Heier Aaron Emt620 At1
Heier Aaron Emt620 At1
Heier Aaron Emt620 At1
Aaron Heier
AT1
Length
Length, as a mathematical concept, is a measurable attribute and refers to the
one-dimensional distance between two points. The attribute of length in different
contexts may describe the magnitude of a straight line end-to-end, or along a path, like
when measuring the length of a bent wire (Battista, 2006). Other terms used to describe
the attribute of length may also refer to the depth, height, width, or longest aspect of an
object (Siemon et al., 2011).
Starting from year three, the Australian curriculum expects students to use
formal metric units in the measurement of length, such as millimetres and metres,
however before that focus is on uniform informal units (Australian Curriculum,
Assessment and Reporting Authority [ACARA], 2014). Informal units appropriate to
year two students could include body parts, such as hand-spans or foot length, or
physical objects, like paperclips or consistent squares of paper (Siemon et al., 2011).
The following assessment directly connects to the Australian curriculum year
two content descriptor compare and order several shapes and objects based on
lengthusing appropriate uniform informal units (ACMMG037), and the proficiency
strand of fluency, which requires students to be able to demonstrate the comparison of
measurements, by using informal units iteratively (ACARA, 2014).
Bush (2009) discusses three of the key conceptual understandings that are pivotal
for students to have a sound understanding of the measurement of length: the use of
identical units, transitive reasoning, and iteration. Firstly, students need to recognise the
importance of the use of identical units when measuring. This reduces ambiguity, allows
for effective communication and comparison, and is a key concept in the measuring of
length whether using standard or non-standard units (Bush, 2009; Siemon et al., 2011).
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Secondly, transitive reasoning, or transitivity, is ability to infer that one object is longer
than another by making direct comparisons of each with a third object (State
Government Victoria Department of Education and Early Childhood Development
[SGVDEECD], 2009, p.2). This aids students to make increasingly complex
comparisons between objects when they cant be compared directly. Lastly is the correct
use of unit iteration, which stems from transitive reasoning, and means children will
recognise that the length of an smaller object can be thought of as part of a whole and
used repeatedly without overlapping or gaps to measure or compare larger objects
(Bush, 2009; Kamii & Clark, 1997).
A student who exhibits sound conceptual understanding of the measurement of
length should be able to display contextually appropriate recognition of the attribute of
length in a variety of situations, whether straight line, path, or one aspect of an object
with multiple dimensions. The student must recognise the need for units of measurement
to be both consistent in length, and be able to use them iteratively, to achieve accurate
measurement in a variety of contexts. They will also be able to adapt what they know to
select appropriate tools, or units, and use those tools to infer length comparisons in
diverse circumstances.
The concept of length has direct and obvious real-world relevance. Battista
(2006) suggests that it adds precision to everyday descriptions and comparisons. For
children who are innately competitive these exact comparisons hold particular appeal,
like accurately measuring the tallest student, or finding out who can jump the farthest.
Misconceptions and difficulties
An accurate concept of length forms the foundation for students progression to
sound measurement ideas of volume and area, but can be prone to misconception and
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difficulty (Siemon et al., 2011). Battista (2006) considers that some of these
misconceptions originate from the variable use of the word length in conversational
English language. To be able to attend to a task a student will need to recognise which
attribute is being described by the word length. Length might be used in the context
of time, it may mean distance, straight line end-to-end or along a curved path, or might
refer to one particular aspect of a three-dimensional object. This ambiguity of definitions
can cause confusion for students, for instance, which attribute to attend to when
measuring the length of a narrow box (Batissta, 2006).
Transitive reasoning can also be a difficult concept for many children to grasp
(Bush, 2009; Hiebert, 1984; SGVDEECD, 2009). This misconception prevents children
from recognising that it is possible to compare the length of two objects, even when it
cant be done directly, by using a third for comparison. It is on the foundation of
transitivity that students learn that objects can be compared using a ruler.
Battista (2006) describes another set of misconceptions that students have with
measurement reasoning. Students regularly have difficulty recognising the need for units
to be of identical length, or to co-ordinate them without overlaps or gaps while
measuring (Batissta, 2006; SGVDEECD, 2009). This gap in conceptual understanding
prevents the accurate description and comparison of the whole length of an object. If
students cant see the inaccuracies of poor iteration, or irregularly sized units, it poses
significant problems in recognising the need for the use of formal measurement units.
Assessment
The assessment is adapted from a task performed by Hiebert (1984), which was
designed in order to observe students having difficulty with transitive reasoning, but also
provides useful information about a students ability to select identical units, use
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iteration, and understand language associated with length. It would be more functionally
effective if the teacher were able to closely observe the actions of smaller groups, so
incorporating it into a rotational set of activities where 6-8 students could be removed at
a time to work in pairs would be ideal. The teacher would construct a simple crooked
road from five 7cm Cuisenaire rods, then pairs would be asked to build a straight road
that was the same length to walk using as many 5cm Cuisenaire rods as they needed
(Hiebert, 1984) (see Appendix A). Instructions would need to include that they may use
anything in the class to help them work out how long their road needs to be, but the
crooked road cant be altered or moved. Because their rods are shorter, a counting
strategy is ineffective, and as the road is crooked, aligning them for a side-by-side
comparison will also be fruitless (Hiebert, 1984). Following students construction of
their roads the teacher will direct a conversation where each pair of students explains
how they worked out how long the road should be and why they used particular
materials to assist them. Students performance is to be recorded against the rubric
below.
Little or no engagement with the task.
Constructed a road the same length as the direct end-to-end distance of the
crooked road.
Counted the number of rods and used the same amount of shorter rods
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The task satisfies Siemon et al.s (2011) requirements for a rich, or authentic,
assessment task. It encourages diversity, by being accessible enough for all ability levels
to make an attempt. Students have an aspect of choice, then an opportunity to share their
justifications for choosing particular units. It presents an unfamiliar context for students
to apply their conceptual knowledge of transitivity, iteration and selection of uniform
informal units. Lastly, the assessment has a sense of authenticity as it shares similarities
with the kind of block road a child might play with.
Learning activities
Those students who were recorded as either 1 or 2 on the assessment rubric may
be having difficulties recognising the correct attribute of length and could benefit from
direct comparison activities to bolster their length vocabulary (Siemon et al., 2011,
p. 444). A task like a length scavenger-hunt could be appropriate (Siemon et al., 2011).
Students would be asked to find items longer, shorter or the same size as a pen within
the classroom. Following this they could be directed to do the same activity with a
length of string. This reinforces the concept that the word length may not always refer to
a straight line, whilst having an object that they can manipulate for a direct comparison.
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The students that scored 3 on the rubric may need some exercises to develop
their transitive reasoning. Siemon et al. (2011) suggests that indirect comparison can be
useful when students are comparing lengths that arent straight. A learning activity that
gives them the opportunity to develop transitive concepts through concrete experiences
would be to draw two curving lines in chalk outside (Siemon et al., 2011). Students
would have lengths of rope made available to them, and without direction have to find a
way of gauging which is longer. Through problem solving and discovery students
identify that objects can be used as a measurement tool for comparison and deduction.
Those students who scored 4 and 5 on the rubric would benefit from some tasks
to help them appreciate the need for uniform units and accurate iteration when
measuring. To improve understanding of the need for identical units Bush (2009)
advocates using concrete tasks using non-standard units. Measuring the length of a piece
of paper using an assortment of paperclips differing in length would provide a good
challenge to their conceptual understanding (Bush). As multiple students engage in the
same activity it can provide the basis for a discussion as to why their measurements
produced different results. Through questioning students can be given the opportunity to
realise that the key factors affecting the difference in results are the size of the paperclip,
and any gaps or overlaps. Hiebert (1984) suggests that an appropriate way to transition
students into accurate unit iteration is through activities measuring distances using their
feet. As students measure heel to toe it is an evolution between filling a length with endto-end units and using a single unit that has to be repositioned repeatedly to measure
length (Hiebert, 1984).
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References
Australian Curriculum, Assessment and Reporting Authority (ACARA). (2014).
Mathematics. Retrieved from
www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1
Battista, M. T. (2006). Understanding the development of students thinking about
length. Teaching Children Mathematics, 13(3), 140147. Retrieved from
www.jstor.org/stable/41198897
Bush, H. (2009). Assessing Children's Understanding of Length Measurement: A Focus
on Three Key Concepts. Australian Primary Mathematics Classroom, 14(4), 2932. Retrieved from
https://login.ezproxy.utas.edu.au/login?url=http://search.ebscohost.com/login.asp
x?direct=true&db=anh&AN=48850964&site=eds-live
Hiebert, J. (1984). Why do some children have trouble learning measurement concepts?
Arithmetic Teacher, 3(7), 1924. Retrieved from www.jstor.org/stable/41192320
Kamii, C., & Clark, F. B. (1997). Measurement of length: The need for a better approach
to teaching. School Science and Mathematics, 97(3), 116-121. Retrieved from
http://ezproxy.utas.edu.au/login?url=http://search.proquest.com/docview/623473
45?accountid=14245
State Government Victoria Department of Education and Early Childhood Development
(SGVDEECD). (2009). Measuring Length. Retrieved from
www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/r
eadmeaslength.pdf
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Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011).
Teaching mathematics foundations to middle years. South Melbourne: Oxford
University Press.
Whitton, D., Barker, K., Nosworthy, M., Sinclair, C., & Nanlohy, P. (2010). Learning
for teaching: Teaching for learning (2nd ed.). South Melbourne: Cengage.
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Appendix A
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