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Marriott_C_18448643_EDP343_Ass1
EDP343
Inquiry in the Mathematics Classroom
Assessment 1
Child Study Report
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Contents
Introduction.........................................................................................1
Rationale............................................................................................1
Diagnostic Assessment Overview......................................................2
Findings and Discussion....................................................................2
Shape.........................................................................................2
Location......................................................................................3
Transformation...........................................................................4
Teaching Plan and Teaching Sessions..............................................4
Summary of Learning and Future Teaching.......................................6
Conclusion.........................................................................................6
References.........................................................................................7
Appendix A ........................................................................................8
Appendix B.........................................................................................9
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Introduction
Geometry forms part of the many components of the Mathematics curriculum and includes the study
of shape, size, position and pattern concepts of 2D and 3D diagrams and objects. It also includes
the relationship of points, lines, angles, and surfaces, as well as the measurement of the area within
a shape, circumference. and perimeter (Australian Curriculum and Reporting Authority [ACARA],
2016a). The spatial features of objects, location, transformation, and geometric reasoning are all
outcomes of Mathematics teaching and learning (Department of Education [DOE], 2013) and it is
recommended students have a sound conceptual understanding of shape and space to enable them
to move from the concrete to the semi-concrete to the abstract levels of thinking (Reys et al., 2012).
The van Hiele model explains the movement of students through 5 distinct stages of geometric
reasoning; visualisation; classes of shapes, analysis; properties of shapes, informal deduction;
relationships between properties, deduction; deductive systems of properties, and rigor; analysis of
deductive systems (Van Der Walle, 2013), however, these stages are not dependent on age or year
level but rather by the student's own experiences with shapes and their properties (University of
Illinois, 2017) and a typical students move from considering only the visual appearance of shape to
considering the properties of shape through the 5 stages. Geometry is an important strand of the
Mathematics Curriculum and is a precursor to high school subjects such as calculus and
trigonometry, and builds understandings of how to follow directions and reasoning mathematically,
as well as helping to develop the concept of spatial awareness (Booker, Bond, Sparrow & Swan,
2010). Geometry also supports understandings of other mathematical concepts such as
multiplication, fractions, and algebra (Reys et al., 2012).
Rationale
The diagnostic assessment process provides teachers will an understanding of a student’s prior
knowledge to assist in planning an appropriate starting point of teaching and gain valuable insight
into a student’s conceptual thinking (Burns, 2010). A teacher's Mathematical Content Knowledge
[MCK], although important, is not enough because teachers also need to know how best to help
students understand content and concepts. In the Geometry Child Study, an interview style
assessment was used to ask questions, allow time for a Year 4 student to explain their thinking and
reasoning, and identify the strategies the student was using to solve geometric problems. Common
misconceptions were determined before designing the diagnostic interview [DI] and some of these
included the incorrect categorising of shapes by their appearance rather than their properties
(Oberdorf &Taylor-Cox, 1999), and location and spatial relationships misconceptions (Reys et al.,
2012). DI questions related to both the Year 2, 3 and 4 Geometry strands and sub-strands because
it was important to determine gaps in learning and misconceptions from earlier years that may still
exist.
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Diagnostic Assessment Overview
During the interview, a combination of open-ended questions and probing questioning were used to
determine the student's reasoning behind their thinking and related to the representation of the
location of shapes, the representation of shapes, and the representation of transformation.
Additionally, 2D and 3D concrete materials were used to allow the student to visualise the problem.
For example, when determining if the student could compare and describe two-dimensional shapes
by combining and splitting (ACARA, 2016b), Tessellations were used to enable the student to
manipulate shapes, problem solve, and show the answer by ‘doing’ the tessellation. Similarly,
during a symmetry task, the use of Magformers provided an opportunity to find out if the student
could explain why some shapes could tessellate while others could not. In all, 5 tasks were
developed relating to the content strands of shape, location, and transformation. The duration of the
interview was 25 minutes and notes together with an audio recording of the interview were taken to
ensure an accurate analysis of the results. Particular care was also taken during the interview when
making the distinction between a mistake, due to lack of attention, and a misconception where an
idea or concept was repeatedly misapplied.
Findings and Discussion
Shape
The student needed to know how to describe and draw 2D shapes with and without digital
technologies; ACMMG042, describe the features of three-dimensional objects; ACMMG043), make
models of three-dimensional objects and describe key features; ACMMG063, compare the areas of
regular and irregular shapes by informal means; ACMMG087, and compare and describe two-
dimensional shapes that result from combining and splitting common shapes, with and without the
use of digital technologies; ACMMG088 (ACARA, 2016c). The student was able to name and sort
most common regular 2D shapes, however, was unable to name and sort irregular 2D shapes, and
demonstrated a reluctance to categorise them. The student also incorrectly named a three-sided
shape with two concave sides a triangle, and a shape with 3-line segments that did not join also as a
triangle. Furthermore, the student was unable to name a right-angled triangle, naming it "half a
triangle" in one orientation and “half a square” in another, a common misconception according to
Oberdorf &Taylor-Cox (1999). The students also lacked the ability to identify less common 2D
shapes such as the heptagon, rhombus, and kite from the Year 2 curriculum. Although identifying
quadrilaterals was not contained in the Year 2, 3 or 4 curriculum content descriptors, a Year 4 work
sample portfolio (ACARA, 2016d) included the measurement of the area of this shape, so it was
included in the DI, however, the student was unfamiliar with this term or how it related to 2D shapes.
The student's responses suggested that, although a number of facts had been memorized about
specific attributes of shapes, the student had limited experience with the broader concepts of
shapes, and was perhaps inexperienced in the kinaesthetic exploration through play (Oberdorf & 2
Figure 1. Magformers Figure 2. playdough
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Taylor-Cox, 1999). For example, the student assigned one label to one specific shape and
orientation, demonstrating a limited applicability and necessary conceptual understandings (National
Council of Teachers of Mathematics, 2017).
The student was able to visualise and represent a very limited number of composite and compound
2D shapes, and was also unfamiliar with the language, demonstrating gaps in understanding which
could has the potential to create difficulties with other mathematical concepts such as measurement,
fractions, and calculating the perimeter and area of composite figures in the future (Reys et al.,
2012). However, when comparing areas of irregular shapes using 1cm grid paper and 1 cm cubes,
the student was able to accurately calculate the areas of rectangles and arrow shapes, as well as
estimate the area of an irregular leaf shape without the use of the 1 cm cube manipulatives despite
the limited knowledge of composite and compound shapes. Interestingly, the student demonstrated
the strategy of multiplying 6 x 4 to equal 24 squares in a rectangle, demonstrating a successful
transition from additive to multiplicative reasoning (Van de Walle, 2013).
The student demonstrated less difficulty with 3D shapes, and was
able to name regular 3D shapes, identify which shapes could roll,
matched most 2D shapes with 3D solids, identified the 2D shapes
that formed the 3D faces. Furthermore, the student was able to
make 3D models; including cubes, pyramids and prisms, using play
dough and Magformers, as seen in Figures 1
and 2, and also successfully identified the top
view of 3D shapes. However, the student was using 2D language to explain 3D properties; using
the word "corners" instead of vertices and "sides" instead of edges. According to Oberdorf & Taylor-
Cox (1999), this is also a common error with students as language is developmental, however, the
student also confused pyramids and prisms which suggested that some consolidation of the
geometric language was required. Despite some difficulties and misconceptions of shape, the
student demonstrated many of the skills and understandings typical of a student in the First Steps in
Mathematics [FMiS] Analysing Phase of development, the middle stage of Key Understanding 2 for
Shape (Department of Education [DOE, 2013) and working between growth point 1 and 2 of the Van
Hiele’s model (Reys et al., 2012).
Location
The student needed to know how to create and interpret simple grid maps to show position and
pathways; ACMMG065, and use simple scales, legends and directions to interpret information
contained in basic maps; ACMMG090 (ACARA, 2016c). During the DI the student could identify
objects using the legend; library, police station, and school, could use specific vocabulary to explain
how to travel from point A to point B; including the use of street names and turns. In terms of object
positions, the student could recognise when an object was moved from one position to another on 3
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the map, could answer questions relating to coordinates using the grid reference in the correct
format; following and naming the horizontal letter before the vertical number on the DI map, and was
able to identify the location of a concrete object when given a verbal grid reference to follow.
Despite the map used in the DI not displaying a compass reference or distance scale, the student
was able to estimate distances explaining that "the police station was closer to the school than the
library". The student was assessed as competent with FMiS KU1 as well as a number of the
pointers for KU2, working in the Analysing Phase of development (DOE, 2013) and working between
growth point 1 and 2 of the Van Hiele’s model (Reys et al., 2012).
Transformation
The student needed to know how to explain the effect of one-step slides and flips with and without
digital technologies; ACMMG045, identify symmetry in the environment; ACMMG066, and create
symmetrical patterns, pictures, and shapes with and without digital technologies; ACMMG091
(ACARA, 2016c). It was evident during the DI that the student had a sound understanding of
transformational concepts as the student was able to identify, demonstrate and explain the slide, flip
and turn of 2D and 3D shapes. Symmetrical understandings were also sound as the student was
able to identify and create symmetrical pictures and patterns, explain why a line of symmetry was
not accurate in some pictorial examples, and demonstrate multiple lines of symmetry in scenes and
shapes (Reys et al., 2012). The student was able to discuss how shapes tessellate using
statements such as “join” and “fit together” and provided real-world examples such as brick paving,
however, was unable to use the appropriate language to explain why heptagons and pentagons
could not tessellate; using statements such as “this bit here” and “it leaves a gap”. The student also
needed to use a trial an error approach rather than being able to distinguish if a shape could
tessellate by analysing shape properties. The student’s responses to the DI tasks indicated the
student was working at the middle stage of FMiS Key Understanding [KU] 2 and the Analysing stage
of development for Transformation (DOE, 2013) and working between growth point 1 and 2 of the
Van Hiele’s model (Reys et al., 2012).
Teaching Plan and Teaching Sessions
It was decided that the earliest point of need was to be able to describe and draw 2D shapes with
and without digital technologies; ACMMG042, describe the features of three-dimensional objects;
ACMMG043 (ACARA, 2016c). Therefore, initial teaching focussed on sequencing lessons on 2D
and then 3D shapes, together with the teaching of the correct vocabulary for space and shape up to
the Year 4 level, supporting the Literacy general capability of the Australian Curriculum (ACARA,
2016a). Formative assessment was undertaken during each session by way of a recording checklist
[Appendix B] to guide future planning for teaching, and sessions included scaffolding at all points of
need using strategies such as explicit instruction, modelling, repetition, recasting, questioning, and
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Figure 3. Geoboards
Figure 4. Anglegs
Figure 5. Tessellations
Figure 6. 3D Models
Figure 7. Geoflip Chart
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elaboration (McDevitt, Ormond, Cupit, Chandler & Aloa, 2013). In using a constructivist approach
the student was guided through the inquiry process and was free to explore and make connections
to prior learning. Tutoring was planned for 2, forty-minute sessions per week, for 3 weeks.
Using understandings of knowledge processes, and the constructivist
learning theory, activities were designed that provided opportunities to
explore concrete and semi-concrete materials to classify, construct and
manipulate shapes in new ways and develop
new conceptual understandings. For
example, when working with 2D shapes,
logic rings were used to demonstrate how to
use Venn diagrams to classify shapes
according to their properties, while
cardboard cut-outs allowed the student to pick up and explore
shapes. New vocabulary was also
introduced including quadrilateral, polygon, convex, concave, parallel,
and congruent, and the student's answers were paraphrased with the
correct mathematical language until the student became more fluent.
Aglegs, as demonstrated in Figure 4, were used to make models of
2D hinged shapes for the purpose of teaching the transformation from
regular 2D shapes into irregular 2D shapes by applying new
knowledge, as well being used to model the concept of parallel and
non-parallel lines and changing angles. Geoboards, as demonstrated in Figure 3, also proved to be
a useful tool for teaching congruency, composite shapes, and compound shapes, while
Tessellations, demonstrated in Figure 5, consolidated understandings of angle properties as well as
composite and compound shapes; a task that had proven difficult during the DI.
In the final 3D shape tutoring sessions, concrete materials such as
plasticine and straws, depicted in Figure 6, were used to construct 3D
prisms and consolidate the change in vocabulary from 2D ‘sides’ and
‘corners’ to 3D ‘edges’ and ‘vertices’
together with the introduction of the
concept of depth. Magformers also
provided a useful tool for the teaching of
3D shapes as the magnetic pieces were
able to be constructed and deconstructed
quickly as the student made discoveries about pattern and
relationships, exercising critical and creative thinking; another general
capability of the Australian Curriculum (ACARA, 2016a). 3D geometric translucent models were
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used to teach the difference between pyramids and prisms, and a Geometric Flip Chart, depicted in
Figure 7, consolidated understandings about 2D and 3D relationships and 3D object properties.
Origami provided the hook for a geometric investigation where the identification of various shapes
and angles could be discussed as a 3D cube was constructed. Demonstrative of constructivist
principles, concrete manipulatives used during the tutoring sessions were left with the student
between sessions to allow for knowledge to be built on through play experiences.
Summary of Learning and Future Teaching
The student demonstrated new understandings of the shared properties of 2D and 3D shapes and
made connections between various geometric concepts during the tutoring sessions when
answering questions, solving problems and making models. The student also demonstrated more
confidence and accuracy when classifying and manipulating shapes and made some unexpected
discoveries. For example, the student found that a 6-pointed star could be constructed in 8 different
ways using Tessellations, whereas only one configuration of composite shapes could be identified
during the DI. The student did not have an opportunity to draw 2D shapes with digital technologies
so it would be beneficial if the future teaching included the skill of using computers and graphic
software. 3D shape nets would also be something to include in the next phase of geometry learning
for the student.
Conclusion
It is clear that students need opportunities to continue to explore 2D shapes beyond Year 2 as they
may not have fully developed the necessary conceptual understandings in a single year. Despite
not being included as a sub-strand of the Geometry and Measurement Year 3 curriculum, re-
teaching aspects of 2D shapes at the beginning of Year 3 before introducing 3D shapes would
consolidate prior learning and enable teachers to fill important gaps in understanding. Similarly,
although 3D shapes are not included in the Year 4 curriculum, it is clear that concepts of shape
should not be taught in isolation in any one year and instead revisited often and integrated in other
learning areas such as The Arts and Design and Technology so students have a balance of
experiences in the classroom to help consolidate ideas before applying and transferring knowledge.
It would seem that the types of learning experiences linked to Geometry are only limited by a
teacher’s imagination and the curiosity of the students.
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References
Australian Curriculum Assessment and Reporting Authority. (2016a). F-10 Curriculum: Mathematics General Capabilities. Retrieved from http://www.acara.edu.au/verve/_resources/Mathematics_-_general_capabilities_learning_area_specific_advice.docx
Australian Curriculum Assessment and Reporting Authority. (2016b). F-10 Curriculum: Mathematics Structure. Retrieved from http://www.australiancurriculum.edu.au/mathematics/structure
Australian Curriculum Assessment and Reporting Authority. (2016c). F-10 Curriculum Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1
Australian Curriculum Assessment and Reporting Authority. (2016d). Work sample portfolio Year 4. Retrieved from https://acaraweb.blob.core.windows.net/curriculum/worksamples/Year_4_Mathematics_Portfolio_Satisfactory.pdf
Booker, G.; Bond, D., Sparrow, L., & Swan, P. (2014). Teaching primary mathematics. Frenchs Forest, NSW: Pearson Australia.
Burns, M. (2010). Snapshot of student misunderstandings. Retrieved from https://lms.curtin.edu.au/bbcswebdav/pid-4155240-dt-content-rid-23568886_1/xid-23568886_1
Department of Education. (2013) First Steps in Mathematics: Space. Retrieved from http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/
McDevitt, T. M., Ormrod, J. E., Cupit, G., Chandler, M., &Aloa, V. (2013). Child development and education. Frenchs Forest, NSW: Pearson Australia
National Council of Teachers of Mathematics. (2017). Principles and Standards for School Mathematics. Retrieved from http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Geometry/
Oberdorf, C, D., Taylor-Cox, J. (1999). Teaching Children Mathematics. Vol.5(6), p.340-45 , Retrieved from http://www.jstor.org.dbgw.lis.curtin.edu.au/stable/pdf/41198858.pdf?refreqid=excelsior%3A218a20a5dba1782fbff16ede0f0b6e8a
Reys, R.E., Lindquist, M.M., Lambdin, D.V., Smith, M.L., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics.(1st Australian edition). John Wiley &
Sons Australia: Milton, Qld.
University of Illinois. (2017). Levels of Mental Development in Geometry. Retrieved from http://www.math.uiuc.edu/~castelln/VanHiele.pdf
Van de Walle, J. A., Karp, K, 1951-; Bay-Williams, J. M. (2013). Elementary and middle school mathematics : teaching developmentally 8th ed. Boston: Pearson
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