illustration itself that caused Espen the difficulties. I suggest that Espen would have calculated 2-1/4 without any difficulty if it had not been for the circles being used to illustrate. Connection There seemed to be a logical coherence in this lesson. There were good links across the lesson with regard to progression and examples and illustrations. Hans started by asking what the goal meant, and went from there to adding two fractions (6/8+5/8) which he had illustrated with rectangles. Then he went on to another addition with fractions (5/8+4/8), illustrated with circles. For the next two tasks (3/5+3/5 and 7/10+5/10), he suggested ‘trying with out illustrations’. Then he went on to subtraction and chose to illustrate with circles again. This task was not only different from the others in terms of not being addition. Now he started from a whole number, 2, taking away a unit fraction (1/4). He also chose to illustrate with circles. There may seem to be a gap between the addition tasks and the subtraction task. The subtraction task was not only another calculation it also involved an integer which none of the addition tasks did. Contingency There were four contingent moments in this lesson on which I will comment. First, Jens’s answer to Hans’ question ‘how much is it all together’ when having shaded five in one and four in the other. Why did Jens answer ninths, and how did Hans respond to it? I suggest that Jens saw the nine shaded pieces, five in the first and four in the second circle as a whole, and that was why he answered 5/9 when Hans broke the task down asking how many are shaded in the first. From Jens’ point of view 5/9 was correct. It did not seem that Hans understood how Jens was thinking. He responded to Jens as if Jens had done an unintended error (counted From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 70 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 9 pieces in the circle instead of 8). Hans did not invite Jens to explain why he kept saying ninths. He let another pupil say eight, and he never came back to Jens and his answers. In this case there was a lack of compatibility between the teacher’s and the pupil’s thinking. The teacher responded to this contingent moment without incorporating it further into the lesson. Second, I will comment on Petra’s input when she asked how it was possible to become nine eighths; when there were eight bits and then take one more. That the question was unexpected to Hans and that he was not prepared to answer it, were mirrored in how he responded to it. He first repeated the question, then he stumbled before Mads inserted ‘you have more pizzas you know’, Hans confirmed saying yes, and when Petra again asked: ‘One pizza, and there are eight pieces of that’, Mads again emphasised his view by saying ‘and then you take a new pizza’. In this case, Hans acknowledged Petra’s question, but left it for another pupil to explain. An unanswered question is if he did that on purpose, or if the pupil’s input helped him out of a problem which he was not able to answer right on his feet. The third contingent moment in this lesson which I will discuss is Espen’s first response to 2- ¼ . Obviously, the way Espen first worked out the task was due to that he perceived 2 as the whole and he took ¼ away from that. He took away ¼ of 2 which is ½. Also here, as in the case with Jens, there was a lack of compatibility between the teacher’s and the pupil’s thinking. However, this time the teacher acknowledged the pupil’s contribution and challenged him on how he had been thinking. Hans demonstrated an open way of asking the child, acknowledging his thinking and incorporated it into the lesson. In the last contingent moment on which I will comment, Hans acknowledged Ella’s question why the 1 was there in 1 ¾. However, he did not incorporate Ella’s thinking in his answer. He worked out the task over again posing questions and funnelling Ella through his doing. Also when Ella repeated her question twice (‘but why is the one number there?’) Hans carried on with his explanation. When Ella still did not understand, she asked ‘but why can it not just be deleted then?’ Like in the case with Jens, Hans broke the task down, concentrated upon one of the circles and 4/4 in that being the same as one whole. Then Ella expressed her understanding. Discussion Hans responded differently to the contingent moments discussed above. I suggest that the reason why he responded differently to different pupils was based on his knowledge about these pupils’ different mathematical abilities. As with Espen, who was one of the best pupils in class, Hans knew that Espen neither was making an occasional error, nor that he did not understand. Therefore he challenged Espen. But did Hans understand why Espen had done what he did? In Hans’ lessons there was a confident atmosphere. Due to the atmosphere, the pupils were actively participating by asking questions and commenting on what their classmates were doing on the board. This confident atmosphere was created by the teacher through the way he invited the children’s contributions. Hans’ knowledge about the pupils’ different abilities and the confident atmosphere of his lessons are not aspects of a teacher’s knowledge as part of the knowledge quartet. However, analysing a lesson through the use of the knowledge quartet made these aspects of a teacher’s knowledge visible and has contributed to draw an even broader picture of how different aspects of a teacher’s knowledge surfaced in this lesson. Finally, I want to discuss the contingent moments which I suggest mirrored some common difficulties the children had in dealing with improper fractions. I have suggested that Espen’s way of interpreting the task was grounded in the way it was represented with circles. Although the teacher kept reminding the pupils what the whole was, pupils seemed to have From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 71 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 difficulties with it. Jens looked upon the shaded pieces as the whole. Petra did not understand how it was possible to take nine bits when eight was the whole. And Ella argued why the whole had to be there. This suggests a shortcoming with regard to the foundation and transition aspects of Hans’ knowledge which incorporates research about what factors that have shown to be significant with regard to pupils’ understanding or lack of such of fractions bigger than one. He chose to use the same illustrations of improper fractions and mixed numbers as he had done when dealing with fractions smaller than one, always emphasising that fractions are parts of a whole. In this matter the illustrations he used seemed to cause more difficulties than help in pupils’ work with fractions bigger than one. This suggests that emphasising fractions as part of a whole and illustrating fractions as such, can explain some of the difficulties revealed in this lesson. References Anghileri, J. 2000. Teaching number sense. London: Continuum. Askew, M. 2000. What does it mean to learn? What is effective teaching? In Principles and Practices in Arithmetic teaching, ed. J. Anghileri, Buckingham Open University Press. Dickson, L., M. Brown and O.Gibson. 1984. Children Learning Mathematics A teacher’s guide to recent research, 134-146. Oxford: The Alden Press Ltd. Novillis, C. 1976. An Analysis of the Fraction Concept into a Hierarchy of Selected Subconcepts and the Testing of the Hierarchical Dependencies. Journal of Research in Mathematics Education, 7:131-144. Rowland, T., P. Huckstep and A. Thwaites. 2005. Elementary Teachers' Mathematics Subject Knowledge: The Knowledge Quartet and the Case of Naomi. Journal of Mathematics Teacher Education, 8(3): 255-281. Shulman, L. S. 1986. Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2):4-14. Shulman, L. S. 1987. Knowledge and Teaching: Foundations of the New Reform. Harvard Educational Review, 57(1). Turner, F., and T. Rowland. 2008. The Knowledge Quartet: A Means of Developing and Deepening Mathematical Knowledge in Teaching? http://www.mathsed.org.uk/mkit/MKiT5_Turner&Rowland.pdf From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 72 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Post-16 maths and university courses: numbers and subject interpretation Peter Osmon Department of Education and Professional Studies, King’s College London The low take-up of mathematics post-16 and consequences for the traditional STEM (science, technology, engineering, and maths) subjects in higher education are well known. The effect on the newer IT-based subjects, like computing and communications engineering, and the commerce-based subjects, like business and management, economics, and finance is less widely recognised but is at least an equal cause for concern. Most university courses in these subjects are populated with students with no maths beyond GCSE, despite the evident need for better mathematical foundationsperhaps a year of post-16 maths. The scale of this effect and the consequences for these subjects in many university courses are described along with potential implications for the AS-level curriculum. Introduction: quantitative subjects, courses and professions My experience during a lifetime in HE doing teaching and research in physics, then in electronics, and then in computer science, and then working in academic management has led me to identify a collection of subjects where students need at least one year of post-16 maths. I call these the quantitative subjects but this does not imply mere familiarity with numerical ways of working: it is intended to mean some degree of general mathematical competence is required. Quantitative courses teach the various quantitative subjects and these courses tend to be gateways to various quantitative professions. In fact the route comprising post-16 maths, then quantitative course and then quantitative profession is so well trodden as to imply that setting students on this path is the main role of post-16 maths. However, many students without post-16 maths also progress to quantitative courses and thence to the quantitative professions. Quantitative subject groups The quantitative subjects are separable into groups, as follows: A. Traditional STEM: Mainly physics based (plus maths of course), includesMaths, Stats, Phys, AerEng, CivEng, MechEng, ElecEng, and also Chem, ChemEng; B. Post-IT revolution: Computer Science and Electronics-Optics technology based, includesCS, InfSc, AI, Electronics, Communications Technology; C. Commerce based: include Economics and Actuarial Sciences (relatively traditional subjects) and others (whose development is heavily indebted to IT) including Accountancy, Finance, Bus&Mgt, D. Bio-Tec based: Biology in combination with subjects from the other groups, includesBioChem, BioEng, BioInf; M. Miscellaneous: includes Architecture and Medicine. The allocation of subjects to these groups is not absolutely clear-cut. Thus, there are subjects, besides those in group D, that cross group boundaries. But the above separation into groups, which also corresponds roughly to the temporal sequence A, then B, then C, and then D in which subject groups have developed, albeit with considerable time overlap, is proving From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 73 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 helpful when considering the mathematical needs of the various quantitative subjects. However group D is at an early stage of development and so its needs are not discussed. UCAS data on admissions to quantitative courses My appreciation of maths requirements for quantitative courses, derived from my own experience and observations and discussions with colleagues, has been greatly extended and quantified by studying the data published by UCAS (UCAS, 2009). In this paper I use 2008 data for national student numbers accepted onto HE courses, and maths requirements for admission in 2010, across a sample of universities, across subject groups A, B, C. The chosen sample universities are ones where I have had some form of inside knowledge and are approximately one per decile of university ranking, where the ranking is according to average UCAS points score as given in The Times Good University Guide for 2010 (The Times, 2009). The UCAS data, while immensely helpful, is not published in precisely the categories needed for my purpose- for example electronics and electrical engineering courses are lumped together. Also, the courses data is complicated by the existence of combined subject courses. Where these combinations fall within my groups, for example Economics & Management, the overall picture remains clear but where they cross groups, for example Maths & Management, they blur the picture somewhat. Organisation and presentation of the data So as to give as clear picture as possible of the maths knowledge universities are requiring for their quantitative courses and how this varies according to university ranking and subject group I have summarised and transcribed the published data. (See Table 1 below). Table 1 Maths qualifications for entry to Quantitative Subject HE courses ** For clarity the Table is divided into three parts- one for each subject group (A), (B) and (C). The columns divide subject groups into subjects and there is a row for each From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 74 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 university in the sample. Table entries are the corresponding maths knowledge level required for entry. The Table is the most recent snapshot. Student numbers by subject For most subjects the picture is confused by the many distinct codes for subject variations and course modes. It is apparent that lower ranking universities offer more combined subject courses, for example Maths with Stats and OR, occasionally to the exclusion of a core course, and also subject combinations which cross my subject groups. In my opinion these are generally marketing devices to attract better qualified students. Student numbers given for each subject are summed across all the variations (which may give rise to some double counting). Maths entry requirements For clarity, so far as possible, I have focussed on the core code for each subject- single subject, three year full-time course. The requirements shown are minimum entry requirements. “-“ means the subject is not offered. “?” means a minimum maths knowledge for entry is not specified. Where a subject is only offered in combination with an adjoining subject in the Table the requirement is shown straddling the two subjects, thus “.GCSE.” For clarity the only maths requirements shown are GCSE and A-level. (Equivalent IB, Scottish Higher, etc, are not mentioned.) GCSE maths grades: GCSE means Pass at gradeC, higher grades are shown explicitly. AS-level maths: not present in the Table- presumably because the majority of students go on to full A-level so it is not used as an entry condition. Full A-level maths: the required grade eg “B” is shown. Commentary on the Table content Subject group (B) The table shows only CS courses in this group. This is a consequence of the way university courses are presented and UCAS data is collated: specifically courses in Electrical Engineering and Electronics are lumped together, and courses focussing on communications are also recorded there or else under CS, rather than as a separate subject. These (CS, Electronics, and Communications) are subjects where the author has first hand teaching experience and is aware that the mathematics needs differ from Electrical Engineering. Correlation with university rank Subjects offered. Chemistry and Physics tend to be offered only by high ranking universities and these tend not to offer Business. Maths needed for particular subjects. Some variation of maths grade is to be expected, but where the same subject, eg Economics or Computer Science, requires A-level grade A at a high ranking university and GCSE grade C at lower ranking universities, then it is remarkable. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 75 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Student numbers in the subject groups The numbers of students in each of the three subject groups are comparable. The combined numbers in the newer groups (B) and (C) are double the number in the traditional STEM group (A) and this is justification for emphasising the hitherto neglected maths needs of these groups. Low maths requirements turn quantitative subjects into qualitative ones The inevitable consequence of having a low maths hurdle for entry, observed at first hand by the author, is that subjects which are essentially quantitative have to be interpreted qualitatively. (The alternative for universities would be to close most of their groups B and C courses!) This in turn means the quantitative professions are fed graduates who are inappropriately prepared. In the opinion of the author this fact implies a significant downgrading effect on the national skills-base. Mathematical needs of the quantitative subject groups The government sponsored STEM initiative (STEM, 2007) has drawn attention to the national importance of the traditional STEM subjects (group A) and the (full) A-level curriculum seems to be associated with the maths needs of this group. In this paper I am more concerned with the maths needs of the newer quantitative subjects, groups B and C, where most of the quantitative student numbers are to be found and where most courses are only viable because they are filled with students without post-16 maths knowledge. The table shows that in quantitative subject groups B and C most of the students are on courses that require no more than GCSE grade C maths. From direct experience of teaching group B courses and discussion with colleagues who teach group C courses it is clear that these students need 1 year of post-16 maths to engage effectively with their subject. (This contrasts with the group A courses which generally require 2 years of post-16 maths.) Remedial/foundational maths classes during the first year of the university course are not very successful. This can be attributed to two factors: (a) the students had their last (GCSE) maths class more than two years previously, so any mathematical knowledge and thinking habits they may have acquired are very rusty, (b) these students typically disliked maths at school and resent being given a dose of it at university when they really want to be getting on with learning their major (which no-one told them is a quantitative subject!) The remedial/foundational maths typically occupies about 25% of the first year of the university course. 25% of one year is about the time that would be allocated to an AS level at school. But it is noticeable that the maths being taught is rather different from the A-level curriculum. It s also noticeable that the maths taught in group B courses is different from the group C maths. The following two questions emerge. (a) Can an AS-level maths curriculum be devised that meets the one-year of post-16 maths requirements of groups B and C and also the first year of the two-year maths requirement of group A? (b) Can this AS-level, somehow become a requirement for entry to quantitative courses? The bigger picture The information in the Table is a snapshot taken during a particular year. But quite big changes in quantitative subjects are occurring over time: there seems to be a pattern whereby new subject groups develop and demand for courses in older ones declines relatively. The timescale for significant change seems to be about a decade. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 76 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Thus, Traditional STEM subject have been haemorrhaging numbers since the 1970s evidenced, for example, by closure of Chemistry and Physics departments at City and more recently Chemistry at QML, as well as closure of many engineering departments. The students who might otherwise have populated these departments appear to have migrated over time in the direction left to right across the table: from group A to group B to group C. . The first group B courses demanded A-level maths. But, as is well known, the number of student with post-16 maths has not increased in step with the general expansion of HE and hence much of the growth of group B and then group C numbers, except at the higher ranking universities, has depended on students with GCSE maths only, as the Table shows. From their inception most group C courses generally expected no more than GCSE maths from their students- presumably because the large cohorts of students wanting admission to these courses had no more maths to offer. If, as seems plausible, the currently emerging group D grows rapidly during the next decade- following the growth pattern of the earlier groups B and C- then presumably this will be at the expense of all three older groups and presumably, unless there are dramatic changes in take-up of post-16 maths, these courses too will mostly be populated with students having GCSE maths only. Besides the development of new disciplines, which may be more attractive to students than established ones, there are other mechanisms governing the changes in quantitative subject provision. Universities are now conscious of being in a league table- ranked according to the average (across all courses) UCAS score of their undergraduate intake. Universities can optimise their scores by allocating students numbers to courses according to the UCAS points they can attract, leading to closure of some courses and departments. (Courses with small intakes have always been at risk of pruning, but the league table provides a spur and a metric.) University departments then market their courses to attract students with maximum UCAS points- points scored becoming more significant than subjects studied- which can be rationalised by treating UCAS points as a measure of ability. This in turn implies a more foundational role for the first year of university courses. In their turn secondary students will come to recognise this situation and since they are free to choose the subjects they study post-16 and, since they are likely to perform best in the subjects they enjoy most, this state of affairs seems purely benevolent. But what of mathematics, which is not generally well liked (Brown, 2008) and maybe harder (Smith, 2004) than most subjects? The above scenario suggests fewer students will study maths post16- unless something changes so that more they find they like doing it! Conclusion The government sponsored STEM initiative has drawn attention to the number of students gaining full A-level maths and recent improvement in this number, emphasising its importance for the future of STEM subjects and the national skills-base. In this paper I have attempted to shift the spotlight towards the newer quantitative subjects (groups B and C) where the student numbers are greater than in traditional STEM (group A) - twice as many in fact- and where most courses are only viable because they are filled with students without post-16 maths knowledge. Students acquire a dislike of maths at school and so generally don’t study it post-16. Schools then pass the problem up to the universities to deal with. For their part universities fail to clarify the importance of post-16 maths knowledge for studying quantitative subjects. Lack of mathematical preparation weakens the courses so that students enter the quantitative professions with only qualitative understanding of their subject. The problem would be eased if courses in these newer subjects required their intakes to have one year of post-16 maths From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 77 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 (AS level). But of course they won’t be able to do this until large numbers of students choose to take AS level maths- which seems unlikely until it becomes a course requirement: chicken and egg! Perhaps the way to improve take-up of AS-level maths is by devising an AS-level curriculum that meets the one-year of post-16 maths needs of groups B and C and can also serve as the first year of the two-year maths requirement of group A. References Brown, M., P. Brown, and T. Bibby, 2008. “I would rather die”: Reasons given by 16-yearolds for not continuing their study of mathematics. Research in Mathematics Education 10, no. 1: 3-18. Smith, A. 2004. Making mathematics count. London: The Stationery Office. STEM, 2007. http://www.dcsf.gov.uk/stem/ The Times, 2009. http://extras.timesonline.co.uk/tol_gug/good university guide UCAS 2009. http://www.ucas.ac.uk/ From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 78 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The role of proof validation in students' mathematical learning Kirsten Pfeiffer, School of Mathematics, Statistics and Applied Mathematics, NUI Galway The study of proofs is a major obstacle in the transition from school mathematics to university mathematics. Given the importance of argumentation and proof in the spectrum of mathematical activities, the incoming students' understanding, appreciation and knowledge of the nature and role of proof must be considered. I describe the results of an exploratory study of first year mathematics undergraduates' criteria and learning process when validating mathematical arguments or proofs. The study is based on a series of written tasks and interviews conducted with first year honours mathematics students at NUI Galway. I presented the whole class with numerous proposed proofs of mathematical statements, and asked them to evaluate and criticize those. The first year students' written comments on different and partly incorrect 'proofs' of mathematical statements revealed some information about their criteria when validating mathematical arguments. In recently held interviews with eight randomly chosen students I focussed on the learning experience during the process of proof validation. Considering the observed learning effect and its large potential extent during the process of proof validation I propose its practice in the teaching of mathematics. Keywords: proof, proof validation, transition to University mathematics In the first part of this article I explain what is meant by proof validation and consider various aspects of this activity, including how and why mathematicians validate mathematical arguments. I will then describe our experiments and findings, and consider the undergraduates' validation skills and practices in relation to previously characterized aspects of proof validation. I emphasize the learning effect during the process of proof validation and finally argue for explicit inclusion of its practice in the teaching of mathematics, as the development of validation skills not only improves the practice of validation itself, but also the ability to construct proofs, the understanding of mathematical context, the knowledge of proving strategies and the links between different areas of mathematics. On proof validation. Before considering how students validate proofs I discuss the nature of proof validation; I further distinguish it from other types of reading and from construction of proof. Selden and Selden (1995) call the readings to determine the correctness of mathematical proofs and the mental processes associated with them “validations of proof”. I extend this description of proof validation considering that mathematicians, advanced students or maths teachers validate not only to determine the correctness of an argument. Discussions with a number of experienced mathematicians suggest that they also wish to reach an understanding of why a mathematical statement is true and often to understand the content and the position of the proved statement in a wider context. Factors that experienced mathematicians might consider while validating proposed proofs include: whether the argument provides the reader with understanding,transparency and quality of proof idea and strategy, clarity of the structure, whether the reasoning is precise, correct and sufficient, and whether the argument is From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 79 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 convincing. In Section 2 below I report on my investigation of the criteria that students consider as essential for a valuable mathematical argument. The activity 'proof validation'. Validation, in comparison to the reading of non-mathematical texts, requires the reader to put some additional effort into understanding of the reasoning. Validation usually takes more time, the validator might consider the whole proof or parts of it several times and might be more inclined to write a few notes checking deductions, verifying justifications, etc. According to Selden and Selden (2003) the mental process when validating proofs can include for example asking/answering questions, constructing subproofs or recalling other theorems and definitions. It is well documented that construction of proof is a major obstacle for students. Selden and Selden (2003) describe how the ability to validate proofs relates to the ability to construct them. On the one hand proof construction and proof validation are different. Proof construction requires 'the right idea' at the 'right time'. The validation process can usually be managed in a linear order, unlike construction of proof. On the other hand proof construction and proof validation entail each other as one considers during the process of proof construction how that proof would be validated, and as validation of a proof is likely to require the construction of subproofs. I summarize this relationship in the following diagram. Figure 1: Construction related to validation of proof Considering my comments above on the nature of proof validation, I extend this diagram to highlight the extent of the learning effect through the process of proof validation. Figure 2: Proof Validation in the process of learning about mathematical proof I summarize my hypothesis: The ability to validate proofs can improve the ability to construct proofs, develop deeper understanding of the meaning and significance of the proved theorem and develop knowledge of proving methods or strategies. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 80 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The experiment. The study is based on several tests and interviews conducted with first year honours mathematics students at NUI Galway. The students' way to validate mathematical proofs first caught my interest when I was analysing their responses to a written exercise that I held in May 2008, at the end of their first year at University. My research has focussed on that topic since. Therefore the design of further research instruments concentrated on proof validation. The test for the new incoming students (D-test08), held in September 2008 with 103 participants, included tasks designed to give me some insights into their proof validation skills. Based on the findings of the analysis of the written exercises I designed interviews to be held with a smaller number of students. The aim of these oral exercises was to get a deeper insight in the students' validation processes. The interviews were held with eight undergraduates in March 2009. I report below on findings arising to date from all three of these exercises. Analysis of the data is ongoing. Some of the questions had been adapted (with permission) from the Longitudinal Proof Project which ran 1999 until 2003 in the U.K. Considering the previously described aspects of the process of proof validations I investigate how our first year students validate mathematical proofs, which criteria they use to decide whether an argument is correct or not and in the learning process during the exercise of validating and comparing different mathematical arguments. Written exercises. The test held in May 2008 was attended by 37 participants. The students were presented with six attempts to determine, with proof, whether the statement “When you add any two even numbers, your answer is always even” is true or false. For each of the attempts, I asked the students to give a mark out of five and a line of advice. Observations from written exercises: students' criteria when validating proofs. The prevalence of some expressions indicates what the students found essential in a good proof. I list below a number of themes that caught my interest, either because they appeared quite often or because they surprised me. The role of examples. When confronted with a few examples to show the truth of the statement, I found that most students recognized the necessity of rigour and commented on that. "He has just given examples, this is not a proof" or "Not a general solution" are typical remarks. Some of the students' comments indicate that "proof by example" might be seen as another type of proof, just not as good as one including a general formula: "It's not bad. But it's only proved by example.", "Although this does prove the statement, it only does so for a few egs...". On the other hand, students do give examples a surprisingly high if not essential value within a proof. They often deduct marks on the basis of absence of examples. Some students comment on the absence of examples to explain the answers ("Give an example to show ...") but most criticize the lack of examples as an essential part of the reasoning. Some students request examples in (correct) answers to "back up proof" or to "finish the proof". One student criticizes a correct answer because the statement is "not proven by numerical example". Without any examples students don't seem to be satisfied by an argumentation. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 81 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 "Not mathematical enough" The following are a few selected comments on a correct answer that was expressed purely in the form of text: "The proof makes sense but she could have used a more mathematical approach", "Good intuitive answer but needs a mathematical proof”, "Correct answer but show mathematically", "The proof should be shown mathematically as well as in words". The argument formulated as text was "not mathematical enough" to most of the undergraduates. In comparison to another more algebraic looking approach one student comments: "Although Cathy's answer is true there is too much English and does not mathematically prove it unlike Aoife". These comments raise the question of what the students associate with the term "mathematical". Their comments indicate that "mathematical" appears to mean including formulas ("Try to come up with formula"), algebraic equations ("Give clear equation to support your answer", "Would like to see this expanded with a general equation") and mathematical notation ("Use mathematical notation to show this", "Cathy's answer is well written and ... although she should sum her answer up .. using formal notation"). Consequently another answer, which is fundamentally and irreparably incorrect but includes algebraic equations and mathematical notation, seemed basically correct to more than half of the students. Fewer than 30% of the students recognized that this answer was wrong. A visual approach to prove the statement is generally not accepted by the students. The proposed answer consisted of a diagram showing how an even number can be represented by two rows of dots, and how addition of even numbers can be interpreted as concatenation of two such representations. Most students interpreted the answer as just one example, visualized in a diagram. "Again Finn's answer only covers 1 solution. He needs to give a general statement.", "This proves that it works for 12+8. It doesn't prove for all cases.", "Not a proof, just an example" are a few typical answers. 27% recognized the idea behind the illustration, but didn't acknowledge a graphical representation of the correct idea as a mathematical proof: "Good visual proof but use mathematics", "Good visual representation but needs notational explanation". A proof without numbers and words can't be sufficient: "There are no words in this proof", "Proof is illustrated using graphics rather than numbers", "This does not prove anything, words and numbers are needed", "There are no words in this proof". "it doesn't explain..." Some mathematical educators argue that whether or not an argument is accepted as a proof depends not only on its logical structure, but also on how convincing the argument is. That aspect seems to play a role in the students' proof evaluation as well. "Nice pictures, you could have written a line explaining it though", "Intuitively correct but needs to explain why the answer means the statement is true", "Need more explanation" or "She should explain what she is doing". The positive comments on the highly marked answers often include a note about the good explanations: "Aoife has a very clear and straightforward answer", "Well explained answer" or "Aoife is using clear and simple language to get her answer across..." are a few comments on the students' favourite answer. Those comments indicate that just having a good idea to prove a statement is not sufficient for the students. The skills of convincing and explaining ideas to others matter to them. I summarize that after their first year in university most students are aware that checking the truth of a statement for a few examples is not sufficient to prove the statement. On the other hand examples play an important role in mathematical argumentation to students. Even after accepting the correctness of an argument they are not convinced until it is shown with a few examples. Overall the students' picture of proof seems to be vague. To them a valuable proof should have a certain structure, starting with a definition, followed by some algebraic From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 82 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 equations or formulas, and finishing with some examples. Structure and formalism seem to be more important to the students than the idea behind the proof. If these requirements are met most of the students give at least a few marks regardless of the correctness of the particular steps or whether the overall idea makes sense to them or not. It seems, a good idea to prove a statement is not being valued as highly as the structure and formalism of a proof. Beside structure and formalism the quality of explanations played a role in the students' proof evaluations. It seemed that if an answer didn't show attempts to convince the reader of an argument, most of the students would deduct marks. Oral exercises. In March 2009 I held interviews with eight randomly chosen students who had attended the written exercise in September as well. The aim was to get a deeper insight into • students' opinions about valuable proofs. What do students mean when they use the term “mathematical”? Do they ask for examples in order to understand the given reasoning or because they consider them as essential part of proofs? • students' validation process. How do they attempt the task of validation? For example, do they read the proposed proofs line by line? Do they write notes, verify the arguments? • the learning effect during the validation process. To facilitate comparison of the results one of the statements chosen for consideration in the interviews was similar to the one in the written exercise, but a bit more difficult. The squares of even numbers are even, and the squares of odd numbers are odd. The other statement was different from the exercises the students had performed in the context of this study so far. Let f be a quadratic function, f(x) = ax² + bx + c with a,b,c є R and a>0. Show: f can't have more than two common values with its derivative f'. Again the students were confronted with five or six different arguments, some incorrect or partly incorrect, and asked to comment on them and finally rank them. Some of the proposed proofs were algebraic, some visual, some written in text, others wrong but expressed using “typical” mathematical notation. Observations from the oral exercises: the learning process when validating and comparing different mathematical arguments. A detailed analysis of the interviews is in progress. In structuring and partly transcribing them a remarkable pattern caught my attention. The students were very quick in deciding whether they liked an answer or not and their first opinions and comments were similar to those in the written exercises. During the meeting though I could observe a process of understanding when spending more time with the task, comparing ideas with some proofs they have seen somewhere else, and sometimes even questioning their own criteria. Especially when ranking the different answers the students reconsidered their opinions and sometimes changed their minds about certain answers. These findings are in line with Selden/Selden and Alcock/Weber: • Selden and Selden describe that students' performance in distinguishing valid from invalid arguments improved dramatically through the reflection and reconsideration during the interview. (Selden and Selden, 2003) • “Our results suggest that many of the students in our study could perform this task competently, but did not do so without prompting.” (Alcock and Weber, 2004) From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 83 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 The fact that the students were forced by the ranking task to spend some time on their reflections encouraged a learning process. I conclude that the understanding of mathematical concepts can improve considerably during the process of careful proof validation. Conclusion The findings show on the one hand that undergraduates have a vague yet inflexible picture of valid proofs. Structure and 'mathematical' looking formalism seem more important to them than the idea behind its appearance. On the other hand I discovered that reflection during the process of proof validation encourages a learning process about the nature of mathematical proofs. Recalling the discussion in the first part of this article about the nature of proof validation, in particular the relation between construction and validation of mathematical proof and the attainment of understanding through validation, I conclude that practice of proof validation can not only improve students' validation skills but can also lead them to a better understanding of mathematical content and to improved appreciation of deductive reasoning. References Alcock, L. and K. Weber. 2005. Proof validation in real analysis: inferring and checking warrants. Journal of Mathematical Behaviour 24: 125-134. Selden, A. and J. Selden. 2003. Validations of proofs written as texts: can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education 34(1): 4-36. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 84 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 An exploration of mathematics students’ distinguishing between function and arbitrary relation. Panagiotis Spyrou, Andonis Zagorianakos Department of Mathematics, University of Athens, Greece This paper focuses on students’ awareness of the distinction between the concepts of function and arbitrary relation. This issue is linked to the discrimination between dependent and independent variables. The research is based on data collected from a sample of students in the Department of Mathematics at the University of Athens. A number of factors were anticipated and confirmed, as follows. Firstly, student difficulties involved vague, obscure or even incorrect beliefs in the asymmetric nature of the variables involved, and the priority of the dependent variable. Secondly, there were some difficulties in distinguishing a function from an arbitrary relation. It was also thought that additional problems occur in the connotations of the Greek word for function, suggesting the need for additional research into different linguistic environments. Introduction The concept of function is essential in the understanding and learning of mathematics. It is considered to be the most important concept learnt from kindergarten to college or university (Dubinsky & Harel 1992). The difficulties students experience with this concept can only be understood in relation to its definition and the appearing of cognitive obstacles. Several researchers found that in the early stages of function teaching in secondary schools that natural models dominate using mainly 1-1 (one-to-one) functions. (Evagelidou, Spyrou, Gagatsis, & Elia 2004; Elia & Spyrou 2006). The reliance on the natural models means that the connection between the dependent and independent variable is emphasized rather than focusing on the priority of dependent to the independent variable. Furthermore, the natural models which are offered to the pupils are idealized, distant from the realities from which they were created and described in analytical formula, thus making it “difficult for the students to distinguish between relationships discovered by experience and the mathematical models of these” (Sierpinska 1992, 32). This approach results in a difficulty in realizing that the dependent variable is a magnitude which is used to estimate a measurement and that the independent variable is the means for this particular purpose, with or without an analytical formula. The etymology of the Greek word for “function” introduced a note of caution. The root of the Greek word for function (“synartisi”) is different from the origin of its Latin equivalent which is mainly operational. In colloquial Greek when a person or abstract phenomenon such as time, speed or measurement has a functional relation (“synartate”) with another person or abstraction, the effects tend to be symmetrical. A bond is implied, whether active or inert, which is triggered when “one side” (usually either side) is altered, evoking a change in the “other side”. Therefore, the common perception of the Greek word for “function” implies the symmetry of the function variables. This symmetry might create a difficulty in understanding the difference between the variables in the mathematical definition, i.e. which is the means and which is the one to measure. Sierpinska (1992) recognized this difficulty as the obstacle, “regarding the order of variables as irrelevant” (p. 38). This definition of the obstacle is the starting point of this paper. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 85 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 To conclude, the over reliance on one-to-one correspondence in function teaching and the common perception of function creates an obstacle which may persist throughout Higher Education. It was therefore decided to research the persistence of this obstacle in the thinking of students in Higher Education. Theoretical framework The definition of function went through several stages until it reached its present form. This progressive development has given rise to a number of epistemological obstacles. The first definitions of the concept of function by Bernoulli, Euler and Cauchy were not complete. This is because they saw the symmetricality of the dependent and independent variable, in the context of a relationship. Sierpinska marks this “moment” in the concept’s theory as an epistemological obstacle: EO(f)–5 (Unconscious scheme of thought) “Regarding the order of variables as irrelevant” (1992, 38). It was Dirichlet in 1837 that used his study on Fourier series and the conditions under which a Fourier series converges, and formulated a general definition of function: "if a variable y is so related to a variable x that whenever a numerical value is assigned to x there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x” (Boyer 1968, 600). Dirichlet’s definition of function is still in use and his main conclusion states the necessity of the dependent coordinate being uniquely determined and not always the inverse. Therefore, the Dirichlet definition expressed precisely, for the first time, the notion of a mediated measure in the concept of function. That is to say: to estimate the dependent variable y, and to achieve it although there is no immediate access to y, is to measure it through x. Therefore the independent variable is the mediating variable which gives access to the dependent variable, resulting in the priority of the latter. The literature on the study of the epistemological obstacles that occurred through the development of the definition of function is particularly rich (e.g., Freudenthal, 1983; Sfard, 1992; Dubinsky & Harel, 1992; Sierpinska, 1992; Even and Tirosh, 1995). However, it is difficult to find any research on the particular obstacle which is the subject of this research. As discussed in the introduction, the teaching of function during the first years of school is oriented towards a common perception of function, emphasizing the relation between the dependent and the independent variable, disregarding the priority of the dependent variable. Furthermore, the dominant use of one-to-one functions makes it harder for students to recognize the importance of the dependent variable, which is the target of the measurement. In addition, the focus on the relational aspect of biunique functions conceals the richness of the applications which the definition allows. Finally, the common perception for teaching function in school can be seen as a collection of habits, whereby “the character of a habit depends on the way in which it can make us act” (Peirce 1958, Vol. V, par. 18). It is this habitual comprehension of function, diverging from the formal definition, which results in mathematical obstacles and difficulties. Thus, we formed the following research questions (accompanied by their relation to the questions of the questionnaire that was given to the students —the justification for which is given in the next section): [1] Can the students recognize the difference between the concepts of function and of an arbitrary relation? (1st, 2nd question of the questionnaire) [2] Can the students distinguish the order of the variables x and y, and the asymmetry that they have? (3rd, 4th question of the questionnaire) From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 86 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Methodology The methodology arose from the theoretical framework of this paper and sets out to test the persistence of the mathematical obstacle described above, in students in Higher Education. It was anticipated that students would encounter problems in recognizing the priority of the dependent variable and in distinguishing between a function and arbitrary relation. Within this theoretical framework it was decided to design a survey in two phases. Phase 1 included the completion of a 4 questions questionnaire by the students (see Figure 1 below). The format of the questions in the questionnaire included two types of questions: 1] Crosscheck items corresponding to questions: (key Q=questionnaire) A) (Q1) Students were given 3 correspondences on graphs and another 3 with table values. They were asked to find out which represented functions. B) (Q3) Students were given 4 functions on a graph. They were asked which of the 4 would still be function if the lines on the graph were rotated by 90°. 2] Open-type questions with short answers such as: A) (Q1) Students were asked to make the necessary changes to the graphs and table values to change the arbitrary relations into functions. B) (Q2) Students were asked to give two examples of (arbitrary) relations which were not functions, one algebraic and one represented graphically. C) (Q3) Students were asked to justify whether the 4 functions on the graph remained functions when turned 90°. D) (Q4) Students were asked in which case(s) the 90° rotation of a function’s graphical representation represents a function and to give a general rule. Figure 1 The Questionnaire given to the students ❶Which of the following relations are function relations? Make the necessary corrections to the rest of them, in order to transform them into functions. x -1 0 1 2 3 y 0 1 2 3 4 x -3 -2 -1 0 1 y 2 2 2 2 2 x 5 -3 5 0 -3 y 3 2 1 1 6 ❷Give two examples of arbitrary relations that are not functions (one described graphically and one analytically (with an algebraic formula)). ❸What happens to the graphical representations of the following functions if the line on the graph is rotated by 90°? Are they still functions? Give a short justification in your answer. ❹In which situation(s) does a 90° turn of a function’s graphical representation represents a function? Which general rule would you use? The questions in the questionnaire were designed to correspond to the research questions posed in the theoretical framework, as follows: A) The 1st question asks students to distinguish which correspondences are functions and which are arbitrary relations. The 2nd question asks them to give two examples, one graphical From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 87 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 and one with an analytical description, of an arbitrary relation, which does not fit the function definition. The answers to both of these questions correspond to the 1st research question. B) The 3rd question asks students to distinguish when a function remains a function on a graph when the graphical representation is rotated by 90°. In the 4th question they were asked to interpret this movement and to justify their answers with reference to the application of the general rule. This 90° turn on the graph occurs as a consequence of the permutation between the dependent and independent variables. Therefore, the students are tested in their ability to realize the asymmetry of the x and y variables, i.e., the 2nd research question. The questionnaires were distributed to 17 students who attended lessons in “Epistemology of Mathematics”, in April 2009, in the Mathematics Department of the University of Athens. The course was chosen because of the accessibility to a wide range of students on different courses, theoretical and applied, and at different stages of study. Student participation was voluntary; the questions were answered without time constraints, taking them approximately half an hour to complete. Phase 2 of the research took place after about 3 weeks (May 2009) and included semistructured interviews, with 13 students out of the 17 who participated in the questionnaire. Students were fully informed of the objectives of the research and gave their permission for their interviews to be recorded. In all interviews the interviewer emphasized that the purpose of the interview was not to examine the students but was to explore what they thought when they answered the questionnaire, regardless of whether the answers were right or wrong. Each interview aimed to clarify the answers given on the questionnaire and the problems the students had encountered completing it. Students were asked additional questions in order to start a general discussion and explore their understanding of function with regard to their school and university education. They were asked about (a) their experience of the concept of function at secondary school, (b) their acquisition of the concept of function at the university, and (c) what they thought the use of function was outside the mathematical frame. All the interviews were audio-recorded and listened to, in order to assess them against the research questions. This assessment showed a consistency in the interview results. Four representative interviews were selected, using the following criteria: (i) Variety of students’ responses to the research questions as shown in both the questionnaire and the interview. (ii) The interviews comprised different levels of understanding: 1 high (Georgia), 1 moderate (Diana), 1 low (Iris). The 4th interview highlighted the findings of the research (Thanos). The most relevant parts of the interviews for the research were transcribed. These were then divided into 5 minute intervals or divided according to the research questions, and accompanied with short comments. Due to the limitations of space this paper contains just a few but characteristic dialogues from the interviews. Description of the results — Discussion Summarizing the main findings of our research, we observed the following: R1) All students interviewed, except one, gave only examples of 1-1 (biunique) functions. They said they had used examples they recalled from learning functions at school. R2) All students interviewed, except one, had difficulties in giving good explanation, or any explanation, for the “many-to-one” condition in the definition. R3) The students used mnemonic rules: A) Seven students out of 17 (41%) gave the same example (the circle example) as a graphical representation of an arbitrary relation that is not a function (Q2). Moreover, all the From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 88 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 interviewed students except one used inclusively 1-1 examples of functions, an attitude that shows the strong connection that the students still have with their early function experience. This connection was admitted extensively in the interviews. B) Extract from Diana’s interview, typical of the definitions used as mnemonic rules and the confusion that follows: D: I try to recall the definitions, as far as I can. When I cannot, I “put my mind to work". R: Does your “mind” ... agree with the definitions? D: On this occasion it agrees, not always. R4) 29,5% of the students (5/17) gave the wrong answers to the 1st as well as the 2nd question of the questionnaire, where they were asked to recognize the difference between the concepts of function and arbitrary relation. R5) 47% of the students (8/17) gave the wrong answer to the 3rd question of the questionnaire, revealing a difficulty in distinguishing the order of the variables x and y, and their asymmetry. R6) 64,7% of the students (11/17) gave the wrong answers to the 4th question of the questionnaire, revealing a difficulty in distinguishing the order of the variables x and y, and their asymmetry. R7) Most of the students interviewed have only been concerned with functions in the context of their school and university education. Nevertheless, their education did not equip them with the necessary tools for interpreting the concept of function. Iris is a typical example: although she reported that she always thinks of graphical representations when she deals with functions, she still had difficulties in giving examples of arbitrary relations which are not functions. She did not understand what a 90º rotation of the graph meant despite knowing the formal definition of a function and applying it correctly in the 1st question. R8) The interview with Thanos is indicative of the students’ confusion with the “many-toone” condition of the definition. For instance, concerning the use of functions, he reported as follows: “Wherever I want to put factors say, the x and y are essentially factors. (For example), x is able to measure temperatures and y to count days. Or x to count children and y to estimate the tax return. That is, apart from the fact that we put it in a two-dimensional frame and take a mathematical perspective; the two dimensions (the 2-axes coordinate system) are essentially two parameters. The three dimensions are three parameters, and so on.” It is apparent that he has misunderstandings concerning the priority of the dependent variable. He also considers x1, x2, x3, as parameters of the function f(x1,x2,x3), which “lead” to the y variable, thus showing that he is confusing the function of many variables with the “many-to-one” condition of the definition. The results from the questionnaires and interviews confirmed the problematic areas anticipated at the outset of the research. It is evident from the results of the questionnaires and the interviews that students experienced difficulties in answering all four questions. The most difficult question is question 4 (R6) where 2 out of every 3 students experienced difficulties. When incorrect answers for question 4 were combined with the incorrect answers for question 3, where almost half were wrong (R3), they provided evidence of the difficulties students have in distinguishing the order of variables x and y, and their asymmetry. Although there is a smaller percentage (29.5%) of wrong answers for the 1st as well as the 2nd question (R4) their weakness indicates the persistence of the problem. They show From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 89 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 difficulties students have in recognizing the difference between the concepts of function and of arbitrary relations. The results from the interviews (see R1, R2, R3, R7 and R8) provided data that revealed more than the questionnaires. The majority of the students had separate ideas about the definition and its application. All of the students knew a formal definition. Only 2 students gave examples of “a single-valued but not uniquely invertible function”. Many students still experienced difficulties when asked about this specifically in their interview. The dominance of the biunique functions is further evidence of the dominant influence of the first years of function teaching. From the evaluation of the questionnaire data we conclude that there are difficulties in students’ abilities to recognize the difference between the concepts of function and of arbitrary relation. However, most difficulties occur in students’ abilities to distinguish the order of the variables x and y and their asymmetry, confirming Sierpinska’s (2002) claim (EO(f) – 5). In our opinion, the etymology of the Greek word has an additional negative impact in students’ ability to overcome the mathematical obstacle, by encouraging a common perception in favour of the biunique function. This common perception is handed down through a school’s teaching methods, often remaining unchallenged. We think it would be worth testing the same mathematical obstacle in different environments, to isolate the influence of the etymology of the Greek word and its’ part in the persistence of the obstacle. References Peirce, C. S. 1958. Collected papers of Charles Sanders Peirce, Volumes I-VI (C. Hartshorne & P. Weiss, Eds.). Cambridge: Harvard University Press. Boyer, C. 1968. A history of mathematics. New York: Wiley & Sons. Freudenthal, H. 1983. The didactical phenomenology of mathematical structures. Dordrecht: Reidel. Dubinsky, E., and G. Harel. 1992. The nature of the process conception of function. In E. Dubinsky and G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy 85-106. Washington, D.C.: Mathematical Association of America. Sfard, A. 1992. Operational origins of mathematical objects and the quandary of reification—The case of function. In E. Dubinsky & G. Harel (Eds.), The concept of function, Aspects of epistemology and pedagogy 59-84. Washington D.C.: The Mathematical Association of America. Sierpinska, A. 1992. On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.), The concept of function, Aspects of epistemology and pedagogy (pp. 25-28). Washington D.C.: The Mathematical Association of America. Even, R., and D. Tirosh. 1995. Subject-matter knowledge and knowledge about students as sources of teacher presentations and subject matter. Educational Studies in Mathematics 29: 1-20 Evagelidou, A., P. Spyrou, A. Gagatsis, and I. Elia. 2004. University students’ conception of function. In M. J. Høines and A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education 2: 251-258. Bergen University College, Norway. Elia, I., and P. Spyrou. 2006. How students conceive function: A triarchic conceptualsemiotic model of the understanding of a complex construct. The Montana Mathematics Enthusiast 3(2): 256 -272 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 90 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Identifying and developing the mathematical apprehensions of beginning primary school teachers Fay Turner Faculty of Education, University of Cambridge In this paper I present a summary of a four year study into the development of mathematical apprehensions in beginning elementary teachers using the Knowledge Quartet as a framework for reflection on, and discussion about, mathematics teaching. The term mathematical apprehension is used as an inclusive term to cover both mathematical content knowledge and conceptions of mathematics teaching. Evidence from three case studies suggest that focused reflection using the Knowledge Quartet facilitated the development of mathematical content knowledge and promoted positive changes in conceptions about mathematics teaching. Experience and working with others in classrooms and schools were also seen to influence development and change in the teachers’ apprehensions. However, individual reflection was found to have a mediating role on the influence of these two social factors. Introduction The way in which teachers teach mathematics is influenced both by their mathematical content knowledge (Ball, 1988) and by their conceptions about mathematics teaching (Thompson, 1992). The mathematical content knowledge of elementary teachers has been found to be insufficient for teaching (Brown, Cooney and Jones, 1990; OFSTED, 2000). Researchers have also found that elementary teachers often have unhelpful conceptions about mathematics and mathematics teaching (Brown, McNamara, Jones and Hanley, 1999). Initial teacher education courses alone seem unable to produce necessary developments in mathematical content knowledge (Carré and Ernest,1993; Williams, 2008) or to promote sustained positive changes in conceptions of mathematics teaching (Brown et al, 1999) in beginning teachers. The aim of this study was to investigate the effectiveness of a sustained approach to developing the mathematical apprehensions of beginning elementary teachers. The study began with the conjecture that, supported reflection on the mathematical content of teaching might promote developments in mathematical content knowledge and changes in conceptions of mathematics teaching in beginning teachers. The Knowledge Quartet framework (Rowland, 2008) was used to facilitate such reflection both as a means to, and as a measure of, professional development. Three theoretical frameworks underpinned the study. Theoretical frameworks Mathematical content knowledge for teaching A theoretical framework for consideration of teachers’ mathematical content knowledge was derived from the seminal work of Shulman (1986; 1987) and from the work of Deborah Ball and colleagues in Michigan. Shulman’s three ‘knowledge bases’ which relate specifically to the content of teaching; subject matter knowledge (SMK), pedagogical content knowledge (PCK) and curriculum knowledge (CK) provided a foundation. The division of SMK into substantive and syntactic knowledge (Schwab, 1978) also informed my research as did From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 91 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 refinements of Shulman’s categories by Ball, Thames and Phelps (2008). They identified common content knowledge (CCK) and specialized content knowledge (SCK) as subdivisions of SMK, and knowledge of content and learners (KCL) and knowledge of content and teaching (KCT) as subdivisions of PCK. The Michigan group also include the category of knowledge on the horizon as an aspect of SMK and knowledge of the curriculum as an aspect of PCK. The Table 1 below illustrates the relationship between the categories of Shulman and of the Michigan group and represents the model used in this study as a framework for investigating the teachers’ mathematical content knowledge. Table 1 Mathematical Content Knowledge Subject Matter Knowledge (SMK) Pedagogical Content Knowledge (PCK) Common Specialist Knowledge onKnowledge of Knowledge of Knowledge of Content Content the Horizon Content and Content and the Curriculum Knowledge Knowledge Teaching Learners (CCK) (SCK) (KCT) (KCL) Conceptions of mathematics teaching The models of teachers’ conceptions of mathematics teaching identified by Kuhs and Ball (1986) provided the basic framework for this study although other models, such as those of Ernest (1989) and Askew, Brown, Rhodes, Johnson and Wiliam (1997), were drawn on where appropriate. Kuhs and Ball identified four dominant models: a classroom-focused view; a content-focused with an emphasis on performance view; a content-focused with an emphasis on conceptual understanding view and a learner-focused view. Ernest (1989) identified six models which were very similar to those of Kuhs and Ball, but included two extra categories combining characteristics from Kuhs and Ball’s models. This refinement was found to be generally unnecessary in my research although I drew on it where appropriate. I also drew on the work of Askew et al (1997) which identified three orientations in teachers’ conceptions about mathematics teaching; a transmission orientation, a connectionist orientation and a discovery orientation. An approach to developing mathematical content knowledge and changing conceptions of mathematics teaching My approach was based on the model of professional development through reflection in and on practice (Schön, 1983). However, the role of reflection was investigated in relation to ideas from socio-cultural theory models of professional development. In social theory knowledge is seen as situated in social situations and the development of knowledge as resulting from enculturation or socialisation into the professional culture (Cobb, Yackel and Wood, 1991; Lave and Wenger, 1991; Wenger, 1998). I recognised the role of socio-cultural factors in my study and expected teachers’ apprehensions, as revealed through observations and discussions of their teaching, would be contingent on the context of their teaching. I also recognised that the individual reflection of participants would be made within the communities of practice of schools, and would reflect the role and relationships of the participants within schools. Jaworski (2007) suggested that communities of practice (Wenger, 1998) became communities of inquiry when teachers worked collaboratively to reflect on and develop their practice. It was hoped that the participants in my research group would become such a community of inquiry. The rationale for this study was therefore that teachers should From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 92 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 be supported to use reflection on their mathematics teaching within the social contexts of their teaching and also of the research project. The Knowledge Quartet framework The KQ framework was developed by mathematics educators at the University of Cambridge from observation and videotaping of mathematics teaching (Rowland, 2008). Analysis of this teaching produced 18 ‘emergent’ codes (Glaser and Strauss, 1967) of situations in which mathematical content knowledge of teachers was made visible, e.g. ‘concentration on procedures’ and ‘making connections between concepts’. These were later classified into four ‘superordinate’ categories based on associations between the original codes. These categories make up the four dimensions of the Knowledge Quartet; foundation, transformation, connection and contingency. The foundation dimension includes the propositional knowledge of SMK and PCK that teachers draw on in their practice, as well as their beliefs about mathematics and mathematics teaching. Transformation encompasses the ways in which a teacher’s own knowledge is transformed to make it accessible to learners and connection includes issues of sequencing and connectivity as well as complexity and conceptual appropriateness. The final dimension of the KQ, contingency, could be described as ‘thinking on one’s feet’ and is concerned with the way teachers respond to unexpected student responses. The study The study began with 11 student teachers from the 2004-5 cohort of elementary (5-11 years) postgraduate pre-service teacher education course at the University of Cambridge reducing, as anticipated, to 4 in the fourth and last year of the study. Data came from observation and analysis of teaching using the KQ as well as from post-lesson reflective interviews, group and individual interviews and participant written reflective accounts. Transcripts of interviews and written reflective accounts were all systematically coded using the qualitative data analysis software NVivo 7. A grounded theory approach (Glaser and Strauss, 1967) was used which led to the emergence of a hierarchical organisation of codes into a number of themes. Case studies were built from the analysis of observed teaching as well as from analysis of interviews and the participants’ reflective accounts. Six themes in the development of the participants’ mathematics teaching emerged from the NVivo analysis. These were, beliefs, confidence, subject knowledge, experience, reflection and working with others. The KQ analysis of observed teaching provided a ‘spine’ for presenting findings in relation to the development of participants’ mathematical content knowledge. This was supported by data from interviews and reflective accounts organised under the themes of subject knowledge and confidence. Findings in relation to changes in conceptions about mathematics teaching drew primarily on data organised under the theme of beliefs, and were supported by lesson observation data. Data from the themes of experience, reflection and working with others gave insight into the influences on developments in the participants’ mathematical content knowledge and into influences on changes in their conceptions of mathematics teaching. Findings Development of mathematical content knowledge Looking at the teachers’ content knowledge in relation to the foundation dimension of the KQ suggested that development in propositional PCK, or knowledge of content and From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 93 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 learners, knowledge of content and teaching and knowledge of curriculum, were greater than in SMK, or common content knowledge (CCK), specialised content knowledge (SCK) and knowledge on the horizon. Where development in SMK did take place, it was in relation to SCK rather than CCK. Reflection on practice helped the teachers identify areas of their SCK that needed development. Development in these areas was achieved through support from me, attendance at in-service training or through self-study. It is unlikely that such developments would have occurred through reflection alone. The teachers’ active PCK, as revealed through the Transformation, Connection and Contingency dimensions of the KQ, was also seen to have developed over the study. The three teachers appeared to focus on different aspects of their Transformation knowledge, and this focus was reflected in the apparent developments in their practice. All three teachers made more effective use of demonstrations but this was most apparent in Amy’s teaching. A more considered use of representations was a strong feature of Kate’s practice and Jess showed particular development in relation to her use of examples. All the teachers considered making connections to be important for effective teaching, and demonstrated this throughout the study. Focusing on connections developed the teachers’ practice in different ways. Amy concentrated on making connections to individual children’s understanding and interests, Kate became more likely to make connections between mathematical ideas and Jess increasingly focused on the connections between calculation operations. The ability to act contingently also became a focus for the teachers, and they all increasingly considered this to be integral to effective teaching. Responding to children’s needs and ideas became central to Amy’s early years practice. Kate became particularly proficient at acting contingently to find alternative representations to address misconceptions or lack of understanding and Jess became more likely to respond to children’s errors and to try to understand their mathematical thinking. Changes in the teachers’ conceptions of mathematics teaching Each of the teachers held complex views of mathematics teaching, incorporating elements from all four of Kuhs and Ball’s (1986) views of mathematics teaching. Although the balance of these elements varied, there appeared to be a pattern in the direction of change in the three case studies. The teachers moved towards a focus on developing conceptual understanding and towards a learner-focused view of mathematics teaching. All the teachers had elements of a classroom-focused view at the beginning of the study but this quickly diminished. Throughout the study, the three teachers demonstrated content-focused conceptions of mathematics teaching with differences between them in the balance between an emphasis on performance and an emphasis on conceptual understanding. However, for all three, the direction of change over the four years was towards a greater emphasis on conceptual understanding. Amy demonstrated some emphasis on performance at the beginning but moved to a strong emphasis on conceptual understanding. Kate and Jess increasingly appeared to emphasise conceptual understanding although both retained strong elements of emphasis on performance. Amy began with a learner-focused view of teaching and this appeared to be strengthened over the study. Kate and Jess also appeared to focus increasingly on the needs of their pupils. However, the way in which they interpreted these needs differed. Amy’s interpretation was the most consistent with constructivist ideas inherent in a learner-focused view of mathematics teaching. Kate also increasingly tried to understand children’s mathematical thinking but remained focused on helping them achieve pre-determined products or processes. Jess appeared to interpret the needs of her pupils in terms of their ‘ability’ and the need for them to achieve success in order to develop confidence. Amy increasingly took an inquiry or problem solving approach to her teaching, consistent with a From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 94 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 learner-focused view of mathematics teaching. approach, but found it more difficult to adopt. Influences on development and change Kate and Jess also moved towards this Experience of teaching mathematics, and working with colleagues, were found to be important influences on the teachers’ mathematical apprehensions. However, reflection mediated the effects of these two influences. It was reflection on experience which catalysed developments in mathematical content knowledge, and the focus of the teachers’ reflection on experience which reinforced or changed their conceptions about mathematics teaching. Working with colleagues was also found to be an important influence in developing mathematical content knowledge and conceptions of mathematics teaching. This influence was quite different for each teacher. Amy’s learner-focused view of mathematics teaching was shared by her colleagues and reinforced this view. However, Kate questioned some of the practices in her school and Jess was uncomfortable with the way in which mathematics was taught in her first post. All three teachers reflected on the principles and practices of their schools facilitating alignment (Wenger, 1998) for Amy, leading to critical alignment (Jaworski, 2006) for Kate and to non-alignment with her first school for Jess. Reflection using the KQ to focus on the mathematical content of teaching helped the teachers to focus on their own and on their pupils’ understanding of mathematical concepts, and this supported developments in their SCK. Their reflection also informed the teachers’ thinking about how to transform their own understanding in order to make it accessible to their pupils, and this supported developments in their PCK. Where the teachers focused on the effectiveness of their teaching their KCT was enhanced, and where they focused on the understanding of learners, their KCL was enhanced. The KQ focused the teachers’ reflection on the content of the mathematics and on the children’s engagement with, and learning of, that content. This supported movement towards conceptions of mathematics teaching that emphasised conceptual understanding, and which were learner focused. References Askew, M., M. Brown, V. Rhodes, D. Johnson, and D. Wiliam. 1997. Effective Teachers of Numeracy. Report of a study carried out for the Teacher Training Agency 1995-96 by the School of Education King’s College London. Ball, D. L. 1988. Unlearning to Teach Mathematics. For the Learning of Mathematics, 8(1): 40-48. Ball, D. L., M. H. Thames, and G. Phelps. 2008. Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5): 389-407 Brown, S. I., T. J. Cooney, and D. Jones. 1990. Mathematics teacher education. In Handbook of Research on Teacher Education , ed. W.R. Houston, 639-656. New York: Macmillan. Brown, T., O. McNamara, L. Jones, and U. Hanley. 1999. Primary student teachers’ understanding of mathematics and its teaching. British educational Research Journal, 25(3): 299-322. Carré, C. and P. Ernest. 1993. Performance in subject-matter knowledge in mathematics. In Learning to Teach, eds. N. Bennett and C. Carré, 36-50. London: Routledge. Cobb, P., E. Yackel, and T. Wood. 1991. Curriculum and teacher development: Psychological and anthropological perspectives. In Integrating research on teaching and learning mathematics, eds. E. Fennema, T. P. Carpenter and S. J. Lamon, 83- 120. Albany, NY: SUNY Press. Ernest, P. 1989. The knowledge, beliefs and attitudes of the mathematics teacher: a model’, Journal of Education for Teaching 15: 13-33. Glaser, B. G. and A. L. Strauss. 1967. The discovery of grounded theory: Strategies for qualitative research. New York: Aldine de Gruyter. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 95 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Jaworski, B. 2006. Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning teaching. Journal of Mathematics Teacher Education 9: 187-211. Kuhs, T. M. and B. L. Ball. 1986. Approaches to mathematics: Mapping the domains of knowledge, skills and dispositions. East Lancing:Michigan State University, Center on Teacher Education. Lave, J. and E. Wenger. 1991. Situated learning: Legitimate peripheral participation. New York: Cambridge University Press. Office for Standards in Education (OFSTED). 2000. The National Numeracy Strategy: The First Year: A Report from HM Chief Inspector of Schools. London: HMSO. Rowland, T. 2008. Researching teachers’ mathematics disciplinary knowledge. In International handbook of mathematics teacher education, eds. P. Sullivan and T. Wood, 1: 273-298. Rotterdam: Sense Schön, D.A. 1983. The Reflective Practitioner, New York, Basic Books. Shulman, L. S. 1986. Those who understand, knowledge growth in teaching. Educational Researcher 15 (2): 4-14. Shulman, L. S. 1987. Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1): 1-22. Schwab, J. J. 1978. Education and the structure of the disciplines. In Science, Curriculum and Liberal Education, eds. I. Westbury and N. J. Wilkof, 229-272. Chicago: University of Chicago press. Thompson, A. G. 1984. The Relationship of Teachers’ Conceptions of Mathematics and Mathematics Teaching to Instructional Practice. Educational Studies in Mathematics, 25(2): 105-127. Wenger, E. 1998. Communities of practice: Learning meaning and identity. Cambridge: Cambridge University Press Williams, P. 2008. Independent review of mathematics teaching in early years settings and primary schools, Department for Children, Schools and Families (DfCSF). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 96 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 What Might We Learn From the Prodigals? Exploring the Decisions and Experiences of Adults Returning to Mathematics Robert Ward-Penny Institute of Education, University of Warwick This paper reports on a research project which explored the decision making process and the experiences of adults who had returned to mathematics after a significant period of time away. Data was gathered using a combination of a questionnaire and follow-up interviews with selected participants. This paper presents some of the key findings together with some examples from the stories of these learners. Finally, it argues that these ‘prodigals’ offer a vivid reminder of the role of mathematics as cultural capital, and an additional perspective on many issues of current interest in mathematics education. Introduction and Background At the time of writing, the twin issues of disaffection and underachievement appear prominently in many pieces of research (for example, Nardi and Steward 2003) and published reviews of teaching and learning (for example, Smith 2004). Against this rather pessimistic background there exist a significant number of individuals who voluntarily return to learning mathematics in one form or another. Having met a number of these individuals personally, I questioned how we might best consider their actions against the wider background; do they contradict our conceptualisations of disaffection and underachievement, or are they the ‘exceptions that prove the rule’? In order to focus this research around this apparent contradiction, this research was designed to include only a subset of adult learners: those whose initial schooling had taken place within the United Kingdom, and who had spent some time away from formal education before beginning their current study of mathematics. This last criterion gave rise to the collective moniker ‘prodigals’, after the biblical story of the prodigal son, and excluded learners who had begun adult learning courses directly from school. The three key research questions can therefore be posed using this new term: firstly, who are the prodigals? What are their demographic characteristics, and is there sufficient diversity within the group to suggest that there are different types of prodigal learners in mathematics? Secondly, what motivates the prodigals to return to the study of mathematics? How might we conceptualise their decision-making process? Finally, how do the prodigals’ experiences of learning mathematics as an adult compare to, and contrast with, their experiences of learning mathematics at school? By combining the answers to these three questions, we can begin to consider what we might learn from the prodigals. Construction of the Sample Constructing a sample that accurately reflects the prodigal community is problematic; in the words of Coben (2003, 73), “experience tells anyone who has ever worked with adults that there is no such thing as a generic adult learner of numeracy.” Furthermore, there is considerable diversity in the prodigals’ experiences, and some sub-groups were difficult to isolate, such as those undertaking informal, community-based courses, as well as learners who were studying using independent tutors and entering exams as private candidates. The sample for this initial exploratory research was drawn from learners in official programmes, From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 97 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 connected to a school or further education institution. Although this was primarily due to matters of convenience, there is some evidence to support this approach: the 2006 report into the Skills for Life programme reported that “more than two million of the 2.4 million people taking up courses by July 2004 undertook them in further education.” (House of Commons Committee of Public Accounts 2006, 6) In an attempt to increase the range of backgrounds included in the study, participant groups were selected from two contrasting regions, one centred on a major city, and another which was largely non-metropolitan. In fact, the eventual results demonstrated little or no differences between regions. The groups for inclusion were selected through a process of negotiation with staff working in adult education and a consideration of each group’s background; groups where the majority had been educated outside of the United Kingdom were excluded for the reasons discussed above. Participant groups were also chosen so as to encompass both adult numeracy courses and GCSE (General Certificate of Secondary Education) or GCSE-equivalent courses. Methodology The research was conducted in two stages. The first stage consisted of a questionnaire. After a promising pilot, this was distributed to participants (n=66). It contained a combination of open and closed questions, which are discussed below, together with the results. After some preliminary analysis, the questionnaire was followed up by six face-to-face semi-structured interviews, and one interview conducted by e-mail. The questions in these interviews were designed not only to help support the generation of a narrative of each individual’s experience of learning mathematics, but also to support the development of a convergent validity regarding common responses. This concern also influenced the choice of participants – for example, since a number of questionnaires had mentioned that ‘memory’ was a concern when learning mathematics as an adult, at least one participant who had included this response on their questionnaire was selected for interview. The focus on the individual places this research firmly within an interpretive paradigm, drawing on some of the ideas of grounded theory. It was conceived and conducted as an exploratory study, and an attempt was made to minimise dependence on any one theoretical framework during the study; this is due in part to the nature of the research, but it is also consequent of the objects of study; Coben (2003, 110) comments that “explicit reference to a theoretical frame is constrained by the under-theorised state of the field.” The analysis of the interview data drew on the phenomenographic tradition, managing the transcripts by sorting features qualitatively into broad categories which allowed for subsequent analysis. All participants volunteered to take part in the research. Its aims and purposes were explained to them, anonymity was ensured and the option to withdraw was offered to participants at each stage of the research. Questionnaire Results: Demographic Characteristics The first section of the questionnaire concerned the participant’s gender, age, and previous educational history. The majority of the sample (80%) was female, a tendency which has been observed elsewhere in adult education contexts (for example, Benn and Burton 1994; Coben et al. 2007). This gender bias was statistically independent of course type (χ2 = 0.310, df=1, p=0.578). The age distribution of the participants was roughly normal, centred at about 35. However, there was a highly statistically significant relationship between age and course type From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 98 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 (χ2 = 12.810, df=3, p=0.005). The age profile of the adult numeracy learners demonstrated a negative skew, and the age profile of the GCSE learners demonstrated a positive skew. One contributing factor towards this skew could be the presence of learners who were retaking their GCSE within a decade of leaving school. This suggestion is supported by the information gathered regarding participants’ previous educational history. Of 39 GCSE-level participants, 17 had already attained a GCSE grade in mathematics and were seeking to better it. (This proportion increases to more than half if CSE and O-level grades are included.) This high proportion could be interpreted as a consequence of the role of the GCSE qualification as a gatekeeper. Interestingly, 12 of the 26 participants on adult numeracy courses had also previously achieved a GCSE or CSE grade in mathematics. This belies the existence of a uniform pathway through adult qualifications in mathematics. Although some learners might begin with a numeracy qualification and then move onto GCSE, this model does not fit all learners. Another possible stereotype challenged by the results of this section was that of adult learners of mathematics possessing a low general level of education. There was a huge range of responses in terms of previous educational history, ranging from no qualifications to a degree in Fine Art. Finally in this section, 84% of those taking an adult numeracy course indicated that they had undertaken a previous adult education course, compared to 25% of the GCSE level learners (χ2 = 21.467, df=1, p<0.001). Questionnaire Results: The Decision to Return to Mathematics The second section of the questionnaire concerned the participant’s decision to return to studying mathematics. Issues of motivation are, of course, difficult to assess and summarise; writers such as Hamilton and Hillier (2006) suggest that the decision to return to mathematics should be considered as both gradual and sudden, and this dual perspective was followed up in the interviews. Despite its basic approach, however, this section of the questionnaire yielded interesting results. Firstly, a question about the timescale of the decision suggested that most learners had been considering taking the course for a substantial period of time; the modal response was ‘for significantly longer than a year’. (This tendency seemed much more pronounced in the females than the males, but the difference was not statistically significant.) Next, participants had to indicate their motives for returning to the course. Initially they were required to tick which motives from a list of twelve (drawn from the literature, and together with a thirteenth ‘other’ option) they felt were relevant to them. They then had to indicate which motive was the single most important one. Whilst space constraints prevent a full summary of the results here, intrinsic motives such as confidence and personal development scored highly with the numeracy learners, whilst extrinsic motives focused on the GCSE qualification itself scored more highly with the GCSE learners. This became especially obvious when the most important reason was selected – 22 out of 31 GCSE level learners selected “I need a qualification to help me get onto another course.” No other reason scored more than 3 votes. Questionnaire Results: Learning Mathematics as an Adult The third and final section of the questionnaire encouraged the participants to compare their experiences of learning mathematics as an adult with their school experiences. This was done through a combination of paired Likert scales and open-ended questions. Typically, the adult experience was portrayed much more positively, with ‘as an adult’ responses tending to be about two grades higher on a five-point Likert scale than their From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 99 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 corresponding ‘at school’ responses. The validity of these increases was supported by the accompanying open-ended responses. These comments related to a number of clear themes; for example, the role of the teacher in the learning process, and the use of contexts and applications. “I didn’t have things explained at school, work was put in front of you and you were expected to do it.” (GCSE Learner) “Hated my school maths teacher. He didn’t make maths fun. My current maths tutor is fantastic, if you don’t understand she will explain it in as many different ways as possible until you do. She also relates all our maths to everyday life.” (GCSE Learner) Some of these were inevitably shaped by the structure of the questionnaire, but in other places issues which I had presumed to be relevant (such as the use of ICT) were ignored, and others became apparent instead. These themes continued to be present in the interviews. Although some participants were less enthused about learning mathematics than others, (and one openly resented being forced to take the course in order to progress in their wider education,) the overwhelming tone of the comments was very positive, both in terms of the participants’ attitudes towards mathematics and their own self-confidence. One participant wrote that “it has been rewarding to change past negative messages”; another rejoiced in “the fact that after 30 years of thinking I can’t do maths I can!” Summary of the Interview Dialogues As discussed above, the interviews served to explore issues such as the decision making process in a finer level of detail, as well as offering me an opportunity to clarify meaning. Many of the earlier findings were confirmed or further exemplified in the individual narratives that emerged. For example, the role of mathematics as a gatekeeper qualification was again evident, with many participants needing a qualification for a promotion or for entrance to another course. The role of the teacher also continued to be prominent in many stories, both at school and as an adult: “It was far more informal – there was no sort of, ‘yes sir, no sir, three bags full sir’, you know, if I had a question I could ask *****. It was far more informal, almost on a sort of friendly level – you know, just a friend who could do maths, whereas before it was sort of, you know, Mr. Such-a-body…” (Adult Numeracy Learner) A related concern was the issue of fault and the apportioning of blame. It was interesting to see how different participants interpreted their experiences and allocated responsibility for their perceived ‘failure’, both onto themselves and to others, and this might offer some insight into the reliability of prodigals as narrators. Other themes that re-emerged included the issue of explanation and context, and the role of mathematics as a functional toolkit: “There’s still things I can’t do, and don’t understand, but in general I’m a lot happier… for instance, the course is sort of designed around real life, so if I went into a shop and it said that there was seventy-five percent off, I’m now able to stand there and work out how much I am actually saving. So yeah, there are lots of situations in life where I am now using number, where, I feel a lot happier… yeah, definitely.” (Adult Numeracy Learner) As discussed above, some issues became prominent at the interview stage that had not been directly examined in the questionnaire. One of these was the effect of ability grouping on achievement and motivation. Interestingly, negative experiences were reported by prodigals who had been placed in bottom sets, middle sets and top sets: From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 100 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 “Yeah, because I think the highest I could get was a ‘D’ anyway, and I didn’t even get that, I got an ‘F’, so it was almost pointless really… I think because I couldn’t do it I just lost interest.” (GCSE Learner – Bottom Set) “It was like they were constantly pushing you. And the way they’d split the classes up, so you had bottom maths, middle maths, higher maths, it was like – the person in higher maths was… ‘oh yeah, but I’m better at maths than you, I’m in a higher group than you’, and all that sort of stuff, and we never had that in college, because you’re all in the same group.” (GCSE Learner – Middle Set) “I was put in the wrong group really, because I was put in the top group, and I had no idea what he (the teacher) was on about most of the time. So I switched off, really. I think if I’d started in a lower group I would have found my feet and then maybe been able to progress up. But at the top group there was all the really bright kids, and I had no idea what I was doing.” (GCSE Learner – Top Set) Issues of space prevent a full discussion of the accounts that made up the interviews, but other key issues raised included the role that some of the participants’ desire to help their children played in their decisions, and the influence that learning mathematics and conquering certain fears has had in developing a wider academic self-confidence. Discussion Each of the three research questions identified above has been answered in part, but each also lends itself to further study. Preliminary demographic features of the prodigals have been identified, and these have some resonance with pre-existing research into the general adult education community. However, both the demographic results and the differences revealed in the decision making process point towards there being at least two different types of prodigals, which are connected to the two different types of course considered. It would be interesting to extend this research to prodigals undertaking A-level courses, or distance learning degrees in mathematics, and see how their responses compared to the two groups explored above. This study has also produced some preliminary conclusions about the motivations behind the prodigals’ decisions, and also explored how their experiences of learning mathematics as an adult compare (generally favourably) to their experiences at school. However, the question still remains: what might we learn from the prodigals? Perhaps the most obvious, and encouraging thing that we can take from this group of learners is proof at the level of the individual that negative attitudes towards mathematics can be changed, and negative experiences can be overcome. Moreover, there is some evidence that whilst an improvement of one’s general confidence and academic self-concept is not always an explicit motive for returning to study mathematics, it is often a consequence. It is possible for learners to improve what has been termed as a ‘mathematical trajectory’ (Noyes 2007), often with great benefits to the individual concerned, and sometimes with a refreshing enthusiasm. “For the first time in my life,” one learner wrote, “I actually enjoyed solving maths problems. Weird or what?” Beyond this, the results outlined above form a vivid reminder of the role of mathematics as cultural capital in the sense of Bourdieu (1973). A proper deconstruction of the meaning of this term as it applies to mathematics, and the relevance this has to the stories detailed above lies outside the scope of this summary, but the role that mathematics plays as a gatekeeper is indisputably clear throughout both the questionnaire and interview results. Finally, it is also striking how often, and how strongly, the narratives gathered touch on issues that relate to current discussions in mathematics education, such as the consequences of ability grouping in the classroom, or the place of contexts in demonstrating relevance. Whilst this is undoubtedly partly an artefact of the methodology, the frequency and From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 101 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 nature of the occurrences suggests a genuine phenomenon. Possibly, then, the prodigals offer an additional perspective on these issues. Although their viewpoint in undoubtedly biased, it could be challenging in a healthy way; perhaps they are the ‘exception that proves the rule’. Robert Ward-Penny holds an ESRC sponsored scholarship at the University of Warwick. References Bourdieu, P. 1973. Cultural Reproduction and Social Reproduction. In Knowledge, Education and Cultural Change, ed. R. Brown. London: Tavistock. Benn, R. and R. Burton 1994. Participation and the Mathematics Deterrent. Studies in the Education of Adults 26(2): 236-249. Coben, D., ed. 2003. Adult Numeracy: A Review of Research and Related Literature. London: NRDC. Coben, D., M. Brown, V. Rhodes, J. Swain, K. Ananiadou, P. Brown, J. Ashton, D. Holder, S. Lowe, C. Magee, S. Nieduszynska and V. Storey 2007. Effective Teaching and Learning: Numeracy. London: NRDC. Hamilton, M. and Y. Hillier 2006. Changing Faces of Adult Literacy, Language and Numeracy – A Critical History. Stoke on Trent: Trentham Books. House of Commons Committee of Public Accounts 2006. Skills for Life: Improving Adult Literacy and Numeracy – Twenty-First Report of Session 2005-06. London: HMSO. Nardi, E. and S. Steward 2003. Is Maths TIRED? A Profile of Quiet Disaffection in the Secondary Mathematics Classroom. British Education Research Journal 29(3): 345367. Noyes, A. 2007. Rethinking School Mathematics. London: Paul Chapman Publishing. Smith, A. 2004. Making Mathematics Count. London: HMSO. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 102 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Design Decisions: A Microworld for Mathematical Generalisation Eirini Geraniou, Manolis Mavrikis, Celia Hoyles, Richard Noss Institute of Education, London Knowledge Lab This paper provides the preliminary analysis of a study in which year 7 students interacted with eXpresser; a microworld designed to support students’ transition from the ‘specific’ to the ‘general’ by constructing figural patterns of square tiles and finding rules to describe their model constructions. We present evidence that support three key design decisions of eXpresser and discuss how these features facilitate students’ expression of generalisation. Introduction Utilising young students’ natural algebraic ideas and developing them through carefully designed digital media appears to be a fruitful avenue to explore in teaching of algebra. Students can verbalise algebraic rules in natural language but struggle to use mathematical language (Warren and Cooper 2008). They often fail to see the rationale, let alone the power, of generalisation. The MiGen project4 is tackling this problem by supporting 11-14 year old students in their problem-solving during generalisation tasks, and providing them with a rationale for finding and checking general constructions and rules. The core of the MiGen system is a microworld, named eXpresser, in which students can build figural patterns of square coloured tiles and express the rules underlying them. The eXpresser seeks to provide students with a model for generalisation that could be used as a precursor to introducing algebra, to help them develop an algebraic ‘habit of mind’ (Cuoco et al. 1997). In this paper, we will focus on student interactions with specific functionalities of expresser arising from three of the key design decisions (all of the design decisions are extensively described in Noss et al. 2009, Geraniou et al. 2009). The Microworld, the eXpresser This section provides a short description of expresser (for a detailed description the reader is referred to Noss et al. 2009, Geraniou et al. 2009). Students are presented with tasks such as the one shown in Figure 1. The pattern is animated and the figure number changes accordingly. Students then build constructions for the patterns by expressing what they ‘see’ as the structure of the pattern, making explicit any of their rules, and finally using the relationships to obtain the number of tiles needed in the pattern using the metaphor of colouring the right number. Figure 2 shows a snapshot of the microworld. The student Figure 1. Train-track activity has just finished constructing the given pattern using a C-shaped building block (shown in A). She associated the Figure Number of the task with the number of ‘holes’ in the pattern and used a variable with this name. To create the pattern, the building block is repeated as many times as the value of the variable ‘holes’ (B), in this case 3. In 4 The project is funded through the Technology Enhanced Learning Phase (ESRC/EPSRC-TLRP- RES-139-250381). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 103 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 every repetition the block is placed two squares across (C) and zero places down (D). To complete the pattern the student needed a ‘line’ of three tiles at the end. Finally the student has to colour the pattern by allocating to it the exact number of coloured tiles. In the case of the sub-pattern made of C-shapes, this required 15 tiles, i.e. 5 * holes (E). As students build their constructions in ‘My World’, a ‘General World’ can be seen alongside it (right window). This world exactly mirrors ‘My World’ until the student has unlocked a number (i.e. created a variable), at which point eXpresser randomly Figure 2. Constructing and describing with rules a pattern in the changes its value in the eXpresser. Letters highlight the main features: A) base shape to be ‘General World’. The idea of repeated to make a component of the pattern, B) number of repetitions ‘locked’ and ‘unlocked’ (in this case the value of the variable `holes’) C) Number of squares to move to the right after each repetition (in this case 2) D) Number of numbers was introduced to squares to move down after each repetition E) Units of colours required allow students to specify to paint the pattern component F) Any variable used in the construction whether a number should stay takes a random value in the ‘General World’ (G) A rule for the total the same (locked, i.e. number of units of colours required to paint the whole pattern in a constants) or could change general way. (H) Patterns can be animated when the system changes the value of the variables and are coloured if the rule is correct. (unlocked, i.e. variables). In ‘My World’ students can edit the unlocked numbers whereas in the ‘General World’, eXpresser chooses random values for the ‘unlocked’ numbers. For example, in the snapshot the value of `holes’ is 5 resulting in a different instance of the pattern (F). The pattern in the ‘General World’ is coloured only when students express correct general rules for the pattern. Students cannot interact with the ‘General World’ except by clicking the play button (H) to animate their patterns and test its generality. Methodology Throughout the development of eXpresser, we have followed an iterative design process, interleaving software development phases with pilot studies with students of our target age (11-14 years old). We have also integrated feedback from teachers and teacher educators as well as the students who participated in the studies. The study presented here comprised three activities: the first introduced the functionalities of the microworld through a series of eight video-tutorials, the second where students used eXpresser to build a simple pattern, and the third, called Train-track (as presented in Figure 1) in which students were asked to find a rule that gave the number of green tiles for any figure number. Sixteen students who participated in this study were asked questions throughout their interactions designed to reveal their comprehension of the system and the particular design features under study. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 104 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Students’ Interactions with eXpresser This section focuses on three key design features and student reactions to them. Animated task presentation to provide a rationale for generality All patterns were presented to students animated changing at fixed time intervals showing a different instance of the pattern each time. This made it hard for students to count the number of green tiles (e.g. Figure 1), while allowing them to see the variant and invariant parts of the pattern. This presentation provided a rationale for deriving a rule that output the number of green tiles for any instance of the pattern, i.e. a ‘general’ rule giving concrete instantiation to the meaning of ‘any’. In addition, they were given the chance to see their constructions animated by clicking the play button in the General World (Figure 2H) that allowed them to validate their model’s generality. One student when asked to describe the way she saw the task in Figure 1 replied: Ann: ‘These are flashing green squares and it [referring to the figure number] changes number’. At the end of the session, she was asked: Why do you think we presented the task like this (animated)? Ann: because you can’t make it move on paper. […] it’s not just one number and it doesn’t stay the same. Besides the dynamic potential the technology offers in comparison to conventional paper presentation, Ann seemed to have gained a more ‘general’ perspective after interacting with eXpresser. Ann expressed the notion of a variable in her own words as ‘it doesn’t stay the same’. We could also see how she switched to paying attention to the value of the number of tiles rather than the changing shape – a rather clear instance of differentiating (or at least seeing the importance) between the object and its value- and in so doing sees the connection between the changing pattern and the figure number. Working on a specific case ‘with an eye’ on the general The most crucial, yet difficult, step for students is to distinguish between what stays the same and what changes between different instances of a pattern. Students tend to work with a particular example and struggle to find the solution for an infinite number of invisible and unspecified cases. We therefore designed the system with two separate, yet linked, windows as described earlier. If students make two or more sub-patterns (e.g. see figure 3) to build their pattern, they have to express the necessary relationship(s) between the unlocked numbers if their construction is not to be messed-up in the General World (random numbers are chosen for each unlocked number). This happened with Nancy who made the train-track pattern using two sub-patterns of vertical lines and vertical with a gap but any relationship between them. She was surprised when she noticed the messed-up pattern in the General World, but this helped her realise that she had to express that the number of repetitions of the building block A1 as one more than the number of the building block B1. Researcher: what happened there [pointing at the General World]? Nancy: It is not joined up ... because you only have 1,2,3 of them and it’s got to be , one more extra. [...] If there is 1,2,3,4,5 of these, that means that you always have to add one more. You might have 5 then you need to add an extra line, so that’s 6. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 105 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 By using the word ‘always’, Nancy revealed her ability to generalise in natural language and eXpresser allowed her to express this relationship by using locked and unlocked numbers. She was then able to construct the general rule for her model (as mentioned later in this section). Figure 3. Nancy’s construction from two sub-patterns. The first (A1) is made by repeating a building block of 3 tiles five times. Since the number of repetitions was unlocked, it was changed randomly to 3 in the General World (A2). The second sub-pattern (B1) is made by repeating a block of 2 tiles four times. In the General World (B2) the unlocked number was changed to 6. This helped her realise that she had to express the number of repetitions of the building block A1 in terms of the number of repetitions of B1. A powerful description of the General World and its overall purpose was given by Kathy: Kathy: in My World is like a plan and in the General World is what comes to life and actually moves. It [General World] makes it live and animates it. […] The only difference is that on My World you put the figure number and it doesn’t change, whereas in the General World, the figure number changes. Kathy succeeded in seeing the rationale of the General World as a window that provided different instances of her pattern and therefore a window on to her own generalisations. She also had a rather powerful metaphor of a ‘plan’ for the construction of a rule via the specific case, one which could be ‘implemented’ (our term) in the general case. Mutually supportive model construction and rule construction In eXpresser, colouring a pattern requires an expression (a rule) that allocates the correct number of tiles in all cases. Based on the constructed pattern, students need to find the exact number of coloured tiles for their pattern. For local patterns (figure 2E) this can be expressed as a multiplication of the number of repetitions and the number of tiles in the building block. We were convinced that this design decision would give a direct experience with the number of tiles needed to construct the building block and encourage students to look at the structure of their pattern. It would further support them to construct a rule that calculates the number of coloured tiles using the number of tiles of the building block as a coefficient. For their complete construction and colouring of the pattern, they need to provide a combination of local rules. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 106 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Nancy, for example, when constructing the model in Figure 3, derived two rules for her two sub-patterns. Once she had coloured the individual patterns, it only required a small step to understand that the general rule can be constructed by adding the two together (see Figure 4). It seemed, therefore, that for most students the process of model construction supported the construction of a final rule for the pattern. In addition, when asked to show how they worked out their rule, they usually employ elements of their model to describe it (e.g. see figures 5 and 6). Figure 4. Nancy’s rule Figure 5. Henry’s rule for train-track task Figure 6. Henry’s demonstration of his rule based on his model. Conclusion Our attempt to build a system that gives students the opportunity to work with the particular and general at the same time is still ongoing. The approach we chose through the three key design ideas is not the only existing one, but has advantages compared to the conventional methods with paper-and-pencil. We have therefore some provisional evidence interaction with eXpresser as a model of generalisation engages students, provokes them to think about generalisable structures and helps them to make the transition from numbers to variables in a way that is meaningful. In the interviews, all students used their rules to give the right number of green tiles for different figure numbers using structural reasoning and not pattern spotting. Kathy, for example, having found a similar rule to Henry, said: Kathy: my rule was 5 green times the figure number add 3. Researcher: so for figure number 6, what would it be? How many would there be? Kathy: it would be… 5 times 6…30…it would be 33. Researcher: for 12? Kathy: it would be 5 times 12…60…add 3 ….63. Researcher: for 600? Kathy: 5 times 6…300…and then 3… It seems that interaction with eXpresser discouraged students from calculating and spotting patterns at the expense of expressing structure (see, for example, with reference to this type of task, Noss et al. 1997, Healy and Hoyles 2000, Küchemann and Hoyles 2009). Paper and pencil approaches tend to lead to the referents of the relevant variables becoming obscured, thus limiting students’ propensity to conceptualise relationships between variables, to justify and use them in a meaningful way. With eXpresser, students have to construct models employing the structures they see and find rules with generality in mind. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 107 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 REFERENCES Cuoco, A., E. P. Goldenberg and J. Mark. 1997. Habits of Mind: an organizing principle for mathematics curriculum. Journal of Mathematical Behavior 15(4): 375-402. Geraniou, E., M. Mavrikis, K. Kahn, C. Hoyles, and R. Noss. 2009. Developing a Microworld to Support Mathematical Generalisation. In PME 33: International Group for the Psychology of Mathematics Education, 49-56. Thessaloniki. Healy, L., and C. Hoyles. 2000. A Study of Proof Conceptions in Algebra. Journal for Research in Mathematics Education 31(4): 396-428. Küchemann, D. and C. Hoyles. 2009. From computational to structural reasoning: tracking changes over time In Teaching and Learning Proof Across the Grades K-16 Perspective, ed. D.A. Stylianou, M.L. Blanton and E.J. Knuth. Lawrence Erlbaum Associates. Noss, R., L. Healy, and C. Hoyles. 1997. The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic. Educational Studies in Mathematics 33(2): 203-233. Noss, R., C. Hoyles, M. Mavrikis, E. Geraniou, S. Santos and D. Pearce. 2009. Broadening the sense of `dynamic': a microworld to support students' mathematical generalisation. Special Issue of The International Journal on Mathematics Education (ZDM): Transforming Mathematics Education through the Use of Dynamic Mathematics Technologies 41(5): 493-503. Warren, E. and T. Cooper. 2008. The effect of different representations on year 3 to 5 students’ ability to generalise. ZDM Mathematics Education 40: 23-37. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 108 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Do students’ beliefs relating to the teaching of primary mathematics match their practices in school? Caroline Rickard Initial Teacher Education, University of Chichester, UK This paper reports research findings from the second year of a small-scale, longitudinal case study undertaken with undergraduate students at the University of Chichester. This four year project seeks to explore the impact of the beliefs of students in Initial Teacher Education (ITE) upon their teaching of primary mathematics, noting placement constraints. Data collection involved observations and interviews, and took place in the latter half of a six week block of school experience. Keywords: Mathematics; Beliefs; Primary; Teacher Education; School Experience Aims and introduction Working in ITE since 2001, I seek to engage students in mathematics in a way which reflects my beliefs about how mathematics should be taught to primary age children, noting that as teachers we “convey messages about the nature of mathematics by the way we teach it” (Nickson, 2004: 43). Interviewing ten first year undergraduates after a six week introductory mathematics module established, unsurprisingly, that their views about how primary mathematics should be taught broadly reflected my own (Rickard, 2008). A clear question remained however: • Would clearly articulated beliefs relating to the teaching of primary mathematics match the students’ practices on school experience? The purpose of this next phase was thus to observe some of the students’ mathematics teaching in order to investigate the transferability of their beliefs into the school context; in particular whether their beliefs were resilient enough to withstand the demands made upon student teachers, and whether a particular school placement might constrain or shape the student teacher’s views. Various sources (see for example Brown and Borko, 1992) caution that classroom pressures may mitigate against application of perceived ‘best practice’ however “early and continued reflection about mathematics beliefs and practices, beginning in teacher preparation, may be the key to improving the quality of mathematics instruction and minimizing inconsistency between beliefs and practice” (Raymond, 1997: 574). Williams supports this view suggesting that involving students in “articulating and discussing beliefs and practices associated with mathematics” (2001: 447) is likely to result in more effective practices. Methodology Data was gathered through observation and interview in the spring of 2009 with a triangulation of methods chosen because, as Elton-Chalcraft et al (2008: 79) point out, “observations can be useful for overcoming the difference that can exist between what people say and what they do in practice”. Holding interviews directly after the observations also gave an opportunity to discuss any additional beliefs, noting that not all beliefs were necessarily going to be explicitly incorporated in any single observation. Three opening questions were identified in advance: From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 109 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 1. What would you have done differently in that lesson given the chance? (Incorporating self-evaluation and leading to exploration of whether anything prevented students from doing what they wanted, asking them for example to reflect on pedagogical beliefs they felt they had to put ‘on hold’ during the placement.) 2. Since the initial interview asking you about your beliefs as to how primary mathematics should be taught, are you conscious of any changes to those beliefs? (Including describing the new beliefs and what students felt had caused them to alter.) 3. Are there any other examples you want to give of mathematics lessons you’ve planned and taught which you feel would help to exemplify your beliefs? (An opportunity to explore beliefs and practices over the slightly longer term.) Four of the original research group were observed; students who knew me and were willing to participate in the interviews in their first year and then available in year 2, and in this respect a climate of trust (Hopkins, 2008) was easily established. One participant, Jenny, was taking a secondary rather than a primary placement (names have been changed to preserve the anonymity of the students). Observations were undertaken on a non-participant basis. Figure 1 shows the responses relating to the most commonly expressed beliefs in the original interviews in the first year of the research (n=10) and these features were developed into a simple observation schedule for use in the second phase. ETHOS... there should be a positive atmosphere in the mathematics classroom; the 70% word ‘fun’ was mentioned a lot, and a desire to avoid mathematics being perceived as boring. DOING... an emphasis on practical mathematics lessons using a variety of 70% resources, with the phrase ‘hands-on’ used several times. CONTEXT... referred to in various ways including mathematics linked to themes, 60% cross-curricular opportunities and real world mathematics. DISCUSSION... opportunities for children to talk about their mathematics, often 50% linked to implications for working in groups. Figure 1. Interval sampling helped to focus my attention on how frequently (if at all) I was seeing the sorts of practices that reflected my students’ original beliefs. Having predetermined categories for observation may have helped to address issues of reliability to some extent, but personal expectations may have been a potential source of bias: “Put simply, you are more likely to see the things you expect to see, and hear the things you want to hear” (O’Leary, 2004: 176). The schedule was accompanied by a page for notes about the context of the observation, such as the number and age of the children, organisation of the classroom, and of course detail of the mathematical focus. I also recorded information about the setting and/ or ability grouping of the children. As a result of piloting the observation schedule, systematic recording in the ‘ethos’ column was abandoned, value judgements being too subjective. It was however pleasing to note that students praised children regularly, congratulating them on their effort and achievement. Findings Four observations took place in May of 2009 in different schools in West Sussex and the background information is summarised in figure 2. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 110 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Janey Y1 (n=27) Amelia Y1 (n=27) Y3 (n=31) Jill Jenny Y10 (n=14) Mixed ability class organised into three ability groups. Lower ability set. Counting on, including crossing a tens boundary. Lower ability set organised into ability groups. Average ability set n an all boys school. Multiplication relating to multiples of certain creepy crawlies (e.g. spiders) with certain numbers of legs. Collecting data from peers including tallying. Revision of geometrical constructions. Figure 2. In the interviews which took place after the observations, three of the four students reported that they would have liked to have done things differently given more freedom. Interestingly the fourth student, Amelia, would herself have made minor changes but none of these related to a lack of freedom, or in fact to the improvements her mentor felt were required. Whereas the others spoke a lot about the children’s learning in reflection on their lessons, Amelia’s focus was associated more closely with things she would do differently, for example not talking over the children (waiting for quiet) and making greater use of a puppet. Several themes emerged from the analysis of the data; these will be reported under the same broad headings as those taken from the original research, ignoring ‘Ethos’ however as mentioned above. Doing This category encompassed the idea of practical activity and the use of resources, and issues associated with equipment use came to light even before the visits took place! In emails prior to my visit Jill wrote about being asked to teach without using resources, something which the class teacher had suggested would help the Y3 children to better develop their mental skills. Jill felt she was in an awkward position as she privately disagreed, feeling that a number of the children were struggling and failing to make progress as a result. She was however ‘allowed’ to use resources for her observation, which suggests that this was a ‘special’ lesson for my visit, something which Bryman refers to as ‘reactive effects’ (2008: 266). O’Leary (2004) warns that any person who knows they are being observed is likely to alter their resulting behaviour and my students’ awareness of the research focus was unavoidable having already asked them to articulate their beliefs about the teaching of primary mathematics in year 1. As the schools were oblivious to my research focus however, I hadn’t, perhaps naively, anticipated any alteration to school practices on account of my visit. Whilst Jill felt that this lesson had been more successful as a result of the use of resources, she went further to say that she would still have done things differently had the class teacher been in agreement, for example providing photocopies of the minibeasts on the children’s tables so that to start with they could count the numbers of legs if they needed to. Interestingly, she was the only one of the four research participants who had not explicitly mentioned practical activity through use of resources in her original interview, and yet it is clearly something she now feels strongly about. Jill’s lesson also provided an explicit attempt to have children ‘doing’ as she had given out mini whiteboards which were used for 17% of the lesson. Janey felt that I would have seen a greater level of interaction in other lessons, for example when she made a broom handle into a physical/ visual numberline and through her regular use of stories and songs. She reaffirmed her belief that mathematics should be taught using a creative approach; she said that she would have liked to take the children outside or From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 111 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 into the hall for more of their mathematics lessons, but was concerned that she would find it difficult to control her “bubbly” class. In this lesson examples of practical activity were fairly minimal such as inviting children to hold items or stick things up on the board. Typical to any secondary placement, Jenny was working across different classes and with different members of staff, and she cited a particular member of staff, an ex-student of Chichester, whose classroom and teaching style she felt were noticeably different to the other practitioners, incorporating lots of engaging resources. In Jenny’s original interview she had talked about the benefits of practical activity using resources, and whilst this lesson necessarily met this condition (geometrical constructions using a ruler and pair of compasses) she would like to have made greater use of resources in her other lessons. General expectations of silent working were also mentioned by Jenny in relation to constraints on doing more “interesting things”. The opportunity for children to engage in practical activity was most extensive in the lesson taught by Amelia as the children were circulating the room to gather data. In many ways, however, the fact that all four lessons were on different themes makes it difficult to compare them effectively. Context The value of context related to the teaching of mathematics was mentioned by Janey, Jill and Amelia in the original interviews and all three lessons included some reference to a context. The strongest link was in Amelia’s lesson but as mentioned above, comparison is affected by differences in focus; this was a data handling lesson and data has to be about something! Jill’s lesson made effective use of bugs (the focus of about 35% of the lesson) to exemplify multiplicative structure (numbers of legs) and Janey told a short story about a character who was scared of crossing the tens boundary, but links made to this focus were only briefly maintained. She did however speak articulately about making greater use of links to real life in other lessons. Jill stated that she had become far more conscious of the importance of context as a result of this placement; she reported that the focus had been predominantly upon pure calculation, and that the children had then struggled with some complicated word problems which she had been given to use with them. Jenny didn’t mention context in her original interview and the observed lesson wasn’t linked to any aspect of real-life or similar. Discussion Talk opportunities were mentioned as a feature of best practice by Jenny and Amelia in the original interviews. Classroom talk in all four observations was predominantly interactions between the teacher and the children, and only Amelia explicitly directed children to talk to their peers about the mathematics, although Jenny did invite pupils to the board to explain their work. Directed discussion occurred in 24% of Amelia’s lesson; 4 minutes being allocated to the discussion of something specific with a partner, and 9 minutes of collecting data from peers, an activity which necessitated discussion. In reflecting on her lesson Jill mentioned that she thought her teaching was dominated by teacher talk, and this was indeed borne out in practice. The emphasis in all four lessons was upon the teacher asking and the children answering questions, and the giving of instructions for work, behaviour reminders etc. There was also self-initiated discussion between pupils in all four lessons; many of Jill’s Y3 children discussed the mathematics between themselves whilst they were working, and this focus on mathematical talk continued until nearly the end of the lesson. With both Y1 classes however, the talk more often seemed to relate to off-task topics of discussion. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 112 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 In addition to the three themes explored above, two new themes also emerged: issues associated with lesson structure; and opportunities for children to think for themselves. Lesson structure Volunteering reflections on lesson structure both Janey and Jill stated clearly that they would have liked a better link between their mental and oral starters and the rest of their lessons, but were constrained by school planning. Jill stated that hers was “nonsense in relation to lesson” and she could suggest alternatives that would have better supported the children in preparation for the lesson ahead. Lesson structure can also be linked to notions of pace; in another email communication from Jill, also prior to the visit, she expressed considerable frustration. She was concerned that the pace of the teaching was leaving children behind, writing “I have a feeling that the children are sometimes lost with the concepts and pace”. In her follow-up interview Jill returned to the lesson structure theme saying that she felt logical steps were sometimes omitted to the detriment of the children’s understanding. Thinking Valuing thinking was clearly a concern shared by both Jenny and Jill; the idea that children should be given ample opportunities and encouragement to think for themselves in mathematics lessons. When comparing her primary and secondary experiences Jenny stated that she really wished “the boys to think for themselves” and to be more independent and Jill made a similar point about her lowest ability group in the set. As these children never worked alone she felt they were not getting “a chance to think for themselves and play around with numbers”. Jenny stated that whilst last year the “kids were willing to try anything”, this year the “children are out of the habit”. Conclusion and future research Returning to the original themes, first identified in 2008, it’s particularly interesting to note that the notion of ‘fun’ in relation to mathematics lessons was not mentioned in the same explicit way in this, the second year of the research. One possibility however is that one doesn’t necessarily think to mention everything, however important, in the course of a single interview. This may particularly be the case with fairly fundamental and deeply ingrained beliefs. Jill failed to mention the use of resources in her initial interview and yet having limited access to them in this year’s placement was quite an issue for her, suggesting that their use may well have been an integral part of her original belief system. Linked to general use of resources is a question over the effectiveness of their use: Jill for example chose to refer to minibeasts, an excellent way of exemplifying multiplicative structures. With three of the four participants articulating very clearly what they would have wanted to do differently, given the chance, it is possible to conclude that they felt somewhat constrained by their particular school placement. This is good news as it suggests that they will be able to think for themselves and hopefully make some healthy decisions about the way in which they teach primary mathematics in the future. One concern however is the effect of a school experience which is at odds with your own belief system with both Jill and Jenny speaking of doubts; Jill in relation to her own ability to teach, and Jenny the desire to pursue a career in teaching. Reflective awareness of one’s own practice allows us to move forwards; Jill was clearly aware that she was dominating classroom talk and is therefore in a position to try to address this. In fact, a whole new project on how we encourage student teachers to afford the children lots of talk-time in the mathematics classroom would be a very suitable line of From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 113 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 enquiry! This could be linked to Jill and Jenny’s opinion that the children they were working with this year were not being offered nearly enough opportunities to think for themselves. The intention is now to continue to ‘follow’ this small group of students into their third and final year (when they will be placed in different schools), and where possible into their first year of teaching, noting the issues associated with comparing lessons involving different branches of mathematics and classes of children of different ages. Clearly the most useful focus in the long term relates to how we might support students and newly qualified teachers in remaining consciously aware of their own beliefs, continuing to reflect on their own and others’ approaches to the teaching of mathematics, and where necessary to strive for change for the better. Acknowledgements Grateful thanks are owed to the small band of participants who have so willingly allowed me to invade their classrooms. References Brown, C.A. and Borko, H. (1992) Becoming a Mathematics Teacher, in Grouws, D. A. (Editor) Handbook of Research on Mathematics Teaching and Learning, Oxford: Macmillan Publishing Company Bryman, A. (2008) Social Research Methods (3rd Edition) Oxford: University Press Elton-Chalcraft, S., Hansen, A. and Twiselton, S. (Editors) (2008) Doing Classroom Research: A Stepby-Step Guide for Student Teachers, Maidenhead: McGraw Hill Open University Press Hopkins, D. (2008) A Teacher’s Guide to Classroom Research, Maidenhead: McGraw Hill Open University Press Nickson, M. (2004) Teaching and Learning Mathematics: A Guide to Recent Research and its Applications (2nd Edition) London: Continuum O’Leary, Z. (2004) The Essential Guide to Doing Research, London: Sage Publications Raymond, A.M. (Nov 1997) Inconsistency between a Beginning Elementary School Teacher’s Mathematics Beliefs and Teaching Practice, Journal for Research in Mathematics Education, Vol.28, No. 5, pp. 550576 Rickard, C. (2008 unpublished) Developing Beliefs about the Teaching of Primary Mathematics, University of Chichester Williams, H. (2001) Preparation of Primary and Secondary Mathematics Teachers, in Holton, D. (Editor) The Teaching and Leaning of Mathematics at University Level: an ICMI Study, London: Kluwer Academic Publishers From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 114 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 BSRLM Geometry working group: tasks that support the development of geometric reasoning at KS3 Sue Forsythe; Keith Jones University of Leicester; University of Southampton Students at Key Stage 3 (ie aged 11-14) in English schools are expected to learn the definitions of the properties of triangles, quadrilaterals and other polygons and to be able to use these definitions to solve problems (including being able to explain and justify their solutions). This paper focuses on a pair of Year 8 students (aged 12-13) working on a task using dynamic geomtry software. In the research, the children investigated triangles and quadrilaterals by dragging two lines within a shape (ie the diagonals of a quadrilateral, or base and height of a triangle) and noting the position and orientation of the lines which gave rise to specific shapes. Following this, the students were asked to use what they had found in order to construct specific triangles and quadrilaterals when starting with a blank screen. While the research is currently ongoing, and is using a design research methodology, the evidence to date is that the task has the potential to scaffold students’ thinking around the properties of 2D shapes and hence support the development of geometric reasoning. Keywords: dynamic geometry, task, design-based research Introduction The Framework for Secondary Mathematics in England (DCSF, 2008) indicates that Year 8 students (aged 12-13 years) are expected to know and understand the properties of triangles and quadrilaterals, to be able to solve problems using these properties and to classify quadrilaterals according to geometric properties. Yet simply expecting students to memorise such shapes and their properties is likely to be insufficient support for students developing their own meaningful concepts in geometry (Battista, 2002). This paper describes an attempt to devise a task which would encourage students to develop a deeper understanding of how shapes can be defined by considering the properties of two internal perpendicular lines. These lines are the diagonals in the case of certain quadrilaterals and the height and base in the case of triangles. The students explored these shapes in a dynamic geometry environment, specifically The Geometers Sketchpad (GSP) version 4 (Jackiw, 2001). Using tasks to support learning in geometry Open problems in geometry have been shown to encourage children to develop meaningful concepts (eg: Mogetta et al 1999 a). Open problems usually consist of a short statement where students are asked to explore connections between elements of a figure. Open problems do not lend themselves to solution solely through the use of learned procedures; students have to decide how to explore the problem and there may be a number of results that could be reasonable solutions to the problem. The benefit to the students of working through open problems is that the outcomes are meaningful to them and the opportunity to explain their results may be a pre-cursor to being able to prove in geometry (eg: Jones, 2000). Working on a problem in a computer environment also has benefits. Papert (1993) argued that computers encourage concrete thinking and ways of solving problems that involve From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 115 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 finding approximate solutions and then tweaking them until the optimal solution is found. This approach seems to suit children. It is helpful to provide problems which the students find engaging. Ainley, Pratt and Hansen (2006) consider that tasks which involve programming computers can provide purpose and utility, which enriches students’ learning of mathematical concepts; de Villiers (1994) said something similar when he described ‘functional understanding’ ie the understanding of the usefulness or value of doing a specific bit of mathematics, as being just as important as relational and logical understanding which was described by Skemp (1973). The task, described in this paper, was designed to take all of the above into account. It was intended that the task would stimulate the students into thinking more deeply about the methods of constructing specific triangles and quadrilaterals and that they would be able to explain why their methods worked. As Mogetta et al (1999 b) say, switching between what the student notices on the screen (empirical evidence) and the geometrical theory may stimulate the students’ capability to prove as they have to explain the reasons behind what they are observing. Problems can only really be solved when what is observed on the screen is explained geometrically. Dragging and measuring in a dynamic geometry environment With DGS software (such as GSP, Cabri, GeoGebra), an important function is the dragging mode which allows the user to drag geometric objects on the computer screen. The computer interface allows direct manipulation of the drawing on the screen via the drag mode whilst, at the same time, preserving all the geometric properties used to construct the figure (Laborde, 1993). Another function in DGS is the measure menu. Students can measure lines and angles on the diagram and, as the diagram is dragged, the measurements given on the screen are updated continuously. Olivero and Robutti (2007) describe students using the drag mode to adjust a sketch on the screen until the measurements indicate that they have obtained a particular figure from a generic one and they called this an example of ‘guided measuring’. When students check their constructions through measuring and dragging this is called ‘validation measuring’. Design-based research The methodology used in this study was that of design-based research (Brown, 1992; Designbased research collective, 2003). In a design-based research experiment, the researchers aim to study how learners learn by designing tasks and learning situations through which they hope to see improved learning outcomes. Design-based research uses the design experiment to study learning and develop theories about learning in a specific context but which can be extrapolated to theorise about learning in a broader context (Barab and Squires, 2004). In a learning situation, even a simple one where there are two students and one instructor / researcher as in this case, cognition is not separate from the thinker, the task or the environment - these all need to be treated as one complex system (Design-based research collective, 2003). The learning environment is a complex system of inter-relating aspects where one aspect cannot be changed without it affecting all other aspects. Design-based research methodology accepts this as the case and works with it rather than against it (Brown, 1992). Design-based research experiments thus need to take account of all the aspects of the learning situation and how these all work together. Testing, scrutinising and revising a design results in an iterative process over several cycles of the research (Cobb et al, 2003). From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 116 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Clearly there are some challenges in design-based research which need to be considered. The role of the researcher(s) in a design-based experiment means that they have a large influence in how the experiment proceeds (Barab and Squire, 2004). Design-based research experiments can also generate a large amount of data and the researcher must choose which data to focus on in the analysis. The researcher must be aware of the need to be objective and not select data that backs up their own preconceived ideas (Brown, 1992). This shows how important it is for the researcher to try to be objective and for all their activities to be transparent. The experiment Two boys and two girls worked in pairs in the summer term of Year 8 (students aged 13) for two 50 minute sessions. Data was collected from these sessions in the form of an audio tape and a recording of the computer screen using image capture software. The students were given a task which was loosely related to a ‘real world’ situation in that they were asked to imagine a toy kite made of two ‘sticks’ which provide the scaffolding for the kite. The fabric which makes up the kite is imagined to be elastic so that the ‘sticks’ can be moved around to create different shaped ‘kites’. The students were asked to investigate the different shapes that can be made in this way and to describe the orientation of the ‘sticks’ inside each shape. In a later session the students were asked to construct ‘drag proof’ shapes starting with a blank screen. The findings The first session allowed the researcher to assess the students’ prior knowledge of shapes and their properties. The students were presented with a GSP file which contained a 6 cm horizontal bar and 8 cm vertical bar. The students completed the shape by joining the ends of the bars (see figure 1) and then constructed the interior of the shape. Even though the bars could be dragged anywhere on the screen, the students preferred shapes that had vertical symmetry. By dragging the bars they were able to make kites, a rhombus, an isosceles triangle, right angled triangles in different orientations, and concave kites. The students were then asked to measure objects in the shape in order to be absolutely sure that they had made the shape. They did this using the measure facility of GSP, which shows the measurements as text on the screen. Next the students were provided with a GSP file with perpendicular bars of adjustable lengths. With two equal perpendicular bars the students were able to make a square as well as various kites, an isosceles triangle and a right angled isosceles triangle. Each time the students made a shape they were asked to describe the orientation of the bars inside the shape. The students realised that the measures changed as they dragged the bars inside the shapes. The students moved the bars inside the shape whilst checking the measurement given on the screen until the measurements were as close as they could get them. This is an example of ‘guided measuring’ as described by Olivero and Robutti (2007). The students were happy to accept measurements which were close, but not exact, in order to feel confident that they had generated a particular shape. This mirrors the use of measurement in the pencil and paper environment. They also noticed, when prompted, the orientation of the bars inside the shape which would be useful in order to carry out the task in the second session when they would be asked to construct shapes starting with a blank screen. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 117 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Figure 1: the Kite task in GSP For the second session the students were provided with ‘instruction cards’ which explained how to do various operations from the Construct and Transform menus (such as mid-point of a line, perpendicular to a line). These cards were placed on the desk so that the students could choose which ones they needed. The instruction cards served to give hints to the students as to what might be a useful construction. For the first part of the session, the students were asked to make an isosceles triangle. It was clear from observing their attempts that they had remembered the orientation of the bars from the previous session. At first they drew lines to represent the bars ‘by eye’ and then completed the outside edges of the shape. When they discovered that the result could be dragged out of shape they realised they needed to make shapes that were ‘drag proof.’ It was at this point that they looked at the help cards on the desk. The best constructions that the students made were when they chose what they would like to do to their diagram and found ways to get the GSP to achieve that. For example when making the square they first drew a line, then constructed a mid-point and then constructed a perpendicular line through this mid-point. They were perplexed when the perpendicular was an infinite line and they did not know where to place the opposite corners of the square along the infinite line. The following conversation ensued with the girls. Researcher: What would you like to do? If you don’t know how to do it, what would you like to do? And I might be able to tell you how to do it. Girl 1: Find a point here which is, like, the same distance and will make a right angle. Researcher: Well we know if it’s on that line it’ll be a right angle. So what do you think you could do? Girl 2: Is there some way you could, almost, spin it round? Researcher: If you spin it round, what’s that called? Girl 2: Rotate. The students then rotated the first line 90 degrees onto the infinite perpendicular line to find out where the corners of the square would go. As they did in session 1, the students used the measure facility to check that they had made the shapes correctly. An example of a square with measurements is shown in figure 2. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 118 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Figure 2: constructing a square in GSP Discussion The levels of geometrical reasoning devised by van Hiele (1986) are often used to characterise the level of students’ development in geometrical reasoning. If the task used in this research is useful for developing the students’ reasoning then we should see some progress within the levels. The students showed, in the first session, that they were able to understand shapes as being collections of properties, which is evidence of reasoning at van Hiele level 2 (ie shapes as being collections of properties). In the second session the students were able to construct specific shapes starting with a blank screen, namely isosceles triangle, kite, square, equilateral triangle. This indicates that they had learnt something about the properties of the diagonals of the quadrilaterals, or the base and height of triangles, in order to do this. Being able to solve problems using these properties, such as constructing the shapes in a dynamic geometry environment, would indicate progression towards van Hiele level 3. In addition, the provision of ‘instruction cards’ served to support the development of the students’ geometrical vocabulary. The students started to talk about what they were doing using the language on the cards, which is also the language used by the software. Conclusion What has been described in this paper is the first iteration of the design experiment process. The evidence to date is that the task has the potential to scaffold students’ thinking around the properties of 2D shapes but that the task needs to be developed further in order to consolidate and build on what the students have learned. For example, a third session would be useful, where the students might be asked to build a macro which would generate a shape such as a square. This would also encourage the students to find efficient ways to generate shapes leading them, perhaps, to a realisation of the minimum properties required to render a specific shape such as a square. This activity could also suggest to students that they explain why their macro works - which may lead them onto more formal proof and would provide more solid evidence of development towards van Hiele level 3. The use of geometrical language to support learning is an issue which has received only modest attention in this study so far. Future iterations of the work with students could usefully consider the potential that the task and the dynamic geometry software have to develop students’ use of geometrical language. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 119 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Acknowledgements The authors would like to thank the colleagues who attended the BSRLM conference session on 14 November 2009 and especially for their comments and suggestions. References Ainley, J., Pratt, D., and Hansen, A. 2006. Connecting engagement and focus in pedagogic task design. British Educational Research Journal, 32(1), 23-38 Barab, S., and Squire, K. 2004. Design-based research: Putting a stake in the ground. Journal of the Learning Sciences, 13(1), 1-14 Battista, M. T. 2002. Learning geometry in a dynamic computer environment. Teaching Children mathematics, 88(6), 333-339 Brown, A. L. 1992. Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning sciences, 2(2), 141-178 Cobb, P., Confrey, J., diSessa, A., Lehrer. R. and Schauble, L. 2003. Design experiments in educational research. Educational Researcher, 32 (1), 9-13. DCSF 2008. Framework for Teaching Mathematics. http://nationalstrategies.standards.dcsf.gov.uk (accessed Dec 2009) Design-based Research Collective 2003. Design-based research: An emerging paradigm for educational enquiry. Educational Researcher, 32 (1), 5-8 Jackiw, N. 2001. The Geometers’ Sketchpad (version 4). Key Curriculum Press. Jones, K. 2000. Providing a foundation for deductive reasoning. Educational Studies in Mathematics, 44(1-3), 55–85. Laborde, C. 1993. Do the pupils learn and what do they learn in a computer based environment? The case of Cabri-géomètre. Proceedings of ‘Technology in Mathematics Teaching: A bridge between teaching and learning’. Birmingham : University of Birmingham. Mogetta, C., Olivero, F. and Jones, K. 1999. Providing the motivation to prove in a dynamic geometry environment. Proceedings of the British Society for Research into Learning Mathematics, 19(2), 91-96. Mogetta, C., Olivero, F. and Jones, K. 1999. Designing dynamic geometry tasks that support the proving process. Proceedings of the British Society for Research into Learning Mathematics, 19(3), 97-102. Olivero, F. and Robutti, O. 2007. Measuring in dynamic geometry environments as a tool for conjecturing and proving. International journal of Computers for Mathematical Learning, 12, 135-156 Papert, S. 1993. The Children's Machine: Rethinking School in the Age of the Computer. Harvester Wheatsheaf. Skemp, R. 1973. Relational versus instrumental understanding. Mathematics Teaching, 77, 20-26 Van Hiele, P. M. 1986. Structure and Insight: A theory of mathematics education. Orlando, Fla: Academic Press From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 120 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 Working group on trigonometry: meeting 4 Notes by Anne Watson Department of Education, University of Oxford These notes record the discussion at the fourth meeting of this working group. The focus was on the history of trigonometry, and discussing three different approaches to teaching it which have appeared in recent readings. Keywords: congruence, similarity, astronomy, triangle, trigonometry Historical development Leo Rogers had supplied us with a brief overview of the history of trigonometry. Unfortunately he was unable to join us, being occupied with a Working Group on History of Mathematics. There appear to be two strands, the astronomical strand which used circles and arcs as the basic tool to track the position and movement of stars and planets, and the surveying strand which used ratios of sides of right-angled triangles. The sundial uses both. In an earlier meeting we had circulated the suggested teaching approach of Thompson, Carlson and Silverman (2007) for pre-service teachers. They use arcs of circles in order to present sine etc. not as ratios which have to be imagined, but as measures of arcs which can be seen. We referred to this briefly, and the fact that for small angles one can approximate a circle segment as a triangle. Historically there seems to be a progression from approximating a slim segment as one triangle to bisecting the relevant chord to approximate it as a rightangled triangle on the half-chord. We thought that the history of trigonometry includes the history of our exploration of their properties. Two future actions arise from this discussion: • Learn more about the triangle roots of trig, as illustrated in the Nine Chapters to find heights of tall things (Liu Hui); find out what the engineers were doing, as well as the astronomers • Return to reading the Thompson paper to look for analogies. Comparing two approaches We then compared two recent articles on teaching trigonometry (Kemp, 2009; Steer, da Silva and Easton, 2009). Both had appeared in the same issue of Mathematics Teaching without editorial comment. We thought it might be helpful to compare them in the light of some of the analysis we have been doing in the working group. Similarities • • • • both took an approach which focused on properties of triangles both delayed the introduction of technical terms until the relations had been established empirically both depended on learners noticing what stays the same when certain features of triangles are varied both aimed at complex understanding rather than technical procedures From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 121 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 • • • both emphasised the significance of similarity both mentioned ‘SOHCAHTOA’, but we were not sure if this mnemonic was being used to communicate to the reader that formalisation and application of the ideas was the endgoal, or whether it had been introduced to learners in that form – the mnemonic itself being the endgoal both appeared to have resolution of triangles as the goal, rather than a functional understanding. Differences One approach (Steer et al., 2009) depended on dynamic geometry, and was reported a way which emphasised the technology, while the other (Kemp, 2009) used lowtech materials. The paper by Steer et al. described the implementation of ideas developed by Jeremy Burke (2006). It was the third of a series but summarised the overall approach. Burke’s ideas had been circulated to this group earlier, and this article omitted relating trigonometric ideas to the unit circle, as he had suggested. We wondered why this might be so, and concluded that the realities of pressures on teaching may have led them to truncate the proposed sequence. Some readers felt that it dwelled in detail on technology use, rather than relegating software to the position of a tool. Another difference was that, while both used similarity as the central idea, this was approached as a ‘constant binary relation between sides of these triangles’ rather than through ‘preservation of proportion by scaling’ being a central idea that makes trigonometry possible. The scaling idea was more prevalent in Steer et al’s approach than Kemp’s. In the former, similarity arises after consideration of congruence, so the emphasis is on types of sameness, and scaling (multiplication) when triangles share the (a,a,a) characteristic is central. In the latter, similarity arises as a relation among sides of triangles that ‘look the same’ and multiplication is a choice from four binary operations to find one that is constant. Neither approach explains why only right-angled triangles are chosen for this exploration, since all sets of similar triangles have common ratios. Use of a unit circle could have made this clear. Embodiment of trigonometric ideas Finally we read a passage from Lakoff and Nunez (2000) about how trigonometric ideas can arise from blended fundamental conceptual metaphors which relate to how humans are in the world. They claim that there has to be a metaphor to enable us to relate angles to numbers, for which we have some fundamental understandings. They draw on the idea of the unit circle as the appropriate metaphor, and describe this as a blend of circles in the Euclidean plane with the two-dimensional numerical metaphor of the Cartesian plane, and the angle in the Euclidean plane having two legs that delineate the angle. The final blend gives the familiar diagram of the right angled triangle generated by one rotating leg of the angle and the vertical dropped from it. They show that trig ratios and the functions are both represented and generated by this diagram. What they offer is logical, in that these ideas are related mathematically in the way they describe, but incompatible with the historical development we had thought about earlier. We want to think more about how ‘surveying the earth’ and ‘measuring the heavens’ might have been perceived over centuries, and what role the unit circle might have played. Trigonometrical ideas were used long before Descartes offered the ‘metaphorical blend’ for From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 122 Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 number and space. The chronological development does not necessarily match the logical or metaphoric perspective, and does not support the order in which they relate the ‘metaphors’. We observed that what is ‘natural’ is culturally loaded, since for some cultures distance and direction combined is a fundamental way of seeing the world. Further, some relevant static relations, such as our understanding of near and far objects, are also naturally embodied in our ways of being in the world and also contribute to trigonometric understanding. In summary, we could not understand why it is helpful to see these ideas as metaphors, nor could we agree that these were the appropriate metaphors for trigonometric understanding. Future plans We shall next meet at the BSRLM meeting in Summer 2010. This is an open group and all are welcome to join. If you would like copies of earlier readings please contact anne.watson@education.ox.ac.uk. References Burke, J. 2006. Trigonometry: an introduction using dynamic geometry. London, King’s College. Kemp, A. 2009. Trigonometry from first principles. Mathematics Teaching. 215: 40-41. Lakoff, G. and Nunez, R. 2000. Where mathematics comes from: how the embodied mind brings mathematics into being. New York, Basic Books. Steer, J., de Vila, M. and Eaton, J. 2009. Trigonometry with year 8: Part 3. Mathematics Teaching. 215: 6-8. Thompson, P., Carlson, M. and Silverman, J. 2007. The design of tasks in support of teachers’ development of coherent mathematical meanings. Journal of Mathematics Teacher Education. 10: 415-432. From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 123