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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


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


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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http://www.australiancurriculum.edu.au/curriculum/contentdescription/ACMMG091




Figure 3. Geoboards

Figure 4. Anglegs

Figure 5. Tessellations

Figure 6. 3D Models

Figure 7. Geoflip Chart


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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Appendix A

Parent Consent and Tutoring Log

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Appendix B

Tutoring Recording Sheet

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