Investigating Evidence of Girls’ and Boys’ Early Symmetry Knowledge Through Multiple Modes of Assessment

14th International Congress on Mathematical Education July 11-18, 2021, Shanghai Proceedings of the ICME-14 Topic Study Group 1 Mathematics education at preschool level Marja van den Heuvel-Panhuizen Angelika Kullberg Editors TSG 1. Mathematics education at preschool level Chair: Marja van den Heuvel-Panhuizen Nord University, Norway / Utrecht University, Netherlands m.vandenheuvel-panhuizen@nord.no / m.vandenheuvel-panhuizen@uu.nl Co-chair: Angelika Kullberg Gothenburg University, Sweden angelika.kullberg@ped.gu.se Team members: Ineta Helmane University of Latvia, Latvia Xin Zhou East China Normal University, China IPC Liaison person: Marta Civil The University of Arizona, USA 2021, Shanghai, China: ICME-14 ICME-14 TSG 1 Mathematics education at preschool level i CONTENTS Introduction Marja van den Heuvel-Panhuizen & Angelika Kullberg Mathematics education at preschool level Papers on investigations of children’s learning 1. 2. 3. 4. 5. 6. 7. Xiaoting Zhao & Xiaohui Xu Application of number line estimation strategy for 5-6 years old children: Effect of reference point marking Marja van den Heuvel-Panhuizen & Iliada Elia Unraveling the quantitative competence of kindergartners Yuly Vanegas, Carla Rosell & Joaquin Giménez Insights about constructing symmetry with 5-year-old children in an artistic context Joanne Mulligan & Gabrielle Oslington Kindergartners’ use of symmetry and mathematical structure in representing SELF-portraits Nicole Fletcher, Diego Luna Bazaldúa & Herbert P. Ginsburg Investigating evidence of girls’ and boys’ early symmetry knowledge through multiple modes of assessment Fang Tian & Jin Huang 4-Year-olds children’s understanding of repeating patterns: A report from China Insook Chung Investigating how kindergartners represent data with early numeracy and literacy skills through a performance task Papers on investigations of children’s learning environment 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. ii Dina Tirosh, Pessia Tsamir, Ruthi Barkai, & Esther S. Levenson Counting activities for young children: Adults’ perspectives Miriam M. Lüken & Anna Lehmann Asking early childhood teachers about their use of finger patterns Catherine Walter-Laager, Manfred R. Pfiffner, Xin Zhou, Douglas H. Clements, Julie Sarama, Linh Nguyen Ngoc, Lars Eichen & Karoline Rettenbacher Performance expectations in the area of “Shapes and Spaces” of early childhood educators in an international comparison Ronald Keijzer, Marjolijn Peltenburg, Martine van Schaik, Annerieke Boland, & Eefje van der Zalm Mathematics in play Oliver Thiel Does preservice teacher training change prospective preschool teachers’ emotions about mathematics? Audrey Cooke & Jenny Jay Bishop’s (1988, 1991) mathematical activities reframed for pre-verbal young children’s actions Jianqing Wen When math meets games—The active construction of children's core mathematics experience in games Birgitte Henriksen Analysing a Danish kindergarten class teacher’s instructional support in mathematics with the tool Class Øyvind Jacobsen Bjørkås, Dag Oskar Madsen, Anne Grethe Baustad, & Elisabeth Bjørnestad Mathematical learning environments in Norwegian ECEC child groups Ann LeSage & Robyn Ruttenberg-Rozen “More Gooder”: children evaluate early numeracy apps iii 1 6 11 16 21 26 31 36 41 46 51 60 65 70 75 80 89 ICME-14 TSG 1 Mathematics education at preschool level INVESTIGATING EVIDENCE OF GIRLS’ AND BOYS’ EARLY SYMMETRY KNOWLEDGE THROUGH MULTIPLE MODES OF ASSESSMENT ž ž ž Nicole Fletcher Herbert Ginsburg Diego Luna Bazaldúa Fairfield University Teachers College, Columbia University The World Bank Intervention using software to teach symmetric transformations— reflection, translation, and rotation—to young children Prior study (Fletcher, 2015) found intervention group was better able to identify, explain, and create symmetric transformations than control group Purpose of current study: — (1) to explore the convergence of qualitative and quantitative evidence collected in the study — (2) to explore any potential differences between girls and boys in this convergence ž Results: — (1) Moderate positive correlations were found. Children who demonstrated conceptual understanding in explanations of symmetric transformations at post-test also scored higher on paper-and-pencil post-test tasks. — (2) Similar correlations were found for girls and for boys. ž Discussion: — Importance of multiple means of assessment when assessing knowledge of young children — Verbal tasks may reveal understanding not observed in traditional written assessments — Early exposure to symmetry concepts can benefit both girls and boys nsf2109@tc.columbia.edu 21 The 14th International Congress on Mathematical Education Shanghai, 11th ‒18th July, 2021 INVESTIGATING EVIDENCE OF GIRLS’ AND BOYS’ EARLY SYMMETRY KNOWLEDGE THROUGH MULTIPLE MODES OF ASSESSMENT Nicole Fletcher Fairfield University Diego Luna Bazaldúa The World Bank Group Herbert P. Ginsburg Teachers College, Columbia University Symmetry is a fundamental geometric concept that receives minimal attention in early childhood curriculum. This study sought to explore the relationships between children’s accurate identification and explanation of symmetric transformations after a symmetry software intervention and children’s performance on a post-test that asked students to draw symmetric transformations and identify lines of symmetry. This study also sought to explore whether the relationships between these measures differed for girls and for boys. Positive correlations between accurate symmetry identification and the domains assessed at post-test and between explanations of symmetries and the domains assessed at post-test were found for participants overall, and similar patterns were found for girls and for boys. The findings have implications for software learning opportunities and modes of assessment for all children. INTRODUCTION A computer program was developed to expand young children’s understanding of three symmetric transformations—reflection, translation, and rotation. Results from a previous study evaluating the program’s effectiveness showed that children assigned to the experimental condition, in which they used the software, were better able to accurately identify and explain symmetric transformations and had higher post-test translation scores and overall scores than the control group, which did not use the software, controlling for pre-existing ability (Fletcher, 2015). The purpose of the current study is to explore the convergence of the qualitative and quantitative evidence collected in the study and to explore any potential differences between girls and boys in this convergence. To achieve its purpose, this study sought to answer the following questions: (1) Is there a relationship between children’s identification and explanation of symmetric transformations after the symmetry software intervention and post-test performance on key measures of symmetry understanding? (2) Does the relationship between children’s accurate identification and explanation of symmetric transformations, on the one hand, and post-test performance, on the other, differ for girls and for boys? THEORETICAL FRAMEWORK Symmetry is present in everyday life, and the cognitive processes linked to it develop over the lifespan. Children begin to develop the ability to perceive symmetry in infancy (e.g., Bornstein, Ferdinandsen, & Gross, 1981), and symmetry aids adults in processing visual information (Bornstein and StilesDavis, 1984). Preschool-aged children often experiment with ideas of shape, pattern, and spatial relationships in their play. Such activities in early childhood help to build a foundation for concepts important in later mathematics (Clements and Sarama, 2007). Despite symmetry’s relevance across mathematics, learning standards addressing symmetry do not appear in the United States’ Common Core State Standards until grade four (National Governors Association, 2010). ICME-14 TSG 1 Mathematics education at preschool level Fletcher, Luna Bazaldúa, and Ginsburg Because geometry is often taught in a cursory manner in early childhood (Clements, 2004), children’s experiences with symmetry often occur in informal contexts. Gender differences favoring boys in symmetry-related spatial tasks such as mental rotation have been documented (e.g., Maeda & Yoon, 2013), but some researchers argue that these gender differences are likely substantially attributable to socio-cultural or experiential factors (Fennema & Sherman, 1977; Terlecki & Newcombe, 2005) rather than biological factors. Certain informal play experiences may help children develop understanding of symmetry, but gender differences in the frequency of these types of play may contribute to gender differences in symmetry understanding. METHODS Materials: Symmetry Software A computer program was designed to teach three symmetric transformations—reflection, translation, and rotation—to young children. Cognitive principles for the design of mathematics software for young children (Ginsburg, Jamalian, & Creighan, 2013) guided the software development. The program’s exploratory mode allows children to move, stretch, or shrink shapes on the screen and observe corresponding changes in the symmetric transformation. In prediction mode, the program prompts the child to place a shape on the screen to create a specified symmetric transformation. Visual and audio feedback identify mistakes and provide solution strategies. Setting and Participants A study of the computer program’s effectiveness was conducted in one urban public elementary school in the Eastern United States. Participants included 86 first and second grade children—43 were randomly assigned to the experimental group (24 females and 19 males) and 43 were assigned to the control group (21 females and 22 males). Participants’ ages ranged from 5.8 to 7.8 years. Research Design and Procedure The study utilized a pre- and post-test between-subjects randomized experimental design with exposure to the symmetry software as the treatment condition and a control group that experienced non-symmetry math software and was conducted over 9 weeks. A video transformation task was implemented at post-test to assess children’s ability to identify and explain symmetries. The treatment condition consisted of 9 symmetry software sessions (three each for reflection, translation, and rotation); the control condition had 9 sessions using a non-symmetry-related mathematics software. Measures A paper-and-pencil instrument designed by the primary investigator to measure children’s understanding of reflection, translation, and rotation was used for pre- and post-testing. The instrument included items that asked students to draw symmetric transformations and identify lines of symmetry. The instrument included explanations or examples for the symmetric transformations to ensure that it assessed symmetry concept understanding rather than familiarity with relevant vocabulary. A task designed by the primary investigator to measure participants’ ability to identify and explain reflection, translation, and rotation was implemented at post-test. Participants were shown six short videos of symmetric transformations of shapes. Participants were then asked to identify the symmetric transformation and explain their reasoning for selecting the stated symmetry (“How do you know?”). ICME-14 TSG 1 Mathematics education at preschool level Fletcher, Luna Bazaldúa, and Ginsburg Participants’ identification of the symmetric transformations in each video were scored for accuracy (0 for incorrect or 1 for correct). Participants’ explanations of each symmetric transformation were coded for the presence of words or phrases indicating conceptual understanding of the symmetric transformation. This emergent coding scheme was developed by reviewing children’s explanations and identifying words and phrases appearing in student responses that indicated conceptual understanding of that symmetric transformation. Explanations were scored 1 for indicating conceptual understanding or 0 for not indicating conceptual understanding of the symmetric transformation. RESULTS Previous research found that there was a statistically significant treatment effect when controlling for pre-intervention symmetry knowledge (Fletcher, 2015). Previous research also found that there was a statistically significant difference benefitting the treatment group in accurate identification and explanation of symmetric transformations at post-test. In order to explore the relationship between children’s identification or explanation of symmetric transformations and post-test performance, the accuracy and explanation scores were correlated with the mean post-test scores on the paper-andpencil instrument. With the exception of rotation post-test scores, there are moderate positive correlations between accuracy or explanation and the domains assessed by the post-test, indicating that children who accurately identified symmetric transformations or provided explanations indicating conceptual understanding of the symmetric transformations also tended to reach higher scores on posttest reflection and translation tasks as well as post-test total score (see Table 1). A similar relationship between children’s accurate identification and explanation of symmetric transformations and post-test performance was found for girls and for boys (see Table 1). Accuracy Post-Test Total Score Post-Test Reflection Post-Test Translation Post-Test Rotation Explanation All (N = 43) 0.423* Girls (N = 24) 0.418* Boys (N = 19) 0.430* All (N = 43) 0.365* Girls (N = 21) 0.383* Boys (N = 22) 0.342* 0.360* 0.342* 0.381* 0.317* 0.314* 0.308 0.475* 0.483* 0.469* 0.449* 0.489* 0.421* 0.243* 0.351* 0.107 0.057 0.208 -0.088 Table 1: Correlations among dependent variables and post-test scores by group (Note. * p < .05) DISCUSSION In examining the relationships between post-test performance, accuracy of symmetric transformation identification, and explanations of symmetric transformations, moderate positive correlations were found between post-test reflection and translation scores and accuracy or explanations on the video transformation task. Children who achieved higher scores on reflection and translation items at posttest also tended to identify symmetric transformations accurately and demonstrate conceptual understanding in their explanations. Because prior research has found gender differences favoring boys ICME-14 TSG 1 Mathematics education at preschool level Fletcher, Luna Bazaldúa, and Ginsburg in symmetry-related spatial tasks, we separated the data by gender to see if the pattern we discovered in our results was being driven by boys’ performance on the measures. We found the relationships between post-test scores, accurate symmetry identification, and conceptual understanding of symmetric transformations conveyed in explanations to be similar for girls and boys. The convergence of the post-test scores, accuracy of symmetry identification, and explanations of symmetries points to the importance of multiple means of assessment—and in particular, the use of interview-like questions—when assessing the skills and knowledge of young children. While some children may feel comfortable completing a paper-and-pencil task, others may not yet have the skills or confidence to accurately complete a written task. Including verbal tasks that allow children to explain their thinking when assessing children provides an important opportunity to reveal understanding that may not observed in traditional written assessments. In this study, qualitative and quantitative evidence of girls’ and boys’ symmetry knowledge shows similar patterns of convergence, suggesting that early exposure to symmetry concepts can benefit both girls and boys. Teaching symmetry concepts to young children — either as part of the standard curriculum or through supplementary educational tools such as the computer program utilized in this study — allows teachers to build on children’s natural interest in symmetry and prepare children for success in higher level mathematics and career opportunities both in and out of mathematics. References Bagirova, R. E. (2012). Teaching of symmetry at mathematics lessons in the first forms of Azerbaijan primary schools. Asian Social Science, 8, 285-291. Bornstein, M. H., Ferdinandsen, K., & Gross, C. G. (1981). Perception of symmetry in infancy. Developmental Psychology, 17, 82-86. Bornstein, M. H., & Stiles-Davis, J. (1984). Discrimination and memory for symmetry in young children. Developmental Psychology, 20, 637-649. Clements, D. H. (2004). Geometric and spatial thinking in early childhood education. In D. H. Clements & J. Sarama (Eds.), Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp. 267-298). Mahwah, NJ: Lawrence Erlbaum Associates. Clements, D. H., & Sarama, J. (2007). Early childhood mathematics learning. In F. Lester, Jr. (Ed.), Second handbook on mathematics teaching and learning (pp. 461-555). Fennema, E., & Sherman, J. (1977). Sex-related differences in mathematics achievement, spatial visualization and affective factors. American educational research journal, 14(1), 51-71. Fletcher, N. (2015). 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The relation of computer and videogame usage to gender differences in mental rotation ability. Sex Roles, 53(5), 433-441. ICME-14 TSG 1 Mathematics education at preschool level