newcombe, d edp443 assignment 1 · 2018-05-09 · newcombe, d edp443 assignment 1 5 child study...

Newcombe, D EDP443 Assignment 1

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Student details

Name Darrien Newcombe

Course Bachelor of Primary Education

Unit details

Unit code EDP443

Unit name Pedagogies and Planning for Mathematics

Unit lecturer or tutor Reid Hamilton

Assignment details

Topic Child Study Report

Due date 15/04/2018 Word count 3297

Extension granted □ No □ Yes Extension date

Is this a resubmission? □ No □ Yes Resubmission date

Declaration

I certify that the attached material is my original work. No other person’s work or ideas have been used without acknowledgement. Except where I have clearly stated that I have used some of this material elsewhere, I have not presented this for assessment in another course or unit at this or any other institution. I have retained a copy of this assignment. I have read and understand the Curtin University of Technology document Academic Integrity at Curtin: Student guidelines for avoiding plagiarism.

Name/signature Darrien Newcombe Date 14/04/2018


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Contents

A) The Rationale of Teaching Fractions .............................................................................................. 3

B) Child Study Intervention Report .................................................................................................... 5

1.0 Introduction ................................................................................................................................... 5

2.0 Diagnostic Assessment .................................................................................................................. 5

3.0 Tutoring Overview ......................................................................................................................... 7

3.1. Lesson series structure and focus ......................................................................................... 7

4.0 Tutoring Sessions: Developing Angle ............................................................................................ 8

4.1. Understanding the part-whole relationship ......................................................................... 8

4.2. Relating the number of parts to fraction size ....................................................................... 9

4.3. Modelling and representing fractions ................................................................................... 9

4.4. The numerator-denominator relationship .......................................................................... 10

5.0 Future Learning............................................................................................................................ 12

6.0 Conclusion.................................................................................................................................... 12

7.0 References ................................................................................................................................... 13

8.0 Appendices .................................................................................................................................. 15


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The Rationale of Teaching Fractions

The ideas associated with fractions are among the most complex concepts that students will encounter

(Hurrel & Day, 2014). It is vital to place emphasis on the meaning of fractions so that students are able to

use number-sense and proportional-reasoning, with being numerate in everyday life as the ultimate goal

(Clarke et al., 2008; Siemon, 2004). This paper will summarise the importance of fraction understanding

(including key ideas), how it is linked to other Mathematics strands and the sequence of teaching fractions.

Charles (2005) advocates teaching and assessing fractions through key concepts known as ‘big ideas’ that

connect linear curriculum strands into a coherent whole (Hurst & Hurrel, 2015). Fractions are the

foundation of probability, algebraic-thinking, decimals, percentages and ratios (Clarke, Roche and Mitchel,

2008). Fractions also support the conceptual links in learning areas of Measurement and Geometry

(Australian Curriculum Assessment and Reporting Authority [ACARA], 2018). For example, shaded area

region problems relate to multiplication, area, fractions and percentages (Siemon, 2004). Consequently, a

constructivist approach informed by ongoing diagnostic and formative evaluation, rich in manipulatives is

vital (Clark et al., 2008; Hurst, n.d.).

Fractions is a challenging subject, largely due misconceptions formed during early learning of whole

numbers that often focusses heavily on instrumental understanding (Skemp, 2006; Clarke, Van de Walle,

2006). One of the challenges is the introduction of the five fractional constructs: part-whole, measure,

quotient, operator and ratio (Clarke et al., 2008). For example, an assessment task requiring a student to

represent a fraction such as ½ through symbols, manipulatives and words, would require extensive

procedural and conceptual understanding. Fractional understanding should be assessed by the construction

of fraction models and diagrams as well as naming, recording comparing, ordering, sequencing and

renaming common fractions (Siemon, 2004). Students must be exposed to a variety of models and

constructs that emphasise fractional meaning, starting with part-whole relationships (Clarke et al., 2008;

Reys et al., 2016). Flexible materials that allow students to craft their own manipulatives such as clay,

paper-strips and string to gain a deeper understanding of equal sharing (Way, 2011; Clarke et al., 2008).

The ‘Quantifying phase’ involves constructing fair shares into equal parts without altering the total quantity

(Department of Education Western Australia [DoEWA], 2013). Initial fraction concepts begin with

manipulatives that represent ones or a ‘whole’ (Booker et al., 2014). Paper-folding exercises breaking the

whole into simple fractions such as halves, thirds and fifths are a vital foundation for fraction language

development expected by the end of Year 3 (Clarke et al., 2008; ACARA, 2008 ). Partitioning exercises

should move onto concepts of measure that do not rely on symmetry and promote estimation, such as

shading, cutting and sharing a variety of shapes in order progress through the ‘Partitioning phase’ (DoEWA,


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2013). This can correct the misconception that equal parts come only from halving often encountered

during folding activities (Booker et al., 2014).

When an understanding of the ‘whole-unit’ exists, it is pertinent to cultivate the connection between

fractions and division (Booker et al., 2014). Teaching fractions as division is an ideal segue into introducing

equivalent fractions. Initially this can start with counter sharing before moving onto groups of varied

objects such as identifying ‘what fraction of a collection of blocks are blue?’. This is an ideal opportunity to

introduce common and equivalent fractions through generalisable rules of the numerator/denominator

relationship, improper fractions and subsequently ratios (Clarke et al, 2008; Van de Walle et al., 2013). For

example, if a is the number of parts and b is the name or size of the part, this rule effectively illustrates

proper or improper fractions (Clarke et al., 2008).

In the ‘Factoring phase’, students can typically use visual models (such as arrays) to compare and order

fractions with the same denominator (3/5 and 2/5) or simple equivalences such as ½, 2/4, and 4/8 (DoEWA,

2013). Dot paper activities encourage estimation and verification of equivalency without manipulatives

(Reys et al., 2016).

Circular pie pieces are ideal for modelling division, where sets are separated into equal parts. Such tasks

can be extended to focus on operations using common fractions, improper fractions and equivalences

(Clarke et al. 2008; DoEWA, 2013). Division, addition and subtraction using counters during the ‘Operating

phase’ can be explored through splitting and combining common fractions (DoEWA, 2013). Using visual and

manipulative constructs and ordering activities on number-lines link concepts of fractions as division,

decimals, ratio and percentages (DoEWA, 2013). Number-lines also enable students to gain a deep

understanding of the density of rational numbers as they learn that fractions are infinite (Clarke et al.,

2008).

Towards the end of ‘Operating phase’ students gain confidence connecting fractions with concepts of

decimals and percentages as representations of the same number. This enables solving of operational

problems such as addition or subtraction of related fractions and making connections between

multiplication and division of decimals (ACARA, 2018). This highlights the importance of learning fractions

in sequence as students progressively learn to operate with rules that facilitate everyday numeracy; such as

connecting fractions with percentages to calculate discounts, or prioritising items of value in a retail-sale

using multiplicative comparisons (DoEWA, 2013).


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Child Study Intervention Report

1.0 Introduction

For the purpose of this study, the subject was a 9 year-old boy in Year 3, who will be referred to by the alias

of ‘Ben’. The aim of this study was to determine Ben’s conceptual understanding of fractions through

diagnosis of any misconceptions in-line with the First Steps in Mathematics (FSiM) Key Understandings

(DoEWA, 2013) and the proficiencies expected of the Australian Curriculum (Australian Curriculum

Assessment and Reporting Authority [ACARA], 2018). This was followed by six tutoring sessions that

addressed these misconceptions. A pre-interview discussion with Ben’s mother indicated that Ben was not

enjoying Mathematics. Ben’s Year 2 report was graded slightly below standard for Mathematics, with

Measurement and Geometry, Money and Financial Mathematics (ACARA, 2018) identified as areas of

concern. Consequently, Ben had been placed in a small maths intervention group. Ben’s attention span

when given a challenging task was becoming an issue, therefore a constructivist pedagogy influenced by

Hurrel & Day’s (2014) interpretation of the Concrete, Representational Abstract (CRA) approach was

observed throughout the study.

2.0 Diagnostic Interview

A diagnostic interview was delivered to capture Ben’s fraction understanding to gain insight into

misconceptions using activities derived from the First Steps in Mathematics (FSiM) Number – Book 1

(DoEWA, 2013) . Thorough questioning was used throughout the interview to provide deep insight into his

thought process (Booker et al., 2014). The interview covered questions across the first five of FSiM’s

(DoEWA, 2013) fractional numbers Key Understandings, concentrating on the Fractions and Decimals

Mathematics sub-strand of the Australian Curriculum. (ACARA, 2018).

Ben appeared confident with the underlying concept of halving seen in Key understanding 1 (FSiM, 2013).

However, he relied purely on symmetry and folding strategies which were ineffective in cases requiring

mental partitioning or complex non-symmetric shape and dot configurations. (Key Understanding 1, Level

2). This limited range of strategies was apparent when asked if varied shapes divided into quarters were of

the same size, fraction or quantity. Ben unsuccessfully attempted to manipulate or fold the fractions of

multiple coloured circles to assist in partitioning and argued that, “equal parts must be the same” as well as

having the same size or amount of the relevant quantity (Key Understanding 2, Level 1). Challenging this

misconception is vital as it forms the basis of using common denominators to find equivalent fractions, as

well as ordering, comparing and calculating with them (Booker et al., 2014; DoEWA, 2013).


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Ben was able to effectively partition when asked to use manipulatives such as plastic counters. In this case

he was much more successful with partitioning, although he was unfamiliar with both fraction naming and

notation. When dividing a set of 15 counters into equal shares, he was able to do this, however did not

understand that fractional language such as a third meant splitting the whole into a three-parts. He was

unable to identify a half when the fraction was represented by 4 alternating shaded-cells on card, but

indicated equal portions when shaded-cells were linked together (Key Understanding 3, Level1)(DoEWA,

2013).

When testing Key Understanding 4, Ben struggled to recognise that every fraction can be represented in an

infinite number of equivalent forms and did not believe that 2/4 and 4/8 were effectively the same share of a

whole (DoEWA, 2013). When physically folding fraction shapes, he was able to partially grasp this

relationship, but insisted that ½ was not the same as 2/4. Ben did not fully understand the meaning of

fractions, which is vital before expecting him to be able to perform complex or simple computation (Clarke

et al., 2008).

When comparing and ordering fractional numbers (Key Understanding 5)(DoEWA, 2013), Ben appeared to

be guessing. Ben considered only the denominator and ignored the numerator, highlighting limited

exposure to fraction language and notation. For example, he considered 1/10 as bigger than ½.


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3.0 Tutoring Overview

3.1 Lesson series structure and focus

Having diagnosed that whilst Ben was confident using strategies of symmetry to determine halves and

quarters, he appeared most challenged when asked to use other strategies. Therefore, he appeared to be

transitioning through the ‘Quantifying phase’ (DoEWA, 2013). The key gap was that Ben could not grasp

that a fraction is a part of a whole. This suggested he would need further emphasis on the meaning of

fractions before focus was placed on the procedures of complex operation (Clarke et al., 2008).

Thanheiser’s first principle suggests that Ben first understands the underlying concept as a foundation,

before addressing the mathematical procedure (Thanheiser et al., 2013). Given the importance of this

fundamental, only the first lesson was planned, however an outline was prepared that mapped out what he

should be able to achieve based on FSiM and the Australian Curriculum (ACARA, 2018). Ben was close to

achieving a Year 2 understanding of fractional numbers, however the first few lessons would ensure a full

understanding existed for the meaning of halves and quarters as a part-whole relationship (KU1) (DoEWA,

2013). Subsequently, the intent was to transition into activities linking the language of partitioning to

fractional language and re-arranging quantities (KU2) (DoEWA, 2013). This simulated Booker’s (et al., 2014)

sequence for developing the initial fraction concept as follows: First demonstrating simple partitions as per

the Australian Curriculum Achievement Standards of Year 2 (ACMNA032), then moving into Year 3

standards exploring modelling and representation of halves, quarters, thirds, fifths and their multiples to

complete the part-whole relationship understanding (ACMNA058) (ACARA, 2018).

http://www.scootle.edu.au/ec/search?accContentId=ACMNA032

http://www.scootle.edu.au/ec/search?accContentId=ACMNA058


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4.0 Tutoring Sessions: The Initial Fraction Concept

4.1 Understanding the Part-Whole Relationship

Beginning with the understanding of the part whole relationship, Ben needed further use with the part-

whole construct to complete the foundational meaning of fractions he was already forming (Van de Walle,

2013). As per the Fractions Key Understanding 1 of FSiM – Numbers (DoEWA, 2013), instruction would

further explore Ben’s concept of a whole using a variety of concrete manipulative strategies. Ben

continued to display a preference towards symmetry strategies for creating partitions with fraction shapes.

However, over-reliance on such strategies was holding back his concept of division (Booker et al., 2014).

When asked to produce two halves from a ball of clay, he understood that both pieces should be equal. He

rolled the clay into a line, then attempted to cut along the line of symmetry. When prompted to try another

strategy such as measuring, he did so with his fingers before partitioning. When the halves were split

again, creating quarters or eighths; Ben identified these also as halves, which is a common misconception

for his age (Booker et al., 2014). When two quarters were combined into a single ball, he incorrectly

declared this as ‘one whole with two halves’. When asked to create divisions of 4, 6 and 8 equal shares of

clay, Ben was able to do so using finger measuring techniques, recognising partitions were not equal in size.

Ben had an emerging concept that all parts of a fraction must be equal, however this needed further

exploration through different models and strategies to ensure he was not left with the misconception that

fractions are only achieved from halving or doubling (DoEWA, 2013).

Using Lego blocks and the part/whole chart (Fig 1.) enabled Ben to display an understanding of this as he

was extended into using both counting and non-standard measurement strategies to divide various mixes

of blocks and identify equivalence (Disseler, 2016; DoEWA, 2013). Ben was counting the studs on each

brick, grouping them and identifying the large brick as the whole (Fig 1). He was encouraged to continue

using such counting and measuring strategies as they enabled him to grasp more complex tasks involving

area, length and quantity (Reys et al., 2016).


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4.2 Relating the number of parts to fraction size

During the diagnostic assessment, Ben was challenged in identifying halves on a 4x1 line with alternate cells

shaded. He also incorrectly related the number of parts to the size of the fraction based on shapes or

appearance. As per Key Understanding 2, activities were chosen involving the comparison of halves by first

starting with shape-cutting and then by using grid-shading to provide an opportunity to estimate and

measure (DoEWA, 2013).

When Ben was presented with a range of squares, all with half the area coloured in various configurations,

he was asked to specify which were indicative of ‘halves’. Ben identified only the symmetrically divided

squares with a single line through them as ‘halves’ and instead counted the pieces created by the bisecting

lines to classify them as ‘not-halves’. When Ben cut the squares into parts and reconfigured them into

different shapes, he then realised that each colour provided represented equal shares which he understood

as a ‘half the square’ (Fig 2.). By then reverting to the counting strategy used to count segments, he then

stated, ‘I don’t need to fold it to make half,’ suggesting he was beginning to grasp that symmetry was not

required to make a half (Reys et al., 2016). Sharing activities using these same pieces highlighted that Ben

was able to identify that if he were to share the coloured segments between 2 people, both would have

half the ‘whole’ square.

4.3 Modelling and representing fractions and their multiples

Ben’s emerging concept of the part-whole relationship and the number of parts within a fraction, suggested

he was confident to extend beyond simple halving and partitioning into models that represent a wider

variety of fractions (ACARA, 2018). As he was linking his language of half meaning to represent 2 equal

portions, he was beginning to grasp a half as being a quantitative unit and could then begin to explore

fractions as words (DoEWA, 2013). His concept of equivalence was evident by his awareness that all parts


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of a fraction must be equal, although ongoing focus was required to ensure he understood that varying

shapes could still contain the same area or volume and therefore represent an equivalent partition (Booker

et al., 2014).

As Lego activities had proven ideal for defining a whole and representing parts of different sized wholes

(Disseler, 2016), pieces were used to represent a ‘large chocolate bar’ for length and area models. Ben was

asked to split a variety of ‘chocolate bars’ first into three shares, then thirds, four shares, fourths, quarters

and so forth. Ben’s progress was evident by his ability to correctly complete this, however required some

prompting with the fractional language of quarters. The identified misconception of equivalence where

Ben failed to recognise that ½ and 2/4 are effectively the same was also approached (KU4) (DoEWA, 2013).

By taking the opportunity to use blocks of various sizes, Ben measured using the dot counting strategy and

stated that fractions of ½, 2/4 and 4/8 were “all half of the chocolate” (Fig 3.). However, when this same

process was explored using counters, Ben found it challenging to acknowledge a collection of discrete

objects as a whole, which suggested further opportunity to work with a wider variety of manipulatives to

reinforce this concept (Reys et al., 2012).

4.4 The Numerator Denominator Relationship

Ben was introduced to the Numerator/Denominator relationship in the aforementioned exercise where he

indicated he had encountered a similar representation in a recent class at school. (Figure 3). The

misconception encountered during the diagnostic survey implied that Ben determined a notated fractions

size purely by its denominator (KU5) (DoEWA, 2013). This concept was explored through ordering exercises

involving fraction strips, before folding and labelling of paper strips emphasising the denominators meaning

for the size of each part (Clark et al., 2008). This provided a concrete and representational way for Ben to


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estimate and explore that the more shares something is split into, the smaller each share is, yet the whole

remains the same (Booker et al., 2013). Attempts were made to extend Ben’s emerging understanding of

fractional language towards the abstract by having him label and cut his folded paper strips with a range of

simple fractions with common numerators. Ben was able to correctly order the fractional pieces on a

number-line with extensive modelling support. When the task was made abstract by placing fraction tokens

on a number-line without the aid of manipulatives, a number errors were made (Figure 4). Ben’s

explanation suggested he was still ordering fractions by their numerator and with the abstract

representation of half fractions such as 2/4 and 4/8, he was failing to recognise equivalence as he did with

earlier representational tasks. This suggested a greater need for additional concrete and representational

tasks, before further attempts are made at abstract exercises that focus purely on fractional notation

(Peltier & Vannest, 2017).


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5.0 Future Learning

Ben displayed a robust foundation for the part-whole relationship and a stronger understanding of the

meaning of fractions that was impeding his capacity to manipulate them (Clarke et al., 2008). However,

there is further opportunity to explore his assumption that fractions must appear equal as seen in the clay

exercise through region or area models such as geoboards, with ongoing emphasis on non-symmetrical

partitions (Reys et al., 2016).

Ben demonstrated a disposition towards inferring that the number of parts was the only mode of

determining the size of a fraction and was inclined to not view 2/4 as being identical to ½ in abstract

exercises involving notation. Concrete and representational sharing activities that utilise circular pie pieces

or lollies such as M&M’s would be an ideal next step to challenge this concept before revisiting fractional

notation (Booker et al., 2014).

6.0 Conclusion

A closing review of the tutorial series highlights the importance of using diagnostic analysis to deliver

tailored experiences that develop learning in a rational continuum. The diagnostic interview identified that

whilst Ben had an apparently sound understanding of partitioning objects into equal-sized parts (KU2)

(DoEWA, 2013), he was of the misconception that folding and symmetry were the only methods in which

this could be achieved. Consequently, the series needed to begin at the Australian Curriculum’s Year 2

descriptor for ‘fractions and decimals’. Efforts were made to deliver the series using Hurrel & Day’s (2014)

interpretation of the CRA approach, however, Ben’s co-operation proved challenging when attempting to

draw out deep reasoning. Ben’s evolving fluency was implied through his ability to link that equal shares

are derived from splitting a whole into equal parts associated with their fractional words (ACARA, 2018).

He had made progress through Level 3, moving towards the ‘Partitioning phase’ in his understanding

(DoEWA, 2013). He was more confident moving on from strategies of symmetry and displayed attempts at

problem-solving using other schemes such as measuring and counting. Ben exhibited an affinity toward

Lego manipulative tasks that required explanation and application of geometric reasoning. It is therefore

important that Ben continues to use manipulatives that engage through simulation of real-world problems,

encouraging a variety of strategies to be employed.


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7.0 References

Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2018). The Australian curriculum

v8.3. Sydney, NSW: ACARA.

Booker, G., Bond, D., Sparrow, L., & Swan, P. (2014). Teaching Primary Mathematics (5th

Ed.). Frenchs Forest, New South Wales: Pearson Australia.

Charles, R. (2005). Big ideas and understandings as the foundation for early and middle school

mathematics. NCSM Journal of Educational Leadership, 8(1), 9-24

Clarke, D. M., Mitchell, A., & Roche, A. (2005). Student one-to-one assessment interviews in mathematics: A

powerful tool for teachers. Brunswick, VIC: The Mathematical

Clarke, D., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make

sense. Mathematics Teaching in the Middle School, 13(7), 373-380

Department of Education WA [DoEWA]. (2013). First steps in mathematics: Number - Book 1. Perth, WA:

Department of Education WA

Disseler, S. (2016). Teaching fractions using lego bricks. Brigantine Media

Hurrell, D., & Day, L. (2014). The importance of fractions in being a successful mathematics

student. Mathematical Association of Victoria Annual Conference, 51.

Hurst, C. (n.d.). In defence of constructivism. Retrieved from: https://lms.curtin.edu.au/bbcswebdav/pid-

5153606-dt-content-rid-28636050_1/xid-28636050_1

Hurst, C., & Hurrell, D. (2015). Developing the big ideas of number. International Journal of Educational

Studies in Mathematics, 1(1) 1-17.

Peltier, C. & Vannest, K.J. (2017). Using the concrete representational abstract (CRA) instructional

framework for mathematics with students with emotional and behavioural disorders. Retrieved

from: https://doi.org/10.1080/1045988X.2017.1354809


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Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, M. L., Rogers, A., Falle, J., Frid, S., &

Bennett, S. (2016). Helping Children Learn Mathematics. John Wiley & Sons Australia: Milton, QLD

Siemon, D. (2004). Partitioning – The missing link in building fraction knowledge and confidence. RMIT

University: Victoria

Skemp, R.R. (2006). Relational Understanding and Instrumental Understanding. Mathematics

Teaching in the Middle School, 12(2), 89-95. Retrieved from:

http://www.jstor.org.dbgw.lis.curtin.edu.au/stable/pdf/41182357.pdf

Thanheiser, E., Philipp, R.A., Fasteen, J., Strand, K., & Mills, B. (2013). Preservice-Teacher Interviews: A

Tool for Motivating Mathematics Learning. Mathematics Teacher Educator (1)2, 137-147.

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and Middle School Mathematics:

Teaching Developmentally 8E. Frenchs Forest, New South Wales: Pearson Australia.

Van de Walle, J.A., & Lovin, L.H. (2006). Teaching student-centred mathematics. Boston:

Pearson.

Way, J. (2011). Fractions: Teaching for Understanding. Retrieved from:

www.aamt.edu.au/content/download/19932/273057/file/tdt_F_way1.pdf


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8.0 Appendices

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