PROMOTING THE FOUNDATION CONCEPTS OF

PROPORTIONAL REASONING: A CASE STUDY OF

P-3 TEACHERS

Angela K. Drysdale

GradDipEd(EC), DipT, ASDA

Dr Sonia White PhD

Professor Joanne Lunn

Submitted in fulfilment of the requirements for the degree of

Master of Education (Research)

Office of Education Research

Faculty of Education

Queensland University of Technology

2018

i

Keywords

Case study, early years, foundation concepts, mathematics, additive thinking, multiplicative

thinking, proportional reasoning

ii

Abstract

Proportional reasoning is a sophisticated way of thinking which is used in

everyday living, including workplace and educational contexts. Its application is

central to many domains throughout mathematics curriculum, as it encompasses many

interconnected concepts (e.g. ratio, decimals, and fractions). The foundations of

proportional reasoning are established in the early years of school. However, the link

between foundational concepts and proportional reasoning is not overtly stated in

mathematics curriculum documents.

This research aimed to investigate the foundation concepts of proportional

reasoning and the promotion of these concepts in the early years of school. The

research questions guiding this investigation were:

1. What do Preparatory to Year 3 teachers recognise as foundational concepts

of proportional reasoning?

2. How do Preparatory to Year 3 teachers promote foundational concepts of

proportional reasoning?

An exploratory case study design was adopted for this study (as per Yin, 2009).

Multiple sources were used for data collection, including focus groups, discussions,

concept mapping and interviews. The case study took place in one south east

Queensland primary school and the participant group consisted of five teachers - one

teacher from each Year level from Preparatory to Year 3 and the participant researcher.

The data were analysed using template analysis, which began with pre-determined

codes relevant to each research question (King, 2012). As required throughout the

analysis, the templates were adapted to represent the collected data.

The findings highlighted that the foundational concepts of proportional

reasoning evolve from three stages of development, which included additive thinking,

transition to multiplicative thinking, and multiplicative thinking. The promotion of

these concepts was found to take place through the provision of environmental

resources, problem solving, and language. Throughout the data collection the teachers

reported that their own knowledge grew and this was reflected in discussions which

highlighted a synthesis of the foundational concepts and the promotion of proportional

reasoning.

iii

Implications of this research highlight the significance of teacher knowledge. A

teacher needs a deep knowledge of the foundational concepts which underpin a

student’s development of higher level concepts (additive thinking, transition to

multiplicative thinking and multiplicative thinking). The Australian Curriculum:

Mathematics requires interpretation for planning and teaching and a deep curriculum

knowledge of how foundational concepts, that is lower level concepts relate to higher

level concepts. Teacher knowledge has the potential to grow when a teacher is

involved in teacher research within the context of a teachers’ own school setting.

iv

Table of Contents

Keywords ........................................................................................................................ i

Abstract ......................................................................................................................... ii

Table of Contents .......................................................................................................... iv

List of Figures ............................................................................................................... vii

List of Tables ................................................................................................................ viii

Statement of Original Authorship .................................................................................. ix

Acknowledgements ........................................................................................................ x

Chapter 1: Introduction ............................................................................................... 1

1.1 Context for Mathematics Education ................................................................ 2

1.2 Importance of Early years Mathematics .......................................................... 5

1.3 Big Ideas of Mathematics ............................................................................... 7

1.3.1 Proportional Reasoning ...................................................................................... 9

1.4 Chapter summary ......................................................................................... 11

1.5 Overview of the Thesis ................................................................................. 12

Chapter 2: Literature Review ..................................................................................... 14

2.1 Proportional Reasoning is a Big Idea of Mathematics .................................... 14

2.1.1 Additive and Multiplicative Thinking ................................................................ 19

2.1.2 Foundation Concepts of Proportional Reasoning............................................. 21

2.2 Promoting Mathematical Concepts in the Classroom ..................................... 28

2.2.1 Teacher Knowledge .......................................................................................... 30

2.2.2 Classroom Community...................................................................................... 33

2.2.3 Discourse in the Classroom .............................................................................. 35

2.2.4 Mathematical Tasks .......................................................................................... 38

2.3 Chapter Summary ........................................................................................ 43

Chapter 3: Methodology ........................................................................................... 46

3.1 Methodology and research design ................................................................ 46

3.2 Teacher research .......................................................................................... 48

3.3 Research Methods........................................................................................ 49

3.3.1 Research Setting ............................................................................................... 49

v

3.3.2 Participants ...................................................................................................... 50

3.3.3 Data Collection ................................................................................................. 51

3.3.3.1 Focus Group ................................................................................................. 54

3.3.3.2 Discussions ................................................................................................... 55

3.3.3.3 Interview ...................................................................................................... 56

3.4 Data Analysis ............................................................................................... 57

3.4.1 Analysis of Research Question 1 ...................................................................... 58

3.4.2 Analysis of Research Question 2 ...................................................................... 62

3.5 Credibility .................................................................................................... 70

3.6 Ethical Considerations .................................................................................. 72

3.7 Chapter Summary ........................................................................................ 74

Chapter 4: Results ..................................................................................................... 75

4.1 Understandings of Proportional Reasoning ................................................... 78

4.1.1 Prior to Curriculum Investigation Meeting ...................................................... 78

4.1.2 After the curriculum investigation meeting ..................................................... 83

4.1.3 Summary .......................................................................................................... 93

4.2 Understandings of Practices that Promote Proportional Reasoning ............... 94

4.2.1 Prior to lesson implementation ....................................................................... 96

4.2.2 After lesson implementation ......................................................................... 102

4.2.3 Summary ........................................................................................................ 111

4.3 Chapter summary........................................................................................ 113

Chapter 5: Discussion ............................................................................................... 116

5.1 Research question One ................................................................................ 116

5.1.1 Proportional reasoning is based on additive and multiplicative thinking ...... 117

5.1.2 Additive and multiplicative thinking evolves from foundational concepts ... 118

5.1.3 Transition between additive and multiplicative thinking supported by a focus

on specific foundational concepts .............................................................................. 120

5.1.4 Summary ........................................................................................................ 122

5.2 Research Question Two ............................................................................... 122

5.2.1 Promoting proportional reasoning through problem solving ........................ 123

5.2.2 Promoting proportional reasoning through language ................................... 125

5.2.3 Promoting understanding of foundational concepts with resources ............ 126

vi

5.3 Changes in teacher knowledge ................................................................... 128

5.4 Implications ............................................................................................... 130

5.5 Limitations and Future Directions ............................................................... 134

5.6 Conclusion ................................................................................................. 135

Reference List ........................................................................................................ 139

vii

List of Figures

Figure 2.1 Proportional Reasoning Development (adapted from Lamon, 2010) .................. 17

Figure 2.2 Additive and Multiplicative Problems (adapted from Lamon, 2010) ................... 21

Figure 3.1. Initial Template for Research Question 1 (adapted from ACARA, 2016) ............. 60

Figure 3.2. Template 2 (version 2) for Research Question 1 (adapted from ACARA,

2016) ....................................................................................................................... 61

Figure 3.3. Third template (version 3) for Research Question 1 (adapted from Jacob &

Willis, 2004) ............................................................................................................ 61

Figure 3.4. Final Template (version 4) for Research Question 1 (adapted from Jacob &

Willis, 2004) ............................................................................................................ 62

Figure 3.5. First template (version 1) for research question 2 (adapted from Anthony

& Walshaw, 2009) .................................................................................................. 64

Figure 3.6. Second template (version 2) for research question 2 (adapted from

Anthony & Walshaw, 2009) .................................................................................... 65

Figure 3.7. Third template (version 3) for research question 2 (adapted from Anthony

& Walshaw, 2009 and Anghileri, 2006) .................................................................. 68

Figure 3.8. Final template (version 4) for research question 2 (adapted from Anghileri,

2006) ....................................................................................................................... 70

Figure 4.1. Template for Research Question 1 ...................................................................... 78

Figure 4.2. Final template ...................................................................................................... 96

Figure 5.1. Developmental pathway based on Jacob & Willis, 2003 ................................... 119

viii

List of Tables

Table 2.1. Synthesis of research in the development of multiplicative thinking ................... 22

Table 2.2. Common foundational concepts in each stage of development to

multiplicative thinking ............................................................................................ 24

Table 2.3. Elements of practice and principles ...................................................................... 30

Table 3.1.Data sources .......................................................................................................... 52

Table 4.1. Sources for data collection .................................................................................... 76

Table 4.2. Explanation of the labelling system for data references, e.g. T3:PL:56 ................ 77

Table 4.3. Foundational concepts of additive and multiplicative thinking (ACARA,

2016) ....................................................................................................................... 81

Table 4.4. Additive and multiplicative thinking evolves from foundational concepts ........... 83

Table 4.5. Teachers identification of foundational concepts in concept map and

individual lesson topic............................................................................................. 84

Table 4.6. Applications of proportional reasoning ................................................................. 98

ix

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature: QUT Verified Signature

Date: 1 May 2018

x

Acknowledgements

I would like to acknowledge Dr Sonia White and Professor Joanne Lunn for their

constant support, commitment and guidance throughout the research process. I have

appreciated their understanding and encouragement. Thank you to the participants for

their generosity of time and commitment. Also to my two loved ones who left my side

during this process, both valued education and encouraged me along the way. I am

grateful to my family who have given me unwavering support, encouragement and

love throughout this process of professional growth.

1

Chapter 1: Introduction

Early childhood mathematics sets an important foundation for more complex and

abstract mathematics that develops throughout schooling. Proportional reasoning is

one such example of multifaceted mathematics, as it is dependent on thinking

processes and types of understanding. Proportional reasoning is understanding

situations of multiplicative comparison and “refers to a certain facility with rational

number concepts and contexts” (Lamon, 2010, p. 3). Whilst understanding develops

in mathematics, it is also used in other subjects, professions and everyday life (Boyer,

Levine & Huttenlocher, 2008). Examples of proportional reasoning in daily life

include shopping (e.g. estimating the better purchase), cooking (e.g. changing a

recipe), interpreting scales and maps and things like gambling, which involves risk-

taking estimations associated with chance (Dole, 2010). Proportional reasoning

describes a “sophisticated mathematical way of thinking” and it’s considered a big

idea as its development is dependent upon a web of conceptual ideas and strategies

(Lamon, 2010, p. 8). A big idea connects mathematical concepts into a “coherent

whole” (Charles, 2005, p. 10). As a big idea, proportional reasoning connects many

mathematical concepts, for example, ratio, rational numbers or fractions, and

measurement (e.g. area, volume) (Siemon, Bleckley & Neal, 2012b).

Proportional reasoning typically appears in middle school mathematics and

science curricula, such as density, speed, acceleration, and force. The mathematical

concepts that set the foundation for proportional reasoning are established in the early

years of schooling, however they are commonly misunderstood by teachers (Hilton,

Hilton, Dole & Goos, 2016). While the term, early years can encompass birth to eight

years of age, in this thesis there will be specific focus on the school years Prep – Year

3. Therefore the term, early years will be used to throughout this thesis and refers to

the first four years of Primary School - Preparatory (Prep) to Year 3 - and will be

referred to as P-3 throughout this thesis, with Preparatory referred to as Prep hereafter.

This research project was designed to investigate what teachers in the case study P-3

2

classrooms recognised as the foundational concepts of proportional reasoning and

aimed to explore how these concepts are promoted in the early years.

This chapter has five sections. The first section provides the Australian and

international context for mathematics education (see Section 1.1). The second section

addresses the importance of early childhood mathematics (important for the present

study, as it was conducted with teachers of Prep to Year 3) (see Section 1.2). The third

section outlines a discussion of the big ideas in mathematics, including a broad

description of proportional reasoning (see Section 1.3). The fourth section is a chapter

summary (see Section 1.4). The chapter concludes with an overview of the thesis (see

Section 1.5).

1.1 CONTEXT FOR MATHEMATICS EDUCATION

As a nation, Australia is committed to the development of numerate students as

evidenced by The Melbourne Declaration on Educational Goals for Young

Australians developed by the Ministerial Council of Education, Employment, Training

and Youth Affairs (MCEETYA) in 2008 and set for a ten-year period until 2018.

According to MCEETYA (2008), “improving educational outcomes for all young

Australians is central to the nation’s social and economic prosperity and will position

young people to live fulfilling, productive, and responsible lives.” This declaration

aims “for all young Australians to become successful learners, confident and creative

individuals, and active informed citizens” (MCEETYA, 2008, p.6). Central to this aim,

is the belief that success in all learning areas is based on learners having “the essential

skills in literacy and numeracy” (MCEETYA, 2008, p. 8).

Proportional reasoning is regarded as an essential element of numeracy (Hilton

& Hilton, 2016). Numeracy is a broad concept which is described as “the ability to

access, use, interpret and communicate mathematical information and ideas, to engage

in and manage the mathematical demands of a range of situations in adult life”

(Organisation for Economic, Co-operation and Development [OECD], 2012, p. 36).

Numeracy is identified in the Australian Curriculum as one of the seven general

capabilities (literacy, numeracy, information and communication technology (ICT)

capability, critical and creative thinking, personal and social capability, ethical

behaviour, and intercultural understanding) (Australian Curriculum Assessment and

Reporting Authority ([ACARA], 2016), which permeate all curriculum documents to

serve as a reminder to teachers of the interrelationships of content and a child’s

3

development of each of the capabilities. In the Australian Curriculum, numeracy is

described as encompassing “the knowledge, skills, behaviours and dispositions that

students need to use mathematics in a range of situations” ([ACARA], 2016, p.4). A

numerate person is described as someone who has a strong mathematical knowledge

combined with “tools and representations to solve problems in multiple contexts”

(Hilton & Hilton, 2016, p. 32). Therefore, proportional reasoning supports a person to

become numerate, as it is based in mathematical knowledge and assists in the

understanding and ability to solve problems.

The Education Council (2015) recently released a National STEM (Science,

Technology, Engineering and Mathematics) School Education Strategy (2016-2026)

and identified the early years as the starting point for building the foundation. This

national focus on STEM education is intended to provide the momentum to improve

student performance in these areas of learning and for students to have a stronger

foundation in STEM. The strategy acknowledges mathematical thinking as a

fundamental skill and supports the development of students’ problem solving and

reasoning skills (Education Council, 2015). This focus on STEM teaching includes

supporting teachers in their capacity to teach STEM, with a priority on lifting teacher

standards, as it is believed that “some primary school teachers lack confidence in

teaching science and maths, particularly where they have limited expertise in these

content areas” (Education Council, 2015, p. 8). This current study addresses the

teaching of one of the STEM subjects, mathematics, by exploring how, teachers in the

case study school, promote proportional reasoning in their classrooms.

One reason for the government’s commitment to supporting teachers in the

teaching of STEM may have evolved from the macro-level performance of Australian

students in the international testing arena. Trends in International Mathematics and

Science Study (TIMSS) is one such forum that is relevant to primary school teaching

and learning. This international test measures students’ achievements in maths and

science in Year 4 and Year 8. Relevant to this study, in the 2015 TIMSS results the

Year 4 maths score for Australian students “is significantly higher than the

corresponding score in the first test in 1995” (Thomson, Wernert, O’Grady &

Rodrigues, 2016, p.6). However, following a significant increase in 2007, Australia’s

results have stagnated for the past three cycles, including the assessment in 2015, and

declined against international benchmarks (Thomson et al., 2016). High performing

4

Asian countries have made steady improvements in their results, while other countries,

including the USA and some European countries, have overtaken - and now

outperform - Australia.

Under the TIMSS testing regime, the results are measured at four levels:

advanced, high, intermediate, and low international benchmark. “Seventy percent of

Year 4 Australian students achieved the intermediate international benchmark in the

2015 results” (Thomson et al., 2016, p.6). The highest possible benchmark (advanced)

was achieved by 9% of Year 4 Australian students, compared to 50% in Singapore and

27% in Northern Ireland (Thomson et al., 2016). To achieve the advanced

international benchmark, students demonstrated their ability to “apply their knowledge

and understanding in a variety of relatively complex tasks and explain their reasoning”

(Thomson, et. al., 2016, p. 12). This knowledge included understanding of whole

number, fractions, decimals, simple equations and relationships; concepts that are

interconnected with proportional reasoning.

Students are tested in content and cognitive domains. The content domains

include number, geometric shapes and measures, data, and display. Year 4 Australian

students’ results were weaker in number, which comprised 50% of the test. The latter

domains (geometric shapes and measures, data, and display) were significantly higher.

The cognitive domain consisted of knowing, applying, and reasoning. ‘Knowing’

covers facts and procedures, ‘applying’ is the ability to apply conceptual

understanding, and ‘reasoning’ involves complex contexts and multi-faceted problems

(Thomson et al., 2016). The Year 4 Australian students achieved significantly higher

results in ‘applying’ and ‘reasoning’, but weaker in ‘knowing’ (Thomson, Wernert,

O’Grady & Rodrigues, 2017, p.17). Interestingly it could be inferred that the Year 4

students may have greater conceptual understanding than procedural fluency;

however, a “blend of conceptual understanding and procedural fluency is required” to

develop a deep understanding of mathematics (Hurst & Hurrell, 2016, p. 35).

The Australian Year 4 student results in TIMSS have implications for both

policy makers and this study. With a view to learning from these international

outcomes, Australian policy makers have looked at the curricula in high performing

countries, to compare them with the Australian curriculum. In 2015, the Australian

state and federal education ministers endorsed a revised Australian Curriculum. The

support materials for the revision, Review of Australian Curriculum Supplementary

5

Material (Stephens, 2014) made comparisons between the USA, Japanese and the

Australian mathematics curriculum. These countries were selected for comparison as

they rated similarly to Australia in 2011 TIMSS scores for Year 4. The review

recommended that content elaborations of key mathematical ideas should be included

in the curriculum so teachers know when to introduce them as a “foundations for future

learning”, and when they should be addressed with more rigour (Stephens, 2014, p.

53). Key mathematical ideas are indicative of the concept of big ideas, which involves

interconnected mathematical concepts contributing to an overarching mathematics

concept (i.e., a big idea). The review recommended integrating number and algebra

in the early years as a foundation for the “key ideas of equivalence, functional

relationships, ratios and variables in the upper primary years” (Stephens, 2014, p.55).

This recommendation, to establish number and algebra in the early years as foundation

for the development of interrelated mathematical concepts, could be interpreted as a

recommendation for the inclusion of proportional reasoning as a big idea. The

mathematics concepts, equivalence, functional relationships, ratios, and variables all

contribute to proportional reasoning. The Review of Australian Curriculum

Supplementary Materials (Stephens, 2014) and, in turn, the development of the current

version of the Australian Curriculum: Mathematics (Version 8.3) ([ACARA], 2016)

saw limited changes to the Australian Curriculum: Mathematics. This international

testing highlights the need for teachers and, in turn, students, to possess a deep

understanding of mathematics. Development of a deep mathematical understanding

begins in the early years and its development is reliant upon quality programs that

acknowledge the importance of early years mathematics education.

1.2 IMPORTANCE OF EARLY YEARS MATHEMATICS

Children need the knowledge and skills to fully participate in the 21st century

life and this is acknowledged by governments worldwide (Australian Association of

Mathematics Teachers [AAMT], 2014). Policy-makers recognise that quality early

childhood education and care creates the foundations for lifelong learning. For

example, the Organisation for Economic Co-operation and Development (OECD)

early this century reviewed early childhood education in its report Starting Strong,

acknowledging that “childhood (is) an investment with the future adult in mind,”

([OECD], 2001 p. 38). Australia has responded to these calls by prioritising the

educational programs prior to school and in the first year of primary schooling. The

6

development of the Early Years Learning Framework (Department of Education,

Employment and Workplace Relations [DEEWR], 2009), the National Quality

Standard for early childhood provision (Australian Children’s Education and Care

Quality Authority [ACECQA], 2011) for childcare and kindergarten years and the

inclusion of the Foundation Year (preparatory and preschool) in the Australian

Curriculum. The development of the Australian Curriculum, not only prioritised a

national curriculum, but also acknowledged the importance of the year prior to Year 1

by creating a curriculum for this year, identified as the Foundation Year. The

development of these documents (Early Years Learning Framework, National Quality

Standard) and the inclusion of the Foundation year in the Australian Curriculum are

each testament to the nation’s commitment to early childhood education and

acknowledgement of the importance of providing a quality education for children in

these age groups (prior to school and first year of school). These documents

acknowledge the need to provide quality mathematics experiences children in these

early years (Clements & Sarama, 2016).

An ever-growing body of knowledge, related to young children’s capabilities,

has impacted on our understanding about the education of this age group and young

children’s mathematical proficiencies. Previous thinking included the view that young

children have little or no knowledge of mathematics. In the sixties, the work of Piaget

recognised that children were mathematically curious and able to actively construct

mathematical knowledge, but young children were viewed as “incapable of abstract

and logical thinking until concrete-operational stage around age seven” (Hachey, 2013,

p. 420). A growing body of research has moved away from the belief that young

children, because of their stage of development, have limited capacity to learn

mathematics (Askew, 2016). Recent research has found that children can show

mathematical competencies “that are either innate or develop in the first and early

years of life” (Clements & Sarama, 2016, p. 77). Moreover, explicit quantitative and

numerical knowledge in the years before formal schooling has been found to be a

stronger predictor of later mathematics achievement and school success than tests of

intelligence or memory abilities (Claessens, Duncan & Engel, 2009; Duncan et al.,

2007; Krajewski & Schneider, 2009). There is some discrepancy in the research about

when children have the capability to reason however, there is agreeance that children

in these early years are capable learners and they need opportunities to develop their

mathematical proficiencies (Boyer et al., 2008).

7

The capabilities of students in the early years and the provision of learning

opportunities are relevant to this study, as it investigates mathematical concepts that

will contribute to sophisticated thinking. Early years teachers need to ensure all

students have access to rich and challenging mathematical experiences. Research has

found that the skills that normally develop during adolescence can be developed, with

purposeful teaching, in younger students in school settings (Hilton & Hilton, 2013).

For example, young children have the capacity to share equally and have the capacity

to solve simple problems that are proportional in nature (Fielding-Wells, Dole &

Makar, 2014). Therefore, teachers need to offer mathematically stimulating and

focused activities, “based on research-informed knowledge of children’s mathematical

development” (Bruce, Flynn & Bennett, 2016 p. 543; Siemon et al., 2012b).

This study investigated the learning experiences that can contribute to P-3

students’ understanding and development of the foundational concepts of proportional

reasoning. To support P-3 students in this understanding, teachers need to have

knowledge of proportional reasoning: that it is a big idea with many foundational

concepts that contribute to its development. Therefore, this study also investigated the

foundational concepts of proportional reasoning.

1.3 BIG IDEAS OF MATHEMATICS

A big idea links numerous and interrelated mathematical understanding into a

whole, which is central to the learning of mathematics (Charles, 2005). It has been a

long-held view of the seminal work of Vergnaud (1983) that it is important for

educators to understand the significance and range of big ideas, so interconnected

mathematical concepts are acquired with understanding rather than in isolation of each

other (Vergnaud, 1983). The notion of big ideas in mathematics education was

identified as a way to classify mathematical information differently from traditionally

represented content areas or strands (Steen, 1990). The concept of a big idea moves

the focus from viewing the content in isolation to seeing its relationship to a larger

mathematical idea. Steen (1990) identified six broad categories of big ideas: pattern,

dimension, quantity, uncertainty, shape, and change. In contrast, Charles (2005)

offered 21 big ideas which he believed were connected in understandings and permeate

across year levels. Eight of the 21 big ideas focus on chance and data, space and

measurement. The remaining 13 big ideas relate to number development and include:

numbers, numeration, equivalence, comparison, operation meanings and relationships,

8

properties, algorithms, estimation, patterns, variable, proportionality, relations and

functions, equations and inequalities. Concepts embedded in each of these 13 number

big ideas can contribute to proportional reasoning, but the most relevant is

proportionality, as “proportional reasoning is a prerequisite for understanding contexts

and application based on proportionality” (Lamon, 2010, p. 3).

Two key aspects of big ideas of mathematics, in the Australian context, are

identified. First, the Australian Association of Mathematics Teachers (AAMT) (2009)

built on the work of Steen (1990) and developed the Discussion Paper on School

Mathematics for the 21st Century. This discussion paper identified dimension,

symmetry, transformation, algorithm, patterns, equivalence, and representation as

relevant big ideas. Second, Siemon et al., (2012b) identified big ideas, but unlike the

AAMT, identified big ideas that focussed only on number, on the basis that teachers

find number a difficult aspect of teaching and learning in mathematics. These big ideas

included trusting the count, place-value, multiplicative thinking, partitioning,

proportional reasoning, and generalising. These number concepts were selected as

big ideas as they are regarded as the most important components of number

development “without which students’ progress in mathematics would be severely

restricted” (Siemon et al., 2012b, p. 24).

Siemon et al., (2012b) identified Year 8 as the time students should think and

work proportionally. Preceding its attainment, three big ideas are expected to be

attained which is indicative of a hierarchy of big ideas of number leading to

proportional reasoning: trusting the count (Foundation Year), place value (Year 2) and

multiplicative thinking (Year 4). As Siemon et al., (2012b) suggests, these big ideas

are developed in the early years and therefore the teachers of these year levels are in

the prime position to lay the foundation for the development of proportional reasoning.

The three aforementioned big ideas are relevant to this study because they expected to

be achieved in the early years of primary school (by Year 4), which is the focus of this

study.

While the discussion of big ideas has occurred internationally in the wider

educational research context, it has not yet become part of teaching or lesson planning

in mainstream classrooms (Charles, 2005). Clarke, Clarke and Sullivan (2012)

surveyed Australian teachers about their understanding of mathematical ideas or big

ideas. As part of the study, teachers were requested to identify the important

9

mathematical ideas (big ideas) that would contribute to the teacher’s next teaching

topic. The survey results indicated that teachers lacked an understanding of the

connection of mathematical concepts and how they contribute to bigger mathematical

ideas. The teachers’ lack of knowledge about big ideas in Australia may reflect the

fact that they are not evident in the Australian Curriculum: Mathematics. This will be

discussed in detail in the literature review in Section 2.1. Proportional reasoning is

considered a big idea of mathematics and its relevance and importance will be

discussed below.

1.3.1 Proportional Reasoning

Proportional reasoning has been acknowledged as a big idea that is fundamental

to mathematics knowledge and, therefore, it is important that it is promoted by

teachers. It is vital that the foundational concepts are developed so students can access

the understanding and thinking processes attached to proportional reasoning. In

comparison to other mathematics concepts, the way in which proportional reasoning

is developed has only recently been identified (Lamon, 2010). This may be why there

is no universal definition of proportional reasoning or agreement as to the foundational

concepts of proportional reasoning (Lamon, 2010). Proportional reasoning is allusive

and is best defined using an example of a problem:

Mr Short is 6 paperclips tall. When you measure his height in buttons, he is 4

buttons tall. Mr Short has a friend named Mr Tall. When you measure Mr Tall’s

height in buttons, he is 6 buttons tall. What would be Mr Tall’s height if you

measured in paperclips? (Lamon, 2010, p. 14).

This example highlights the complexity of a proportional reasoning problem as a

solution to the problem is not in a rule or symbol, but rather requires sophisticated

thinking. Research consistently highlights that students experience difficulties with

proportion and proportion-related tasks and applications. (e.g. Ben-Chaim, Fey,

Fitzgerald, Benedetto, & Miller, 1998; Lo & Watanabe, 1997). Therefore, students

need to be supported by teachers who can analyse student responses to problem

situations to make informed decisions for further instruction (Lamon, 2010).

Proportional reasoning is important in academic and everyday living and is

“considered so central to mathematics thinking” (Boyer et al., 2008, p. 1) that a

directive was given by the body of mathematics teachers in America that every effort

“must be expended to assure its careful development” (Boyer et al., 2008 p.1).

10

Notwithstanding the importance of proportional reasoning in mathematics and daily

life, Lamon (2010) estimates over 90% of adults do not reason proportionally, which

suggests that the understanding of proportional reasoning - and how to teach it - are

significant to educators (Hilton, et al., 2016). Thus, it is imperative that early years’

teachers have a sound understanding of proportional reasoning to teach young students

competently and establish a strong foundation for their future learning. While research

has explored middle school teachers’ understanding of proportional reasoning (Dole,

2010), little is known about teaching and teachers’ understanding of the foundational

concepts in the early years of primary school just as little is known about students’

knowledge of proportional reasoning (Lobato, Orrill, Druken & Jacobson, 2011).

The setting for this study is a Primary School within a P-12 school and I hold a

leadership position within this school. As Head of Primary School, I oversee all

aspects of curriculum and pastoral care development for Prep to Year 6 students and

staff. A culture of professional development and growth is encouraged within the

school by offering to the teaching staff, external and school based professional

development and annual invitations to be involved in action research projects. My role

in curriculum leadership includes responding to the needs of the staff and developing

wider school initiatives.

The genesis of this study into proportional reasoning emerged from a previous

action research project that found students had difficulty in developing automaticity of

multiplication facts. The same action research project also indicated that this may be

associated with teacher understanding of the development of number concepts and its

impact on student development of automaticity of multiplication facts. Additionally,

Year 3 teachers identified that their students experienced difficulty with the problem-

solving aspects of NAPLAN testing, relative to other aspects of the tests. These two

areas were the catalyst to investigate and develop teacher knowledge of number

development and problem solving to help students improve their understanding of

number concepts and problem solving.

As a big idea, I saw proportional reasoning as providing a context that involved

both number concepts and problem solving. The foundation concepts of proportional

reasoning were selected because of their relevance to P-3 classrooms. Given my

curriculum leadership role, I saw working with P-3 classroom teachers as the best

starting point to address the issue of student performance on important aspects of

11

mathematics. Working with the P-3 teachers I wanted to understand what they

identified as the most important foundation concepts of proportional reasoning, and

how they promoted those concepts in their respective early years (P-3) classrooms.

Considering my professional rationale for this study, my school setting as the

context and working with the teachers, I chose a teacher research approach to this

study. The teacher research approach and term aligns with the work of Cochran-Smith

and Lytle (1999). Using this approach allowed for reflective dialogue amongst the

teachers in the study, it provided a link between research and the classroom and

employment of systematic data collection methods (Campbell, 2013; Cakmakei,

2009;). This is further discussed in Section 3.2.

1.4 CHAPTER SUMMARY

The early years of schooling (P-3) are intended to create the foundations for

future mathematics learning. Proportional reasoning is a big idea that permeates

mathematics, other learning areas, and daily living. Early childhood teachers can lay

the foundation for the development of proportional reasoning to support students’

future understanding and success in understanding mathematics. Early childhood

teachers teach many mathematical concepts that provide the foundation for

proportional reasoning. However, the research indicates that the teaching of the

foundational concepts in P-3 is not always explicitly connected with the development

of proportional reasoning as teachers’ understanding of proportional reasoning is

limited (Dole, 2010). This may be because proportional reasoning is a big idea of

mathematics that is intertwined with many mathematical “concepts, operations,

contexts, representations and ways of thinking” (Lamon, 2010, p. 9).

The aim of this study was to understand and identify the foundational concepts

and teaching practices associated with promoting proportional reasoning in the early

years of primary school. This aim led to two research questions:

1. What do Prep to Year 3 teachers recognise as the foundational concepts of

proportional reasoning?

2. How do Prep to Year 3 teachers promote the foundational concepts of

proportional reasoning?

The research was conducted as a case study bounded by one school, (including

five teachers from Prep, Year 1, Year 2, and Year 3 and the researcher). The approach

12

was informed by a teacher research approach, which provides the opportunity for the

researcher to understand and improve teaching practice alongside other teachers

involved in the study. This study employed a qualitative methodology to explore

teachers’ understandings of the foundational concepts of proportional reasoning and

how these were promoted in the classroom. A variety of research methods, in keeping

with case study methodology, were used to explore teacher understandings. These

included focus group interviews, group planning discussions, and individual

interviews. For each research question, a template analysis technique (as per King,

2012) was used to analyse the data.

1.5 OVERVIEW OF THE THESIS

This thesis is organised into five chapters. Chapter 1 has provided an orientation

to the broad context of mathematics teaching in the national and international arena.

It also highlighted proportional reasoning as one of the big ideas of mathematics and

outlined its relevance. Chapter 1 has established that early childhood education is an

important area of schooling and the year levels that the foundations are established.

Embedded in this foundation are concepts that contribute to big ideas of mathematics,

particularly, proportional reasoning. Chapter 2 reviews relevant literature. While

proportional reasoning is identified as a big idea of mathematics, the foundational

concepts associated with additive and multiplicative thinking emerge from the

literature as the foundation for the development of proportional reasoning in the early

childhood years. Therefore, the associated concepts of these two types of thinking are

addressed in terms of their development. The way in which they are promoted in the

early childhood years is explored.

Chapter 3 describes and justifies the selection of case study as an appropriate

research approach for this study. This chapter provides detail of how the research was

managed including the research design from a case study perspective; participants and

the role of the researcher informed by the teacher research approach; and data

generation and analysis using a template analysis technique.

The results are presented in Chapter 4, structured around the two research

questions. In Chapter 5, a discussion of the results is presented in relation to the

identification of foundation concepts, ways to promote proportional reasoning and the

implications of these two areas.

13

14

Chapter 2: Literature Review

Mathematics taught in the early years is critical to a child’s mathematical

development as it lays the foundation for more complicated and abstract mathematics.

Proportional reasoning is a complex big idea of mathematics, explicitly taught in

middle secondary school; however, the fundamental principles and a strong foundation

are established through instruction in the early years of primary school (Lamon, 2010).

The purpose of this chapter is to critically review the literature in relation to the

developmental concepts of proportional reasoning and identify ways teachers can

promote an understanding of proportional reasoning in the classroom.

Proportional reasoning is regarded as a big idea in mathematics as it comprises

interconnected mathematical concepts. The first section of this chapter focuses on this

aspect of proportional reasoning (see Section 2.1). This section will also review the

research in terms of proportional reasoning as a big idea. The different types of

thinking (additive and multiplicative) proportional reasoning is based on the

foundational concepts that contribute to the development of proportional reasoning.

The second section will review the literature to identify ways teachers can promote

and build student understanding of the foundational concepts of proportional reasoning

(see Section 2.2). This section will also review teacher knowledge, classroom

community, discourse in the classroom and mathematical tasks. The final section of

this chapter contains the chapter summary (see Section 2.3).

2.1 PROPORTIONAL REASONING IS A BIG IDEA OF MATHEMATICS

Proportional reasoning is a big idea. It signifies more complex and higher

mathematics, as it is “foundational to the development of algebraic reasoning”

(Langrall & Swafford, 2000, p. 6). It is regarded as one of the most important skills

to be developed in middle schooling and is the key to understanding and functioning

in many content strands of the Australian Mathematics: Curriculum in the year levels

four to nine. Proportional reasoning development consolidates knowledge and

mathematical understandings established in primary school (Lamon, 2010; Langrall &

Swafford, 2000).

15

Although big ideas in mathematics have been discussed in the literature for some

time, there has been little evidence to suggest that such concepts have been applied in

the classroom (Siemon, 2006; Steen, 1990; Vergnaud, 1983). As Charles (2005)

highlighted that big ideas and curriculum have not reached a juncture as it, “had not

become part of the mainstream conversations about mathematics standards,

curriculum, teaching, learning and assessment” (p. 9), is consistent with discussions in

the Australian educational context. The absence of mathematical discussion may be

due to the divergence within the literature regarding big idea topics and how they can

best be represented in the teaching and learning of mathematics (Clarke et al., 2012;

McGaw, 2004; Siemon et al., 2012). To address this issue, Siemon (2013) defines big

ideas as needing “to be both mathematically important and pedagogically appropriate

to serve as underlying structures on which further mathematical understanding and

confidence can be built” (p. 40). This suggests that big ideas are significant in their

own right, but also provide a foundation for conceptual understanding of mathematics.

A knowledge of, and focus on, big ideas deepens teacher understanding and, in turn,

develops student understanding of the maths concept under exploration (Charles,

2005).

Neither the concept of big idea nor big ideas are expressly stated in the

Australian Curriculum: Mathematics ([ACARA], 2016). However, the document

describes mathematics as “interrelated and interdependent concepts and systems,”

which are applied beyond the classroom (p. 4). Four proficiency strands are identified:

fluency, understanding, problem solving, and reasoning. Proficiency strands provide

the broad context for how the content should be explored through the relationship of

procedural fluency and conceptual understanding and their contribution to

mathematical problem solving and mathematical reasoning. These strands are how

students “think and act mathematically” (Stephens, 2014, p. 1) and are suggestive of

an alignment between big ideas and the mathematics curriculum. Therefore, if content

descriptions, are taught in conjunction with these proficiency strands, as they are

intended, student understanding of the big idea will be strengthened (Siemon, 2013).

Further alignment of big ideas and Australian Curriculum: Mathematics could

be inferred with this document’s numeracy learning continuum. This continuum

identifies six broad elements; within each element, sub-elements are offered and

student expectations for each year level are identified. Each of the six elements could

16

be regarded as a big idea and viewed with an “interpretative lenses through which skill-

based content descriptions can be examined in more depth” (Siemon, 2013, p.40). The

most relevant element to this study is “using fractions, decimals, percentages, ratios

and rates element” with two sub-elements identified as “interpret proportional

reasoning” and “apply proportional reasoning” ([ACARA], 2015b, p.1). The number

continuum characterises concepts in an integrated fashion, rather than discrete isolated

concepts as they are presented in the Australian Curriculum: Mathematics.

Proportional reasoning is explicitly mentioned in the Australian Curriculum:

Mathematics ([ACARA], 2016) for Year 9. Prior to Year 9, Australian Curriculum:

Mathematics ([ACARA], 2016) does not reference proportional reasoning nor are the

foundational concepts of proportional reasoning identified in the content descriptions

in the preceding year levels. In an earlier iteration (version 7.5) of the Australian

Curriculum: Mathematics ([ACARA], 2015a), proportional reasoning and related

concepts were considered significant enough to be identified in summary statements

of key content for year level groupings. In the Foundation to Year 2 (F-2) statement

([ACARA], 2015a, p. 7), the key message was the identification of F-2 students

accessing mathematical concepts that provide “a foundation for algebraic, statistical

and multiplicative thinking that will develop in later years.” The Years 3-6 statement

([ACARA], 2015a, p. 7) highlighted the need for students to develop conceptual

understanding of fractions, decimals, and place value “to develop proportional

reasoning.” These two statements highlight the importance of laying the foundations

for proportional reasoning, but they do not extend to identifying the mathematical

content that contributes to proportional reasoning development. In the most recent

version (8.3) of the Australian Curriculum: Mathematics ([ACARA], 2016), these

summary statements were not included in the document.

Proportional reasoning is complex, as it is a manifestation of foundational

mathematics and the associated developmentally appropriate experiences (Langrall &

Swafford, 2000). There is no definitive list of foundational concepts of higher level

math however, multiplicative thinking, partitioning, extended multiplication, division,

rate, ratio, and percent are identified by Siemon et al., (2012) as concepts

interconnected with proportional reasoning. Hilton et al., (2016) identified fractions,

decimals, multiplication, division and scale as the foundational concepts of

proportional reasoning developed in the middle years of schooling. Lamon (2010) in

17

her seminal work, describes proportional reasoning as, “concepts, operations, contexts

representations and ways of thinking” which are reflected in seven mathematical areas

(p.9). She represents these seven mathematical areas or topics as an interconnected

web (see Figure 2.1), rather than being linear in nature, because as a student develops

one topic, other topics are affected.

Relative thinking, in Figure 2.1, also called multiplicative thinking, is the most

relevant topic to the current study as multiplicative thinking is a direct stepping stone

to proportional reasoning (Hilton et al., 2016). Multiplicative thinking is an advanced

level of thinking that involves multiplication and division (Siemon, 2013). It is a

cognitive function which describes the ability to analyse changes of quantities in

relative terms. That is, it alters the original amount by a quantity relative to that

amount (for example, 20% of an amount) (Langrall & Swafford, 2000; Pitta-Pantazi

& Christou, 2011). Multiplicative (relative) thinking involves recognising, and

working with, relationships between quantities that are relative to each other. It is a

more powerful way of thinking and more complex than additive thinking.

Multiplicative thinking builds on additive thinking, or absolute thinking, as illustrated

in Figure 2.1 (Lamon, 2010). In its most basic definition, additive (absolute) thinking

involves addition and subtraction of number and builds on the big ideas, trust the count

and place value as is evident from Figure 2.1. Additive thinking is not considered a

big idea, but builds upon these two big ideas (trust the count and place value) (Siemon

et al., 2012). Multiplicative thinking, additive thinking, trust the count, and place value

will be reviewed in Section 2.1.2.

Figure 2.1 Proportional Reasoning Development (adapted from Lamon, 2010)

18

The development of proportional reasoning has been shown to emerge from

age eight to ten years through to the middle years of schooling (eleven to fourteen

years of age) (Kastberg, D’Ambrosio & Lynch-Davis, 2012). Students in the early

years have “strong intuitive notions which can provide a foundation from which to

develop their understanding” of proportional reasoning (Hilton & Hilton, 2016, p.

38). Even before the formal school years, there is evidence of students’ capacities to

think proportionally. For example, in the kindergarten years (prior to school), when

children participate in sand and water play, they investigate how many cups are used

for one situation compared to another situation and demonstrate the use of relative

thinking (Fielding-Wells, et al., 2014). Therefore, in developing proportional

reasoning, students are on a learning trajectory from qualitative reasoning, additive

and, in turn, multiplicative thinking (Steinthorsdottir, 2005). It is vital for teachers to

know about this learning trajectory and the factors that support its development so

that students will be able to reason proportionally in the future. Thus, in the context

of this study, early primary school teachers’ knowledge of the components of

proportional reasoning, the learning trajectory, and how teachers support this

learning are of interest. In the context of this study, P-3 teachers’ knowledge of these

components of proportional reasoning will be explored.

The interconnected topics, identified by Lamon (2010) and represented in Figure

2.1 are formally introduced from Year 4 onwards, but students have capacity and

capability to experience some of these topics (rational number interpretations,

unitizing, covariation, and relative thinking) in the P-3 years and this is acknowledged

in the Mathematics and Science curriculum documents. Rational number

interpretations (Figure 2.1) are considered fractions, decimals and partitioning and

these concepts are introduced from Year 1 the Australian Curriculum: Mathematics

([ACARA], 2016). Unitizing is the basis for multiplication and the place value system

(e.g. ten units as one ten). Multiplication is introduced in Year 2 and place value is

developed from Year 1 by partitioning numbers using place value. In the science

curriculum, students from P-3 are offered learning experiences that draw on

covariation (e.g. lower temperature or more/thicker clothing), scale and relative

thinking for timelines (Hilton & Hilton, 2016). For these curriculum experiences to

be most effective in enhancing and contributing to children’s understandings of

proportional reasoning, teachers need to be cognisant of these critical and complex

19

components in the development of proportional reasoning (Pitta-Pantazi & Christou,

2011).

The Australian Middle Years Numeracy Research Project (Siemon, Virgona &

Corneille, 2001) attributes mathematical performance differences as “almost entirely

due to difficulties with large whole numbers, decimals, fractions, multiplication and

division, and proportional reasoning, collectively recognised as multiplicative

thinking” (Siemon et al., 2012b, p. 23). This statement suggests that proportional

reasoning is multiplicative thinking (Siemon et al., 2012b). It also indicates that,

together, those mathematical terms are multiplicative thinking, and that a lack of

understanding of some or all the concepts will have an impact on a student’s

mathematical performance, therefore it is vital for teachers to also have this

understanding to support the students, this study contributes to this understanding. The

thinking that is involved in proportional reasoning and multiplicative thinking is so

closely aligned that in some literature the terms are interchanged (Bright, Joyne &

Wallis, 2003; Van Dooren, De Bock & Verschaffel, 2010). In this study,

multiplicative thinking is viewed as the foundation on which proportional reasoning is

developed (Siemon, 2006). The development of multiplicative thinking is based on a

foundation of additive thinking, which is developed from a strong sense of number in

regard to trust the count and place value (Siemon, et al., 2013).

2.1.1 Additive and Multiplicative Thinking

Additive thinking occurs when an amount (the original amount) is changed by

an absolute or fixed amount (Langrall & Swafford, 2000). Students in the P-3 years

are more likely to use additive thinking because of the cognitive capabilities of

students in this age group (Kastberg et al., 2012; Lamon, 2010). For example, students

are given a graph of the amount of different types of fruit eaten by children. Questions

might include,

How many different berries were eaten? How many more green fruits (12) than

red fruits (8) were eaten?

Additive thinking is the understanding of number that is the connection between part-

part-whole and place value ideas with counting that takes the addition and subtraction

beyond an algorithm to a higher level of thinking (Siemon et al., 2013). These

concepts will be further explained in Section 2.1.2 (foundation concepts of

proportional reasoning). Students’ ability to think multiplicatively relies on students

20

having a competent understanding of additive part-whole strategies (Young-

Loveridge, 2005).

Multiplicative thinking requires a higher level of thinking about number and a

solid understanding of multiplication and division (Ell, Irwin & McNaughton, 2004;

Jacob & Willis, 2001). The required understanding of number is developed through

additive thinking, as additive thinking builds on trust the count and place value (Figure

2.1). In the learning trajectory of proportional reasoning, multiplicative thinking builds

upon and moves beyond counting and absolute or additive thinking (Lamon, 2010).

In the development of proportional reasoning, children move from additive to

multiplicative thinking, but this does not happen automatically (Hurst, 2015; Van

Dooren et al., 2010). Teachers support children in their development of multiplicative

thinking by supporting two key areas (Jacob & Willis, 2001); first, the development of

additive and multiplicative thinking as separate entities and second, supporting the

transition between additive and multiplicative thinking. The development of both

additive and multiplicative thinking emerges from a well-developed sense of number,

a deep understanding of operations (multiplication and division) and the ability to

apply thinking to generate solutions to problem situations (Ell et al., 2004; Siemon,

2013). These two key areas are important to this study as it will how investigate how

teachers support students in the development of the key areas, additive and

multiplicative thinking and ability to apply thinking.

As part of their literature review, Ell et al. (2004) reported that children begin

with “counting strategies, progress to strategies based on repeated addition and lastly

use of features of multiplication to solve problems” (p. 199). Features of

multiplication include commutative properties; that is, the order does not matter and a

knowledge of known basic facts. Problem situations provide a way for students to

apply their thinking and for teachers to observe and categorise children’s responses in

terms of their understanding and thinking (Ell et al., 2004). The difference between

these two types of thinking is illustrated in the two problems presented in Figure 2.2.

Students move beyond multiplicative thinking to proportional reasoning when they

can differentiate between additive and multiplicative situations (Lamon, 2010). This

will be discussed in Section 2.2.4. The foundational concepts on which additive

thinking is built, and multiplicative thinking, will be discussed in the next section.

21

Figure 2.2 Additive and Multiplicative Problems (adapted from Lamon, 2010)

2.1.2 Foundation Concepts of Proportional Reasoning

The foundational concepts of proportional reasoning are, therefore, based on the

development of additive and multiplicative thinking. To identify these foundational

concepts, a number of different studies were reviewed (Downton, 2010; Jacob &

Willis, 2001, 2003; Siemon et al., 2012; Young-Loveridge, 2005) and will be

discussed in this literature review with key features of each study summarised in Table

2.1.

Jacob and Willis conducted two studies in relation to additive and multiplicative

thinking. In their second study, Jacob and Willis (2003) identified four stages

multiplicative thinking: additive, transitional, beginning to think multiplicatively, and

multiplicative thinking. These form the row labels for Table 2.1. The subsequent

columns in Table 2.1 identify other research and their alternative descriptions

(Downton, 2010; Jacob & Willis, 2003; Siemon et al., 2012; Young-Loveridge, 2005).

The progression described by these four studies has been mapped against the initial

stages framework of Jacob and Willis (2003) (additive, transitional phase, beginning

to think multiplicatively, and multiplicative thinking). The research findings of

Downton (2010), Jacob and Willis (2003), Siemon et al. (2012) and Young-Loveridge

(2005) are reviewed in the subsequent paragraphs.

Tom has 6 snake lollies, Alice has 9 snake lollies. Who has more lollies? (additive or

absolute)

How many more lollies does Alice have? (additive or absolute)

How many times would you have to join up Tom’s lollies to get the same length as

Alice’s joined up snakes? (multiplicative or relative)

22

Table 2.1.

Foundation concepts that align with stages of thinking

Stages Phases (Jacob &

Willis, 2003)

Strategies

(Downton,

2010)

NZ Number

Framework

Big Idea

(Siemon et

al., 2012)

Additive One to one

counting

Transitional Emergent

One-to-one

counting

Additive

Composition

Building up Count from one

with materials

Count from one

using imaging

Advance

counting

Trust the

count

Transitional

phase

Many to one

counters

Doubling and

halving

Early additive

part-whole

Place value

Beginning to

think

multiplicatively

Multiplicative

relations

Multiplication

calculation

Advanced

additive part-

whole

Multiplicative

thinking

Operate on the

operator (fully

multiplicative

thinkers)

“Wholistic”

thinking

Advanced

multiplicative

and

proportional

part-whole

Multiplicative

thinking

Each of these studies (Table 2.1) share common findings; these include the

notion that multiplicative thinking is developmental and there are different stages to

its development. Through a synthesis of literature, Jacob and Willis (2003) identified

five broad phases of development (one to one counting, additive composition, many

to one counters, multiplicative relations, and operate on the operator). Within each of

these phases Jacob and Willis (2003) found number concepts that support the

development of each phase and provide the understandings to move to the next phase

and these will be addressed later. They attributed to each of the phases, the

developmental stages of thinking, additive, and multiplicative. Importantly, they

found in their synthesis that there is a transitional phase between additive and

multiplicative thinkers and that when children first begin to think multiplicatively,

23

their understanding is limited compared to more advanced multiplicative thinking

(Jacob & Willis, 2003). This, therefore, is identified as a stage (beginning to think

multiplicatively), which presents approximately in Year 3. It is the first four stages

which are relevant to this study.

Downton (2010) conducted a study of Year 3 students to investigate the

strategies these students would use across a range of multiplication word problems.

Whilst this study evolved from children completing multiplication problems, there

were similarities with Jacob and Willis (2003) as the strategies matched the stages of

development. Five strategies were identified: transitional, building up, doubling and

halving, multiplication calculation, and wholistic thinking. Downton found that Year

3 students move from the concrete to the abstract as they transition from using additive

to multiplicative thinking (Downton, 2010).

Young-Loveridge (2005) cites The New Zealand (NZ) Number Framework

(Ministry of Education, 2001), which identifies eight stages of development as

“multiplicative thinking builds on additive thinking, and in turn provides the

foundation on which proportional reasoning can be built” (p. 34). The first four stages

are counting based strategies and the remaining four are based on part-whole or

partitioning strategies. She found that learners need to have a good grasp of additive

part-whole strategies to reason multiplicatively. At early additive part-whole stage

recognised numbers can be treated as wholes or partitioned and recombined (e.g. 8+5,

37+9, 36 -7). At the more advanced stage students choose from a repertoire of part-

whole strategies and see numbers as whole units with the possibilities of recombining

(e.g. 52+62+18; 122-67) (Ministry of Education, 2001).

Siemon et al., (2012) offer a sequence of big ideas (trust the count, place value,

and multiplicative thinking) that are hierarchal in their development and contribute to

proportional reasoning. Trust the count and place value are important foundational

concepts, as additive thinking builds on these two big ideas. These will be discussed

in subsequent paragraphs. The development of additive thinking is formative to the

development of multiplicative thinking and, in order “to become confident

multiplicative thinkers, children need a well-developed sense of number (based on

trust the count, place value and partitioning) and a deep understanding of the many

different contexts in which multiplication and division can arise (e.g. sharing, equal

24

groups, arrays, regions, rates, ratio and the Cartesian product)” (Siemon, 2013, p. 43).

The relevance of multiplication will be discussed in subsequent paragraphs.

Each of the studies described and outlined in Table 2.1 identify number concepts

that contribute to the development of proportional reasoning. The common

foundational number concepts identified in each of the studies are outlined in Table

2.2. As each of the different studies identify these concepts as relevant and important

they can be described as the foundational concepts of proportional reasoning. The

foundational concepts (identified in each of the studies) align with an expected stage

of thinking as outlined in in Table 2.2 and will be described below.

Table 2.2.

Common foundational concepts in each stage of development to multiplicative

thinking

Additive Stage Transitional Stage Beginning to think

multiplicatively

One to one counting

Trust the count

Subitise

Equal groups

Repeated addition

Skip counting

Multiplicand (group of

equal size)

Attends to multiplier

(number of groups) and

multiplicand

Partitioning with

doubling and halving

Additive part-whole

strategies

Knows multiplicative

situations, multiplicand,

multiplier, total amount

(product)

Part-part whole

(advanced additive)

Each of the concepts will be described and discussed within the stage of thinking.

Additive Stage - One to one counting occurs when a child calls number values by name,

knowing the last number named is the total or the cardinality and answers a how many

question (Reys et al., 2012).

Trust the count is when children understand that the count is a permanent

indicator of quantity and can be rearranged but will remain the same.

Subitise is closely aligned with trust the count is when children recognise small

collections of numbers (1-5) and can do so without counting; that is, it said they can

subitise.

25

At the early stage of additive thinking, children do not trust the count, understand

that skip counting tells how many and grouping makes no sense to them (Jacob &

Willis, 2003). To be successful in this additive thinking stage (Table 2.2), students

need to develop each of these concepts as one to one counting and trust the count are

key foundational concepts relating to the first stage of thinking (Siemon et al., 2012).

As the child progresses through this stage, the child recognises equal size groups or

the multiplicand, but does not understand that the number of groups can be counted

and the role of number of the groups in multiplication (Jacob & Willis, 2003). The

student progresses from one-to-one counting to trust the count, and can count using

repeated addition and skip counting.

Transitional stage - Repeated addition, in its simplest form, is a counting based

strategy of counting in groups especially counting with materials (Siemon et al., 2011).

If it is used with a multiplication, it involves adding of the same number, e.g. if Tom

has 3 bags with 5 marbles in each, Tom would count: 5+5+5

Skip counting is a higher form of repeated addition, and relies on knowing

number-naming sequences and instead of counting by 1’s, counts by 2’s, 5’s, 10’s or

other amounts.

Partitioning is the physical separation or renaming of a collection in term of its

parts. It can occur in additive terms (e.g. 8 is 5 and 3) in multiplicative terms, parts are

equal double (8 is double 4) or half a quantity or collection into equal parts (with

concrete objects).

Place value is a means of giving a value to a digit based on its position in a

number (e.g. 2 in 256 and 562 represents different values). The position of the number

denotes the value of the unit eg. 356 is 3 hundreds, 5 tens and 6 ones.

Additive part-part-whole is the knowledge of numbers to 10 (8 is 2 and 6). Also,

in relation to larger numbers; that is, those numbers of which it is a part (part-whole)

(e.g. 6 is 1 less than 7, 4 less than 10, half of 12).

Part-part-whole with place value is used when needed to reconfigure numbers in

different ways to suit addition or subtraction calculations (e.g. to calculate 37 and 26,

you might add 37 + 3 = 40, plus 23 more is 63). The computation is facilitated by the

aggregation and disaggregation of collections, through the understanding of part-

26

part-whole and place value knowledge (Siemon, 2012). This is regarded as advanced

additive part-whole (Ministry of Education, 2001).

The transitional thinking stage (Table 2.2), is the stage where the child begins to

move between additive and multiplicative thinking, and attends to the multiplier

(number of groups) and the multiplicand (size of group). At this stage, the child can

partition and understand part-part-whole concept for 1 to 10 and larger numbers and,

combined with place value, can use this concept for calculations.

Beginning to think multiplicatively - 3 aspects of multiplication. These relate to

groups of equal sizes (the multiplicand), number of groups (the multiplier), and a total

amount (the product). For example, in the number facts, 4 × 2, 4 × 3, 4 × 4, the

multiplicand is 4, the multiplier is the number of groups (2,3,4), and the answer is the

product (4 × 2 = 8). The stage at which a child begins to think multiplicatively (Table

2.2) is when a child knows three components (multiplicand, multiplier, and product)

constitute a multiplicative situation and they understand both multiplication and

division are grouping structures (Jacob & Willis, 2003). At this stage, the child

automatically recalls known multiplication facts and can use them to find solutions for

other facts (e.g. 8 times 10 is 80, so I just add an 8 to get 88 and that’s 11 eights)

(Downton, 2010).

The foundational concepts identified in Table 2.2 and described above contribute

to a well-developed sense of number, which is necessary for the development of

additive and multiplicative thinking. The transition from additive to multiplicative is

not a straightforward process (Siemon, 2013). Ell et al., (2004) conducted a study of

Year 3 students to resolve how children manage the transition between additive and

multiplicative thinking. They did this by observing the strategies adopted by the

students and the change of strategies when solving multiplication problems. The

results from this study support Siegler’s (2000) overlapping waves theory, which

suggests that strategies are not replaced, but acquired strategies remain available, or

co-exist, as children progress in their learning of new strategies. In relation to this

study, Siegler’s (2000) research suggests students do not just transition from one type

of thinking to another - additive to multiplicative - as could be interpreted by the linear

stages identified in Table 2.2. Rather, the transition is a switch in the dominance of

additive to multiplicative thinking allowing for a co-existence of each type of thinking.

27

The development of the co-existence of additive, and multiplicative thinking is

supported by other research which suggests “that there is a parallel path of

development of multiplicative thinking based on a child’s development of the

conceptual understanding of multiplication and addition” (Siemon, 2013, p. 45). At

the same time students can follow both a counting-based path and a path is based on

“splitting (multiple versions of a collection or the fair share of a collection)” (Siemon

et al., 2012, p. 357). Traditionally, it has been regarded that children need to be

experienced in counting-based strategies and multiplication is taught as repeated

addition with a focus on equal groups (which is characteristic of additive thinking).

This is reflected in the Australian Curriculum: Mathematics when multiplication is

introduced in Year 2 as repeated addition. However, the research suggests there is a

parallel pathway to the development of multiplicative thinking based on share equally

(Siemon, 2013).

Multiplicative thinking is linked to the development of the concept of

multiplication and the way it is taught has undergone review in the literature.

Multiplication is more complicated than repeated addition, and teaching it as a bi-

product of addition does not “provide children with important multiplicative

structures” (Jacob & Willis, 2001, p. 306). Siemon et al., (2012) views this number

naming sequence and counting all groups as “an additive, counting based approach to

multiplication” (p. 365). Such an approach can be detrimental to students’ ability to

think multiplicatively and potentially delay the transition from additive to

multiplicative thinking, as children are more likely to remember multiplication as

repeated addition when first trying to think multiplicatively (Hurst, 2015). Students

need to know the differences and understand that adding is the combination of parts

and multiplication is the outcome of scaling some quantity, which is key to the

development of multiplicative thinking (Hurst, 2015).

To support children’s development of foundation concepts, and transition of

additive and multiplicative thinking, teachers also need to know and recognise the

difference between additive and multiplicative thinking. Hilton et al., (2016) found

that teachers often lack knowledge of concepts and reasoning and “struggle to teach

the conceptual underpinnings of multiplicative thinking” (p. 195). Despite the

suggestion that teachers struggle with multiplicative thinking there are limited studies

aimed at teacher understanding of this topic (Lobato et al., 2011). This project

28

addresses this gap, as central to it is the development of early years teachers’

understanding of the foundational concepts that contribute to multiplicative thinking

and in turn proportional reasoning.

Proportional reasoning is a big idea of mathematics as it encompasses many

mathematical concepts. It is not a concept that is normally associated with the maths

in P-3 classrooms, but it has the potential to be developed in early childhood, as the

foundation is established in these year levels. The development of proportional

reasoning is a learning trajectory from additive to multiplicative thinking, developed

through a strong sense of number. In this section, the stages of a developmental

continuum of multiplicative thinking were identified with the mathematical concepts

attached to each stage as the foundational concepts of proportional reasoning. For

children to develop an understanding of these concepts, teachers need to find ways to

promote the foundational concepts that enhance student understanding and these will

be discussed in the next section.

2.2 PROMOTING MATHEMATICAL CONCEPTS IN THE CLASSROOM

The mathematical concepts that contribute to the development of proportional

reasoning, as discussed in the previous section, will lie dormant on the pages of

curriculum documents unless they are promoted by teachers. The ways in which

concepts are promoted are informed by the teacher’s belief of mathematics pedagogy

(Anthony & Walshaw, 2009a). Pedagogy is defined as the “knowledge and principles

of teaching and learning” (Siemon et al., 2011 p. 111). Siraj–Blatchford (2010)

expands upon this broad description to include the learner and defines pedagogy as,

“instructional techniques and strategies that enable learning to take place for the

acquisition of knowledge, skills, attitudes and dispositions” (p. 149). It is these

instructional techniques and strategies or practice that are relevant to the promotion

of mathematical concepts and relevant to this study as it investigates how teachers

promote proportional reasoning.

This study is underpinned by the work of Anthony and Walshaw (2009a) and

Muir (2008), who both proposed a set of principles for teaching mathematics

effectively. Based on a synthesis of literature, Anthony and Walshaw (2009a)

identified ten principles of effective pedagogical practice, which included “teacher

knowledge and learning, an ethic of care, arranging for learning, building on students’

thinking, mathematical communication, mathematical language, assessment for

29

learning, worthwhile mathematical tasks, making connections and tools and

representations” (p.148).

Muir (2008) interpreted two existing research studies to identify “six principles

of practice which she believes encapsulates effective teaching of numeracy

(mathematics)” (p. 80). These included “make connections, challenge all students,

teach for conceptual understanding, purposeful discussion, focus on mathematics and

positive attitudes” (p. 80). Then Muir (2008) designed and conducted her own case

study and further determined six teacher actions that also contribute to effective

pedagogical practice. The teacher actions identified included “choice of examples,

choice of tasks, teachable moments, modelling, use of representations, and

questioning” (Muir, 2008, p. 86). Muir (2008) cautions that these actions should not

be implemented simply, but rather be integrated with each other and teacher beliefs

and knowledge. Therefore, these actions will be addressed within elements of practice

to be discussed in Section 2.2.1 Teacher Knowledge, and Section 2.2.4 Mathematical

Tasks.In terms of elements of practice, Askew (2016) offers three key elements (talk,

tools, tasks) which he believes “can maximise the likelihood of mathematics emerging

from lessons” (p.115). He identifies these three elements as a teaching tripod, these

three elements of the tripod align with the principles, discourse and mathematical tasks

and therefore will be discussed within these sections.

The principles proposed by Anthony and Walshaw (2009a) and Muir (2008) are

presented in Table 2.3. Table 2.3 also includes elements of practice, which were

identified by Anthony and Walshaw (2009a) as a way of grouping the ten principles

that underpin effective pedagogy in mathematics. These elements (teacher learning

and knowledge, classroom community, discourse in the classroom, and mathematical

tasks) and the associated principles are identified in Table 2.3. These elements of

practice provide a useful framework in which to address ways to promote

mathematical concepts and these will be addressed in the next four sub sections (2.2.1

Teacher Knowledge, 2.2.2 Classroom Community, 2.2.3 Discourse in the Classroom).

30

Table 2.3.

Elements of practice and principles

Elements of Practice

Anthony & Walshaw

(2009a)

Principles

Anthony & Walshaw

(2009a)

Principles of Practices

Muir (2008)

Teacher Learning &

Knowledge

Teacher Knowledge Teach for Conceptual

Understanding

Classroom Community An ethic of care

Arranging for learning

Building on students’

thinking

Positive attitudes

Challenge all pupils

Discourse in the classroom Mathematical

communication

Mathematical language

Assessment for learning

Purposeful discussions

Mathematical Tasks Worthwhile Tasks

Making Connections

Tools and Representations

Focus on Mathematics

Making connections

2.2.1 Teacher Knowledge

Teacher knowledge influences the teaching and learning of mathematical

concepts and how they are promoted. Both Anthony and Walshaw (2009a) and Muir

(2008) acknowledge the importance of teacher knowledge in relation to responding to

student needs and the provision of classroom activities and regard this as a principle

of practice (See Table 2.3). Muir (2008) also identifies teacher knowledge as a factor

that influences her principles of practice. Both studies are influenced by the work of

Shulman (1986, 1987). Muir (2008) acknowledged teacher knowledge guided by

Shulman’s seven types of knowledge as a factor influencing her principles (Muir,

2008). Shulman’s content, and pedagogical content knowledge, informed one of

Anthony and Walshaw (2009b)’s principles, “teacher knowledge” (p. 25).

Teachers’ knowledge of mathematics content and its importance to the teaching

of mathematics has received attention, and been the subject of debate, since the

foundational work of teacher knowledge by Shulman (1986, 1987) was first proposed.

31

Shulman (1986, 1987) proposed that a teacher’s knowledge base comprises seven

different types: content knowledge; general pedagogical knowledge; curriculum

knowledge; pedagogical content knowledge; knowledge of learners and their

characteristics; knowledge of educational contexts; and knowledge of educational

ends, purposes and values and their philosophical and historical grounds (Ben-Peretz,

2011; Shulman, 1986).

Motivated by Shulman’s proposal, researchers have investigated the significance

of these types of knowledge. It is acknowledged that knowledge is an “important part

of an individual’s identity” (Bennison, 2015, p. 10) There is a community of

researchers (e.g. Watson, Callingham, & Donne, 2008) who acknowledge the

complexity of each knowledge type. They also recognise an interaction between the

different types of knowledge which contribute to the implementation by teachers of

mathematics programs. The promotion of mathematical concepts is based on teacher’s

content knowledge (CK) of mathematics and pedagogical content knowledge (PCK)

as the teaching of mathematics involves “a blend of mathematical knowledge and

pedagogical knowledge that only a mathematics teacher would need” (Siemon et al.,

2012, p. 55). Differences in knowledge and the significance of teacher CK, PCK and

more recently the development of knowledge of students as learners into knowledge

for teaching mathematics have been the focus of further research (Ball, Thames &

Phelps, 2008; Shulman, 1986). CK and PCK will be discussed in relation to teacher

mathematics knowledge.

Curriculum Knowledge (CK) includes knowledge of the subject and its

organising structures (Shulman, 1986, 1987; Wilson, Shulman & Richert, 1987). In a

more recent study, Ball et al., (2008) have reported that a teacher’s knowledge of

mathematics for teaching is multi-dimensional, and consists of both general

knowledge of content and more specific domain knowledge, that is specialised CK.

CK is considered a deeper knowledge of specific content, such as number and

operations, and includes knowledge of student misconceptions and analysing unusual

procedures or algorithms. A strong understanding of CK does not necessarily correlate

with a well-developed PCK as CK alone is not enough to provide high quality teaching

(Lee, 2010).

Pedagogical Curriculum Knowledge (PCK) is regarded as lying at the heart of

effective teaching. It is the knowledge and skills that an individual teacher carries with

32

the cognitive demands of teaching. Not only does it represent teachers’ understanding

of the subject matter, as a key contributor to effective teaching, but it also influences

a teacher’s ability to make appropriate instructional decisions for learners (Anthony &

Walshaw, 2009b; Bobis, Clarke, Clarke, Thomas, Wright & Young-Loveridge, 2005;

Lee, 2010). PCK is a content-based form of professional knowledge that informs

“synthesis of actions thinking, theories and principles within classroom episodes”

(Walshaw, 2012, p. 182). A teacher’s capacity to synthesise the pedagogy on the spot

is a significant contributor to children’s mathematical achievements, as the teacher is

knowingly able to respond to children’s learning to support or extend the child’s

understanding (Anthony & Walshaw, 2009b).

Optimal learning environments are ones where the teacher possesses both deep

and connected knowledge of subject matter (CK) and methods of teaching

mathematics (PCK) (Commonwealth of Australia, 2008; Ediger, 2009). Teachers with

limited subject knowledge tend to focus on a “narrow conceptual field” rather than

build wider connections between mathematics concepts (Walshaw, 2012, p. 182). An

understanding of the broader concept, or big ideas of mathematics, affords teachers the

opportunity to make the connections between mathematics concepts and help children

to build an understanding to generalise and, in doing so, “they build up networks of

big ideas” (Askew, 2013 p. 7). This aligns with Muir’s (2008, p. 81) principle ‘teach

for conceptual understanding’, which highlights the importance of making the

conceptual connections between important ideas found in mathematics curriculum.

In relation to this study, a teacher’s content knowledge and pedagogical content

knowledge is significant for a teacher to support young children’s development of the

big idea, proportional reasoning. Understanding of proportional reasoning (CK) and

understanding of effective teaching strategies (PCK) are both critical components to

helping children’s understanding of proportions. Fernandez, Llinares and Valls (2013)

investigated “how pre-service primary school teachers notice students’ mathematical

thinking in the context of proportional and non-proportional problem solving” (p. 441).

They found that some preservice teachers (a) displayed weaknesses in their own

understanding of additive and multiplicative situations (related to the difference

between proportional and non-proportional relationships), and (b) others who could

recognise the difference, but could not justify the mathematical elements in relation to

the students’ answers.

33

Another study (Pitta-Pantiazi & Christou, 2011) of preservice teachers,

investigated prospective kindergarten teachers’ knowledge of proportional reasoning.

The study found that the teachers in the research group had a limited understanding of

proportional reasoning, and lacked relative and multiplicative thinking. Pitta-Pantiazi

and Christou (2011) argued that if a teacher was well informed about the

interconnected concepts of proportional reasoning, then “students’ proportional

abilities will be appreciated, attended to, explored and expanded” (p. 164). If early

childhood teachers are to enhance their students’ thinking to develop proportional

reasoning, it is essential that the teachers enhance their own knowledge of proportional

reasoning and, in particular, multiplicative thinking (Pitta-Pantiazi & Christou, 2011).

The two studies described each had a different focus. However, their results

shared a common feature: both found that the teacher’s knowledge of proportional

reasoning has implications on the teacher’s ability to teach this mathematics concept

(Fernadez et al., 2013; Pitta-Pantiazi & Christou, 2011). While these studies were in

relation to pre-service teachers, it highlights the need for a study to investigate

experienced teachers’ understanding of proportional reasoning and this was the focus

of this study.

The seminal work of Shulman (1986) has informed further studies, which have

identified that it is essential that teachers possess a content knowledge of mathematics

- in this case, the foundational concepts of the proportional reasoning. This content

knowledge is further enhanced by a teacher’s pedagogical content knowledge, which

informs the practices employed by the teacher to expose students’ thinking and ability

(Siemon et al., 2012). These knowledges sit within the broad context of two principles:

classroom discourse and mathematical tasks; however, the success of these rely on a

positive classroom community.

2.2.2 Classroom Community

The teacher has a vital role to create a safe psychological environment to achieve

student engagement (Anthony & Walshaw, 2009b). The teacher establishes this

environment through an ethic of care, arranging for learning and building on students’

thinking through a constructivist approach to teaching. A constructivist approach to

learning, allows learners to “actively interact with their environments: physical, social

and psychological” (Siemon et al., 2011, p. 34). The constructivist paradigm informs

34

the teaching-learning process of effective mathematics teaching and influences a

positive classroom environment.

Such an environment is also created by establishing a positive relationship

between teachers and students and is a powerful influencer of student achievement

(Hattie, 2009). In such a classroom, an ethic of care is created by teachers encouraging

a harmonious environment through both the teacher and student sharing a positive

attitude towards mathematics. A teacher’s possession of a positive attitude to

mathematics evolves from their own confidence in knowledge of mathematics (Muir,

2008). Similarly, a positive attitude towards mathematics is encouraged in students by

developing their confidence “in their capacity to learn and to make sense of

mathematics” as their mathematical identity emerges (Anthony & Walshaw, 2009b, p.

8).

The way in which a classroom is arranged for learning reflects the classroom

community. In a positive classroom environment, the teacher provides students with

opportunities for students to make sense of mathematical ideas through independent,

collaborative, and whole class groupings (Anthony & Walshaw, 2009a). The manner

and approach in which the teacher interacts with students in these different groupings

also contributes to the psychological environment and this will be addressed in the

following section, Discourse in the Classroom (Section 2.2.3).

In a classroom community that builds on students’ thinking, the teacher plans

with the student’s proficiencies at the centre of the learning through planning

sequential, developmentally meaningful contexts or mathematical tasks that builds on

student’s prior knowledge and contributes to a child’s understanding of the

mathematical concept (Hattie et al., 2017). By contrast to planning for student

thinking, teachers need to also capitalise on teachable moments to help the student

make mathematical connections. These teachable moments are dependent on the

teacher possessing the knowledge to connect learning to what students are thinking

and at the same time take the opportunity to challenge all pupils through extending

their thinking (Muir, 2008, p. 81).

For students to confidently explore mathematics the teacher structures the social

climate of the classroom so that students can learn to communicate their ideas to others

and construct their knowledge (Fast & Hankes, 2010). The learning occurs as students

actively construct their own mathematical knowledge through exploration, making

35

hypotheses, discoveries, reflecting, and drawing conclusions to construct shared

knowledge through a constructivist approach (Beswick, 2005; Fast & Hankes, 2010).

In such an approach, the teacher is cognisant that students “will construct their own

knowledge and understandings” and, therefore, will adapt their instruction to cater for

the student’s learning and thinking (Siemon et al., 2011, p. 37). A classroom

community that encourages and plans for student thinking is relevant to this study.

Problem solving provides opportunities for teachers to build on student thinking

as they allow students to explain their thinking while facilitating student learning from

a constructivist perspective (O’Shea & Leavy, 2013). Problem solving will be further

discussed in, Mathematical Tasks (Section 2.2.4). In a positive classroom community,

teachers establish classroom practices which support the child’s thoughts and

understandings (Anghileri, 2006). Much of this support is established through

classroom discourse, that is, mathematical communication.

2.2.3 Discourse in the Classroom

Discourse, that is communication in the mathematics classroom, is an

instructional technique or strategy for articulating understanding. The elements of

practice identified by Anthony & Walshaw (2009b) include mathematical

communication (revoicing and argumentation), mathematical language and

assessment for learning (questioning, dialogic teaching). These will now be discussed.

Mathematical communication is productive when there is a focus revoicing and

argumentation (Anthony & Walshaw, 2009b). Revoicing, or elaborating on student

talk, is a valuable prompt for understanding and discourse (Anthony & Walshaw,

2009b; Hattie, Fisher, & Frey, 2017). Discourse that encourages inquiry includes a

focus on “mathematical argumentation” (Anthony & Walshaw, 2009b, p. 19). In these

purposeful discussions, teachers challenge students through “explaining, listening and

problem solving” (Muir, 2008, p. 81). The development of argumentation is

particularly useful in working with problems that do not possess an obvious structure

and such an approach encourages students to explain their understandings (Fielding-

Wells et al., 2014). It is constructivist in its approach as it helps build a deep conceptual

understanding of mathematics and requires teachers to be skilful in their questioning

and guiding as students construct their own knowledge (Fast & Hankes, 2010).

Mathematics is a unique form of communication, as it has its own verbal and

written language. Through rich classroom discourse using mathematical language

36

during mathematical experiences, children learn to “talk mathematics” (Askew, 2016,

p. 147). The verbal communication of mathematical language consists of the words

through which meaning is developed, by “giving voice to tasks and tools” (Askew,

2016, p. 147). Its written language requires a specific knowledge and interpretation of

symbols and diagrams, for which students need support to understand the meaning and

to make connections with the information (Siemon et al., 2012). The significance of

making connections is highlighted by Muir (2008), who identified it as a principle of

practice, as the teacher makes the connection “using a variety of words, symbols and

diagrams” (Muir, 2008, p. 80). Such a teacher is regarded as having connectionist

orientation and is a distinguishing factor for a highly effective teacher (Askew, Brown,

Rhodes, Johnson & William, 1997). A teacher with connectionist orientation is

characterised as one who uses prompts, such as “background knowledge, process or

procedure, reflective, heuristic” (Hattie et al., 2017, p. 95) to help children make

connection with previous knowledge, which can be forgotten when presented with new

concepts and one who helps the child to move forward in their learning.

Assessment for learning supports mathematical understanding through

questioning and is a contributor to effective mathematics teaching (Anthony &

Walshaw 2009b). While prompts can be regarded as questions, there is a difference

between prompts and questioning. Prompts encourage students to engage in cognitive

processing by clarifying their learning; questioning, on the other hand, is a way a

teacher can check for student understanding when students “explain and justify their

mathematical thinking” (Siemon et al., 2011, p. 62). Asking open-ended questions

and listening to student responses allows teachers to draw on their mathematical CK

and, in turn, help students build their own knowledge (Muir, 2008; Siemon et al.,

2012). Such focusing questions “allow students to do cognitive work of learning by

helping to push their thinking forward” and teachers can better understand the level of

student understanding (Hattie et al., 2017, p. 89). On the other hand, funnelling

questions are too specific as they guide the student to find one answer and do not

promote dialogic communications.

Dialogic teaching allows teachers scaffold and facilitate classroom dialogue and

purposeful discussions through dialogic teaching to effectively promote mathematical

concepts (Bakker, Smit & Wegerif, 2015). In dialogic teaching approach children

listen to others and respond. Dialogic teaching provides the teacher with the

37

opportunity to understand, be responsive to student thinking, and identify the student’s

level of sophistication when thinking about mathematical concepts. Each of which

informs classroom instruction to faciliate student learning (Bright et al., 2003). In such

discussions, teachers ensure the dialogue between classroom participants provides the

opportunity for students and teacher to engage in productive talk and is dual in its

approach, “teaching for dialogue as well as teaching through dialogue” (Bakker et al.,

2015, p. 1047). That is, classroom participants can engage in dialogue with each other,

or by listening to others. In this way, students learn to think rather than participating

in a discussion that can have elements of trying to win or impose their own opinion

(Askew, 2016). Such an approach supports the premise that students will learn not

just from the teacher, but students will “learn to ask open questions and learn new

things for themselves through engaging in dialogic inquiry” as they are supported by

a constructivist approach (Bakker et al., 2015, p. 1048).

Dialogic teaching allows for student and teacher interactions that are richer and

this can be achieved through scaffolding practices, as they allow for a more structured

questioning approach through a process of support. This process includes the teacher

summarising what she believes is shared knowledge, followed by “focussing joint

attention on a critical point not yet understood” (Anghileri, 2006, p. 36). Anghileri

(2006) proposes a scaffolding model which summarises a hierarchy of different levels

of scaffolding strategies. At the most basic level of scaffolding she identifies

“environmental provisions”, which allows learning to take place “without direct

intervention of the teacher” (p. 39); that is, artefacts are conceptualised as scaffolding

(Bakker et al., 2015). At the next level, scaffolding involves interactions between

teacher and students; that is, “explaining, reviewing and restructuring” (Anghileri,

2006, p. 41). Central to this type of scaffolding is “show and tell and explaining”

scaffolding approaches that aligns with the traditional approach of teaching. The

teacher is in control with a show and tell approach and with little input from the

students. Similarly, explaining is conducted by the teacher and can inadvertently

constrain students’ thinking and does not allow for insight into the student’s

understanding.

Within this same level, “reviewing and restructuring” are described as

interactions that are more effective (Anghileri, 2006, p.41). Within the reviewing, five

scaffolding interactions are identified: “getting students to look, touch and verbalise

38

what they see and think; getting students to explain and justify; interpreting students’

actions and talk; using prompting and probing questions; and parallel modelling”

(Anghileri, 2006, p. 41). While reviewing, the teacher-student interaction encourages

reflection and clarification. Restructuring is a higher level of interaction as the teacher

supports the student through revoicing to assist in “taking meaning forward”

(Anghileri, 2006, p. 44). The highest level of scaffolding encourages engagement in

conceptual discourse (Anghileri, 2006). Scaffolding is effective in different groupings

with the teacher and the student in one to one, group, and whole class settings.

Collaborative learning between students encourages students to scaffold each other’s

learning (Bakker et al., 2015). This review of literature in relation to mathematics

communication highlights its importance in promoting mathematical understanding,

and is therefore has relevance to this study. Discourse in the classroom can be further

enhanced by the provision of mathematical tasks which will be discussed in the next

section.

2.2.4 Mathematical Tasks

Mathematical tasks for learning should be engaging to ensure students develop

“ideas about the nature of mathematics and discover that they should have the capacity

to make sense of mathematics” and promote a constructivist approach to learning

(Anthony & Walshaw, 2009b, p. 13). These tasks are regarded as worthwhile tasks by

Anthony and Walshaw (2009a). Mathematical tasks can be achieved through teachers

supporting students in making connections as they challenge students’ mathematical

thinking extend their understanding through problem solving and through the

presentation of tools and representations to support their mathematical

understandings.

Worthwhile tasks need to have a focus on mathematics and need to be clear and

purposeful to allow for different “possibilities, strategies, and products to emerge”

(Muir, 2008, p. 82). Tasks need to allow students to engage in mathematical ideas

coupled with a high level of teacher engagement that is broad and challenging. As

Muir (2008) found in her case study, higher level cognitively demanding tasks need to

be matched with high-level instructions to develop conceptual understandings.

Students should not come to expect practice and drill tasks, but rather tasks that

challenge thinking (Muir, 2008). In particular, Hattie et al., (2017) identifies two main

types of tasks. The first type of task helps students become more competent with what

39

they already know. The second presents new ideas and challenges and deepens

understanding, where the solution is not immediate. This falls within the child’s zone

of proximal development; that is, tasks that can be successfully completed with

scaffolding or support from someone more able (Reys et al., 2012). Offering problem

solving tasks helps students use mathematical processes and make connections

between diverse approaches for solving problems, this will be discussed next.

Making connections is a key element of practice identified by both Muir (2008)

and Anthony and Walshaw (2009a), in their synthesis of research. They both agree

that students need support in making connections across different areas of mathematics

and by supporting students in “creating connections between different ways of solving

problems.” (Anthony & Walshaw, 2009a p. 15). Through a constructivist approach to

problem solving, teachers can offer experiences for students to make their own

connections as they construct knowledge as they “work collaboratively and are

supported as they engage in task-orientated dialogue” (O’Shea & Leavy, 2013 p. 296).

Problem solving is at the heart of the Australian Curriculum: Mathematics, and

as a proficiency strand is a common thread through the document with students

encouraged to interpret, investigate problem situations and communicate ([ACARA],

2016). It allows mathematics to move beyond another proficiency strand, fluency, that

is procedural fluency to encompass understanding (another proficiency strand) so that

“learners are active constructors of knowledge not passive recipients of it” (Askew,

2016, p.123). Problem solving contributes to understanding proportional reasoning

as proportional reasoning capability is evident in “the ability to solve a variety of

problem types and the ability to discriminate proportional from non-proportional

situations” (Fernandez, Llinares, & Valls, 2013, p. 1). Therefore, the provision of

problem solving tasks (that have a more open approach) will enable teachers to

determine the developmental stage of the children (additive and/or multiplicative)

based on how they engage with the task. Providing opportunities for students to explain

their ideas, justify and share their strategies for solving mathematical problems in a

collaborative way supports “learning from a constructivist perspective” (O’Shea &

Leavy, 2013 p. 312).

A crucial issue in teaching multiplicative thinking is that students need to be able

to identify and distinguish between multiplicative and additive situations. Lamon

(2010) suggests that while most Year 3 students will be additive thinkers, it is

40

important to present students with both additive and multiplicative tasks in different

contexts. This aligns with Langrall and Swafford (2000) who identify that students

need to recognise the difference between additive and multiplicative thinking as a

prerequisite component for proportional reasoning and is the key to identifying student

thinking (Lamon, 2010). Students need to experience different problems with initial

emphasis on, not solving the problems, but on the discussion of classification of

thinking and, at the same time, providing rich discourse for learning (Van Dooren et

al., 2010). Research (e.g. Bright, Joyner & Wallis, 2003; Jacob & Willis, 2001;

Lamon, 2010) has reported the use of problem solving tasks as an appropriate

mechanism for teachers to check student understanding. An example of a task to

assess student understanding might be,

A recipe for a cake requires 4 cups of sugar and 6 cups of flour. However,

we want to make a larger cake. If we use 10 cups of sugar, how much flour

will be needed? (Hilton, et al., 2016, p.210)

A child who thinks additively may give the answer 12 cups of flour (sugar

increased by 6 cups so must the flour). A multiplicative thinking child thinks that the

amount of sugar has been increased by a factor of 2.5, so the flour must also be

increased by this factor, so the answer is 15 cups of flour. In addition to the provison

of these types of problems students, teacher understanding of the difference between

additive and multiplicative thinking is paramount. That understanding enables the

teacher to support the student in their understanding and transition from additive and

multiplicative thinking.

Teachers also need to possess instructional approaches, and strategies for

enhancing student thinking and understanding. Instruction was found to be significant

in the study conducted by Ell et al., (2004). In their study on how students approach

multiplication problems, they found that students take a pathway that is exclusive to

them, but the journey is “determined not only by developmental trends, but by

instruction, expectations, interpretations and prior experience” (p. 204). This finding

of teacher instruction can affect student approaches to problems (Ell et al., 2004).

Students need to be given the opportunity to experience a collaborative approach to

solve problems, clarify their ideas and offer their strategies to others, thus enabling

student learning from a constructivist standpoint (O’Shea & Leavy, 2013).

41

The impact of teacher and student expectation was also evident in a study

conducted by Downton (2010). The results show students need opportunities to

participate in activities that challenge their thinking through posing “problems some

of the time that extend children’s thinking beyond what appears to be commonly the

case” (Downton, 2010, p. 176) as it will help teachers understand the student’s thinking

capacity that might otherwise not be evident when offered problems at a more simpler

level.

In summary, problem solving is an important mathematical task as it permeates

mathematics. By offering problem solving through a constructivist approach, it creates

a classroom community that is collaborative and allows for a stimulating environment

which involves discourse that encourages debate, justification and argumentation as

described in Section 2.2.3. Finding solutions to problems and enhancing student

understanding of mathematical concepts can also be supported by mathematical tasks

that are inclusive of tools and representations as described below.

The terms, tools and representations have different meanings in the literature,

with some viewing them as the same and, therefore, interchanging the terms, or

viewing them separately (Anthony & Walshaw, 2009b). Geiger, Goos and Dole

(2011) see tools as a key element of their numeracy model and use the term to describe

“materials, representational and digital” (p. 298). Askew (2016) uses the term tool

when the experienced user engages with the artefacts (concrete materials, pictures,

diagrams, and mathematical symbols) “to serve the purpose and meaning with which

experienced users imbue them” (p.132).

The literature, however, is in agreeance over what is considered a mathematical

representation or tool (Anthony & Walshaw, 2009b; Askew, 2016; Siemon et al.,

2012). These include “traditional representations for example; number lines, arrays,

models, language and symbols” (Siemon et al., 2012, p. 100). The presentation of

concrete materials, and student interaction with these materials, is described by

Siemon et al., (2012) as concrete thinking. The purpose of offering different

representations is to help learners gain insight into mathematical structure and is

considered part of the “pedagogical furniture” (Askew, 2016, p. 146). Perhaps,

Anghileri’s (2006) suggestion that the act of providing artefacts is a form of teacher

scaffolding, it can “have a significant impact on learning” (p. 40) as learning takes

place through the interaction between the child and the artefact. However, Muir (2008)

42

suggests otherwise, as she sees the mere inclusion of representations in a lesson as not

enough; it is how they are “utilised that can facilitate student understanding” (Muir,

2008, p. 97).

Representations that support a student’s understanding and ability to solve the

mathematical problems are regarded as “instructional representations” (Siemon et al.,

2012, p. 99). These representations can take two forms: one of which is imparted by

the teacher and the other by the student. While the teacher communicates through

examples and models, the student develops a representation to make meaning, to find

a solution to a problem, and communicate their ideas. Similarly, Askew (2016)

suggests that artefacts start life as “models of” when the teacher uses the model to

make the mathematical concept explicit. Through this approach, and over time, a

student begin to use the teacher’s model, so it becomes a “model for”, and eventually

becomes “a tool for thinking” for the student (Askew, 2016, p. 132). An example of

this type of approach to move from the teacher’s use of a ‘model of’ to the student’s

use of a ‘model for’ is the representational bar model. A bar model is a pictoral

representation of mathematical quantities (known and unknown) and their

relationships (part-whole and comparison) given to the problem and are very

supportive of helping students understand additive and multiplicative problems (Liu

& Soo, 2014). Drawing the bar model allows students to transpose their thinking into

a graphic representation. Initially, the teacher exposes the students to the model of

thinking with the intention that eventually it will become the student’s model for

problem solving and in turn become their ‘tool for thinking’. The bar model is a

representation that clarifies student thinking to find solutions to additive and/or

multiplicative based problems.

Understanding of the foundational concepts of proportional reasoning can be

promoted by the use of arrays, which are powerful tools or models for supporting the

development of multiplication and, in turn, multiplicative thinking. Arrays are physical

representations, or rows and columns and provide a visual of the thinking associated

with groups. The ten frame is a simpler version of the array and helps young students

develop additive reasoning. The ten frame can enhance student early understanding

of the difference between additive and multiplicative thinking and flexible partitioning

of numbers. Importantly, arrays focus children’s attention on the three quantities

involved in a multiplicative situation (Jacob & Mulligan, 2014; Hurst, 2015). Integral

43

to the introduction of the array, young students need to understand the “collinearity –

that is, the recognition and coordination of rows and columns are equal sized spacing”

(Jacob & Mulligan, 2014, p. 37). More meaning is added to an array if the children

are provided with situations where arrays are meaningful. For example,

There are three chairs in a row. There four rows. How many chairs altogether?

Here, students use counters to model the situation, which leads to an array

representation. Students can find awareness of arrays in the environment e.g. egg

cartons, chocolate box, muffin tin.

Representations and tools are traditionally part of a mathematical instructional

technique because they provide the concrete representation of thinking. However it is

in the selection and use of tools and representations that teachers needs to be discerning

(Askew, 2016). Askew (2016) suggests that children benefit from a limited number

of models with intentional and consistent use; that is, children need many experiences

with particular models to make sense of the mathematical concept in order for them

to become “tools for thinking” (Askew, 2016, p. 146). Representations and tools reach

their potential when embedded in worthwhile mathematical tasks, problem solving can

provide a context for proportional reasoning, both are powerful for helping develop

understanding for the learner. Therefore, in the context of this study, the worthwhile

tasks, problem solving and representations and tools have the potential to help

students’ understanding of proportional reasoning and so is significant to this study.

2.3 CHAPTER SUMMARY

As a background to the teaching of proportional reasoning, the importance of

early childhood mathematics and key issues that relate to the teaching and learning of

proportional reasoning have been raised. Firstly, proportional reasoning encompasses

many interconnected mathematical concepts and is regarded as a big idea of

mathematics. It can be difficult for teachers to teach proportional reasoning as it is not

explicitly stated in the Australian Curriculum for years Prep to Year 3, even though

the foundational concepts are developed in these years. Teachers need to have a

knowledge and understanding of proportional reasoning to connect concepts within

this big idea and provide the best practice for teaching its foundational concepts.

Studies have found that foundational concepts evolve from additive thinking and

in turn multiplicative thinking (Jacob & Willis, 2003; Ministry of Education, 2001;

44

Siemon et al., 2012). There are elements that impact on the development of

multiplicative thinking and, in turn, the development of proportional reasoning. These

include ensuring students have developed a deep understanding of the foundational

concepts that contribute to additive and multiplicative thinking to allow a child to

transition between additive thinking and beginning to think multiplicatively (Jacob &

Willis, 2003). The way a child is taught multiplication can impact on the child’s ability

to develop multiplicative thinking and there is limited research linking the teaching of

multiplication in the early years and the impact this has on a child’s ability to think

multiplicatively (Hurst, 2015).

A teacher’s knowledge (both content knowledge and pedagogical content

knowledge) will impact on a teacher’s ability to promote the foundational concepts of

proportional reasoning. In addition to teacher learning and knowledge, three other

elements of practice were identified as ways to promote proportional reasoning. These

included classroom community, discourse in the classroom, and mathematical tasks.

Classroom community addressed the safe psychological environment that needs to be

established for students to feel confident to engage in mathematical experiences

through different arrangements for learning, especially a constructivist approach to

learning. Discourse in the classroom is supported by effective teachers who facilitate

classroom discussions, takes dialogic approach to teaching and through revoicing,

questioning, and argumentation, teachers help children make meaning of mathematics

promote understanding (Anthony & Walshaw, 2009b; Hattie, et al., 2017). The

provision of mathematical tasks highlights the need for teachers to help students make

connections between mathematical concepts, provide problem solving situations, and

the provision of tools and representations. Teachers support students “in creating

connections between mathematical concepts, ways to solve problems, and between

representations” (Anthony & Walshaw, 2009b, p.15). The provision of problem

solving situations helps identify the strategies children use to solve additive and

multiplicative problems and give insight into a student’s thinking. Tools and

representations can help students’ understanding, knowledge, and skills of

mathematical concepts and is another way to promote understanding of proportional

reasoning. Chapter 3 will identify the methods and approaches used to collect and

analysis the data collected of this study.

45

46

Chapter 3: Methodology

The purpose of this study was to investigate the foundational concepts of

proportional reasoning and how teachers promote them. Specifically, this study

investigated the following research questions:

1. What do Prep to Year 3 teachers recognise as foundational concepts of

proportional reasoning?

2. How do Prep to Year 3 teachers promote the foundational concepts of

proportional reasoning?

This chapter begins with a discussion of the methodology and research design

that was employed in this research (Section 3.1). The teacher research approach is

addressed in Section 3.2. This chapter addresses the research methods (Section 3.3),

which includes a description of the research setting, the participants, and data

collection, followed by a description of data analysis for each research question

(Section 3.4). The fifth section addresses issues of credibility (Section 3.5) followed

by the ethical considerations (Section 3.6) of the study. The final section summarises

the methodology chapter (Section 3.7).

3.1 METHODOLOGY AND RESEARCH DESIGN

This study is an exploratory case study of how Prep to Year 3 teachers in a single

primary school recognised and promoted proportional reasoning. It is an exploratory

case study because it investigated a phenomenon for which there was no predetermined

outcome (Yin, 2009). That is, there was no predetermined expectation of the level of

teacher recognition of the foundational concepts of proportional reasoning or the ways

for promoting the foundational concepts of proportional reasoning in this case. As part

of the case study design process, it is recommended that boundaries are placed on the

case that “indicate the breadth and depth of the study” (Baxter & Jack, 2008, p. 547);

that is, the case is “separated out for research in terms of time, place or some physical

boundaries” (Creswell, 2008, p. 476). The case in this study involved Prep-Year 3

teachers, the boundary being a single primary school, and the research project was

conducted over a period of two months.

47

A case study should be explored from different perspectives using a variety of

methods to collect evidence of the phenomenon (Simons, 2009; Thomas, 2012). It is

“not explored through one lens, but rather a variety of lenses, which allows for multiple

facets of the phenomenon to be revealed and understood” (Baxter & Jack, 2008, p.

544). The purpose of a case study is to use multiple methods “to generate in-depth

understanding of a specific topic (as in a thesis), programme, policy, institution or

system to generate knowledge and/or inform policy development, professional

practice and civil or community action” (Simons, 2009, p. 21). The multiple collection

methods were used in this study, which included focus groups, discussions and

interviews.

This qualitative case study is underpinned by a constructivist paradigm, which

is “built upon the premise of a social construction of reality” (Baxter & Jack, 2008, p.

545) in exploring teachers’ understanding and promotion of foundational concepts for

proportional reasoning. The exploration involved teachers telling their stories about

the foundational concepts and how they teach those foundational concepts in each of

their classrooms. The analysis and interpretation of the participants’ stories and views

enabled the researcher to develop a greater appreciation of their understanding and

promotion of, proportional reasoning. A constructivist paradigm, is a paradigm that

suits this case, as it allows for subjectivity. The data is regarded as subjective because

the researcher analysed and interpreted teachers’ thoughts, feelings and actions, which

ultimately provided understanding and insight into the case study (Simons, 2009).

Simons (2009) cautions that single cases, with qualitative methods, require that

the effect of the researcher on “the research process and outcome” is monitored

(Simons, 2009, p. 4). In this type of case “the ‘self’ is more transparent” (Simons,

2009, p. 4) so the researcher should acknowledge values and biases that she possesses

and the impact those characteristics may have on her interpretation of the data. In this

case this requires reflexivity on the part of the researcher both during the research

process and during the analysis of the data (Simons, 2009; King, 2017). In this study,

the researcher is a participant and is also in a leadership position at the school and,

therefore, needed to adopt a reflexive approach to data collection. This case study was

underpinned by a teacher research approach to the case study design.

48

3.2 TEACHER RESEARCH

Teacher research refers to “research carried out by teachers to seek practical

solutions to issues and problems in their professional and community lives”

(Cakmakci, 2009, p. 40). Cochran-Smith and Lytle (1999) defined it as “all forms of

practitioner inquiry that involve systematic, intentional and self-critical inquiry about

one’s work” (p.15).

Three significant characteristics of teacher research have relevance for this

study, being the process of data collection, the impact on a teacher’s classroom practice

and the collaborative process of the research (Roulston, Legette, Deloach & Pitman,

2005). First, collecting and making use of qualitative data (journals, oral inquiries,

and observational data) is common practice in teacher research (Roulston et al., 2005).

It has been suggested that collecting data is common practice for good teachers, but

the teacher research movement argues that the approach advocated by the movement

takes data collection to the next level. “Data collection becomes systemized, reflection

is built into practice, findings are analysed, and discoveries are disseminated”

(Campbell, 2013, p. 2). The research methodology should draw on multiple sources

of data. The methodology should also ensure consistency and transcend bias that could

be perceived because “the teacher is functioning as a researcher within his or her own

school or classroom setting” (Cochran-Smith & Lytle, 1999, p. 20).

Second, teacher research contributes to teacher practice through a professional

culture of exploration of current research findings and, in turn, its impact in the

classroom (Cakmakci, 2009). It has been found that teacher researchers are “likely to

become more reflective, critical and analytical in their teaching” with the possibility

of teachers using the “findings from literature to improve their classroom practice”

(Cakmakci, 2009, p. 41). Finally, the collaborative process is an important aspect of

the teacher research model as it encourages co-construction of knowledge and

curriculum through “constructivism and reflective practice” (Campbell, 2013, p. 2).

The process of working collaboratively as part of teacher research was found by

Roulston et al., (2005) to be personally valuable to the participants involved in the

teacher research process.

Teacher research challenges the perception of the teacher’s role in the research

process and advocates for dissonance and questioning in an inquiry community to be

viewed as a positive “sign of teachers’ learning rather than their failing” (Cochran-

49

Smith & Lytle, 1999, p. 22). Therefore, teacher research is a conduit between teaching

and research as it alters the relations of knowledge and power and highlights the need

to identify the dual role of participant/researcher and “what it means to expose

intentionally one’s own biases, assumptions, and purposes” (Cochran-Smith & Lytle,

1999 p. 22). This had implications for this study as I was both the researcher and

teacher participant and held a position of leadership in the school. My position within

the school meant I needed to be aware of perceived or potential power imbalance. The

process of teacher research allows boundaries to be blurred between the different

participants as it advocates for equality amongst the participants involved in the

research. This was evident in this study as the participants had equal say in the

direction of the study and this will be indicated throughout the description of the

research methods.

3.3 RESEARCH METHODS

This section will discuss the research methods used in this case study. It includes

the research setting, a description of the participants and researcher. Each data

collection method (focus group, discussions, interview) used in this study will be

discussed.

3.3.1 Research Setting

This research was undertaken at Brightwater School (pseudonym), a

denominational private school in South East Queensland. This single sex girls’ school

(P-12) is situated in an upper socio-economic area of the city. At the time of the data

collection there were approximately 300 students in the primary school section of the

large school. The school has one class of Prep, Year 1, 2, and 3; two classes of Year

4; and 3 classes of Years 5 and 6. Brightwater School has a whole school commitment

to providing the best educational learning experiences for their students. This is clearly

articulated in the school mission statement, strategic plan and the school’s teaching

and learning framework (no citation provided to protect identity of the school). The

school has a commitment to teacher professional growth which is supported by

mentoring, professional learning (both in-house and externally), curriculum

committees, and school-based research. As part of the classroom timetable, there is

an across-year-level (P-2, 3, 4 and 5, 6) teaching structure for approximately 20% of

the timetable. The purpose of this program is to provide learning experiences that are

targeted at the children’s development, rather than the year level.

50

The research setting was selected due to the professional development approach

to teaching and learning used in the school, and due to its convenience for me. Further,

it was hoped that any knowledge gained by the participants and through the study, may

prove beneficial for the teachers and students at the school. The professional learning

program encompasses weekly interactions (in year level and/or across year level)

where the teachers and the Head of Primary as leader of curriculum reflect on student

learning, data-informed practice and decision making, as well as engage in lesson

reflection and curriculum unit planning. This approach to professional learning is

focussed on a professional culture of exploration of best practice, which provides a

strong platform for teacher research. The findings of this research study will contribute

to the decisions the school makes relating to teacher practice, students’ learning and

the sharing of information and results between teachers which is reflective of teacher

research (Campbell, 2013).

3.3.2 Participants

In this study, the teachers were selected and invited based on being a teacher of

each year level involved in the study (P-3). Using a focus group to commence the data

collection process was useful in establishing the parameters and the pathway for how

the study was planned to proceed, but with a clear understanding that the participants

could take a different pathway during the process.

The participants included four female classroom teachers, who each taught one

class of Prep, Year 1, Year 2, and Year 3 as well as myself as a participant researcher.

This was a convenience sample, as the participants were invited to participate were all

the teachers of these year levels (P-3), and therefore were invited for their accessibility,

and their willingness to be involved (Cohen, Manion & Morrison, 2011). These

teachers had a strong working relationship with each other, as they were physically

located in the same building. Based on the existing professional learning program in

the school, the P-2 teachers and I, as Head of Primary, met regularly to plan across

year level literacy and numeracy teaching. Furthermore, the Year 3 and Year 2

teachers and I met regularly to review the curriculum as preparation for the students

transitioning from Year 2 to Year 3. Some of these teachers have also been involved

in previous school based action research projects.

Four of the five teachers in the study were Bachelor of Education (Early

Childhood) (Prep, Year 1, Year 3 and the participant researcher). The Prep teacher

51

had an additional degree in Psychology (Honours). The Year 2 teacher was a primary-

trained teacher with a Master’s Degree in Educational Studies. At the time of the data

collection, each of the teachers had over five years’ experience in teaching early

childhood and the participant researcher had been Head of Primary for more than ten

years. That leadership position includes a focus on teaching and curriculum with the

participants accustomed to working alongside each other and the participant researcher

in professional learning and school research projects. They were used to expressing

their views and making contributions in relation to school based projects. Therefore,

this study resembled a similar approach to the professional learning regularly

undertaken in the school; an approach that included collaborative planning and

research, which also shares characteristics with teacher research.

Although the participant group of teachers had a history of collaborative

planning and research, I sought to ensure that any power imbalance created by my

position within the school did not impact on the voluntary nature of the teachers’

consent and the collaborative research process. The power relationship was discussed

with the participants, emphasising the relationship of the participants would be based

on a collaborative research process. An example of a collaborative research approach

was evident when the participants decided as part of the research process that I would

teach a lesson while the teachers observe how the lesson worked to promote

understanding with the students. This allowed them to feel more comfortable and for

me to effectively manage the power relation. I also needed to ensure subjectivity did

not impact upon data collection and analysis of the findings as it is not sufficient to

only acknowledge the subjectivity but rather a researcher should “systematically

identify their subjectivity throughout the course of their research” (Peshkin 1988 p.

17). Simons (2009) suggests exploring subjectivity throughout the research process

by documenting conscious biases and reflecting on procedures to reduce them.

Examples of the procedures to reduce subjective bias included checking interactions

with all participants individually, ensuring fair treatment, and allowing equal time for

each participant, so they could each contribute to the different data collection

discussions.

3.3.3 Data Collection

Several methods of data collection were implemented to address the two research

questions. The data collection took place over a period of two months at the end of

52

term three of the academic year. The sources of data were a focus group, discussions

and interviews. The data collection also included the teachers individually completing

a concept map about the sequence of the foundational skills and practices for

promoting proportional reasoning. Table 3.1 provides a sequence of the data collection

sources. The process of data collection will be discussed, followed by a description of

each type of data. Audio recordings were made of each of the data sources.

Table 3.1.

Sequence of data sources

Timeline Data Sources

4 weeks

1.

2.

3.

4.

Focus Group (FG)

Discussion: Curriculum Investigation Meeting (CI)

Discussion: Planning Meeting (PM)

Discussion: Concept Map

LESSON PLANNED BY GROUP - IMPLEMENTED BY RESEARCHER

2 weeks 5. Discussion: Post Lesson Discussion (PL)

INDIVIDUAL LESSON IMPLEMENTATION

2 weeks 6. Individual Interviews (II)

First, a focus group commenced the data collection process. As the participant

group of teachers were used to working collaboratively with each other, they were

cooperative with each other and this contributed to yielding the best information

(Creswell, 2008). The participant researcher adopted a similar approach to that used

in existing professional learning sessions prior to the research beginning. As a

participant researcher, I presented the proposed research structure: a cyclical process

of planning, teaching, and reflection of four lessons. I highlighted to the other

participants at the selection stage and at the time of the focus group, that they were

able to contribute and alter the direction of the study, therefore reducing the potential

of a power imbalance between the participant researcher and the other participants.

During the focus group, the participants and the researcher shared their understandings

of proportional reasoning and discussed how best to plan a lesson. An investigation

of the Australian Curriculum: Mathematics, as suggested by a participant, informed

the direction of the study; the other participants (n=4) supported this suggestion and

each participant investigated their year level curriculum before the second meeting.

53

Second, the teachers engaged in the Curriculum Investigation Meeting (Table

3.1 - Data Collection Point 2). At this meeting, following a review of the Australian

Curriculum: Mathematics document, each teacher presented her findings, focusing on

content strand, Number and Algebra and proficiency strand statements for their year

level. Each of the participants identified the relevant content and associated teaching

tools and made suggestions for possible lesson topics. The data collection process

changed with the introduction of a document by the Year 2 teacher, Proportional

Reasoning Lesson Study Toolkit (Mills College Lesson Study Group, 2009). This

document outlined a process for collaboratively planning a lesson for teaching

proportional reasoning. This process aligns with teacher research by “interrogating

one’s own and other practices and assumptions” (Cochran-Smith & Lytle, 1999 p. 17).

The process commenced with curriculum discussion, followed by the collaborative

planning and teaching of a lesson by one teacher and with other teachers observing.

The participants decided to read and review the document as a possible lesson planning

process. I also presented readings including the article, The development of

multiplicative thinking in young children (Jacob & Willis, 2003), as background

reading.

Third, there was a planning meeting discussion which was the juncture at which

the process took a new trajectory (Table 3.1 – Data collection Point 3). As an outcome

of the teachers reading the documentation presented at the curriculum investigation

meeting, the teachers decided to use the planning process outlined in the lesson study

tool kit. The process commenced with a curriculum discussion, followed by the

collaborative planning and teaching of a lesson by one teacher and with the other

teachers observing the student for understanding. This planning process provided

direction for the study. The participants decided only two class lessons would be

conducted: a lesson planned by the group and implemented by the researcher and an

individual lesson (planned individually and taught by that participant in their own

class). The lesson study toolkit provided a framework and process for collaboratively

planning a lesson in which proportional reasoning was contextualised into

mathematical problems. A feature of lesson study is that the collaboratively planned

lesson is taught by one person and observed by the others. Observations of the lesson

should focus on the thinking demonstrated by the students as they interact with the

problems in small groups. The participants unanimously agreed that the researcher

should teach the lesson, whilst the participants observed.

54

Fourth, as part of the cyclic planning process, the toolkit included concept

mapping exercise that allowed participants to individually map their understanding of

the sequential development of proportional reasoning in addition to tasks and

experiences that help students develop these understandings (Table 3.1 – Data

collection Point 4). The teachers drew on this information during the planning and

post lesson discussions. Fifth, a discussion was held after this lesson (group planned

lesson) implemented by the researcher. The teachers’ observations of the lesson were

then captured in the post-lesson discussion which informed the results (Table 3.1 –

Point 5). Finally, each teacher chose one of the maths concepts as the basis of an

individual lesson to be conducted by that teacher in their own classroom. The

individually planned and implemented lesson was discussed during the participant’s

individual interview with the researcher (Table 3.1 – Point 6).

The focus group and each discussion were audio recorded, and transcribed

verbatim. Notes were taken by the researcher during the individual interview. A

review of each technique and its application to the study is explored in the next section.

3.3.3.1 Focus Group

Focus groups are a form of group interview that rely on the researcher acting as

leader of the discussion and stimulating discussions among the participants (Cohen et

al., 2011). Focus groups are considered an appropriate method of exploring

participants’ experiences in depth. Kandola (2012) suggests a successful focus group

relies on the facilitator, or in this case, participant-researcher, and that “credibility,

confidence, confidentiality” is established through communication (Kandola, 2012, p.

262). These will be discussed in Section 3.5. Communication is a key feature of the

focus group process. Through communication, participants need to be informed about

the purpose of the study, process for selection of participants (as identified in Section

3.3.2.), and the process of data collection. Focus groups were used in this study to

communicate this information to the participants.

A focus group approach to data collection follows a process of questioning by

the researcher, who asks several questions based on different categories of questions:

opening, introductory, transition, and key questions (Krueger & Casey, 2009).

Generally, a focus group commences with an opening question aimed at helping

participants feel comfortable and open to talk. As the participants in this study knew

each other, and were comfortable in professional learning interactions, this was a

55

simple process that allowed the researcher to move to the next stage of the process. At

this stage, the participants talked generally amongst themselves about proportional

reasoning, discussing how they had conducted research using Google to find out more

about the topic. Their conversation was vivacious and upbeat. The researcher

explained that the session would be audio recorded only with the consent of the

teachers. The participants’ discussion seemed to become stilted once recording

commenced. The researcher shared her own personal fear, to expose her vulnerability

about hearing herself on a recording and having others hear it. The researcher

reassured the participants that the recording would be stored securely, no names would

be mentioned, and the recording would be transcribed and then destroyed. The

participants were satisfied with this explanation and were happy to be recorded

throughout the data collection process.

The researcher asked the opening question, ‘Would you like to share your ideas

about proportional reasoning’ (Table 3.1 – Data collection Point 1). The participants

were offered a topic question with the purpose of connecting the participants with the

topic (Krueger & Casey, 2009). The researcher asked, ‘What is the first thing that

comes to your mind when you hear the term proportional reasoning?’ This was

followed by transition questions, which were broad questions around the participants’

knowledge and understanding of proportional reasoning. Based on the interviewees’

responses, these questions were narrowed as the focus group progressed, for example,

‘How do the number activities relate to the development of proportional reasoning?’

As the questions narrowed the participants were given time to fully discuss their

answers. These questions were characteristic of key questions, which is those that

drive the study and the answers are the ones that require “the greatest attention in the

analysis” (Krueger & Casey, 2009, p. 40). The focus group concluded with ‘ending

questions’ which not only bring closure to the focus group, but allow participants to

reflect, clarify, and identify what is most important and what needs to be explored

further (Krueger & Casey, 2009, p. 40). This was important to the study as it allowed

the participants to provide direction for the next stage of the study: the curriculum

investigation meeting.

3.3.3.2 Discussions

Group discussions formed the data collection process for the curriculum

investigation discussion, planning meeting, concept map discussion, and the post

56

lesson discussion (see Table 3.1). For each of these data collection points the

researcher commenced the discussion with a question, based on a questioning protocol

(Krueger and Casey’s 2009):

Begin with an easy question to ensure it is available to everyone

Allow the conversation to flow by asking sequenced questions that arise

Starts with general questions and narrows for specific and important

questions (p. 38).

The opening question for each of the discussions provided the stimulus for

conversations to flow naturally. The opening questions for each of the data sources

were as followed:

Curriculum investigation discussion – What mathematical content can you

identify for your Year level that is foundational to proportional reasoning?

Planning meeting discussion – What content have you identified that could

be the basis of a lesson? How would you promote this content?

Concept map discussion – What sequence of understandings did you

identity? What were the common features across each person’s concept

map? What tasks or experiences did you identify to promote understanding?

Post-lesson discussion – What did you observe with the children when they

were engaging with the tasks?

Whilst the researcher asked the opening questioning, she ensured that the she did

not dictate the line of discussion or dominate the discussion. This purposeful

approach was to ensure there was no power imbalance and that the researcher

was considerate of her subjectivity both with her role in the school and as a

participant and, therefore, endeavoured to not allow any of her biases to

influence the discussion.

3.3.3.3 Interview

Interviews are an important way to gather data in a case study. The interviews

operate on two levels, following the researcher’s “line of inquiry” and asking questions

in a conversational, friendly manner (Yin, 2009, p. 106). In this study, interviews were

used at the end of the data collection process to capture participant’s perspectives about

the individual lesson taught, the understandings of foundational concepts, and the way

57

the participant had promoted the foundational concepts in the classroom. The data

collection processes before the interviews had created the opportunity for teachers to

feel comfortable with the research process. Through ongoing discussions with other

participants and the researcher, the participants built a level of comfort and trust in the

collaborative nature of the research process and so were more likely to be open in their

responses during the individual interview. This level of trust and comfort was also

supported through the process of providing each participant with the interview

questions prior to the interview. Some participants chose to prepare for the interview

by recording their answers to the questions and using these written responses as

prompts during the interview process. The interview provided the opportunity for each

participant to discuss individually the foundation concepts each used in the class lesson

and to also share the pedagogical practices used to promote student understanding.

This systematised data collection - incorporating focus groups, discussions, and

interviews - is a characteristic of teacher research, which distinguishes it from other

types of self-study (Campbell, 2013). Each of these data sources contributed to each

research question. In the data collection process, there was a juncture where there was

change in the data collected. This juncture was at different points for each of the

research questions. For research question 1 (identifying the foundational concepts), the

data changed after the curriculum investigation meeting. Whilst identifying ways to

promote proportional reasoning (the focus for research question 2), the change

occurred after the lesson was implemented by the researcher. This change is reflected

in the development of the template for each question and this process of template

development will be discussed in the following section.

3.4 DATA ANALYSIS

A case study is signified by the analysis of multiple data sources (Baxter & Jack,

2008). “Each data source is one piece of the puzzle, with each piece contributing to

the researcher’s understanding of the whole phenomena” (Baxter & Jack, 2008, p.

554). The data for this study were analysed at the end of the data collection period

using template analysis. Template analysis is a process “that involves the development

of a coding template” (King, 2017 p. 1). It is used for analysing data thematically by

using a coding template (Waring & Wainwright, 2008). The identification of codes

can be deductive; that is, before data analysis the researcher selects codes, based on

theory or existing research, which have the potential to be appropriate to the analysis.

58

These codes are called a priori codes (King, 2012). A unique feature of template

analysis is that a priori codes can be modified or deleted, allowing for inductively

derived codes to emerge (King, 2012).

In this study, a priori codes were used to form an initial template for each

research question. Starting with a priori codes enabled the first coding phase of

analysis to be accelerated which otherwise could be time consuming (King, 2017).

King (2012) suggests organising the template with the codes in a ‘hierarchical’ order

with overarching broad themes or categories as the heading and subcategories and/or

codes categorised under these headings. In each research question, the initial template

was applied to the data, and codes were either modified and/or new codes emerged

inductively from the data as “expressed in the discourse and language” of the

participants (Waring & Wainwright, 2008, p. 90). For each research question, the first

version of the template a priori categories were applied to the first data source. I was

open to new codes emerging inductively from this data source and the others therefore

making way for new versions of the template. That is, if the template version did not

work, it continued to be revised as I applied it to more data sources. The final version

of the template was defined and then applied to all data sources. I continued to analysis

the data until the final template or version was defined and then all transcripts, concept

map and interview were coded to it. Four versions of the template evolved for both

research questions, until the fourth and final template for each research question was

developed. A more detailed explanation of the process of template development for

each research question follows (see Section 3.4.1 and Section 3.4.2).

The final templates (one for each question) were applied to the data. The data

analysis involved reading and coding the data according to the template. The data was

collated by constructing data tables of coded excerpts to monitor the frequency of

specific codes, and determine key themes from the data. Through the peer debriefing

process, each code and coding of data was checked by the university supervisors to

enable the data to tell the story.

3.4.1 Analysis of Research Question 1

The identification of the foundational concepts of proportional reasoning was

the focus of research question one: What do Prep to Year 3 teachers recognise as

foundational concepts of proportional reasoning? In this section, the development of

the template for data analysis of research question one will be discussed. The first

59

version of the template used a priori codes which were drawn from the content

descriptions, number and place value, in the Australian Curriculum: Mathematics

because of their relationship to the development of proportional reasoning. The initial

template is shown in Figure 3.1.

1. Foundation Year

1.1 The language and processes of counting

1.2 Connect number names, numerals and quantities

1.3 Subitise small collections

1.4 Compare, order and make correspondences between collections

1.5 Represent practical situations to model addition and sharing

2. Year 1

2.1 Count collections to 100 by partitioning numbers using place value

2.2 Develop confidence with number sequences to and from 100 by ones from

any starting point. Skip count by 2s,5s,10s starting from 0

2.3 Investigate number sequences, initially those increasing and decreasing

by 2s,5s,10s from any starting point, then moving to other sequences

2.4 Recognise numbers 1-100; locate on number line

2.5 Simple addition and subtraction

2.6 Recognise and describe one-half as one of two equal parts

3. Year 2

3.1 Recognise, model, represent and order numbers to at least 1000

3.2 Investigate number sequences, initially those increasing and decreasing

by 2s, 3s, 5s, and 10s from any starting point then moving to other

sequences

3.3 Recognise and interpret common uses of halves, quarters, and eighths of

shapes and collections

3.4 Group, partition and rearrange collections up to 1000

3.5 Recognise and represent multiplication as repeated addition, groups and

array

3.6 Division as grouping into equal sets and solve simple problems using these

representations

4. Year 3

4.1 Numbers to 10 000

4.2. Model and represent unit fractions

4.3 Recall multiplication facts

60

4.4. Represent and solve problems involving multiplication

Figure 3.1. Initial Template (version 1) for Research Question 1 (adapted from

[ACARA], 2016)

The initial template (Figure 3.1) was applied to the first data source but the

organisational structure using year levels of the curriculum (Foundation Year (Prep),

Year 1, Year 2, and Year 3) was not appropriate for the collected data and required a

refinement. This led to template version 2, where the year level headings were

removed and the headings 1. additive thinking, 2. multiplicative thinking were added.

Where relevant, the content descriptions from the initial template, drawn from the

Australian Curriculum: Mathematics, were positioned within additive thinking or

multiplicative thinking. Two extra codes were inductively derived from the data and

added to the template, namely, proportional reasoning is applied to other

mathematical areas and a way of thinking.

1. Additive thinking

1.1 The language and processes of counting

1.2 Connect number names, numerals and quantities

1.3 Subitise small collections

1.4 Compare, order and make correspondences between collections

1.5 Represent practical situations to model addition and sharing

1.6 Count collections to 100 by partitioning numbers using place value

1.7 Develop confidence with number sequences to and from 100 by ones from

any starting point. Skip count by 2s,5s,10s starting from 0

1.8 Investigate number sequences, initially those increasing and decreasing by

2s,5s,10s from any starting point, then moving to other sequences

1.9 Recognise numbers 1-100; locate on number line

1.10 Simple addition and subtraction

1.11 Recognise and describe one-half as one of two equal parts

2. Multiplicative thinking

2.1 Recognise, model, represent and order numbers to at least 1000

2.2 Investigate number sequences, initially those increasing and decreasing by

2s, 3s, 5s, and 10s from any starting point then moving to other sequences

2.3 Recognise and interpret common uses of halves, quarters, and eighths of

shapes and collections

2.4 Group, partition and rearrange collections up to 1000

61

2.5 Recognise and represent multiplication as repeated addition, groups and

array

2.6 Division as grouping into equal sets and solve simple problems using these

representations

2.7 Numbers to 10 000

2.8 Model and represent unit fractions

2.9 Recall multiplication facts

2.10 Represent and solve problems involving multiplication

3. Proportional reasoning is applied to other mathematical areas

4. Proportional reasoning is a way of thinking

Figure 3.2. Template 2 (version 2) for Research Question 1 (adapted from

[ACARA], 2016)

This template (version 2) was applied to the first three data sources (focus group,

curriculum investigation meeting, planning meeting); however, the sub categories

(1.1-1.5 and 2.1-2.9; Figure 3.2) were too specific for the types of comments made by

the teachers. This led to template version 3 where the sub categories for additive

thinking and multiplicative thinking were changed from the specific content offered by

the Australian Curriculum: Mathematics to align with the phases offered by Jacob and

Willis (2003) (Figure 3.3). Another category, transitional thinking, was added, which

reflected the work of Jacob and Willis (2003). The sub-categories Proportional

Reasoning is applied to other mathematical areas and A way of thinking continued to

be useful sub categories and so were retained in template version 3.

1. Additive Thinking

1.1. One to one counting

1.2. Additive composition

2. Transitional thinking

2.1. Many to one

3. Multiplicative thinking

3.1. Multiplicative relations

4. Proportional Reasoning is applied to other mathematical areas

5. A way of thinking

Figure 3.3. Third template (version 3) for Research Question 1 (adapted from Jacob

& Willis, 2004)

62

The third template (version 3) was applied to all six data sources. The

application of this template to the teacher comments resulted in further categories

emerging about each of the sub categories (1.1, 1.2, 2.1, 3.1) which formed the fourth

and final template (version 4) (Figure 3.4). The emergence of these last details could

have been influenced by the teachers’ reading of the Jacob and Willis (2003) article,

but are also generally aligned to the initial template, which focused on the Australian

Curriculum: Mathematics, as the article and curriculum document shared identified

mathematical concepts. This final template (Figure 3.4) was applied to all data

sources, with coding of data verified through the peer debriefing process with the

university supervisors.

1. Additive thinking

1.1. One to One Counting

1.1.1 Count one to one

1.1.2 Can answer, how many

1.2 Additive composition

1.2.1 Trust the count/ Subitising

1.2.2 Additive thinking

1.2.3 Skip counting

1.2.4 Repeated addition

2. Transitional thinking

2.1 Many to one

2.1.1 Keeps track – numbers of groups and total in each group

3. Multiplicative thinking

3.1 Multiplicative relations

3.1.1 Knows multiplicand, multiplier, product

4. Proportional reasoning is applied to other mathematical areas

5. A way of thinking

Figure 3.4. Final Template (version 4) for Research Question 1 (adapted from Jacob

& Willis, 2004)

3.4.2 Analysis of Research Question 2

The identification of ways to promote proportional reasoning was the focus of

research question two, How do Prep to Year 3 teachers promote proportional

reasoning? In this section, the development of the template for data analysis of

research question two will be discussed. In the first step of analysis, a priori codes

63

were drawn from nine of the ten principles of effective pedagogies for teaching

mathematics, derived from the work Anthony and Walshaw (2009b). The tenth

principle Ethic of Care was not included as it addressed the psychological

environment, which did not offer specific ways to promote proportional reasoning.

Template version one is shown in Figure 3.5.

1. Mathematical Language

1.1 Teacher fosters use and understanding of ‘appropriate mathematical

terms’ (p.153) and mathematical meaning symbols and words through

explicit instruction

2. Communication Mathematically

2.1 Teacher explicitly teaches students to communicate mathematically using

oral, written and concrete explanations. Teacher models by revoicing

the students understanding ‘the process of explaining, justifying, and

guiding into mathematical conventions

2.2 Model and encourage mathematical argumentation

3 Assessment

3.1 Assessment for learning – uses a variety of types of assessment ‘to make

students’ thinking visible and support students’ learning.’ (p.154), provide

feedback for students.

3.2 Gathers information about students – observation, in the moment

assessment. Questioning for understanding

3.3 Self and peer assessment

4 Planning for Student Thinking

Teachers plan mathematics learning experiences that enable students to build on their existing proficiencies, interest and experiences.

4.1 Planning for learning –Teacher puts students’ knowledge, interest and

competencies at the heart of their instructional decision making.

5 Groupings

When making sense of ideas, students need opportunities to work independently and collaboratively. Different working arrangements – individual, partner, small group and whole class to allow students to make sense of mathematical ideas.

5.1 Independent thinking time – teacher gives students opportunities to think

and work by themselves.

5.2 Whole class discussion – students are active participants in purposeful,

whole class discussions which provides the opportunity to clarify their

understanding.

5.3 Partners and small groups – helps children see themselves as

mathematical learners. Enhance engagement, facilitate the exchange and

testing of ideas, encourage higher level thinking

64

6 Tools and Representations

6.1 Teachers provide a variety of representations and tools and ensures the

tools are effective to support student’s mathematical reasoning.

7 Making Connections

7.1 Teachers provide opportunities for students to connect in multiple ways

to other mathematical ideas, to listen to others thinking

7.2 To present sequential mathematical ideas

7.3 Make connections to real experiences

7.4 Highlight multiple solutions and representations

8 Mathematical Tasks

8.1 Appropriate tasks to build on students understanding, encourage

mathematical thinking. Open ended tasks

8.2 Provide problem and practice tasks

9 Teacher Learning and Knowledge

9.1 Teachers need a robust knowledge of maths in order to make informed

decisions concerning all aspects of mathematical teaching.

Figure 3.5. First template (version 1) for research question 2 (adapted from Anthony

& Walshaw, 2009)

The initial template (version 1, Figure 3.5) was applied to the focus group

transcript, but not all the subcategories were relevant. The irrelevant subcategories

(Assessment 3.1, 3.2; Groupings 5.1, 5.2; Making connections 7.2, 7.3, 7.4) were

removed. The last category Teacher learning and knowledge was also removed as it

did not provide information about how to promote proportional reasoning. An

additional subcategory was inductively derived from the data and added to the template

under the category of Mathematical tasks. This was 8.3 Using proportional reasoning

in real life and other subjects. These changes led to the development of the version 2

template (Figure 3.6).

1. Mathematical Language

1.2 Teacher fosters use and understanding of ‘appropriate mathematical

terms’ (p.153) and mathematical meaning symbols and words through

explicit instruction

2. Communicating Mathematically

2.1 Teacher explicitly teaches students to communicate mathematically

using oral, written and concrete explanations. Teacher models by

65

revoicing the students understanding ‘the process of explaining,

justifying, and guiding into mathematical conventions

2.2 Model and encourage mathematical argumentation

3. Assessment

3.1 Gathers information about students – observation, in the moment

assessment. Questioning for understanding

4. Planning for Student Thinking

Teachers plan mathematics learning experiences that enable students to build on their

existing proficiencies, interest and experiences.

4.1 Planning for learning –Teacher puts students’ knowledge, interest and

competencies at the heart of their instructional decision making.

5. Groupings

When making sense of ideas, students need opportunities to work independently and

collaboratively. Different working arrangements – individual, partner, small group and

whole class to allow students to make sense of mathematical ideas.

5.1 Partners and small groups – helps children see themselves as

mathematical learners. Enhance engagement, facilitate the exchange

and testing of ideas, encourage higher level thinking

6. Tools and Representations

6.1 Teachers provide a variety of representations and tools and ensures the

tools are effective to support student’s mathematical reasoning.

7. Making Connections

7.1 Teachers provide opportunities for students to connect in multiple ways

to other mathematical ideas, to listen to others thinking

8. Mathematical Tasks

8.1 Appropriate tasks to build on students understanding, encourage

mathematical thinking. Open ended tasks, open-ended

8.2 Provide problem and practice tasks

8.3 Proportional reasoning is used in real life and other subjects

Figure 3.6. Second template (version 2) for research question 2 (adapted from

Anthony & Walshaw, 2009)

The second template (version 2) was applied to the first four data sources (focus

group, curriculum investigation meeting, planning meeting which included discussion

of concept map however, much of the teacher discussions, about ways to promote

promotional reasoning, focused on interactions between the teacher and student. The

codes in template version 2 therefore, did not cater for the nuances of the different

66

teacher-student interactions. Some of the other categories were also redundant as they

were not relevant to the data. The researcher returned to the literature and adapted the

a priori codes based on Anthony and Walshaw (2009b)’s work to include the work of

Anghileri (2006), which offered more specific descriptions of teacher/ student

interactions. The third version of the template was developed (Figure 3.7) and

included the following changes made to the version 2 template:

1. Mathematical language category remained.

2. Communicating mathematically category was replaced by more specific codes

offered by Anghileri (2006) including: 2. explaining, 3. reviewing, 4. restructuring,

and 5. developing conceptual thinking. With each category, subcategories were also

included.

3. Assessment- assessing verbally was able to be addressed through the new categories

(2, 3, 4)

4. Planning for student thinking category remained

5. Groupings – addressed in 1.2.

6. Tools and representations category was replaced with Environmental provisions as

Category 1 (Anghileri, 2006), which included tools and representations but also

offered more details.

7. Mathematical tasks from the first template remained with one sub category retained

(7.4 Proportional reasoning is used in real life and other contexts). One new sub

category emerged from the post lesson discussion. Being 7.3 Provide proportional

reasoning problems.

1. Environmental Provisions

Environmental provisions enable learning to take place without the direct intervention of the

teacher or interactions between the teacher and students

1.1 Artefacts- learning can take place through interaction with artefacts e.g. wall

displays, manipulatives, puzzles, appropriate tools

1.2 Classroom organisation – seating arrangement, lesson structure

1.3 Structured activities – worksheets, directed activities

1.4 Free play – children set their own challenges and learn through feedback

1.5 Self-correcting – further feedback – supports autonomous learning – processes in

finding a solution

67

1.6 Grouping – peer collaboration

1.7 Emotive feedback – gain attention, encourage, approve student activities

2. Explaining

Teacher interactions that are increasingly directed to developing richness in the support of

mathematical learning.

2.1 Explaining – show and tell (traditional teaching)

2.2 Teacher in control

2.3 Structure conversations – next step. Little use of pupil contribution

3. Reviewing

The Teacher refocus student attention and provide further opportunity to develop their own

understanding rather than reliance on teacher.

3.1 Looking, touching, verbalising - getting students to look, touch and verbalise – tell

me what you did – lead student to verbalise their thinking and notice an error –

that they can correct for themselves

3.2 Using Prompting and Probing - promoting questions- support student thinking and

is responsive to student’s intentions. Probing questions – take student

understanding forward

3.3 Interpreting students’ actions and talk

3.4 Parallel modelling- teacher creates and solves a task that shares similar

characteristics of the student’s problem

3.5 Students Explaining and Justifying – teacher promote understandings through

‘orchestration’ of small group and whole class discussions so students participate

by making explicit their thinking and justify their approach to the task.

4. Restructuring

Teacher understanding students’ existing understandings but taking meaning forward.

4.1 Identifying meaningful contexts – moving to abstract understanding through the

introduction of a number of contexts.

4.2 Simplifying the problem – Teacher simplifies a task so that understanding can be

built in progressive steps

4.3 Re-phrasing students’ talk – to make ideas clearer without losing the intended

meaning

5. Developing Conceptual Thinking

Highest level of scaffolding as teachers engage their pupils in conceptual discourse that extends

their thinking. Making connections and generating conceptual discourse, less commonly found

but identified as the most effective interactions

5.1 Developing representational tools – most commonly found where teachers notate

mathematical processes

5.2 Making connections – pupils are challenged to explain their thinking and to listen

to the thinking of others

68

5.3 Generating conceptual discourse – the norms and standards for what counts as

acceptable mathematical explanation and the content of the whole class

discussion

6. Planning for Student thinking

6.1 Planning – in planning for learning, teachers put students’ current knowledge and

interests at the centre of their instructional decision making

7. Mathematical tasks

7.1 Teachers design learning experiences and tasks that are based on sequential,

sound and significant maths

7.2 Proportional reasoning is used in real life and other subjects

7.3 Provide proportional reasoning problems

8. Mathematical Language

8.1 Foster students’ use and understanding of mathematical terminology

8.2 Modelling appropriate terms

8.3 Communicating in ways that students understand.

Figure 3.7. Third template (version 3) for research question 2 (adapted from

Anthony & Walshaw, 2009 and Anghileri, 2006)

The third template (version 3) (Figure 3.7) was applied to all the data sources.

The application of this template resulted in version 4 of the template, which included

two categories Restructuring, and Developing conceptual thinking and their

subcategories (4.1, 4.2, 4.3; 5.1, 5.3) being removed as they were identified as

superfluous, as the data did not reflect these levels of teacher/student interactions. A

new category code Student conceptual understanding emerged from the data (post

lesson discussion) to reflect the participants discussion about the importance of

checking student understanding.

The first sub category (7.1) of Mathematical tasks was removed as it was not

represented in the data. The last two subcategories (1.5. Grouping, 1.6. Emotive

feedback) of Environmental provisions were also removed as there was no evidence

of these in the data. Two subcategories of Reviewing were removed (3.2 Prompting

and Probing). This was similar to the category Student explain and justifying and 3.4

Parallel modelling (no data was reflected this sub category).

The final template (version 4) (Figure 3.8) was then applied to all data sources.

69

1. Environmental Provisions

Environmental provisions enable learning to take place without the direct intervention of

the teacher or interactions between the teacher and students

1.1 Artefacts- learning can take place through interaction with artefacts e.g.

wall displays, manipulatives, puzzles, appropriate tools

1.2 Classroom organisation – seating arrangement, lesson structure

1.3 Structured activities – worksheets, directed activities

1.4 Free play – children set their own challenges and learn through feedback

1.5 Self-correcting – further feedback – supports autonomous learning –

processes in finding a solution

2. Explaining

Teacher interactions that are increasingly directed to developing richness in the support

of mathematical learning.

2.1 Show and tell (traditional teaching)

2.2 Teacher in control

2.3 Structure conversations – next step. Little use of pupil contribution

3. Reviewing

The Teacher refocus student attention and provide further opportunity to develop their

own understanding rather than reliance on teacher.

3.1 Looking, touching, verbalising - getting students to look, touch and

verbalise – tell me what you did – lead student to verbalise their thinking

and notice an error – that they can correct for themselves

3.2 Interpreting students’ actions and talk

3.3 Students Explaining and Justifying – teacher promote understandings

through ‘orchestration’ of small group and whole class discussions so

students participate by making explicit their thinking and justify their

approach to the task.

4. Planning for Student thinking

4.1 Planning – in planning for learning, teachers put students’ current

knowledge and interests at the centre of their instructional decision

making

5. Mathematical tasks

5. 1. Proportional reasoning is used in real life and other subjects

problems

5.2 Proportional reasoning is used in real life and other subjects

6. Mathematical Language

6.1 Foster students’ use and understanding of mathematical terminology

70

6.2 Modelling appropriate terms

6.3 Communicating in ways that students understand.

7. Student Conceptual Understanding

7.1 Supporting students in the development of understanding

Figure 3.8. Final template (version 4) for research question 2 (adapted from

Anghileri, 2006)

The development of the template for the coding of the transcripts provided the

opportunity for the researcher to interpret the data. The transcripts were also analysed

based on the frequency of the codes and this provided further data for interpretation.

3.5 CREDIBILITY

Credibility is offered by Trochim and Donnelly (2008) as a “criteria for judging

qualitative research” (p. 149). It is believed that qualitative research needs its own

criteria because of the investigation of people’s ideas, experiences and points of view

(Kumar, 2014). Credibility is one of the most significant elements for establishing

trustworthiness in qualitative studies (Shenton, 2004; Simons 2009). It is the process

by which a researcher aims to ensure “that their study measures or tests what is actually

intended” (Shenton, 2004, p. 64). As suggested by Shenton (2004), credibility is

established through acknowledging the importance of the participants and role they

play in the study, use of variety of research methods, data analysis supported by peer

debriefing, reflexivity and thick rich descriptions. Each of these will now be discussed.

Prior the commencement of the study, the potential participants were invited to

find out about the study, to see if they were interested and if they regarded the study

as relevant. Other teachers with experience in teaching in early childhood had

expressed a willingness and interest to participate if one of the invitees chose not to do

so and this was shared with the invitees. At this point, I briefly outlined the study in

terms of relevance and revealed that the study would include a collaborative approach,

which allowed the participants to contribute to the direction of the proposed study with

the research process including different data collection sessions (focus group,

discussions, collaboratively planning, and participating in a one to one interview).

An understanding of and establishing familiarity of the setting is suggested as a

way a researcher can promote confidence in the participants and help establish

credibility. It is suggested that the researcher initially develops an understanding of the

71

organisation and establishes a relationship of trust between the researcher and

participants (Shenton, 2004). In this study, this was already established as the study

took place in my ‘own backyard’ because of the teacher research approach to the study.

I had an existing and trusting relationship with each of the participants through

working collaboratively at the school. However, a concern could emerge for the

participants because of the change of relationship from staff member to participant

researcher. I needed to be cognisant of this to ensure this already established

relationship was maintained by employing strategies to support the participants.

Strategies were also employed to address the process for valuing the participants

and to support honesty in the participants, a tactic suggested as a way to contribute to

credibility (Shenton, 2004). I was aware of risks to the reputation of the participants,

so endeavoured to establish a safe psychological environment. The participants were

made to feel that everyone’s opinion was equally valued and all participants were

given equal time to contribute during the focus group and discussions. The participant

group of teachers encouraged each other to be open and honest from the outset of each

session and reassured each other that all answers were accepted and that there were no

right answers. As a participant in the study I participated in, and contributed to, all

aspects of the study to minimise perception of power imbalance and as part of the

process of teacher research. During data collection and analysis, I adhered to QUT’s

data management procedures. The employment of numerous sources of evidence is

an important characteristic of case study and the inclusion of well-established data

gathering methods and data analysis procedures contributes to the credibility of the

study (Shenton, 2004; Yin, 2009). In this study, credibility was established through

the process of questioning and interaction used in this data gathering sessions and

described in Sections 3.3.1; 3.3.2; and 3.3.3.

Credibility was added to the study through peer debriefing processes carried out

for the duration of the study. To provide credibility, the data analysis was regularly

checked by “someone who is familiar with the research or phenomenon being

explored” as part of this peer debriefing review process (Creswell & Miller, 2010, p.

129). This process was used to develop the template and analysis the data both

contributing to credibility. During the evolutionary process of the development of the

template the researcher and university supervisors, (as part of the peer debriefing

process) evaluated the development of the final template. The application of the final

72

template for each question was applied to the data sets and credibility of codes and

template application was achieved through peer debriefing process of discussion and

collaborative interactions. This process allowed the researcher to justify decisions and

clarify outcomes. Probing from those in supervisory roles “may help the researcher to

recognise biases and preferences” (Shenton, 2004, p. 67).

Reflexivity was gained through the process of template analysis and researcher

reflexivity. The process of template analysis allows the researcher to make explicit,

informed analytical decisions based on the analysis of each transcript. The

development of the successive versions of the templates enabled reflexivity as the

process required the researcher to be “explicit about the analytical decisions” made.

(King, 2017, p. 3). The researcher also adopted a reflexive approach in the context of

the research setting. It is important for validity that the researcher, “self-disclose their

assumptions, beliefs and bias” (Creswell & Miller, 2010, p. 127).

It is also suggested that credibility is established through “thick, rich description”

(Creswell & Miller, 2010 p. 128). In addition to describing the setting and the

participants, the results included detailed and many excerpts of transcripts “to provide

as much detail as possible” (Creswell & Miller, 2010). Measures were instigated

throughout this study to ensure trustworthiness and provide credibility for the

participant group of teachers and the data collected.

3.6 ETHICAL CONSIDERATIONS

Ethical considerations for participants of this study were addressed in two areas:

seeking consent and the possibility of causing harm to the participants (Kumar, 2014).

Firstly, consent was sought by the researcher. The participants were informed at the

outset that participation in this study was voluntary and the teachers were free to

withdraw at any stage without penalty or comment. If the participants chose to

participate they had the freedom to contribute or limit their contribution at the focus

group, discussions, or during collaborative planning. It was clearly communicated that

the contributions could be as little or as much as the participant felt comfortable with.

The participants were clearly informed in writing (participant information sheet) that

they had the freedom to choose if they wanted to be involved and were offered an

opportunity to think about the study and return the consent forms as acknowledgement

of involvement.

73

The participant researcher also needed to consider the possible cause of harm

and ensure that the risk is minimal; that is, “the extent of the harm or discomfort in the

research is not greater than that ordinarily encountered in daily life” (Kumar, 2014, p.

286). The research carried low risk of harm to participants as they were adults and

could ably make an informed decision about being involved in the study. While it is

not possible for the participants to be anonymous to the researcher during the

collection of the data, confidentiality of the school and teachers was ensured using data

collection codes (to de-identify the raw data) and pseudonyms were used in reporting.

Throughout the data collection process, I was cognisant of the complexities of

her involvement from the point of view of the other participants and their possible

perception of power imbalance. There was also a chance of discomfort for the

participants because of the possibility of perceived power imbalance, so every effort

was made to minimise risks through a collaborative approach to the research process

and the inclusion of choice throughout the research process. This was at the forefront

of my mind and I took active steps to reduce this perception. To reduce the possibility

of discomfort because of the perceived power imbalance, the participants were

informed that the data collection process would involve the participants and researcher

working collaboratively and contributing to the direction of the study, which is

reflective of teacher research process. The participants were also informed at the

outset about conduct and expectations; that is, if they decided not to participate or

withdraw from the study at any time, this would have no impact on the future

relationship with me in my position as Head of Primary and/or career progression at

the Primary School. This process was enacted to reduce the possibility of a perceived

sense of coercion and to manage the situational relationship.

In summary, at all stages of the research process, ethical practices were

paramount. This study required ethical considerations in respect of the individuals and

the site. A low risk ethics application was submitted and approved by QUT’s Human

Research Ethics Committee (UHREC), within the Faculty of Education. Ethical

clearance by UHREC for a ‘Negligible/Low Risk’ activity was granted (Approval

Number: 1400000063). Having obtained ethical approval, research commenced with

informed consent from the school and the teachers involved in this case.

74

3.7 CHAPTER SUMMARY

To identify the foundational concepts of promotional reasoning and ways to

promote it in the classroom, this study adopted an exploratory case study in a single

case embedded design (Yin, 2009). The design consisted of data collection from

multiple sources, as per case study design. The data sources included a focus group,

discussions, and individual interviews. Data analysis was conducted using template

analysis and, through the process of analysis, different templates were developed and

applied until two templates were finally developed to address each of the research

questions. These templates were applied to the data in a process that allowed the data

to tell the story. The trustworthiness of the study was executed through peer reviewing

or debriefing. Ethical considerations were addressed by ensuring the participants saw

their participation in the study as relevant, that consent was obtained, and by ensuring

the possibility of causing harm and risk to the participants was minimal. Chapter 4

discusses the outcomes of the methodology in relation to the results.

75

Chapter 4: Results

The research questions in this study endeavoured to identify the foundational

concepts of proportional reasoning and ways in which they were promoted by the

teachers who participated in this study. This study investigated two questions:

1. What do Prep to Year 3 teachers recognise as foundational concepts of

proportional reasoning?

2. How do Prep to Year 3 teachers promote the foundational concepts of

proportional reasoning?

Data were collected using focus group, discussions, one-to-one interviews and written

concept map as detailed in Table 4.1 . The data collection process included two lessons

(one observed by the group and an individual lesson), which informed the post-lesson

discussion and individual interview.

The data collection commenced with the teachers participating in a focus group

(Table 4.1 – Point 1). The focus group was designed to encourage teachers to share

their ideas on the foundational concepts of proportional reasoning and how they

promote such concepts in their classrooms. During the focus group discussion, the

teachers identified the need to engage in independent investigation, which included

reviewing the Australian Curriculum: Mathematics (prior to progressing with the

intended lesson planning). This led into the data collection process of the curriculum

investigation meeting (Table 4.1 – Point 2), where the teacher and researcher reviewed

number and algebra content strand and mathematical proficiency strand. The

subsequent lesson planning meeting (Table 4.1 - Point 3) was informed by a reading

provided by me, this reading was, The development of multiplicative thinking in young

children by Jacob & Willis (2003) which contributed to curriculum background for the

lesson and a lesson study toolkit as detailed in Mills College Lesson Study Group

(2009) provided by a teacher. Both these documents provided background for the

lesson planning. The lesson study toolkit included a concept map exercise (Table 4.1

– Point 4) completed by the teachers individually. The exercise required the teachers

to individually identify a sequence of understandings that they believed contributed to

the development of proportional reasoning across P-3. Each teacher also drew on the

76

concept map for the lesson topic of their individual lesson. The results of these two

activities (identification of foundational concepts, lesson topic) are outlined in Table

4.5.

Table 4.1.

Sequence of data collection

Data Sources

1. Focus Group (FG)

2. Discussion: Curriculum Investigation Meeting (CI)

3. Discussion: Planning Meeting (PM)

4. Discussion: Concept Map

LESSON PLANNED BY GROUP - IMPLEMENTED BY RESEARCHER

5. Discussion: Post Lesson - group discussion (PL)

INDIVIDUAL LESSON IMPLEMENTATION

6. Individual Interviews (II)

Two lessons were taught, the first lesson was planned by the group and taught

by me (Table 4.1 – lesson planned by the group and implemented by the researcher).

The second was an individual lesson, which focused on a foundational concept of

proportional reasoning (Table 4.1 – individual lesson implementation), taught by each

teacher to her own class. The approach to lesson planning for the group lesson was

based on an interpretation of lesson study that was recommended by the Year 2 teacher.

This involved a process where the teachers planned the lesson collaboratively, it was

taught by me and then observed by the other participants. Lesson study approach and

the framework outlined in the document (Mills College Lesson Study Group, 2009)

was adopted by the group as a problem-solving approach that allowed the teachers to

observe the students’ approaches to solving proportional reasoning problems.

Therefore, the teachers decided I should teach the lesson as they felt they wanted to

observe the students, in the observation class, as they interacted with the problem with

their peers. The teachers planned the first lesson collaboratively for the Year 3

students (Table 4.1– Point 3), which was then taught by me (Table 4.1 – Lesson

planned by group implemented by researcher). The lesson was followed by a post-

lesson discussion aimed at capturing key observations of the students from the

participating teachers (Table 4.1 – Point 5). For individual lesson implementation,

77

each teacher selected a foundational concept (Table 4.1. - Point 4) as the focus of their

own individual lesson (Table 4.1 – Individual lesson). Data were collected in relation

to the individual lessons during the follow up individual interviews (Table 4.1 – Point

6). Data collected included the concept taught and the pedagogical practices used to

promote proportional reasoning.

The results are presented in two main sections based on the research questions.

The first section will present the data in relation to the first question - the participating

teachers’ understandings of proportional reasoning (Section 4.1). In this section,

teachers’ understandings of proportional reasoning, in relation to the foundational

concepts, are presented in sections prior to the curriculum investigation (Section 4.1.1)

and after curriculum investigation (Section 4.1.2).

The second section of this chapter focuses on the second question - how the

participating teachers promote proportional reasoning (Section 4.2). The section

presents data collected prior to the group lesson implementation (Section 4.2.1) and

after the group lesson implementation (Section 4.2.2). The results will be supported

by transcripts of the dialogue from each of the participants and will be referenced by

first element = participant, second element = source, third element= transcript

reference (see Table 4.2 for an explanation of the labelling system).

Table 4.2.

Explanation of the labelling system for data references, e.g. T3:PL:56

Labelling System for Data References

First element Second element Third element

TP Teacher Prep FG Focus group Line number

T1 Teacher 1 CI Curriculum

Investigation Meeting

T2 Teacher 2 PM Planning Meeting

T3 Teacher 3 PL Post-lesson group

discussion

II Individual Interview

78

4.1 UNDERSTANDINGS OF PROPORTIONAL REASONING

In relation to the first question concerning teachers’ identification and

understandings of the foundational concepts of proportional reasoning, a change was

evident before and after the curriculum investigation meeting. The final template (see

Figure 4.1 below) was applied to all the data sources.

1. Additive thinking

1.1 One-to-one Counting

1.1.1 Count one-to-one

1.1.2 Can answer, how many

1.2 Additive composition

1.2.1 Trust the count/subitising

1.2.2 Groups

1.2.3 Skip counting

1.2.4 Repeated addition

2. Transitional thinking

2.1 Many to one

2.1.1 Keeps track – numbers of groups and total in each group

3. Multiplicative thinking

3.1 Multiplicative relations

3.1.1 Knows multiplicand, multiplier, product

3.1.2 Inverse relationships

4. Proportional reasoning is applied to other mathematical areas

5. A way of thinking

Figure 4.1. Template for Research Question 1

4.1.1 Prior to Curriculum Investigation Meeting

The participants’ discussion during the focus group and including the curriculum

investigation meeting initially focussed on sharing their general understandings of

proportional reasoning. The results in relation to the data identified in these two data

sources will be discussed according to the codes identified in the template.

Proportional reasoning is applied to other mathematical areas

In relation to mathematics, the participants (n=4) identified mathematical

concepts and areas that related to proportional reasoning. These mathematical areas

79

included measurement (quantities, capacity and time), graphing, probability,

percentage, ratio, fractions, decimals and place value. The identification of these

areas was reinforced by the Year 1 teacher, who acknowledged that proportional

reasoning encompasses many mathematical concepts and understandings,

When you look at and I would think that once you looked at the curriculum and

unpacked it, you’d find that most of it would have proportional reasoning – proportional

reasoning would have relevance to most of those mathematical concepts and skills. (T1:

FG:35)

However, the Year 2 teacher recognised that proportional reasoning was more than

just the mathematics concepts, “I always thought it was fractions moving into decimals

and place value, but it is a lot more” (T2: FG:11). An indication of what ‘more’ might

be was reflected in the discussions in relation to proportional reasoning being a way

of thinking.

A way of thinking

Three of the participants offered a broad description of proportional reasoning

in terms of it being a way of thinking as an adult. The Year 1 teacher stated, “it really

is the kind of maths that you do use as an adult, isn’t it?” (T1: FG:7). Two participants

indicated that this way of thinking is intuitive in adults, “We use it all the time, every

day, but I think up until this point we haven’t realised what it is” (T3: FG:9). Teacher

P built on this concept by acknowledging, “Just being able to know a bit more about

this big concept that seems to have always been there, but not something in my

conscience” (TP: FG:48).

In terms of the type of thinking involved in proportional thinking, the Prep

teacher offered a more specific description to proportional reasoning as a way of

thinking, by saying it is about, “changing and comparing things based on reasoning

and being able to take something that you know, use that knowledge to apply to another

situation” (TP: FG:16). The Year 1 teacher made the link between the teacher’s

thinking and student understanding, “I think that explicitness of it does really impact,

a lot of the time on your thinking as a teacher, but also the children’s learning. It

[proportional reasoning] puts a focus on things” (T1: FG:78). More specifically,

Teacher 3 suggested that this way of thinking, proportional reasoning is developmental

in its understanding, “It’s something you’d have to build on. You would have to start,

obviously, in Prep. Then you work through building on the concepts” (T3: FG:48).

80

Moreover, the participants (n = 4) acknowledged that not all children intuitively grasp

thinking proportionally and that it may be necessary to explicitly teach children this

‘way of thinking’, as is evident in the following dialogue between the participants:

Teacher 2: I think some children do it intuitively.

Teacher 3: Like us- we don’t know, but we are just doing it.

Teacher 3: They’ll make those links

We don’t know and they’re not aware either. They’re just doing

it.

Teacher P: It’s their problem-solving strategy. It’s innate, they didn’t know

how to do it.

Teacher 1: Well that’s what the research says. That 50 percent of adults can

do it and 50 percent obviously can’t.

Teacher 2: They’re just not aware they can. (FG:52-56)

This dialogue highlights that the teachers knew that proportional reasoning could

be intuitive for some and not for others, suggesting that teachers acknowledge that

proportional reasoning needs to be taught so all children - and, in turn, adults - can

learn to think proportionally. A more definitive way of thinking was offered through

the suggestion of the relevance of additive and multiplicative thinking to the

development of proportional reasoning. This dialogue also indicates that the teachers

made the connection between proportional reasoning and problem solving and this will

be discussed in research question 2.

Additive and multiplicative thinking

Additive thinking occurs when an amount – the original amount – is changed by

an absolute or fixed amount (Langrall & Swafford, 2000). In the learning trajectory,

of proportional reasoning, multiplicative thinking builds upon and moves beyond

additive thinking, as it requires a new level of sophistication when thinking about

number and a grounded understanding of multiplication and division (Ell et al., 2004).

Two teachers (Teacher 1 and 2) acknowledged the relationship of additive thinking

and multiplicative with proportional reasoning, “I guess maybe, just that sense, it’s

[proportional reasoning] really about the multiplicative concept, rather than it being

additive” (T1: FG:5). The Year 1 teacher highlighted that proportional reasoning is

about multiplicative thinking, whereas the Year 2 teacher’s comment was made in the

context of Year 2, identifying that she needed to do more work on multiplicative rather

than just additive thinking. “Maybe we do too much of that [additive] and they don’t

81

get to work more on that multiplicative sort of thinking?” (T2: FG:16). The teachers’

statements were indicative of an emerging knowledge of the relevance of

multiplicative thinking to proportional reasoning, additive thinking to multiplicative

thinking.

At this point I made the comment, “You just said maybe rather than just doing

all additive you need to do more multiplicative or make it more explicit, multiplicative.

How would you go about this?” (FG:22). The teachers responded with identifying

ways to promote, for example, real life, language. It wasn’t until the Year 1 teacher

made the connection between the associated thinking of proportional reasoning and

investigating Australian Curriculum: Mathematics did the focus become more about

the number concepts. She stated,

I think that once you looked at the curriculum and unpacked it, you’d find that

most of it would have – proportional reasoning would have relevance to most

of the mathematical concepts and skills. It’s about teaching them the thought

process and then building up those foundation skills to be able to apply it

across all maths areas. (T1: FG:34)

In this statement, the participant initially made the connection between the relevance

of proportional reasoning and “mathematical concepts”, inference however could be

drawn from her statement about teaching the “thought process”, that is additive and

multiplicative thinking. Therefore, the participants decided to collaboratively review

the proficiency strand mathematical reasoning and the content strand Number and

Algebra. The curriculum investigation meeting of the Foundation (Prep) to Year 3

content (Table 4.1 – Point 2) found that neither proportional reasoning, nor the

foundational concepts, were overtly identified in the Australian Curriculum:

Mathematics. However, the participants identified content descriptions in Australian

Curriculum: Mathematics that they believed were in effect the foundational concepts

of proportional reasoning, albeit that they were not explicitly identified as such. I was

able to be categorise the content descriptions according to additive and multiplicative

thinking (Table 4.3).

Table 4.3.

Foundational concepts of additive and multiplicative thinking ([ACARA], 2016)

82

Additive Thinking Multiplicative Thinking

Foundation Year

(identified by Teacher P)

Equal and unequal shares

Year 1

(identified by Teacher 1)

Counting collections using

partitioning

Solving addition and

subtraction

Making groups of and

sharing

Year 2

(identified by Teacher 2)

Group partition and

rearrange collections up to

1000

Representing multiplication

as repeated addition and

groups

Comparing numbers to

1000

Division- sharing in equal

sets

Fractions – recognising

common use of halves,

quarters, eighths of shapes

and collections

Year 3

(identified by Teacher 3)

Partitioning up to 10 000 –

rearranging

Relationship of

multiplication and division

Fractions

In addition to identifying these content descriptions from the Australian

Curriculum: Mathematics, the participants (n=3) acknowledged their role in helping

students to think both additively and multiplicatively. As the Prep teacher suggested,

I think the important part is, I think where we’re going is, there’s a difference

between the additive and the multiplicative thinking of students. That’s where

our curriculum kind of sits and we need to develop those things. (TP:CI:5)

The Year 1 teacher affirmed, “We’re looking at developing the foundational skills that

they need to be able to then think proportionally or multiplicatively” (T1:CI:6). The

Year 2 teacher identified the need to “recognise the difference between absolute or an

additive and relative or multiples of change” (T2:CI:5).

In summary before the curriculum investigation meeting, the teachers’

understanding of proportional reasoning reflected a general understanding; whereas

after the curriculum investigation meeting, the understandings were more specific in

relation to identifying the foundational concepts of proportional reasoning. The main

83

categories from the template that were used to code the participants’ discussions

included proportional reasoning is applied to other mathematical areas, a way of

thinking, additive thinking, multiplicative thinking. As the discussion progressed and

the participants’ comments became more specific with proportional reasoning as a way

of thinking identifying proportional reasoning as involving additive and multiplicative

thinking. The identification of the relationship between proportional reasoning and

multiplicative thinking highlighted the difference between additive and multiplicative

thinking and informed the first finding, proportional reasoning is based on additive

and multiplicative thinking.

4.1.2 After the curriculum investigation meeting

A new trajectory was taken after the curriculum investigation meeting, with the

participants moving from identifying broad categories of thinking (additive and

multiplicative) and content descriptions during the curriculum investigation meeting

(Table 4.3) to identifying more specific foundational concepts. Interestingly, as the

discussions after the curriculum investigation meeting progressed, I did not need to

ask many questions to facilitate the teachers’ discussions. The teachers responded to

each other to keep the flow of the discussion and self-guide the direction of it. This

section looks at the key points identified in the planning meeting (Table 4.1- Point 3)

and outlined in Table 4.4. The foundational concepts identified by the teachers and

relating to additive thinking, transitional thinking and multiplicative thinking will be

presented. This will be followed by a focus on transition between additive and

multiplicative thinking with the sharing of data from the post lesson discussion,

relating to the identification of the foundational concepts, there having been a greater

depth of discussion in relation to this topic at that time.

Table 4.4.

Additive and multiplicative thinking evolves from foundational concepts

84

The main types of thinking identified (additive and multiplicative thinking)

provide an overarching structure for understanding and provide the top level in Table

4.4. The planning meeting discussion was informed by a reading offered by me (Jacob

& Willis, 2003) that identifies phases of development; these phases allow the

overarching thinking to be broken down into smaller components, reflective of early,

then more advanced, thinking as the learners progress through these phases (Jacob &

Willis, 2003). These phases are listed in the second level headings (Table 4.4). During

the discussion and on the concept map, the teachers identified foundational concepts

that contribute to additive and multiplicative thinking. Additive thinking evolves from

the foundational concepts contributing to the first two phases (one-to-one counting and

additive composition). There is a transition between additive and multiplicative

thinking and this is identified as many-to-one phase. Multiplicative thinking evolves

from the foundational concepts of the last phase (multiplicative relations). The

foundational concepts identified by the teachers drawn from the curriculum

investigation meeting and other data collection sources after the curriculum

investigation meeting provide the third level heading in Table 4.4.

As part of the lesson planning process, a concept map (Table 4.1- Point 4) was

created by each participant to identify her understanding of the sequence of concepts

to develop proportional reasoning. This concept map exercise provided the

opportunity for each teacher to identify concepts by drawing on her own knowledge

that she believed would contribute to proportional reasoning, even if they are not part

of a teacher’s year level curriculum. All teachers identified additive thinking and all

ended the sequence with multiplicative thinking. The participants also used this

concept map to identify a topic for the individual lesson. The Year 3 teacher was

unavailable to participate in the individual interview and share the lesson topic,

therefore no results are available in this row (Table 4.1-Point 6).

Table 4.5.

Teacher’s identification of foundational concepts in concept map and individual

lesson topic

85

Table 4.5 identifies the foundational concepts each teacher in each year level

identified in the concept map. It also identifies the topic that the teacher chose to teach

to her class. The foundational concepts identified in Table 4.4, from which additive

and multiplicative thinking evolves, will now be discussed in relation to the data from

the planning meeting, concept map (Table 4.5, concept map), and individual lesson

topic (Table 4.5, lesson topic).

Additive Thinking

The early foundational concepts of additive thinking within the phases (one-to-

one and additive composition) will be discussed. The foundational concepts one-to-

one counting and knowing how to answer, how many questions were named by the

teachers (n=3). One-to-one counting was identified at the beginning of the sequence

of development in each of the concept maps of the Prep, Year 1 and Year 2 teachers.

The Year 1 teacher highlighted this sequence in relation to the development of

multiplicative thinking,

With the understanding that multiplicative thinking leads to proportional reasoning, but

there’s also the progression of additive thinking prior, looking at the one-to-one and

trust the count strategies. (T1:PM:1)

Further, the Prep teacher emphasised the importance of one-to-one phase,

86

I am going back and having a look at that first sort of phase, that one-to-one counting

and trusting the count made me sort of realise that it is embedded into our math lessons

all the time. It’s a very important skill that we focus on for Prep. (TP:PM:21)

The Prep teacher was the only participant to record on her concept map final

number counted=total-how many. The identification of this by the Prep teacher is

reflective of its relevance to this year level as it is an expectation that students in prep

know how to answer the how many question, that is count the objects and final number

stated, is the answer to this question. An understanding of the concepts (trust the count

and skip counting) reflects a more developed additive thinker and with the learner

being in the next phase, additive composition.

The foundational concepts, trust the count/subitising, groups, skip counting and

repeated addition (belonging to the additive composition phase), were most referenced

during the curriculum investigation meeting (Table 4.3), planning meeting, concept

map and lesson topic (Table 4.5). These foundational concepts will now be discussed

in relation to the data collected from the participants.

Trust the count/subitising – The Prep, Year 1 and 2 teachers identified trust the

count/subitising on their concept maps with both the Prep and Year 1 teacher selecting

this foundational concept as the topic for their individual lesson. In total, it was the

most referenced on Table 4.5, indicated by the higher frequency of ticks. However,

the four classroom teachers made comments in relation to the importance of trust the

count in early childhood mathematics. The Prep teacher stated, “So I can see children

who may have trust the count now, where I can take them and what I can move them

towards to get them to that deeper thinking” (TP:PM:22). This was also reinforced by

the Year 1 teacher and by the Year 3 teacher:

When you look at what’s involved in the trust the count strategy, is quite in

depth and there’s a lot that maybe they don’t have a solid understanding in

terms of just the base number. (T1:PM:19)

Even though the kids can trust the count, do they truly have that deep understanding

while they’re going through it. (T3:PM:18)

The ability to trust the count relates to the ability to subitise, a link identified by

the Prep and Year 1 teachers. The Year 1 teacher stated that, “some children still have

a lot of trouble with that in that very deep understanding of trust the count. Of course,

87

that links to subitising” (T1:PM:16). The Prep teacher made the link with looking for

the patterns in the numbers to help with subitising,

They are good at subitising small numbers and we’re working on the higher ones, pretty

much between 6 and 10 actually and taking time to look at the patterns in the number.

(TP:PM:17)

These statements demonstrate the teachers (n=3) acknowledged that trust the

count was more than making sure that children could simply trust the count. It was

appropriate both the Prep and Year 1 teacher noted the importance of student

understanding of trust the count, given its relevance to the curriculum for these year

levels. However, it is interesting to note that the Year 3 teacher also identified the

importance of trust the count and children having a “deep understanding,” suggesting

that this early foundational concept still has relevance to students in Year 3. The link

of trust the count to the development of subitising was also made by the Prep teacher

and the Year 1 teacher. The Prep teacher highlighted that students are confident with

perceptual subitising (numbers 4 or less) but needed to support them in conceptual

subsiting, that is, visual chunking of numbers for larger numbers (4-10).

Groups – The relevance of the concept of groups to additive thinking was

highlighted in the curriculum investigation meeting by the Prep, Year 1 and Year 2

teachers each identifying content descriptions that have a focus on groups. As outlined

in Table 4.3 the concepts equal and unequal shares (foundation/prep), counting

collections using partitioning, making groups of (Year1), group partition, representing

multiplication as repeated addition and groups (Year 2) were identified in relation to

developing understanding of groups. On her concept map, the Prep teacher identified

counting in groups as a foundational concept in the sequence of proportional reasoning

development. An understanding of groups is closely related to repeated addition.

Repeated addition – The foundational concept, repeated addition, was identified

twice by the Year 2 teacher. First, the identification of the content descriptor,

representing multiplication as repeated addition in the curriculum investigation

meeting (Table 4.3). Second, it was highlighted in the planning meeting by the same

teacher that she relied on repeated addition for teaching multiplication, “I probably

focussed too much for a long period on repeated addition because I think repeated

addition seemed to be something that made sense to the girls” (T2:PM:26). The Year

3 teacher also made the same connection, when she identified repeated addition on her

88

concept map in relation to multiplication, suggesting that she also makes the link

between repeated addition to teaching multiplication. The foundational concept,

repeated addition, lays the foundation for the more sophisticated skip counting.

Skip counting – Skip counting was identified on most occasions by the Year 1

teacher, who identified it in her concept map and also discussed it several times in the

planning meeting. The relevance of this foundational concept to Year 1 was

highlighted by the link to the Year 1 mathematics curriculum, “There’s a big focus

which links happily to the Australian Curriculum: Mathematics effective counting

strategies using the skip counting” (T1:PM:8), further noting, “I think with the Year

1s and reference to skip counting that related more specifically to using that for sharing

purposes” (T1:PM:14). Like she did for trust the count, the Year 3 teacher once again

made reference to a foundational concept that was not in the Year 3 curriculum but

one that she felt had relevance to Year 3 students if they didn’t possess a deep

understanding. She highlighted the importance of students having a deep

understanding of skip counting,

Also, to ensure the students are not just reciting their counting, ‘do they truly

understand the skip count or whether they just skip counted by rote? Where

they have just learnt it as a system or whether they truly understand’?

(T3:PM:10)

Building on the linkages of skip counting and its importance, the Year 1 teacher

made a direct link of skip counting to multiplicative thinking “They say that’s where

the multiplicative thinking lies if they’ve got that understanding of skip counting”

(T1:PM:11). Whilst skip counting is foundational to multiplicative thinking, it is

established within the foundational concepts of additive thinking as it is based on

addition. Teacher 3 also made the connection between skip counting and repeated

addition, highlighting the developmental relationship between the two, “Skip counting

starts with repeated addition and then it flows on from there” (T3:PM:9). The Year 2

teacher highlighted the need to ensure students have a strong understanding of the

foundational concepts, “Going back and seeing what additive thinking they may not

have, consolidate and other things” (T2:PM:5). The statement highlighted the

importance of checking to ensure students have an understanding of these concepts.

The Year 2 teacher also made connections between the foundation concepts or

lower level to the higher-level concepts, “I know those things like trust the count, skip

89

counting all those things are important, but I didn’t realise they were tied in with the

concepts so tightly” (T2:PM:25) This statement is indicative of making the connection

between the foundational concepts from which additive thinking evolves in order to

develop multiplicative thinking.

The participants in the study identified that additive thinking evolves from the

foundational concepts (trust the count/subitising, groups, repeated addition, and skip

counting). The consolidation of these concepts is necessary for the development of

additive thinking and are important for the evolution of multiplicative thinking.

Transitional thinking

Transitional thinking or many-to-one phase (Jacob & Willis, 2003) is a more

advanced additive thinking as the learner transitions to multiplicative thinking. More

advanced additive thinking requires the student to keep hold of two of the three

components of multiplication, number of groups and number in each group (Hurst,

2015; Jacob & Willis, 2003). That is, the student can hold two numbers in their head

at once. This is the foundational concept that supports transitional thinking and was

the least identified by the participants but identified mostly by the Year 2 teacher for

which it was most relevant.

The foundational concept number of groups and number in each group, (can

hold 2 numbers) was identified in the concept map by the Year 1 and Year 2 teacher.

The Year 2 teacher was the only teacher who focused on the transition as her lesson

topic (Table 4.5). This could be explained by the fact that these concepts are

mathematically appropriate to Year 2 and that the Year 2 understands, and knows the

relevance of, these concepts for Year 2 students. This was reinforced by the same

teacher, when she highlighted Year 2 as a pivotal year in the transition between

additive and multiplicative thinking as this teacher noted,

I think Year 2 is probably a really pivotal year because I think it’s probably the transition

that moves from additive to multiplicative. So, I think in Year 2 it’s really important to

move from additive to multiplicative. (T2:PM:25)

In the planning meeting the Year 2 teacher twice identified the foundational concepts

a student demonstrates at this phase,

Holding that number in your head or holding those two numbers in your head so that

you can double count. So that you can keep how many groups there are plus how many

are in the group. Which are really high. (T2:PM:25 and T2:PM:9)

90

Like the Year 2 teacher, the Year 1 teacher indicated some of the preparation for

transitional thinking in Year 1,

These are the skills embedded in our curriculum already and the Year 1 curriculum in

number very heavily looks at additive thinking as well as moving into some

multiplicative thinking. (T1:PM:23)

The Year 1 and the Year 2 teacher each made statements during the lesson

planning meeting that identify the foundational concept that belongs to and supports

the transition from additive thinking to early stages of multiplicative thinking. The

acknowledgement by the Year 1 teacher of the preparation for the transition can occur

in Year 1 and the identification by the Year 2 teacher of the foundational concept

number of groups and number in each group align with the expectations of each of the

year levels. The content descriptions identified by the Year 2 teacher, from the

curriculum investigation meeting, were able to be categorised for both additive or

multiplicative thinking (Table 4.3) reflecting the transition or relevance of both

additive and multiplicative thinking to this year level. The content descriptions most

relevant were the representation of multiplication as repeated addition and groups for

the support of additive thinking and division – sharing in equal sets support of

multiplicative thinking. These descriptors also support the inverse relations, another

foundational concept that supports the transition to multiplicative thinking.

In her concept map, the Year 3 teacher did not identify foundational concepts

that support the transition, despite their relevance, as many students are transitioning

between additive and multiplicative thinking in Year 3. The Year 1 gave insight into

a possible reason, why children struggle with the concepts at this stage, “Children in

Year 3 and 4 that we want to move to multiplicative thinking, they really struggle with

many-to-one idea” (T1:PM:7).

Multiplicative thinking

Early multiplicative thinking evolves from foundational concepts that contribute

to the phase, multiplicative relations which follows the many-to-one phase. The

participants identified two foundational concepts that contribute to multiplicative

thinking. First, a more sophisticated understanding of the two components of

multiplication that contribute to transitional thinking to include the total of the groups

(product). It is this understanding of three components of multiplication, groups of

equal sizes (multiplicand), number of groups (the multiplier), and the total amount (the

91

product) that is regarded as one foundational concept. Second, inverse relations, that

is the relationship of numbers in a multiplication situation was also identified as a

foundational concept.

The identification of these two foundational concepts was made by the Year 3

teacher which may be indicative of the relevance of these concepts to the Year 3. She

firstly identified, the three components of multiplication. As she stated,

There’s a lot of research about specific understanding of the multiplied, the

multiplier and the product. So to be at the higher level of multiplicative

thinking, children had to be able to identify all of those three elements

(multiplier, multiplicand, product) very confidently and to understand that

operation correctly. (T3:PM:12)

Developing an understanding of the inverse relations to support a learner’s

understanding of the relationship of groups in multiplication was also identified by the

Year 3 teacher, “I do try to encourage the whole inverse operation or how many threes

might be six” (T3:PM:27). The suggestion of highlighting the inverse of the

components of multiplication is reflective of possessing a sophisticated understanding

of multiplication which is necessary for multiplicative thinking.

In the curriculum investigation meeting (Table 4.3) the Year 3 teacher only

identified descriptors that contribute to multiplicative thinking whereas the Year 2

teacher identified content descriptions that contribute to both additive and

multiplicative thinking. The Year 2 teacher summed up the importance of the

foundational concepts in her statement,

It has highlighted to me what those foundation concepts are that are required

for children to have that high level of understanding. When they get to Year

2, get to Year 3 the emphasis – I guess making sure like you say, being able

to pick out those specific strategies and skills that maybe children have missed

along the way or don’t have a clear understanding of. (T1:PM:24)

The foundational concepts relating to additive and multiplicative thinking that

were identified by the participants and informed the finding that additive and

multiplicative thinking evolves from foundational concepts. For students to develop

multiplicative thinking from additive thinking they need to be supported in that

transition between these two types of thinking.

92

Focus on transition between additive and multiplicative thinking

After the lesson was implemented and the teachers had a post lesson discussion,

there was greater focus on the transition particularly by the Year 3 teacher. Teacher 3

did however focus on teaching doubles for learning tables

Oh they know that straight away that 2 sevens are 14, I can connect to a double. So

that’s the first one I start with all the time, with teaching the times table. Then I go

into the zero’s, the easy ones, ones and then they start doing things like 10’s before

we branch out. (T3:PL:58)

The suggestion by the Year 3 teacher (post lesson discussion) of using doubles

is reflective of the transitional stage moving into multiplicative thinking and is also

suggestive of an increase in teacher knowledge of the relevance to her year level. The

post-lesson discussion followed the group planned lesson, which comprised of

students in small groups completing additive and multiplicative problems. On

completion of the problems, each group of students shared their approach to finding

solutions. Teachers observed the students as they shared their thinking. The teachers

drew on their observations during the post-lesson discussion. The greater focus on the

transition from additive to multiplicative thinking identifies that there was a change in

understanding from prior to lesson implementation and after lesson implementation.

There was also change in the teachers’ descriptions of the foundational concepts as

they were more sophisticated in the post-lesson meeting.

The Year 2 teacher in the post-lesson discussion, identified the foundational

concepts that contribute to a child beginning to think multiplicatively and made the

connection about the shift in teaching to move, that is transition, the children to this

stage,

If you look at those phases, that’s actually – that stage is the sort of almost

where the child – that multiplicative phase. So you know that the children in

that phase know the equal size of a group, the multiplicand and then the

number of the groups is the multiplier and the total number. So that’s sort of

where we want to get them to, but it’s interesting that something as basic as

that take for granted. It shifts the whole focus I was probably always aware

of it, but probably not aware of how important it was in moving them to

multiplicative thinking. (T2:PL:70)

93

She also identified an example of this transition in the context of teaching times-tables.

She highlighted how many groups, which is necessary to support students’

understanding from additive to multiplicative thinking, can be used to change from an

additive to multiplicative approach,

The switch with times tables from additive to multiplicative. So additive

thinking is saying one group of three and multiplicative is three ones, three

twos, three threes, three fours. So the focus is on the number of groups not

what’s in the group. I am just being more aware of that you can see the

difference between – so the additive is focussing on the number in each group,

so 3 plus 3 plus 3 plus 3. Whereas multiplicative you’re focussing more on

how many groups there are. So it’s four threes. So that’s a big switchover.

(T2:PL:45)

The importance of the transition was further highlighted by the same teacher in

her individual interview when she stated verbally and in writing,

Many to one counting phase was the most important as the transition from additive to

multiplicative. This phase applied directly to the majority of students in the Year 2

group.

Whilst the Year 2 may have highlighted the significance of the transition, because of

the relationship to Year 2, this study highlights the importance for teachers of P-3 to

have an understanding so they can support students in their transition. Therefore, an

outcome of these results is that the transition between additive and multiplicative

thinking needs to be supported by a focus on specific foundational concepts.

4.1.3 Summary

The identification of the foundational concepts of proportional reasoning

mirrored the data collection process. Before, and during, the curriculum investigation

meeting the identification of the foundational concepts was limited to 3 three areas.

Initially, proportional reasoning was identified by the participants as a concept that is

applied to other mathematical areas and it was also identified as a way of thinking

which was then identified more specifically as additive and multiplicative thinking.

There was a change in direction in the study when the participants identified

mathematical concepts from the Australian Curriculum: Mathematics which were

reflective of either additive or multiplicative thinking or both for the P-3 Years (Table

4.3).

94

After the curriculum investigation, the understanding that additive and

multiplicative thinking evolves from foundational concepts became evident (Table

4.3). The participants’ identification of the foundational concepts identified included

trust the count/subitising, groups, skip counting, and repeated addition from which

additive thinking evolves. Multiplicative thinking evolves from the foundational

concepts (multiplicand, multiplier, product) and inverse relationship.

There were limited references to the foundational concepts to support the

transition between additive and multiplicative thinking (numbers of groups and the

total in each group). Students need to be supported in the development of these

foundational concepts as these concepts are pivotal for transitioning students from

additive thinking to multiplicative thinking. This gap in the results will be addressed

in the Discussion Chapter 5: (Section 5.1.3).

In summary three key outcomes were identified in relation to the first research

question:

Proportional reasoning is based on additive and multiplicative thinking

Additive and multiplicative thinking evolves from foundational concepts

The transition between additive and multiplicative thinking needs to be

supported by a focus on specific foundational concepts.

The development of the foundational concepts is dependent on the ways teachers

promote their understanding. The ways the participants identified to promote the

foundational concepts of proportional reasoning will be discussed in the next section.

4.2 UNDERSTANDINGS OF PRACTICES THAT PROMOTE

PROPORTIONAL REASONING

In terms of Research Question 2, identifying ways to promote the foundational

concepts proportional reasoning, there was a change in the discussions prior to, and

after, the lesson (that was planned by the group, implemented by me, observed by the

teachers). This section will address the results both prior to the lesson implementation

and after the lesson implementation. The final template for research question 2 (see

Figure 4.2 below) was applied to all the data sources. Each of these categories will

also be discussed.

95

1. Environmental Provisions

Environmental provisions enable learning to take place without the direct intervention of

the teacher or interactions between the teacher and students

1.1. Artefacts- learning can take place through interaction with artefacts eg. wall

displays, manipulatives, puzzles, appropriate tools

1.2. Classroom organisation – seating arrangement, lesson structure

1.3. Structured activities – worksheets, directed activities

1.4. Free play – children set their own challenges and learn through feedback

1.5. Self-correcting – further feedback – supports autonomous learning – processes

in finding a solution

2. Explaining

Teacher interactions that are increasingly directed to developing richness in the support

of mathematical learning

2.1. Show and tell (traditional teaching)

2.2. Teacher in control

2.3. Structure conversations – next step. Little use of pupil contribution

3. Reviewing

The teacher refocuses student attention and provides further opportunities to develop

their own understanding rather than reliance on teacher.

3.1. Looking, touching, verbalising - getting students to look, touch and verbalise –

tell me what you did – lead student to verbalise their thinking and notice an

error – that they can correct for themselves

3.2. Interpreting students’ actions and talk

3.3. Students Explaining and Justifying – teacher promote understandings through

‘orchestration’ of small group and whole class discussions so students

participate by making explicit their thinking and justify their approach to the

task.

4. Planning for student thinking

4.1. Planning – in planning for learning, teachers put students’ current knowledge

and interests at the centre of their instructional decision making

5. Mathematical tasks

5.1. Proportional reasoning is used in real life and other subjects

96

5.2. Provide proportional reasoning problems

6. Mathematical language

6.1. Foster students’ use and understanding of mathematical terminology

6.2. Modelling appropriate terms

6.3. Communicating in ways that students understand

7. Student conceptual understanding

7.1. Supporting student’s in the development of understanding

Figure 4.2. Final template

4.2.1 Prior to lesson implementation

In the focus group, which occurred at the beginning of the data collection, prior

to lesson implementation, the main categories from the template that were used to code

the participants’ discussions were environmental provisions, planning for student

thinking, mathematical tasks and mathematical language. Mathematical tasks address

providing proportional reasoning is used in real life and in other subjects (subcategory

5.1) and provide proportional reasoning problems (subcategory 5.2). Environmental

provisions, planning for student thinking, and mathematical language featured in the

data both prior to, and after, lesson implementation and, therefore, will be addressed

in both sections.

Environmental Provisions

Prior to lesson implementation, environmental provisions were identified in

relation to providing artefacts (subcategory 1.1). In the individual concept mapping,

the participants (n=2) identified concrete materials as a task for promoting

understanding. Wall charts of vocabulary, as described in the previous section, also

offered such an example. The participants (n=3) highlighted the use of arrays to

support understanding of multiplication and groups, for example, “There is a lot of

focus on getting them competent using arrays” (T1:CI:8). Teacher 3 added to this,

They need that understanding of arrays. So, they are able to visualise very firmly in

their head almost like when I say 3 groups of 4, they need to now be able to think into

arrays. (T3:CI:27)

Number expanders were also identified as a way to support understanding of

number,

97

Some girls seem to flow from using concrete materials and manipulating and having

number expanders. Some are able to make that leap straight into abstract. Others seem

to still heavily rely on concrete and not being able to make that abstract. (T3:CI:2)

Diagrams and drawings were also suggested as a practice to support understanding.

The provision of artefacts can be planned for when developing student thinking.

Planning for student thinking

The participants identified planning for student thinking in relation to making it

explicit and planning for using artefacts. In the focus group, the participants (n=3)

shared that proportional teaching needed explicit teaching “to make it more

pronounced” (TP:FG:18) and planned for it so that the learning becomes

developmental, “You see how important it is and how far reaching it is then you look

more carefully to say well, where is it embedded and am I making it explicit?” (T2:

FG:31).

Planning for proportional reasoning will ensure proportional reasoning is taught

and that proportional reasoning is built upon each time it is taught, as suggested by the

Prep teacher, “When you’re planning your lessons, you can see how putting

proportional reasoning into this activity and to make sure you are building on it for the

next lesson” (TP: FG:32). The following quote also suggests that Teacher 3 built upon

this, as she believed that once teachers made this part of their teaching explicit it would

become innate,

I think once you’ve done it a few times it actually then becomes part of you and that’s

how you start to behave all the time within a classroom and you start to use that

language. But initially you’ve got to make that jump. (T3: FG:79)

The Year 3 also teacher discussed planning for student thinking in relation to students’

use of artefacts, “so it’s almost going from concrete to abstract that they just need a lot

more time on the concrete” (T3:CI:2). Specifically, she identified using diagrams to

develop student understanding, “I do a lot of diagrams with them so that they can

literally see how it all works” (T3:CI:2). Therefore, planning for student understanding

includes providing mathematical tasks.

Mathematical tasks

Prior to the lesson being implemented, two types of mathematical tasks were

identified by the participants as a way to promote proportional reasoning. The

participants (n=4) identified that proportional reasoning is used in real life and other

98

subjects (subcategory 5.1) and providing proportional reasoning problems

(subcategory 5.2).

Proportional reasoning is used in real life and other subjects

The Prep teacher identified the practicality of proportional reasoning, “it’s a very

practical skill to have, even in terms of using money to purchase things” (TP: FG:6).

The Year 1 teacher suggested real life as a context for promoting and applying

proportional reasoning.

I guess especially with younger kids, that’s a really important thing to be able

to start getting them being able to think about it and just looking at those

different applications that it does happen in everyday life. (T1: FG:5)

She offered cooking as a context to provide comparative situations, “cooking provides

a context for measuring, being able to make comparisons, if I have a cup of something,

how do I weigh that to grams and kilograms and lots of things” (T1: FG:17). Each of

the teachers (n= 4) referenced the real life and subject applications on more than one

occasion as identified in Table 4.6. In the lesson concept mapping, the participants

(n=3) identified real life experiences as a way to help to develop students’

understanding. The applications, and frequency, of the references by teachers during

the data collection phase are consistent with the research that found that proportional

reasoning has applications in real life and across subjects (Dole, 2010).

Table 4.6.

Applications of proportional reasoning

Applications Number of times mentioned

Cooking 2

Real-life 2

Other subjects

Geography/mapping/plans 4

Science 2

History/timelines 2

99

Provide proportional reasoning problems

Student understanding of proportional reasoning is assessed through presenting

students with proportional reasoning problems. In the lesson planning sheet, the

participants (n=4) individually identified that problem solving was a task for

developing student understanding. During the data collection discussions, prior to the

lesson, the participants (n=3) also suggested providing problem solving or problematic

tasks to develop student understanding. They acknowledged the value of providing

opportunities for students to engage with tasks that allow them to apply and test their

thinking, “I think now I really like the idea of making sure that they’re specifically

proportional reasoning problems” (T2: PM:26).

The Year 3 teacher noted that her students found proportional reasoning

problems the most challenging. These relate to multiplicative problems. She stated,

I’m seeing now that the problem solving seems to be an area that needs a lot of building

on. Towards the end of the NAPLAN test, the girls are unable to answer these questions

correctly, and those problems are proportional reasoning. (T3:CI:6)

Additionally, the Year 1 teacher built upon this by stating,

From my understanding of all the different research that I’ve read, I thought from Year

3 we looked at really embedding the idea of proportional reasoning through using

problem-solving strategies and focusing on that multiplication and division through

problem solving. (T1:PM:2)

These two statements suggest the two participants acknowledged the need for

developing children’s understanding of proportional reasoning from Year 3, as the

Year 3 students have the skills and developmental capabilities to solve proportional

reasoning problems.

The Year 2 teacher suggested a way to promote understanding of problem

solving situations; that is, through collaborative group work

So, some of the reading that I’ve read as well, eludes to the fact that to develop

proportional reasoning in those other grades, in around the Year 3 level and

up, using the investigative problem-solving lessons. So, having a problem,

the children going off, analysing the problem in a group, working out a

solution, bringing back lots of different strategies and working out which

strategies are more effective or efficient and which strategies work. (T2:PM:3)

100

Using problem solving situations as the stimulus for children to share their solution

strategies allows children to make their thinking explicit. To do this, children require

the ability to both use – and understand – the language associated with additive and

multiplicative thinking.

Mathematical language

Mathematical language to promote proportional reasoning was highlighted in all

data collection points, with an increase in its reference as the data collection process

progressed. Therefore, mathematical language will be discussed in both prior to, and

after, the lesson implementation. Prior to lesson implementation, the language focus

was in relation to fostering students’ use of and understanding of mathematical

terminology (subcategory 6.1). In particular, the Prep teacher identified the use of

comparative language and the importance of using it consistently. The Prep teacher

highlighted this by making the link between comparative language and different

subjects to reinforce its use and provide a context for comparative language,

All that comparative language, for me, probably more so in Prep, using that

vocab a lot more when talking about things across subject areas too. So,

there’s that language that carries on. I use the same in science and same in

maths and that’s so they can see that reasoning language flows through

everything that we do. (TP: FG:24)

Two teachers (n=2) also highlighted the importance of identifying

“terminologies” (T2: FG:85) and using the language consistently by modelling, which

aligns with modelling appropriate terms (subcategory 6.2). In terms of consistency of

language, it was highlighted that in the different year levels, students use the same

language, but it becomes more specialised as the students move through the year

levels, “we’re still using the same mathematical language of proportions. We’re still

using the same language throughout. It just becomes more specific” (T1: FG:73).

Additionally, it was suggested that consistency of language could be reinforced

through a teaching chart: “you could probably even, on a wall chart, have your bank

of words, so that you can refer to them and encourage the children, just so you’ve got

some common language” (T3: FG:26). Also, a visual presentation of the words as a

permanent source of language for questioning and consistency of language was

suggested,

101

The words too then would give you (teacher) a link to the type of things that they

(students) might use’ and ‘help you develop your questions better too, if you’re taking

them further with that proportional reasoning, having that visible. (T2: FG:27)

The Year 3 teacher stated,

So when you say to children well, how you worked that out and they say, ‘I don’t know,

I just know the answer’, that sort of thing. So that’s where you would obviously step in

and offer them the words and support them through trying to explain when they’ve just

done. (T3: FG:57)

This suggestion offers a way of modelling appropriate terms (subcategory 6.2) and

communicate in ways that student understand (subcategory 6.3). Modelling and

communicating for understanding was also identified by the Prep teacher, “I just need

to adapt the way I talk about it. The language is already there. It’s just putting the

connectors there and making the links with it” (TP: FG:49). The teacher modelling

and communicating of the language supports students in articulating their

understanding.

Before the lesson was implemented, the teachers identified ways to promote the

foundational concepts proportional reasoning; these included environmental

provisions, planning for student thinking, mathematical tasks (in particular, providing

proportional reasoning tasks in real life and other subjects and providing proportional

reasoning problems). Environmental provisions, planning for student thinking and

mathematical language were identified both prior to, and after, the lesson was

implemented. However, there was a difference between the depth of discussions in

relation to these three areas at the time frames (prior to, and after, the lesson

implementation). Firstly, the results identified environmental provisions that is

providing artefacts as a way to promote proportional reasoning. There were a limited

number of resources identified: using wall charts for vocabulary, number expanders,

and array with limited descriptions. Second, planning for student thinking

encompassed explicitly teaching proportional reasoning and providing resources that

support understanding. Thirdly, the participants identified mathematical tasks, so that

students can see proportional reasoning is used in real life and other subjects and by

providing proportional reasoning problems.

Finally, references to mathematical language were limited to using comparative

language and using consistent language across the year levels. Environmental

102

provisions, planning for student thinking, and mathematical language featured in the

results both prior to, and after lesson implementation. These three ways will also be

discussed after lesson implementation (4.2.2) as the participants also addressed them

at that stage of the data collection. A key outcome of the data collection of this stage

was the relevance of problem solving, as the provision of problems provides the

opportunity for students to apply and test their thinking; therefore, proportional

reasoning is promoted through problem solving.

4.2.2 After lesson implementation

After the lesson the lesson implementation during which the participants

observed the students. The purpose of the observations of the collaboratively planned

lesson was to understand how the lesson works to promote student understanding.

After the lesson was observed, the participants shared their observations in the post

lesson discussion. At this stage, the participants offered more ways to promote

proportional reasoning. Environmental provisions, mathematical language and

planning for student thinking were once again identified by the teachers. The

descriptions and examples, however, were more specific in relation to ways to promote

proportional reasoning. The teachers also identified student interactions, in particular,

by explaining and reviewing. At this stage a new code emerged, student conceptual

understanding, this code was attached to the teachers’ statements made regarding

students understanding of the different aspects of the foundational concepts. The

teachers’ new understanding may have been the outcome of the teachers’ observations

which focused on how the lesson promoted student understanding. Within each of

these ways to promote the foundational concepts of proportional reasoning, the

participants, at times, offered examples that made the link between the foundational

concepts of proportional reasoning and ways to promote them.

Environmental Provisions

Environmental provisions, as a way to promote the foundational concepts of

proportional reasoning was identified in the data both before and after the observed

lesson. The teachers suggested using artefacts or resources (subcategory 1.1) and

classroom organisation (subcategory 1.2) before the lesson. After the lesson, the

teachers (n=3) linked using artefacts or resources (subcategory 1.1) with the teaching

of the foundational concepts.

103

Using resources as a way to promote understanding was most identified by the

Prep teacher which is indicative of the year level. Using resources to support the

understanding of concepts is a practice that is reflective of the age group of Prep

students. The Prep teacher brainstormed the different environmental provisions she

used, “lots of different concrete materials, use of games, number lines, collecting items

for a group. We do lots of songs and chants, finger rhymes using body parts basically

to do counting” (TP:PL:111). In her interview (TP: II), she shared that she also relied

heavily on resources in her individual lesson, as she identified specific artefacts

(flashcards, dice, and ten frames), while the Year 1 teacher just identified using one

artefact (magic counter game) when both describing the way each promoted the

foundational concept in the class lesson during individual interview (T1: II).

The Prep teacher also made the most connections between offering different

resources to support children’s understanding of the foundational concepts of

proportional reasoning. The suggested resources included; flashcards for subitising,

ten frames for skip counting and part-part-whole, dice and dominos for trust the count

and surveys as a way to understand groups,

I’ve always used flashcards and like to try and work on subitising with the students in

Prep. (TP:PL:31)

Skip counting while lining up, using ten frames. (TP:PL:111)

I am using the ten frames a lot more, decomposing numbers (part-part- whole) and doing

a lot of stuff with that and trying to get them to see how numbers can be represented in

different ways. Counting on. (TP:PL:121)

Trust the count: “getting them to realise that they- that the collection remains the same

regardless of where you start counting. That they can count on from that, so we’re

starting to do that additive stuff. We use dice and dominos – they’re great for additive

composition. Collecting items for a group, things like that. (T3:PL:110)

The suggestion of conducting surveys and tally marks, provide a real context for

grouping, was provided by the Prep teacher. The Year 2 teacher made the connection

between using surveys and mathematical concepts associated with proportional

reasoning,

They conducted a survey, then I drew it on the board and they came and added their

tally marks. Someone drew attention, oh, but those tally marks aren’t in groups of five.

Remember we put a line across for five? So we went back and we did they counted in

104

fives and the how many more. So it (survey) is a really great opportunity to do some

groups of. (TP:PL:114)

In relation to the survey, mine [students] have been really interested in the percentage,

so that’s proportion – and probably in the past I would have explained it just quickly.

But for those girls that are kind of getting ready for that idea, because of this study, I

sort of went, oh that’s clearly proportional reasoning and I should really hone in on it.

(T2:PL:115)

By comparison to the number of resources suggested by the Prep teacher, the

Year 3 teacher only suggested one resource (array), to promote understanding. An

array is a resource that supports the understandings of multiplication for students at

this stage,

Understanding arrays, are little groups of. So moving into that whole groups of. Then

representation and presentation of arrays. So if you have got 2 rows of 4 is eight, but

then I can turn that around to 4 rows of two going down like that, is 8. So understanding

the inverse operation as well. (T3:PL:12)

Classroom organisation (subcategory 1.2) was suggested by the Prep and Year

2 teacher. The Prep teacher suggested table work, “If there’s a group of five, how

many do you need and things like that, for small table work” (TP:PL:111). The Year

2 teacher used group work as part of her classroom organisation. In her interview, she

discussed that working in groups allowed the students to learn from each other, to

query others and readjust own thinking according to newly acquired ideas from the

group (T2: II). In her post lesson discussion, she stated,

I probably relied more on starting with an individual. I’ve done more of the group goes

off and works or the two and I do a lot more paired work, starting with a pair or a group,

so that they can cooperatively construct some knowledge around it. (T2:PL:76)

The variety of resources offered suggests an understanding by the teachers (n=4)

of the value of environmental provisions as a way to promote the foundational

concepts of proportional reasoning. The participants however did not indicate the

approaches they would use when offering the resources for students to use.

Mathematical Language

After the lesson observation the teachers made more references to mathematical

language (n= 3). The references to mathematical language related to fostering

students’ understanding of mathematical terminology (subcategory 6.1), modelling

105

appropriate terms (subcategory 6.2) and communicating in ways children understand

(subcategory 6.3). In general terms, the importance of language in promoting

proportional reasoning was acknowledged, “I think the language is really important”

(T2:PL:76).

The teachers identified that they needed to foster students’ understanding of

mathematical terminology (sub category 6.1). First, in relation to problem solving

“having an understanding of the vocabulary that surrounds those problems”

(T3:PL:84). Second, the vocabulary associated with specific foundational concepts,

multiplication and multiplicative thinking: “they need to understand all of those words

like product, arrays, multiplication, commutative properties and things like that and

multiples” (T3 PL:14) and “making them aware of what language you’re using to get

them use to the language of the multiplicative thinking” (TP:PL:62).

Fostering student understanding of terminology by modelling (subcategory 6.2)

was offered by some participants (n=2). The Prep teacher stated, “break apart what

the terminology is have discussions around it and get them to talk to you first so you

can see if there are any misconceptions” (TP:PL:97). The teachers highlighted the

importance of using the language regularly,

I’ve looked at the language that you [teacher] would use, like the product, multiples,

and things like that. So that you’re actually using them all the time and not just maybe

perhaps sparingly, but to keep going through those words. (T3:PL:4)

Then you can work it out, well who already understands and can move on who needs

explicit demonstration of what it means. (T2:PL:97)

It’s a real shift in language and in what you’re [teacher] using in the lesson making them

(students) very aware of what language you’re [teacher] using. (TP:PL:62)

This suggests that the teacher was more aware of promoting the language associated

with proportional reasoning through modelling her own language and making it more

overt.

Communicating in ways children understand (subcategory 6.3) was broadly

offered by two teachers through acknowledgement of explicitly teaching vocabulary,

“I do the same thing explicitly with them so hopefully it’s reinforcing that vocabulary

and terminology” (TP:PL:102) and “I have been more explicit in talking about it”

(T3:PL:116). The Prep teacher was more specific about how she communicates so

106

children understand; that is, by visually showing the word, discussing the meaning,

and putting the word into action,

I will draw attention to a specific word and write it on the board and we will discuss it

first, well what does that mean, what are we going to be doing. Even if they are not

participating in the discussion it’s all the – the talk around. (TP:PL:100)

The teachers identified focusing on relevant language as another way to

demonstrate subcategory 6.3. Comparative language was identified both before, and

after, the lesson implementation, as a way to promote proportional reasoning. The

Prep teacher stated, “we’re talking about in groups here, which led to lots of

comparative language too, as they were talking about collections too” (TP:PL:115).

By encouraging students to put the mathematical terms in their own words, it

helps them understand the meaning, where they take away words and work

out do I know it well? Or have I never heard it before? Working with partners

to talk about what they do know and explain to each other (T2:PL:79).

This suggestion by the Year 2 teacher is reflective of the tenet of constructivism, in

that the teacher encouraged the students to engage in dialogue as a way to support their

learning (Reys et al., 2012).

On several occasions, the teachers (n=3) identified the importance and relevance

of using mathematical language as a way to promote the foundational concepts of

proportional reasoning. These included fostering students’ understanding of

mathematical terminology, modelling appropriate terms, and encouraging students to

communicate and use the language while interacting with peers thus promoting

understanding and a constructivist approach to learning. The use of mathematical

language is significant to teacher/student interactions through explaining and

reviewing, which will be discussed below.

Explaining

Explaining was a way the teachers interacted with the students to promote

proportional reasoning. Explaining is reflected in the subcategories of show and tell

(subcategory 2.1), teacher in control (subcategory 2.2), and structure conversations

(subcategory 2.3) (as per Anghileri, 2006). All three teachers mentioned the role of

explaining and each provided an example of each type. Teacher 3 referred to using

explaining, (subcategory 2.1; show and tell) about proportional reasoning, “I need to

107

really pull this back for some girls and take them back really very carefully and explain

it to them in those terms” (T3:PL:19).

The second subcategory of explaining, (subcategory 2.2; teacher in control), was

evident when the Year 2 teacher suggested being explicit and systematic, “I think we

need to be really systematic about what stage they’re at” (T2:PL:89). The Year 3

teacher’s example of using additive approach to multiplication is indicative of the

teacher in control “this is what you need to keep saying to them, that’s the best way to

solve a problem, because the same number is being added again and again”

(T3:PL:16).

Structured conversations (subcategory 2.3) were evident with the Prep teacher

on four occasions; for example, “I think now I am more mindful of looking at the

patterns in those subitising cards and taking the time to discuss what it looks like”

(TP:PL:32). The Prep teacher also identified how she helped develop understanding

of the foundational concept, trust the count “Getting them to realise that the collection

remains the same regardless of where you start counting. That they can now count on

from that, so we’re starting to do that additive stuff” (TP:PL:110). Whilst the Prep

teacher did not identify how she gets them to ‘realise’, there is inference that it could

be a structured approach to explanation. Beyond the teachers interacting with students,

by explaining, the teachers also shared ways they interact so students have the

opportunity to clarify their learning through reviewing which offered a more

constructivist approach to developing student understanding.

Reviewing

A reviewing approach to student/teacher interactions offers a more responsive

approach to student learning. Reviewing allows the teacher to refocus student

attention and provide opportunities for students to develop their own understanding

(Anghileri, 2006). Reviewing can occur in three ways: looking, touching, verbalising

(subcategory 2.1), interpreting students’ actions and talk (subcategory 2.2) and student

explaining and justifying (subcategory 2.3). Some participants (n= 3) provided

evidence of using reviewing. Teacher P used looking, touching, and visualising;

(subcategory 3.1), “I do spend a lot more time on ‘what did you see’ and sharing ideas

and making that visual” (TP:PL:39). Also, the Year 3 teacher described getting the

students to show their understanding through visual representation, “I actually get

them to also represent, make their own little sentences up and do their own

108

representation” (T3:PL: 8). The Year 2 teacher identified how she encouraged

visualising to help with understanding, “being able to visualise what the question is

asking you” (T2:PL:83).

The Year 2 teacher offered examples using ‘reviewing’ with a combination of

interpreting students’ actions and talk (subcategory 3.2) and students explaining and

justifying (subcategory 3.3). Teacher 2 stated, “So that they [students] can decide

which ones are more efficient, [for example a student might say] ‘I’ve done it in

additive way, my partner said let’s do it in a multiplicative way’” (T2:PL:76). This

example suggests how the teacher provides an opportunity by providing the task and

then allows the students to explain and justify their thinking, which, in turn, allows for

the teacher to interpret their talk. It is a very important approach for teachers as a way

to check students’ understandings of additive and multiplicative thinking.

The Year 3 teacher offered an example of reviewing, where the teacher interacts

with the students to encourage them to explain and justify thinking,

So then to show you, well what’s the repeated addition there? Can you show me that

one? Then show me how you could do it using multiplication? I think I lead the

conversation a bit more in that direction. (TP:PL:36)

Also, Teacher 2 offered an example where the student is encouraged to explain their

own thinking “the girls are very able to explain their strategies for adding and taking

away” (T2:PL:80). The teachers drew three different ways that they promoted

proportional reasoning by interacting with students in a way that encouraged the

students to review which learning. This is reflective of a student creating their

knowledge, that is through a constructivist approach. Reviewing is supportive of the

teacher developing student conceptual understanding.

Student Conceptual Understanding

Building student conceptual understanding of mathematics was a category that

emerged from the data in the post lesson discussion (PL) and relates to the teachers’

identifying ways they support the students’ conceptual understanding of the

foundational concepts of proportional reasoning. There was a crossover in this

category with other categories (language and mathematical tasks) however it was

identified as a category because of the teachers (n= 3) identifying the importance of

developing student’s conceptual understanding. As the Year 3 teacher stated, “I can

109

really focus and see that I really need to make sure that this is quite embedded in them

and they really understand it” (T3:PL:2).

On several occasions, the Year 1 teacher and Year 2 teacher identified the need

to develop the student’s conceptual understandings of proportional reasoning through

language. Conceptual understanding can be supported through building an

understanding of the associated language. This was suggested by two participants on

four different occasions. The Year 2 teacher highlighted the need, “so that the children

do understand what every aspect of the language means. So there is a process where

you go through and decide whether they do understand the language?” (T2:PL:92).

The Year 3 teacher identified the importance of understanding the specific associated

language “they need to understand all of those words like product and arrays and

multiplication and things like that” (T3:PL:14). She also made the link between

language and the mathematical task, problem solving, “having an understanding of the

vocabulary that is surrounding those problems” (T3:PL:84).

Providing challenging tasks through problem solving tasks was offered as a way

to develop understanding by the year 2 teacher, “I think that idea of the way we present

problems that children go away and use and construct their own understanding of it”

(T2:PL:76).

The year 3 teacher also highlighted the value of this approach “So, they really

need to have all that understanding, like when they are presented with a problem, that

they’ve got lots of ways of working it out and they’re still able to explain it”

(T3:PL:15). Therefore, these teachers suggestions of offering problem solving in a

constructivist way, helps children to make sense of the learning and was considered

by them as a valuable approach.

Making connections between problem solving and drawing was also offered as

a way for students to make their understanding explicit, “drawings, so they can make

an abstraction and then they can get an understanding” (T3:PL:82). Highlighting the

difference between rote and conceptual understanding was offered in the context of

learning the times tables,

Yes, it’s important she’ll know that 3 times 5 is 15 eventually by rote fashion, but in her

head she has still seen that it’s additive. So instead of just, oh I know this time tables,

but not really understand what I am doing. (T3:PL:132)

110

This statement made by the Year 3 teacher highlights the importance of teaching times

tables for understanding, and that it is done through teaching an understanding of

multiplication, rather than an additive or repeated addition approach. Developing

student conceptual understanding through acknowledgement of language, problem

solving, and the teaching of multiplication highlights the priority the teachers placed

on it. Conceptual understanding of mathematics supports the student’s ability to think

mathematically and the development of conceptual understanding is supported through

a constructivist approach as the student constructs new mathematical knowledge. This

is supported by a teacher planning for student thinking.

Planning for student thinking

A teacher planning for student thinking allows the teacher to put students’

current knowledge and interests at the centre of instructional decisions. Planning for

student thinking includes both pre-planning and on-the-spot planning, when

identifying teachable moments. Building on children’s thinking was addressed by

Teacher 3 who suggested using ‘teachable moment’ opportunities; that is, the teacher

plans at that point in time when an opportunity arises to promote understanding of

proportional reasoning. As the teacher asserts,

I think now that I know more about proportional reasoning I’m embracing those

opportunities a bit more and seeing where it’s going a bit more. So now I don’t let this

moment pass, this is a teachable moment for multiplicative thinking, so it’s just stay a

bit longer. (T3:PL:118)

and

So I am much more aware, mindful of phases and where it’s going. So I spend, if I can

see that teachable moment arises then I will take the time to do that. (TP:PL:121)

This statement highlights the importance of not only planning for student thinking but

having the knowledge to be able to plan at the point of time, in order to capture

teachable moments. After the lesson implementation, the teacher identified ways to

promote proportional reasoning, that were also discussed prior to the lesson

implementation. These included environmental provisions, mathematical language,

and planning for student thinking. It is the first two that had the most impact on the

findings.

In identifying the data in relation to environmental provisions, the artefacts, or

resources, that were suggested, provided ways to promote proportional reasoning. The

111

Prep teacher identified many resources that are reflective of supporting the

foundational concepts developed in this Year level; in contrast, the Year 3 teacher

identified one resource (array) as it supports the teaching of multiplication. Therefore,

proportional reasoning is promoted through resources.

Mathematical language was addressed through supporting students’

understanding of the associated language. This was identified in relation to the teacher

modelling and communicating the language associated with proportional reasoning.

Therefore, proportional reasoning is promoted through language.

The other ways suggested to promote proportional reasoning included

teacher/student interactions, explaining and reviewing, and students’ conceptual

understanding. Through explaining, the teacher was in control by showing and telling

students; this also included structured conversations. Explaining allowed teachers to

explain the foundational concepts to students. Through reviewing, teachers

encouraged students to verbalise and visualise their thinking through justifying and

explaining the foundational concepts. These interactions were based on teachers

having a strong commitment to developing student conceptual understanding through

language, challenging tasks, and problem solving.

4.2.3 Summary

In addressing research question 2, an analysis of the transcripts at the different

data collection points sought to identify ways to promote proportional reasoning. The

data were analysed at two points prior to and after the lesson implementation. Prior to

the lesson implementation, the teachers offered four ways to promote proportional

reasoning. Prior to the lesson implementation, the teachers offered different ways to

promote proportional reasoning. First, environmental provisions, particularly,

providing artefacts was identified. These artefacts were limited to number expanders,

vocabulary wall charts, and array with limited associated descriptions. Second,

planning for student thinking was identified, which included explicitly teaching

proportional reasoning and providing resources that would support understanding.

Third, the participants identified mathematical tasks that offer proportional reasoning

in real life, and other subjects, and by providing proportional reasoning problems.

Fourth, mathematical language promotes proportional reasoning was offered, but was

limited to using language of comparison and consistency of language across the Year

112

levels. A key outcome of the data collection of this stage was the relevance of problem

solving, promoting proportional reasoning through problem solving.

Mathematical language and environmental provisions featured in the results both

prior to, and after, lesson implementation. After the lesson implementation, there were

a greater number of references to language with a greater variety of ways (terminology,

modelling, and communicating) to use language to promote proportional reasoning,

providing evidence of its importance. Providing resources, that is, environmental

provisions, was offered by the participants as another way to promote understanding

of proportional reasoning. Two teachers contributed to ideas of environmental

provisions, each identifying resources that were reflective of their year level. The Prep

teacher identified many resources that support understanding of the foundational

concepts. The Year 3 teacher identified one resource, array as it supports the teaching

of multiplication and is reflective of the year level content. Mathematical language

and resources were two ways that were identified to promote proportional reasoning.

These ways informed two findings: the promotion of proportional reasoning through

resources and through language.

Communicating with language was a feature of two teacher/student interactions.

At the heart of these interactions was an acknowledgement of the importance of

developing student understanding as critical to proportional reasoning. At this stage of

the data collection, the teachers also identified planning for student thinking to develop

proportional reasoning. This planning included explicit teaching and using resources

to support student understanding. After the lesson was implemented, planning for

student thinking included using teachable moments to support student understanding.

The suggestion of taking advantage of teachable moments aligns with (and requires)

increased teacher knowledge.

In summary, three key ways were identified to promote the foundational

concepts of proportional reasoning.

Promoting proportional reasoning through problem solving

Promoting proportional reasoning through language

Promoting proportional reasoning with resources

113

4.3 CHAPTER SUMMARY

This chapter presented the results of the data collection that sought to identify

what a group of P-3 teachers recognised as the foundational concepts of proportional

reasoning and the ways in which they promoted understanding of proportional

reasoning.

In relation to question one, What are the foundational concepts of proportional

reasoning?, at the beginning stage of the data collection, prior to the curriculum

investigation meeting, the participants identified proportional reasoning as having its

applications in other mathematical areas. Initially, the participants identified

proportional reasoning as a way of thinking; this way of thinking became more specific

when it was identified as additive and multiplicative thinking. This informed the

finding that proportional reasoning is based on additive and multiplicative thinking.

This will be discussed in Section 5.1.1.

The identification of mathematical concepts from the Australian Curriculum:

Mathematics provided direction for the study, as it became evident that additive and

multiplicative thinking evolves from foundational concepts. These will be discussed

in Section 5.1.2. The participants’ identification of the foundational concepts of

proportional reasoning were informed by the curriculum investigation and

foundational concepts attributed to different levels of additive and multiplicative

thinking.

Additive thinking evolves from foundational concepts and these were identified

as trust the count/subitising, groups, skip counting and repeated addition. These

concepts were the most frequently identified and discussed in the data sources after

the curriculum investigation meeting (planning meeting, concept map and lesson

topic) and prior to the lesson implementation. Multiplicative thinking builds on

additive thinking; the foundational concepts from which it evolves are focussed on the

key elements of multiplication (multiplicand, multiplier, product) and an

understanding of the inverse relationship of multiplication and division. A key finding,

therefore, from the data is, additive and multiplicative thinking evolves from

foundational concepts.

Students need to be supported in the development of the foundational concepts

as they are pivotal for transitioning students from additive thinking to multiplicative

thinking. This gap in the results will be addressed in the Discussion Chapter 5:

114

(Section 5.1.3). These foundational concepts support the learner’s development to

begin to think multiplicatively and informed the third finding, the transition between

additive and multiplicative thinking needs to be supported by a focus on specific

foundational concepts.

In identifying the results in relation to question 2, How do Prep – Year 3 promote

proportional reasoning?, the data were analysed prior to lesson implementation, and

after lesson implementation, with some of the ways (planning for student thinking,

environmental provisions, mathematical language) addressed during both data

collection points. Prior to the lesson implementation, teachers indicated that

proportional reasoning can be presented in mathematical tasks that include

applications in real life and other subjects and problem solving. The teacher’s

suggestion of using problem solving to teach proportional reasoning signalled the use

of problem solving as a specific strategy for promoting proportional reasoning, which

will be discussed in Section 5.2.1.

The results indicated that the development of mathematical language was

important to promote understanding and this featured both prior to, and after, the

lesson was implemented. After lesson implementation, a greater number references to

language, with a greater variety of ways, for example using it for modelling and

communicating and offering specific vocabulary, were offered by the teachers, as a

way to promote proportional reasoning, which will be discussed in Section 5.2.2.

Prior to lesson implementation, environmental provisions were identified, using

artefacts or resources as a way to promote proportional reasoning and were very

limited in the number and the descriptions. After the lesson implementation, the Prep

teacher identified many resources that were reflective of supporting the foundational

concepts, in contrast to the Year 3 teacher, who identified one resource (array) as it

supports the teaching of multiplication. The resources suggested and described by the

teachers are reflective of each teacher’s year level content. Therefore, proportional

reasoning is promoted through resources which will be discussed in Section 5.2.3.

After lesson implementation, three new ways to promote proportional reasoning

were offered; two with communication at their heart (that is, explaining or show and

tell, the teacher in control, and structured conversations) and reviewing (that is, helping

the student identify their thinking). Students’ conceptual understanding was the third

115

way to promote proportional reasoning and the teacher identified language and

challenging tasks as ways to check and support student conceptual understanding.

Embedded within the data collection, there was a growth in teacher knowledge.

As the study progressed, so did the teachers’ knowledge. At the key points for each

research question, where the data analysis changed (research question one – after the

curriculum meeting; research question 2 – after the lesson implementation) this change

was also reflected in the teachers’ knowledge. Each of these points, after the

curriculum investigation meeting and after lesson implementation, signalled a change

in the data which was the outcome of an increase in teacher knowledge. As a key

finding, teachers demonstrated a growth in recognition of foundational concepts of

proportional reasoning (Research question 1) as well as identifying more

constructivist ways of promoting proportional reasoning (Research question 2). This

will be addressed in the Discussion Chapter 5.3. Chapter 5 discusses and interprets

these findings in relation to the existing literature.

116

Chapter 5: Discussion

This chapter discusses the data presented in the previous chapter. The first

research question concerned the identification of the foundational concepts of

proportional reasoning. The first three key findings (Key findings 1-3) identified in

Chapter 4: will be addressed in relation to the foundational concepts of proportional

reasoning in Section 5.1. The second research question asked how teachers promote

foundational concepts in the classroom, with the next three key findings identified in

Chapter 4 (Key findings 4-6) discussed in Section 5.2. In Section 5.3, the final key

finding identified in Chapter 4 (Key finding 7), related to changes in teacher

knowledge, is discussed. The remaining sections explore the implications of these

findings (Section 5.4), followed by a discussion of the limitations and

recommendations for future research (Section 5.5) and a conclusion of the thesis

(Section 5.6).

5.1 RESEARCH QUESTION ONE

The first research question posed in this thesis was, What do P-3 teachers

recognise as foundational concepts of proportional reasoning? This section addresses

this question with regards to the following three key findings discussed in Chapter 4:

1. Proportional reasoning is based on additive and multiplicative thinking

2. Additive and multiplicative thinking evolves from foundational concepts

3. The transition between additive and multiplicative thinking needs to be

supported by a focus on specific foundational concepts

These three findings are indicative of the development of proportional reasoning which

also mirrors the teachers’ development of knowledge of proportional reasoning. The

teachers’ knowledge moved from a broad understanding of proportional reasoning to

more specific knowledge by identifying foundational concepts and their importance in

the development of proportional reasoning. Therefore, each of these findings will also

be discussed in relation to the development of the teachers’ knowledge.

117

5.1.1 Proportional reasoning is based on additive and multiplicative thinking

The concept of proportional reasoning is normally associated with secondary

school teaching as it is used and taught to students in these years. This study indicated

that teachers of the early years had a broad understanding of proportional reasoning as

the teachers identified it as a way of thinking and that it was innate. Both

understandings align with the literature and indicate that the teachers started this study

from a broad knowledge base. Identifying proportional reasoning as a way of thinking,

supports the connection between them both with proportional reasoning being a

sophisticated way of thinking (Fielding-Wells, Dole & Makar, 2013). Such a link

between thinking and proportional reasoning highlights that it is more than simply

getting children to apply an algorithm (Langrall & Swafford, 2000). Proportional

reasoning is regarded as allusive and therefore to suggest it was innate, indicates two

possible reasons; first it couldn’t be taught and that was why 90% of adults cannot

proportionally reason or second, that it should be taught so students don’t become one

these adults (Lamon, 2010). The teachers continued in the study indicating they

believed proportional reasoning could be taught.

Consistent with the literature, when recording the sequence of development of

proportional reasoning each teacher highlighted that proportional reasoning is based

on additive and multiplicative thinking. Additive thinking is fundamental to

multiplicative thinking and multiplicative thinking builds on additive thinking and is,

in turn, foundational to proportional reasoning (Young-Loveridge, 2005). Each of the

sequences commenced with additive thinking, included multiplicative thinking and

concluded with proportional reasoning indicating each teacher’s understanding of the

relationship of thinking to reasoning. That is, proportional reasoning is thinking that

evolves, with teacher support, through the sequential development of additive and

multiplicative thinking (Langrall & Swafford, 2000).

The key to the success of student development of additive and multiplicative

thinking and in turn proportional reasoning, is the teacher. A teacher needs to have “an

understanding that proportional reasoning is developmental and how students develop

it” (Hilton et al., 2016). A teacher needs to know the difference between additive and

multiplicative thinking, both in terms of the foundational concepts and in problem

solving situations. (Hilton et al., 2016 p. 193). This will be discussed in Section 5.2.2.

118

At an emerging knowledge stage, two teachers knew of the relevance of additive

and multiplicative thinking, no teachers identified the difference between additive and

multiplicative thinking as relevant. Teacher understanding of that difference is

necessary to support student development of additive thinking, and in turn

multiplicative thinking on which proportional reasoning is based (Hilton et al., 2016).

Broadly defined, additive thinking is regarded as, “the thought processes underpinning

the aggregation or disaggregation of collections or quantities in a wide variety of

contexts” (Siemon et al., 2012 p. 321). That is, it is a combination of splitting whole

numbers through addition and subtraction. On the other hand, multiplicative thinking

requires a “well developed sense of number” and a depth of understanding “of the

many contexts in which multiplication and division can arise” (Siemon, 2013, p. 43).

As the study progressed the teachers’ knowledge grew and they were able to identify

foundational concepts which could be categorised into additive and multiplicative

thinking, and these will be discussed in the next section.

5.1.2 Additive and multiplicative thinking evolves from foundational concepts

Proportional reasoning is based on additive and multiplicative thinking, and the

development of additive and multiplicative thinking evolves from the development of

foundational concepts (Siemon, 2013; Jacob & Willis, 2003). Like the overall

understanding of proportional reasoning, the identification of its foundational concepts

are allusive. This study found that the Australian Curriculum: Mathematics does not

identify foundational concepts of proportional reasoning. The identification instead

relied on the teacher’s own knowledge as the content descriptions relevant to each year

level are presented in isolation from higher-level concepts such as additive and

multiplicative thinking. Siemon (2013) suggests that the content descriptions can

potentially support multiplicative thinking, but the extent to which they do is “heavily

dependent on how the descriptors are interpreted, represented, considered and

connected in practice” (p. 47). Content descriptions identified in this study were

categorised according to additive or multiplicative thinking (see Table 4.2) however

not all the content descriptions that contribute to additive and multiplicative thinking

were identified by the teachers. These results confirm that teachers need a solid

understanding of these higher-level concepts or big ideas to interpret the curriculum

content and an understanding of big ideas. Such an understanding helps teachers (and

students) make connections between related mathematical concepts (Charles, 2005).

119

Therefore, the content descriptions found in the curriculum should not be viewed in

isolation, but rather through the “interpretative lenses” of the higher-level concepts,

that is big ideas (Siemon, 2013, p. 40).

Drawing on other research as part of the teacher research process proved useful

when identifying the foundational concepts. One such study (Jacob & Willis, 2003)

influenced the teachers’ knowledge with the teachers drawing on the developmental

phases and some of the associated foundational concepts found within their study. This

change in teacher knowledge attests to the value of supporting teachers to engage in

evidence based practices by scaffolding engagement with relevant research and

literature. Additionally, the opportunity to discuss relevant research and literature with

colleagues is reflective of teacher research and confirms the value of this research

approach.

In the studies relating to the development of additive and multiplicative thinking,

identified in Table 2.1, only Jacob and Willis (2003) describes transitional thinking as

the link between additive and beginning to think multiplicatively, followed by

multiplicative thinking (see Figure 5.1). The first four types of thinking were relevant

because they are reflective of the year levels in this study, with the expectation that

students should begin to be thinking multiplicatively at Year 3. This developmental

pathway is outlined in Figure 5.1.

Figure 5.1. Developmental pathway based on Jacob & Willis, 2003

The developmental pathway based on Jacob and Willis (2003) identifies each

type of thinking which is denoted by a phase unique to the work of Jacob & Willis

(2003). The foundational concepts identified by teachers in the results are

acknowledged below these phases, as they can be attributed to both the phases and the

different types of thinking. Each of the teachers identified different foundational

concepts (ones that were particularly relevant to the year level they were teaching); for

example, one to one counting, trust the count (Prep and Year 1), keeping track of

number of groups and total in each group (Year 2).

120

The foundational concepts belonging to the additive composition phase were the

most referenced, with the concepts belonging to the multiplicative relations phase

being the next most referenced. The concepts most frequently referenced, throughout

the data collection, were trust the count, counting groups and skip counting. All

teachers (n=4) knew the value of each of these foundational concepts in isolation but

through the process of being involved in the study, one teacher noted that she was now

aware of the importance of these concepts in relation to proportional reasoning.

The foundational concept, counting groups can also be attributed to the strategy

transitional counting and skip counting can be attributed to building up as identified

by Downton (2010). These two strategies are described in relation to multiplication

and have relevance to the next most referenced foundational concept, multiplier

(number of groups) multiplicand (groups of equal sizes) and product (the total

number). An understanding of this foundational concept is reflective of students

beginning to think multiplicatively and is crucial to the development of multiplicative

thinking (see Figure 5.1).

Importantly, the three elements of multiplication and the confidence in

understanding its operation was identified as necessary to be able to think

multiplicatively. This aligns with the research that students’ understanding of

multiplicative situations encompasses the three components of multiplication:

multiplier (number of groups) multiplicand (groups of equal sizes) and product (the

total number) (Hurst, 2015). One teacher attributed her understanding of the

importance of these elements, to research, confirming the benefit of supporting

teachers’ growth with relevant research and literature. The foundational concepts that

support the transition between additive and multiplicative thinking (Figure 5.1) was

limited and this will be discussed in the next section.

5.1.3 Transition between additive and multiplicative thinking supported by a

focus on specific foundational concepts

Teachers play a critical role in supporting students to make the transition from

additive to multiplicative thinking. In this study, the teachers (n=3) were aware of the

need to move a student’s thinking from additive to multiplicative thinking, but were

unable to identify the ways that they could support a child’s transition between additive

and early multiplicative thinking. The foundational concept associated with this phase

is the simultaneous tracking of the number in each group (multiplicand) and the

121

number of groups (multiplier) (Jacob & Willis, 2003). Reference to the foundational

concept of this phase were least referenced by the participants in this study, despite all

participants having the opportunity to identify the concepts.

To transition from additive thinking (number in each group [multiplicand] and

the number of groups [multiplier]) to beginning to think multiplicatively, the

understanding of the third aspect of multiplication: the product is necessary. A child

begins to think multiplicatively when it knows the three aspects of multiplication

(multiplicand, multiplier, and the product) and understands the relationship between

the three (Jacob & Willis, 2003). A child’s understanding of the three aspects of

multiplication is supported by the way the learner is taught the concept of

multiplication (Hurst, 2015).

This study highlighted that the teachers were limited in their knowledge of the

relationship of the three elements of multiplication and its impact on the transition to

multiplicative thinking. Teachers of these early year levels have a strong content

knowledge about the development and understanding of additive thinking. Formal

mathematical teaching in these years builds upon students’ “conceptual underpinnings

that support addition and subtraction” that is established before they come to school

(Siemon et al., 2012, p. 200). However, multiplicative thinking evolves from

multiplication and the evolution of multiplicative thinking is dependent on how

multiplication is taught. The Australian Curriculum: Mathematics attributes the

teaching of multiplication and the introduction of multiplication to students through

“repeated addition, groups and arrays (ACMNA031)” ([ACARA], 2016, p.17). The

teaching of multiplication from an additive perspective with an emphasis on equal

groups and repeated addition is an approach that is discouraged in the research

(Siemon, 2013).

The Year 2 and Year 3 teachers, referenced the mathematics curriculum

suggesting using repeated addition to teach multiplication was suggested by both the

Year 2 and the 3 Year curriculum. Hurst (2015, p. 10) cautions against linking

multiplication with repeated addition at all, as he believes students are “likely to

remember that [repeated addition] to their detriment when they need to be thinking

multiplicatively.” This suggestion of not teaching multiplication through repeated

addition contradicts the Australian Curriculum: Mathematics and provides a quandary

for teachers as to how to teach children to learn to think multiplicatively.

122

The importance of the foundational concepts (multiplicand, multiplier, and the

product) for transition from additive to multiplicative thinking was identified by the

Year 2 teacher when she made the link between teaching the times tables with the

understanding of these three concepts. She identified switching the way she teaches

times tables to a focus on number of groups. This change in approach supports the

conceptual understanding for understanding times tables, rather than teaching them

procedurally or by rote. This aligns with the work of Hurst and Hurrell (2016) who

assert that procedural fluency of number facts must be based on conceptual

understanding.

5.1.4 Summary

In summary, the key findings in relation to research question one was discussed

in this section. The key findings included the notion that proportional reasoning is

based on additive and multiplicative thinking. Whilst the teachers were able to identify

this they did not identify the difference which is important to know in order to teach

students. The second key finding was that additive and multiplicative thinking evolves

from foundational concepts. It is on the developmental pathway that students learn to

think additively and move to beginning to think multiplicatively through the

development of foundational concepts. The teachers identified foundational concepts

that sit either side of transitional thinking; that is, the thinking that supports students

transition from additive to multiplicative. However, the foundational concepts that

support the transition were least referenced and therefore informed the third finding,

that the transition between additive and multiplicative thinking needs to be supported

in order for students to move along the pathway.

5.2 RESEARCH QUESTION TWO

The second research question asked, How do P-3 teachers promote the

foundational concepts of proportional reasoning? This section discusses the next

three key findings identified in Chapter 4 that are related to how teachers promoted

proportional reasoning in their early years’ classrooms.

4. Promoting proportional reasoning through problem solving

5. Promoting proportional reasoning through language

6. Promoting proportional reasoning with resources

123

There was a distinction between the results prior to, and after, the lesson

implementation. Prior to the lesson implementation, the participants acknowledged

that proportional reasoning had applications to real life and problem solving and that

teachers needed to plan for student thinking. This outcome guided the fourth finding

that mathematical tasks, particularly problem solving, are ways to promote application

of proportional reasoning. Mathematical language and environmental provisions of

using resources featured both prior to, and after, lesson implementation indicating the

importance the teachers held for each of these ways of promoting proportional

reasoning. Therefore, promoting understanding through language and resources and

this informed the fifth and sixth key finding. Each of these key findings are now

discussed in turn.

5.2.1 Promoting proportional reasoning through problem solving

Problem solving promotes proportional reasoning through providing a context

to apply associated thinking and construct knowledge (Lamon, 2010; O’Shea & Leavy,

2013). In relation to this study, problem solving was identified as a way for students

to experience, and develop, additive and multiplicative thinking and, in turn, for

teachers to check students’ understanding of additive and multiplicative thinking

situations. To be able to support student understanding, teachers need to be able to

identify the difference between additive and multiplicative thinking (Hilton et al.,

2016).

Proportional reasoning was initially identified as “a problem-solving strategy”

(TP: FG:39) with the teachers in the study (n=3) identifying problem solving as a way

for children to develop proportional reasoning and a way to check student thinking

when solving problems related to proportional reasoning. Problem solving is a way to

promote understanding of proportional reasoning, through the teacher modelling how

to solve additive and multiplicative problems and by allowing children to construct

knowledge by resolving problem situations, which are additive or multiplicative in

nature (Lamon, 2010). Offering problem situations as a model for teaching solutions

to additive and multiplicative problems can be achieved within a context of

“intentional and guided whole class discussion”, which is based on argumentation as

the teacher models and the students are supported to explain their understandings and

to make their thinking visible (Fieldlings-Wells et al., 2014, p. 51).

124

The findings of the study found value in offering either additive or multiplicative

situations to allow students to make meaning of proportional reasoning. Problem

solving was identified as an important way to promote proportional reasoning and

construct knowledge through collaboration with peers in the classroom. This study

found the process of offering a variety of additive and multiplicative problems for

students to solve in small groups and to share their understanding, assists other children

to gain an understanding. This constructivist approach to learning was found to be

valuable for developing student understanding. It is vital that students are afforded the

opportunity to problem solve as the ability to problem solve in different contexts

contributes to being a numerate person (Hilton & Hilton, 2016). Through a

constructivist approach, children can construct knowledge about, and develop an

understanding of, the difference between additive (non-proportion) and multiplicative

(proportional) problems and which thinking to apply to solve the problem. Therefore,

students can construct their understanding by resolving problematic situations (Alsup,

2005).

Students should be given opportunities to solve problems individually or in small

groups as the early stage of a child’s development, this knowledge (understanding the

difference of additive and multiplicative) would be constructed through discussions on

how to solve the problems, rather than a focus on the solution (Lamon, 2010). As

Lamon (2010) suggests, Year 3 students are quite young to be engaging individually

in additive and multiplicative problems, but offering these problems as a basis for

discussion is a valuable way for children to observe other children’s understanding,

especially if it demonstrates multiplicative thinking.

Problem solving was also identified to check student understanding and

thinking by analysing student’s errors and making use of these errors for future

planning to support students’ development of proportional reasoning. Through

offering problems for students to solve, teachers analyse student responses to obtain

evidence of children’s thinking (Ell et al., 2004). More particularly, providing additive

and multiplicative problem-solving situations allows the teacher to check if students

used additive or multiplicative thinking in the correct context. Students need support

in going beyond additive thinking to the more abstract thinking of multiplicative

thinking, “that creates more complex quantities” (Lamon, 2010, p. 32).

125

In order to support students, teachers need to possess a strong understanding of

the difference between additive and multiplicative thinking if they are to use problem

solving in this way and support students in the development of additive and

multiplicative thinking (Hilton et al., 2016). To support students’ shift from one type

of thinking to another, requires knowledge of the differences between additive and

multiplicative thinking, which can often be a struggle for teachers (Hilton et al., 2016).

In their study of prospective kindergarten teachers, Pitta-Pantazi and Christou (2011)

found that many lacked the ability to think additively and multiplicatively. This was

also supported in the initial study of Jacob and Willis (2001) who endeavoured to find

how teachers recognise the difference between additive and multiplicative thinking in

children and they further found it was not an easy task for some teachers. Without an

understanding of the common patterns and differences of these types of thinking

(additive and multiplicative) teachers cannot diagnose children’s thinking and support

them in their development of each as they transition from additive to multiplicative

thinking (Lamon, 2010).

Providing students with the opportunity to articulate thinking and contribute to

discussions about proportional reasoning, provides a context for the development of

the language associated with proportional reasoning, which is discussed in the next

section.

5.2.2 Promoting proportional reasoning through language

In this study, promotion of proportional reasoning was identified through

language by modelling the specific language associated with proportional reasoning

and the foundational concepts and by encouraging students to share understanding

using the language. Teacher’s knowledge of using language changed as the study

progressed. The findings of the study initially identified the use of comparative

language including specific comparative words to promote proportional reasoning

which in the simplest form, is about comparative situations. In particular, comparative

language in proportional situations in mathematics and other subjects (science),

reflects an understanding of the language of comparison as an important aspect of

lower primary curriculum. This aligns with the findings of Hilton and Hilton (2016)

whose audit of the science curriculum found evidence of the comparative language of

proportional reasoning in the lower primary curriculum.

126

It also found that specific language and vocabulary associated with proportional

reasoning and foundational concepts; for example, arrays, product, and multiplication

was imperative for student understanding. A focus on the language associated with

proportional reasoning is regarded as important, particularly as it is a way to explain

the elements of multiplication and contribute to student conceptual understanding

(Hurst, 2015).

The study found two ways the teacher interacted with students, these included

explaining, particularly using show and tell strategies, structured conversations, and

reviewing to support student understanding of proportional reasoning (Anghileri,

2006). Through explaining, the teacher exposed the students to the associated

language of proportional reasoning. A reviewing approach, offers a more

constructivist perspective, as the teacher encourages the child to use language to

explain their own thinking.

Language is vital to the development of student’s conceptual understanding of

proportional reasoning which can be further enhanced while extensively using

materials (Hurst, 2015). Student’s use of the resources will be discussed in the next

section.

5.2.3 Promoting understanding of foundational concepts with resources

This study found the provision of multiple resources was a way to promote

proportional reasoning and scaffold student understanding. Anghileri, (2006) regards

environmental provisions, that is, provision of resources, as a way to scaffold learning

as it can “have a significant impact on learning” (p. 40). In this section, the finding

that foundational concepts can be promoted with the use of a range of resources and

the use of arrays as a particular way to promote proportional reasoning is discussed.

This study found that a variety of resources can promote the understanding of

proportional reasoning. These resources included dominoes, dice patterns, flashcards,

games, songs, rhymes, counting and number lines. In particular, dominoes and dice

were suggested as a way of reinforcing understanding of trust the count/subitising

through a hands-on experience. By seeing the patterns of numbers on a dice or domino,

students are supported to recognise “the numerosity of a small collection without

counting,” which then supports the development of the foundational concepts of

subitising (Siemon et al., 2012, p. 688). The provision of flashcards, through the

speeded presentation of dot patterns promotes the recognition of small number

127

quantities. Resources were also identified to provide opportunities for practice and drill

as a more direct approach to discuss the number patterns and groups. Resources were

also found as a way in which to focus on facilitating students’ understandings of

proportional reasoning. Through utilisation of “appropriate materials and

manipulatives” students are encouraged to reflect upon their involvement which can

be independent or in a collaborative way thus reflecting a constructivist approach

(O’Shea & Levey, 2013). It was found a constructivist approach affords students the

opportunity to explore, hypothesise, think, and socially construct knowledge (Fast &

Hankes, 2010; Reys et al., 2012). In addition to identifying a range of resources to

promote understanding of proportional reasoning, teachers described the modelling of

language associated with the different resources. The teachers discussed the

presentation of resources within the context of explaining (show and tell and

explanation). This is reflective of Anghileri (2006) who identified structured

conversations that guide students to the next stage of understanding, while they

interacted with resources.

This study found the use of arrays as a resource or representation that contributes

to student understanding of the multiplicative situation. In its simplest form, an array

is a model of rows and columns; for example, in a flower bed with 3 rows and 6 plants

in each row. It was identified as a tool to build understanding and proficiency. A more

specific use of arrays was identified as a way to help students to develop mental images

of the multiplicative situation of the three quantities: groups of equal sizes

(multiplicand), number of groups (multiplier), and the total amount (product). This

teacher’s use of arrays to enhance students’ early learning of the concepts associated

with a multiplicative situation to foster multiplicative thinking, aligns with the

literature (Hurst, 2015; Jacob & Mulligan, 2014; Young-Loveridge, 2005). Using the

array to support a student’s understanding of the number of groups and size of groups

supports an understanding of the foundational concepts that support the development

of transitional thinking. By including the third element of multiplication - that is, the

answer or total amount (product) – it helps the student transition to the beginning to

think multiplicatively stage. By exploring with arrays, including drawing arrays,

students construct an understanding of the multiplicative situation.

The study also found that the array could be removed as a strategy to move

students onto the next level of understanding and thinking. The removal supports the

128

view that students need to develop a visualised image of the array in order to solve

multiplicative situations. In particular, Jacob and Mulligan (2014) encourage the

visualisation of arrays so students “can draw on mental images of small number

arrays” (p. 39).

Using arrays to support students’ understanding of the relationship between

multiplication and division as a way to encourage understanding of the inverse

relationship was also identified in the study. Jacob and Mulligan (2014) and Watson

(2016) also advocate for the use arrays to support the understanding of this inverse

relationship. On the other hand, Hurst (2015) discourages this approach for using

arrays, as he believes it is better to use the array only to teach the multiplicative

situation, rather than multiplication and its inverse (division). These contrasting

positions in the literature (the use of arrays and developing understanding) highlight

the importance of teachers possessing a strong understanding of multiplicative

thinking. Teachers can then make an informed decision about how, and when, to use

arrays and how to best support students’ conceptual understanding.

Overall, using resources and arrays to promote proportional reasoning, suggest

the teachers were focussed on supporting children’s understanding of multiplicative

thinking. Their descriptions were predominantly suggestive of a constructivist

approach to building students’ understandings. In particular, the implementation of

arrays contributes to the students’ understanding of foundational concepts, as it enables

students to use their mathematical language to explain models produced and, in doing

so, demonstrate their understanding (Watson, 2016).

5.3 CHANGES IN TEACHER KNOWLEDGE

Throughout the study an interesting finding (Key finding 7) was that teachers

demonstrated a growth in knowledge of the foundational concepts of proportional

reasoning (Research question 1) as well as identifying more constructivist ways of

promoting proportional reasoning (Research question 2). In the initial part of the

study, the teachers were unable to identify specific foundation concepts of proportional

reasoning, but demonstrated a general awareness of proportional reasoning. All

participating teachers (n=4) identified mathematical concepts that draw on

proportional reasoning (e.g. percentage, fraction, ratio, decimals, measurement,

graphing, probability, and place value), that is percentage, fraction, ratio, and decimals

are concepts of which proportionality is a key characteristic. Proportional reasoning

129

permeates measurement, graphing, and probability activities (Lamon, 2010). Place

value was the most relevant as its understanding contributes to additive thinking.

However, the teachers were unable to identify the relationship between these named

mathematical concepts and proportional reasoning. This level of knowledge aligns

with research findings from Fernandez et al., (2013) who, in their research with

preservice primary school teachers, found that these teachers only had a “partial

understanding of proportional reasoning” (p. 164).

There was increase in teacher knowledge as the study progressed, at the

beginning of the study teachers identified foundational concepts for proportional

reasoning simply as a way of thinking. As the study progressed, the teachers were able

to identity the sequence of development of the foundational skills in each of their

concept map. By the end of the study they identified foundational concepts as the

platform for the development of additive and multiplicative thinking. This aligns with

the research that it is vital for teachers to understand and have an in-depth knowledge

of the various aspects of proportional reasoning; know that it is developmental; and be

aware of its foundational concepts (Hilton et al., 2016). The growth of the teachers’

knowledge was aligned with the teachers offering more constructivist ways of

promoting proportional reasoning.

The connection between teachers informed foundational concepts and

constructivist approaches to teaching was particularly evident in relation to embracing

‘teachable moment’ opportunities for promoting proportional reasoning. There was

evidence in the study of the alignment of increased teacher knowledge of proportional

reasoning and taking advantage of teachable moment opportunities that were

constructivist in nature. As knowledge of proportional reasoning increased,

particularly in relation to the phases of multiplicative thinking, there was more

awareness of seizing learning opportunities to reinforce student understanding of this

topic. Two teachers made the connections between their knowledge and the impact

on their teaching, one in relation to their own knowledge and her teaching, “having

now that knowledge, it really helps to fine tune” (T3:PL:19). With the other teacher

sharing a teachable moment opportunity when her students were talking about

percentage, in the past she would have explained it quickly but because of her

knowledge of proportional reasoning as an outcome of the study she “honed in on it”

(T2:PL:115). This example provides evidence that to be able to take advantage of

130

teachable moments requires a teacher to have knowledge of the concepts so that, at

that point in time, the teacher can support the students in their understanding (Muir

2008; Siemon et al., 2012).

It was found to support children in problem solving teachers need to be able to

identify the difference between additive and multiplicative problems but then have the

ability to support student understanding, for example using the bar model as tool to do

this. Fernandez et al., (2013) found in their study, a correlation between the teacher’s

knowledge and the teacher noticing students’ mathematical thinking in problem

solving. These findings suggest that teachers need both CK and PCK to be able to

interpret and model students’ mathematical thinking, particularly in relation to

proportional reasoning (Fernandez et al., 2013).

A growth in teacher knowledge was evident as the study progressed. This

growth was reflected in a better understanding of the foundational concepts and

identification of more meaningful ways to promote student understanding through

problem solving, language and resources each offered from a constructivist approach

to teaching. This was particularly evident in the data collected in the individual

interview when teachers discussed the value of lesson study which is reflective of a

constructivist approach to learning.

5.4 IMPLICATIONS

There are three major implications identified based on the seven key findings in

this study. The first implication that emerges is the importance of teachers possessing

CK of higher level mathematical concepts; in this case, proportional reasoning with

multiplicative thinking, when planning and teaching foundational or lower level

concepts. Secondly, the weakness of the Australian Curriculum: Mathematics

organisation, as it does not identify the development of low level concepts and their

relationship to higher-level concepts. Third, involvement in a teacher research

approach has benefits for teachers’ knowledge and practice, as the approach

encourages teachers to act as observers and learners in their own school setting. Each

of these implications will be addressed in turn. A discussion of these implications will

be followed by the practical outcomes of these implications for me, as a curriculum

leader in the case study school.

131

This study highlighted the importance of teachers CK of higher level

mathematical concepts or big ideas when planning and teaching the foundational

concepts. Through the lens of a big idea, teachers CK would include the content that

contributes to a big idea. Traditionally, teachers in a year level know the content for

their year level and perhaps the content from the year that follows. To have knowledge

of all the content that contributes to the big idea would provide a greater depth of

understanding for a teacher and in turn support a student’s development of the

mathematics, as in this case, proportional reasoning building on multiplicative

thinking. Many of the mathematical concepts taught in Prep to Year 3 are the

foundational concepts (such as one to one counting, trust the count, equal groups and

part-whole) for higher level mathematical concepts. Teachers in these year levels have

a strong understanding of these foundational concepts as they are mathematical

concepts relevant to these year levels. What these teachers do not always know and

have is an understanding of the relevance of those foundational concepts to higher

level mathematical concepts such as multiplicative thinking, and in turn proportional

reasoning. Siemon (2013) supports this view, that teachers need to address the content

descriptions in a way that they are “connected in practice” to support the development

of multiplicative thinking (p. 47). An awareness of the role of these concepts in

forming the foundation for multiplicative thinking and, in turn, proportional reasoning,

needs to be a priority for the mathematical planning and teaching in the early years.

The second implication is that the development of CK is vital as the Australian

Curriculum: Mathematics in presenting the structure of mathematical ideas does not

identify the development of low level concepts, or their relationship to higher-level

concepts. The organisation of the curriculum is arranged in such a way that teachers

view the content descriptions in isolation for the year level and not see them in a

broader context across year levels. The proficiency strands (understanding, fluency,

problem-solving, reasoning) identified in the Australian Curriculum: Mathematics, by

definition, broadly identify how students engage with the content ([ACARA], 2016).

The proficiency strands are the ways learners should act and think mathematically

(Siemon, 2013). However, their focus remains on a year level emphasis of the

curriculum, as they “reinforce the significance of working mathematically within the

content, and describe how the content is explored and developed at each year level

(Stephens, 2014 p. 2). This weakness in the organisation of the curriculum document

132

has implications for teachers’ knowledge of and practices for promoting foundational

concepts.

Finally, teachers have the potential to grow their knowledge when involved in

teacher research process, that is, a process of observing and learning in the context of

their own school settings. Throughout this study, the quality of the professional

discussions used to explore teachers’ views improved as the teachers reflected and

explored the foundational concepts and ways to promote proportional reasoning. These

discussions reflected characteristics of a professional learning community as the

teachers developed mathematics knowledge, improved their professional practice, and

the approach was collaborative in nature (Cherrington & Thornton, 2015). Such

constructivist approaches to research, and reflective practice, align with the research

methodology of teacher research (Campbell, 2013). Throughout the research process

the teachers contributed to the direction of the study; for example, the curriculum

investigation and providing the approach for collaborative lesson planning. This

approach aimed to support and improve the teachers’ professional practice by

discussing readings and their own practice. The most powerful characteristic was the

collaborative approach to learning, with the teachers learning together as they directed

the research agenda and gained greater confidence in their own knowledge of

mathematics. One outcome of the research process was that it allowed teacher

knowledge to grow as they reflected on their own practice and supported the researcher

in the process of all aspects of research, particularly “to be attentive to our data

collection methodology and analysis” (Campbell, 2013, p. 4).

In summary, the three identified implications highlight the significance of

teacher knowledge. A teacher needs a deep knowledge of the low level mathematical

concepts in order to establish a strong foundation for a student’s development of higher

level concepts. Australian Curriculum: Mathematics requires interpretation for

planning and teaching and independently a deep curriculum knowledge of how lower

level concepts relate to higher level concepts. Teacher knowledge has the potential to

grow when a teacher is involved in teacher research within the context of a teacher’s

own school setting.

The genesis of this study, teacher knowledge of number development, was

acknowledged at the beginning of this thesis, provided a rationale for this study and

was the impetus for the two research questions. Through a teacher research approach

133

to find solutions to the research questions, and by identifying findings and

implications, practical outcomes for the future have been identified. For me, in my

curriculum leadership role in this case study school, the three implications described,

inform these practical outcomes. Firstly, in the future, I will have regard to the first

two implications, by giving prominence to the relationship between lower level

concepts and higher-level concepts or big ideas. Encouraging an emphasis on big ideas

for planning and teaching, will provide opportunities for the development of teacher

knowledge and in turn enhanced student understanding. A change to a big idea

approach to planning and teaching would allow teachers to monitor students level of

understanding of the foundational concepts of all big ideas from a different and deeper

perspective. Drawing upon this broader knowledge would help identify a child’s gap

in understanding of the foundational concepts or support a child in extending their

knowledge. A big ideas perspective would also encourage collaborative planning

across year levels as it would be more effective for teachers across year levels to plan

by using mathematical concepts as the starting point rather than compartmentalised

year level content.

This approach would also be supported by viewing the content through the prism

of proficiency strands across year levels, to support the broader development and

understanding of the big idea or higher-level concepts. Using the proficiency strands

as intended, would support students in developing “the relationship between the ‘why’

and ‘how’ of mathematics” rather than just the ‘what’ or content in isolation

([ACARA], 2016 p. 5). This would also have consequences for ways teachers can

promote student understanding of the content.

In terms of the third implication, using the teacher research approach to develop

teacher knowledge will also have practical outcomes for me. In future studies, my

school will encourage teachers to implement their own systematic investigations as

teacher-researchers. Data collection methods will be developed across the school so

that rigour can be embedded within the implemented research projects. Therefore, the

professional development approach currently used will change and be extended to

incorporate two stages. Stage 1 will include research skill development by developing

teacher skills with different research methods, for example template analysis. Stage 2

will be the teachers implementing teacher research approach which will incorporate

the research methods.

134

5.5 LIMITATIONS AND FUTURE DIRECTIONS

The findings of this study are limited to the five participants involved in this case

study, and findings may not be transferable to other Prep to Year 3 teachers. As this

study employs convenience sampling, it is important to acknowledge that the

participants may not be representative of the broader early childhood teacher

population provides a snapshot of the participants understanding at the time the

research was conducted. (Yin, 2009). The overall findings are limited to the details of

this bounded study. Transferability can be difficult to establish because of the

characteristics of the qualitative nature of the study, but it is more likely if the process

is described thoroughly “for others to follow and replicate” (Kumar, 2014, p. 219).

Limitations also relate to the role of the researcher, the duration of the research,

and the methods employed. In relation to the researcher, it is important to

acknowledge the existing relationship between the researcher and teachers

participating in this study (see Section 3.3.2). The researcher is the Head of Primary

at the school and has been for over ten years. Adopting a teacher research approach to

the study allowed all the participants to have equal involvement and provided

opportunities for the participants to collaboratively plan and direct the study. This

approach helped reduce the impact of the relationship between the teachers and the

researcher. It is possible that participants may have felt constrained to provide

responses or may have sought to provide responses that placed the participant in a

more favourable light. The research was designed to build on teachers’ ideas so they

were empowered rather than threatened by the other role of the researcher that is Head

of Primary. The teachers were given every assurance that any response they gave

would have no adverse impact on their reputation, or the respect held for them by the

researcher. On the other hand, the relationship between the researcher and the teachers

may have enabled a freer and more open dialogue because of the existing respectful

and collaborative relationship and the familiarity between the participants and the

researcher.

It is considered that this study, albeit limited in size, has highlighted a number

of areas worthy of further investigation in relation to teacher knowledge of

proportional reasoning. Consideration should be given to when proportional reasoning

is taught. Notwithstanding that proportional reasoning is taught in the later years, early

childhood teachers need to have a strong understanding of proportional reasoning

135

because the ways that teachers promote the teaching of additive and multiplicative

thinking impacts on how proportional reasoning skills of students in the middle years

evolves. Further studies into teacher knowledge of the development of proportional

reasoning could build upon this study. This study also highlights the need for further

studies into the pedagogical practices for teaching the foundational concepts of

proportional reasoning. Ideally this area of investigation would benefit from a

longitudinal study of teacher practice and student outcomes to determine whether a

change in teacher understanding of proportional reasoning and practice in early years

of school would result in better outcomes for students’ acquisition of proportional

reasoning in the early secondary.

This study investigated two broad topics: foundational concepts of proportional

reasoning development and how they are promoted. Identifying the foundational

concepts of proportional reasoning was a large topic because proportional reasoning is

big idea. Further research could investigate teachers’ understanding of the big ideas

of mathematics and how P-3 mathematics contributes to the development

understandings of the big ideas of mathematics. The promotion of big ideas would

provide primary teachers with an in depth understanding of the mathematical

foundational concepts that contribute to the development of each big idea of

mathematics.

The difference between additive and multiplicative thinking is important in

understanding proportional reasoning. The link between addition and multiplication

(and its development), could be investigated and understanding could be tested through

problem situations. Alongside this, the pedagogies for teaching the developing

additive and multiplicative thinking through a constructivist approach could be

identified and investigated either as a part of the suggested study or separately.

5.6 CONCLUSION

Proportional reasoning is a big idea of mathematics. While research has found

that young children have the capacity to understand proportional reasoning in its most

basic form (Hilton et al., 2016), proportional reasoning is not a term normally

associated with primary maths, particularly in the early primary years (P-3). This

study, through the early years (P-3) teachers focused on the identification of the

foundational concepts that found development of proportional reasoning and the ways

such foundational concepts may be promoted. It is vital that children have a strong

136

foundation of understanding to prevent them growing into one of 90% of adults who

cannot proportionally reason (Lamon, 2010). Consequently, it is important for teachers

to understand this big idea of mathematics and the mathematical concepts that emerge

from its development.

The first finding that emerged from the study was that proportional reasoning is

a way of thinking, based on additive and multiplicative thinking. The process of

identifying foundational concepts of proportional reasoning informed the second

finding, that additive and multiplicative thinking evolves from foundational concepts.

The foundational concepts from which additive thinking evolves; trust the

count/subitising, groups, skip counting and repeated addition were the most frequently

discussed in the data collection sources; concept map, individual lesson topic, during

the planning and post lesson discussion. The frequency of the identification of these

concepts may be the outcome of the greater relevance of these concepts to the early

years’ mathematics curriculum. Multiplicand, multiplier, product and inverse

relationship support the learner’s development to think multiplicatively and these were

identified as the foundational concepts of multiplicative thinking. The Year 3 teacher

was the participant who most identified these concepts, which is reflective of the

content of this year level.

The third finding revealed in this study was that the transition between additive

and multiplicative thinking needs to be supported by a focus on specific foundational

concepts. The foundational concepts (numbers of groups and the total in each group)

were identified as the concepts that support the transition between additive and

multiplicative. Teachers need to be aware of the transition from additive to

multiplicative thinking, to enable them to support students in the transition and in turn

be able to develop proportional reasoning.

Teachers in the earlier years have the capacity to support student development

of these foundational concepts, with teachers of year levels after Year 2 helping

children develop these foundational concepts if students are struggling in the transition

to multiplicative thinking. To assist the development of multiplicative thinking, the

teaching of multiplication should focus on the concept (multiplicand, multiplier,

product), rather than a repeated addition approach to the teaching of multiplication as

is traditionally the case.

137

This study identified three ways teachers can promote proportional reasoning.

Proportional reasoning can be promoted through providing mathematical tasks, that

are based on problems, by a focus on mathematical language, and providing

environmental provisions or resources to support the students’ understanding of the

foundational concepts of proportional reasoning. Each of these ways can contribute to

student understanding of proportional reasoning.

Problem solving is important for developing student understanding of

multiplicative thinking. Problem solving tasks can be used in two ways. First, as a

model for teaching solving additive and multiplicative problems and secondly to

understand and check student thinking. An understanding of student thinking (additive

or multiplicative) is gained by presenting problem situations that require the student to

generate solutions and share and explain their thinking. The teacher “acts as a

facilitator to ease the path of the of the students as they attempt to find a solution”

(O’Shea & Leavy, 2013p. 296). Through this constructivist approach the teacher

supports the students to understand the difference between the two types of thinking.

The effectiveness of offering problem solving situations relies on the teacher knowing

the difference between additive and multiplicative thinking so that teachers can

identify how students think and support students in understanding the difference.

Promoting proportional reasoning through language is twofold, both with the

teacher modelling and by student use. The teacher models the language associated

with the foundational concepts and the student uses this language as they interact with

resources and articulate their thinking associated with problem solving. Interactions

with students described by the teachers were reflective of explaining, where the teacher

had more control and reviewing, which encouraged students to articulate their own

understanding. At the heart of these interactions was the acknowledgement of the

importance of developing student understanding as critical to proportional reasoning.

The provision of resources can promote students understanding of the

foundational concepts of proportional reasoning. The prep teacher identified many

resources (e.g. ten frame, dominoes, dice, number line) that can support student

understanding of the foundational concepts while the array can support the transition

from additive to multiplicative thinking when the product or total of multiplication is

the focus. To assist the student to become multiplicative thinkers, the array can

138

enhance student understanding because of the focus on the three concepts (group,

number of groups, product) of multiplication (Hurst, 2015).

Implications for practice arising from this study include the development of

teachers’ knowledge about big ideas of mathematics so that teachers have a strong

understanding of the concepts that lay the foundation and the direction for these ideas.

Teachers need to view the content of Australian Curriculum: Mathematics - not in

isolation - but as part of a bigger picture, which is underpinned by the proficiency

strands: understanding, fluency, problem solving and reasoning. Teachers also need

to interpret the content of the curriculum from a knowledgeable base of understanding

of the foundational concepts of proportional reasoning in relation to teaching

multiplication particularly. The challenge for early year teachers (P-3) is to have an

understanding of the far-reaching impact of the way they teach the foundational

concepts of multiplication albeit that such impact may not be felt by the student until

exposed to the explicit teaching of proportional reasoning several years later.

139

Reference List

Alsup, J. (2005). A comparison of constructivist and traditional instruction in

mathematics. Educational Research Quarterly, 28(4), 3.

Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning.

Journal of Mathematics Teacher Education, 9(1), 33-52. doi: 10.1007/s10857-

006-9005-9

Anthony. G., & Walshaw, M. (2009a) Characteristics of effective teaching of

mathematics: A view from the west. Journal of Mathematics, 2(2), 147-164.

Anthony, G., & Walshaw, M. (2009b). Effective pedagogy in

mathematics/Pangarau: Best evidence synthesis iteration [BES]. Wellington,

New Zealand: Learning Media.

Askew, M. (2016). Transforming Primary Mathematics. Abingdon, OX: Routledge

Publishing.

Askew, M. (2013). Launching futures – you can’t drive by looking in the rear view

mirror. In S. Herbert, J. Tillyer & T. Spencer (Eds.). Mathematics: Launching

Futures, Proceedings of the 24th biennial conference of the Australian

Association of Mathematics Teachers, pp. 36–52. Adelaide, SA: AAMT.

Askew, M. Brown, M., Rhodes, V., Johnson, D, &William, D. (1997). Effective

Teachers of Numeracy. London: School of Education, King’s College

Australian Association of Mathematics Teachers (AAMT). (2009). School

mathematics for the 21st century: A discussion paper. Adelaide, SA: AAMT.

Australian Association of Mathematics Teachers. (2014). Strategic Plan 2014-2016:

AAMT.

Australian Children’s Education and Care Quality Authority (ACECQA). (2011).

Guide to the National Quality Framework. Retrieved from

http://www.acecqa.gov.au/Uploads/files/National%20Quality%20Framework%

20Resources%20Kit1%20Guide%20to%20the%20NQF.pdf

Australian Curriculum Assessment and Reporting Authority (ACARA). (2015a).

Foundation to Year 10 Curriculum: Mathematics (Version 7.5). Retrieved from

http://v7-5.australiancurriculum.edu.au/curriculum/overview

Australian Curriculum Assessment and Reporting Authority (ACARA). (2015b).

Numeracy: learning continuum. Retrieved from http://www.

australiancurriculum.edu.au/generalcapabilities/numeracycontinuum

Australian Curriculum Assessment and Reporting Authority (ACARA). (2016).

Foundation to Year 10 Curriculum: Mathematics (Version 8.3). Retrieved from

https://www.australiancurriculum.edu.au/

Bakker, A., Smit, J., & Wegerif, R. (2015) Scaffolding and dialogic teaching in

mathematics education: introduction and review. ZDM Mathematics Education,

47, 1047-1065. doi: 10.1007/s1858-015-0738-8

140

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching:

What makes it special? Journal of Teacher Education, 59(5), 389–407. doi:

10.1177/0022487108324554.

Baxter, P. & Jack, S. (2008). Qualitative case study: Study design and

implementation for novice researchers. The Qualitative Report, 13(1), 544-559.

Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C. & Miller, J. (1998).

Proportional reasoning among 7th grade students with different curricular

experiences. Educational Studies in Mathematics, 36(1), 247- 273. doi:

10.1023/A:1003235712092

Bennison, A. (2015). Developing an analytic lens for investigating identity as an

embedder-of-numeracy. Maths Ed Res J, 27:1-19. DOI 10.1007/s13394-014-

0129-4

Ben-Peretz, M. (2011). Teacher knowledge: What is it? How do we uncover it? What

are its implications for schooling? Teaching and Teacher Education, 27(1), 3-9.

Beswick, K. (2005). The beliefs/practice connection in broadly defined contexts.

Mathematics Education Research Journal, 17(2), 39-68.

Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, R. & Young-Loveridge.

(2005). Supporting teachers in the development of young children’s

mathematical thinking: Three large scale cases. Mathematics Education

Research Journal, 6(3), 27–57. doi: 10.1007/BF03217400.

Boyer, T. W., Levine, S, C., & Huttenlocher, J. (2008). Development of proportional

reasoning: Where young children go wrong. Developmental Psychology, 44(5),

1478-1490. doi: 10.1037/a0013110.

Bright, G. W., Joyner, J. M., & Wallis, C. (2003). Assessing proportional thinking.

Mathematics Teaching in the Middle School, 9(3), 166-172.

Bruce, C., Flynn, T. & Bennett, S. (2016). A focus on exploratory tasks in lesson

study: the Canadian ‘Math for Young Children’ project. ZDM Mathematics, 48,

541-554 DOI 10.1007/s11858-015-0747-7.

Cakmakci, G. (2009). Preparing teachers as Researchers: Evaluating the quality of

research reports prepared by student teachers. Eurasian Journal of Educational

Research, 35, 39-56.

Campbell, K.H. (2013). A Call to Action: Why We Need More Practitioner

Research. Democracy & Education, 21(2).

Charles, R. (2005). “Big idea” and understandings as the foundation for elementary

and middle school mathematics. Journal of Education Leadership, 7(3), 9–24.

Cherrington, S. & Thonton, K. (2015). The nature of professional learning

communities in New Zealand early childhood education: an exploratory study,

Professional Development in Education, 41(2), 310-328. doi:

10.1080/19415257.2014.986817.

Claessens, A., Duncan, G., & Engel, M. (2009). Kindergarten skills and fifth-grade

achievement: Evidence from the ECLS-K. Economics of Education Review, 28,

415–427. doi: 10.1016/j.econ edurev.2008.09.00

141

Clarke, D. M., Clarke, D. J., & Sullivan P. (2012). Important ideas in mathematics:

What are they and where do you get them? Australian Primary Mathematics

Classroom, 17(3), 13-18.

Clements, D. H., & Sarama, J. (2016). Math, Science, and Technology in the Early

Grades. The Future of Children, 26(2), 75-94.

Cochran-Smith, M., & Lytle, S. L. (1999). The teacher research movement: A decade

later. Educational researcher, 28(7), 15-25.

Cohen, L., Manion, L., & Morrison, K. (2011). Research Methods in Education (7th

ed.). London: Routledge Falmer.

Cohrssen, C., Tayler, C., & Cloney, D. (2015). Playing with maths: Implications for

early childhood mathematics teaching from an implementation study in

Melbourne, Australia. Education 3-13, 43(6), 641-652.

Commonwealth of Australia. (2008). National numeracy review report. Canberra,

Commonwealth of Australia: Commissioned by the Human Capital Working

Group. Retrieved from http://oggiconsulting.com/wp-

content/uploads/2013/08/national_numeracy_review-1.pdf.

Creswell, J. & Miller, D. (2010). Determining validity in qualitative inquiry. Theory

Into Practice, 39(3), 124-130. doi: 10.1207/s15430421tip3903_2.

Creswell, J. W. (2008). Educational research: Planning, conducting, and evaluating

quantitative and qualitative research. Upper Saddle River, N.J: Pearson/Merrill

Prentice Hall.

Department of Education, Employment and Workplace Relations (DEEWR). (2009).

Belonging, being and becoming: The Australian early years learning framework

(EYLF). Canberra, ACT: Australian Government Department of Education,

Employment and Workplace Relations. Retrieved from

https://docs.education.gov.au/system/files/doc/other/belonging_being_and_beco

ming_the_early_years_learning_framework_for_australia.pdf.

Dole, S. (2010, August). Making connections to the big ideas in mathematic:

Promoting proportional reasoning. Paper presented at ACER Research

Conference, Melbourne. (71-74). Retrieved from

http://espace.library.uq.edu.au/view/UQ:227073.

Downton, A. (2010). Challenging multiplicative problems can elicit sophisticated

strategies. In L. Sparrow, B. Kissane., & C. Hurst (Eds.), Shaping the future of

mathematics education: Proceedings of the 33rd annual conference of the

Mathematics Education Research Group of Australasia (pp. 169-176).

Fremantle: MERGA.

Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C.,

Klebanov, P., … Sexton, H. (2007). School readiness and later

achievement. Developmental psychology, 43(6), 1428.

Ediger, M. (2009). Mathematics: Content and pedagogy. College Student Journal,

43(3), 714-717.

142

Education Council. (2015). National STEM school education strategy, 2016–2026.

Retrieved from

http://www.educationcouncil.edu.au/site/DefaultSite/filesystem/documents/Nati

onal%20STEM%20School%20Education%20Strategy.pdf

Ell, F., Irwin, K., & McNaughton, S. (2004). Two pathways to multiplicative

thinking, in I. Putt, R. Faragher and M. McLean (Eds) Mathematics education

for the third Millennium, towards 2010 (Proceedings of the 27th Annual

Conference of the Mathematics Education Research Group of Australasia,

Townsville), 199-206. Sydney, NSW: MERGA.

Fast, G. R., & Hankes, J. E. (2010). Intentional integration of mathematics content

instruction with constructivist pedagogy in elementary mathematics

education. School Science and Mathematics, 110(7), 330-340.

Fernández, C., Llinares, S., & Valls, J. (2013). Primary school teacher's noticing of

students' mathematical thinking in problem solving. The Mathematics

Enthusiast, 10(1/2), 441.

Fielding-Wells, J., Dole, S. & Makar, K. (2014). Inquiry pedagogy to promote

emerging proportional reasoning in primary students. Mathematics Education

Research Journal 26 (1), 47-77. doi: 10.1007/s13394-013-0111-6.

Geiger, V., Goos, M., & Dole, S. (2011). Teacher professional learning in numeracy:

Trajectories through a model for numeracy in the 21st century. In Australian

Association of Mathematics Teachers (AAMT) and the Mathematics Education

Research Group of Australasia (MERGA) Conference 2011 (pp. 297-305).

AAMT and MERGA.

Hachey, A. C. (2013). The early childhood mathematics education revolution. Early

Education and Development, 24(4), 419-230. doi:

10.1080/10409289.2012.756223.

Hattie, J. (2009). Visible Learning a Synthesis of Over 800 Meta-Analyses Relating

to Achievement. Oxford, UK: Routledge.

Hattie, J. Fisher, D & Frey, N. (2017). Visible Learning for Mathematics. London,

UK: Corwin.

Hilton, A., & Hilton, G. (2013). Improving mathematical thinking through

proportional reasoning. Brisbane, QLD: University of Queensland.

Hilton, A., & Hilton, G. (2016). Proportional reasoning: An essential component of

scientific understanding. Teaching Science, 62(4) 32-42.

Hilton, A., Hilton, G., Dole, S. & Goos, M. (2016). Promoting middle school

students’ proportional reasoning skills through an ongoing professional

development programme for teachers. Educational Studies in

Mathematics, 92(2), 193-219. doi: 10.1007/s10649-016-9694-7.

Hurst, C. (2015). The multiplicative situation. Australian Primary Mathematics

Classroom, 20(3), 1-16.

143

Hurst, C., & Hurrell, D. (2016). Multiplicative thinking: Much more than knowing

multiplication facts and procedures. Australian Primary Mathematics

Classroom, 21(1), 34.

Jacob, L. & Mulligan, J. (2014). Using arrays to build towards multiplicative

thinking in the early years. Australian Primary Mathematics Classroom, 19(1),

35–40.

Jacob, L., & Willis, S. (2001). Recognising the difference between additive and

multiplicative thinking in young children. In J. Bobis, B. Perry & M.

Mitchelmore. (Eds.), Numeracy and beyond (Proceedings of the 24th Annual

Conference of the Mathematics Education Research Group of Australasia,

Sydney pp.306-313). Sydney: MERGA.

Jacob, L., & Willis, S. (2003). The development of multiplicative thinking in young

children. In L. Bragg, C. Campbell, G Herbert & J. Mousley (Eds.),

MERINO: Mathematics Education Research: Innovation, Networking,

Opportunity (Proceedings of the 26th Annual Conference of the Mathematical

Education Research Group of Australasia pp. 460-467). Sydney: Deakin

University.Kandola, B (2012). Focus Groups, in G. Symon and C. Cassell (Eds.)

Qualitative Organizational Research Core Methods and Current Challenges.

London: Sage.

Kastberg, S. E., D'Ambrosio, B., & Lynch-Davis, K. (2012). Understanding

proportional reasoning for teaching. Australian Mathematics Teacher, 68(3),

32–40.

King, N. (2012). Doing Template Analysis, in G. Symon and C. Cassell (Eds.)

Qualitative Organizational Research Core Methods and Current Challenges.

London: Sage.

King, N. (2017). Welcome to the Template Analysis website. Retrieved from

https://research.hud.ac.uk/research-subjects/human-health/template-analysis/

Krajewski, K., & Schneider, W. (2009). Early development of quantity to number-

word linkage as a precursor of mathematical school achievement and

mathematical difficulties: Findings from a four-Year longitudinal study.

Learning and instruction, 19(6), 513-526.

Krueger, R. & Casey. M. (2009). Focus Groups – A practical guide for applied

research (4th ed). Thousand Oaks, CA: SAGE Publications.

Kumar, R (2014). Research Methodology a step by step guide for beginners (4th ed.)

Los Angeles, CA: Sage Publishing.

Lamon, S. (2010). Teaching Fractions and Rations for Understanding (2nd ed.).

Mahwah, NJ: Routledge Publishing.

Langrall, C. W & Swafford, J. (2000). Three balloons for two dollars: Developing

proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254-

246.

Lanius, C. & Williams, S. (2003). Proportionality: A Unifying Theme for Middle

Grades. Mathematics Teaching in the Middle School, 8(8), 392-396.

144

Lee, J. (2010). Exploring kindergarten teachers’ pedagogical content knowledge of

mathematics. International Journal of Early Childhood, 42(1), 27-41. doi:

10.1007/s13158-010-0003-9.

Lee, J., & Ginsburg, H. (2009). Early childhood teachers' misconceptions about

mathematics education for young children in the United States. Australasian

Journal of Early Childhood, 34(4), 37-45.

Liu, Y.M. & Soo, V.L., (2014) Mathematical problem solving – The Bar Model

Method. Retrieved from

http://wwwlscholastic.com.au/corporate/pl/assets/pdfs/Bar%20Model%20Metho

d%20MAV%20Article.pdf

Lo, J-J. & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of

a fifth grader. Journal for Research in Mathematics Education, 28(2), 216-236.

doi: 10.2307/749762.

Lobato, J., Orrill, C. H., Druken, B., & Jacobson, E. (2011). Middle school teachers’

knowledge of proportional reasoning for teaching. Paper presented at Extending,

Expanding, and Applying the Construct of Mathematical Knowledge for

Teaching, at the Annual meeting of the American Educational Research

Association, New Orleans, LA. Retrieved from

http://www.kaputcenter.umassd.edu/downloads/products/workshops/AERA2011

/Lobato_Orrill_Druken_Erikson_AERA_2011.pdf

Mills College Lesson Study Group. (2009). Proportional Reasoning Lesson Study

Toolkit. Retrieved from

http://www.lessonresearch.net/NSF_TOOLKIT/pr_maintoolkit.pdf . Accessed

31st October 2017

Ministerial Council of Education, Employment, Training and Youth Affairs

(MCEETYA). (2008). Melbourne Declaration on Educational Goals for Young

Australians. Retrieved from Education Services Australia website

http://www.curriculum.edu.au/verve/_resources/National_Declaration_on_the_E

ducational_Goals_for_Young_Australians.pdf

Ministry of Education (2001). Advanced Numeracy Project: Draft Teachers’

Materials. Wellington: Ministry of Education.

McGaw, B. (2004). Australian Mathematics Learning in an International Context.

Mathematics Education for the Third Millennium: Towards 2010 (pp. 639-673).

Townsville, QLD: MERGA 27.

Mills College Lesson Study Group. (2009). Proportional reasoning lesson study

toolkit. Retrieved from

http://www.lessonresearch.net/NSF_TOOLKIT/pr_maintoolkit.pdf.

Muir, T. (2008). Principles of practice and teacher actions: Influences on effective

teaching of numeracy. Mathematics Education Research Journal, 20(3), 78-101.

doi: 10.1007/BF03217531.

O’Shea, J. & Leavy, A. (2013). Teaching mathematical problem-solving from an

emergent constructivist perspective: the experiences of Irish primary teachers.

Journal of Mathematics Teacher Education, 16, 293-318.

145

Organisation for Economic Co-operation and Development (OECD). (2001).

Starting strong: Early childhood education and care. Paris: OECD Publications.

Organisation for Economic Co-operation and Development (OECD). (2012).

Literacy, numeracy and problem solving in technology-rich environment:

framework for the OECD survey of adult skills. Paris: OECD.

Peshkin, A. (1988). In search of subjectivity one’s own. Educational Researcher,

17(8), 17-21.

Pitta-Pantazi, D., & Christou, C. (2011). The structure of prospective kindergarten

teachers’ proportional reasoning. Journal of Mathematics Teacher

Education, 14(2), 149-169.

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

Frid, S., & Bennett, S. (2012). Helping children learn mathematics (1st

Australian ed.). Milton, Queensland: John Wiley & Sons.

Roulston, K., Legette, R., Deloach, M., & Pitman, C. B. (2005). What is ‘research’

for teacher-researchers? Educational Action Research, 13(2), 169-190.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.

Educational Researcher, 15(2), 4-14.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.

Harvard Education Review, 57, 1-22.

Siegler, R. (2000). The rebirth of children’s learning. Child Development, 71, 26-35.

Siemon, D. (2006). Assessment for Common Misunderstandings - An Introduction to

the Big Ideas. Melbourne, VIC: RMIT University.

Siemon, D. (2013). Launching mathematical futures – The role of multiplicative

thinking. In S. Herbert, J. Tillyer & T. Spencer (Eds.). Mathematics: Launching

Futures, Proceedings of the 24th biennial conference of the Australian

Association of Mathematics Teachers, pp. 36–52. Adelaide, SA: AAMT.

Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2012).

Teaching Mathematics Foundations to Middle Years. Melbourne, VIC: Oxford.

Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the Big Ideas in Number

and the Australian Curriculum: Mathematics. In Goos, M. Atweh, B, Jorgenson,

R. & Siemon, D. (Ed.), MERGA: Engaging the Australian National

Curriculum: Mathematics - Perspectives from the Field (pp. 19-46). Adelaide,

SA: Mathematics Education Research Group of Australasia.

Siemon, D., Virgona, J., & Corneille, K. (2001). The Middle Years Numeracy

Research Project, 5-9. Bundoora, VA: RMIT University.

Simons, H. (2009). Case study research in practice. Los Angeles, CA: Sage

Publications.

Siraj-Blatchford, I. (2010). Learning in the home and at school: How working class

children ‘succeed against the odds’. British Educational Research Journal,

36(1), 463–482. doi: 10.1080/01411920902989201.

146

Steen, L. A. (Ed). (1990). On the shoulders of giants: New approaches to numeracy.

Retrieved from http://www.nap.edu/openbook.php?isbn=0309042348

Steinthorsdottir, O. B. (2005). Girls journey towards proportional

reasoning. International Group for the Psychology of Mathematics Education,

225.

Shenton, A. K. (2004). Strategies for ensuring trustworthiness in qualitative research

projects. Education for information, 22(2), 63-75.

Stephens, M. (2014). Australian Government Department of Education. Review of

the Australian Curriculum Supplementary Material. Retrieved from

https://docs.education.gov.au/documents/review-australian-curriculum-final-

report.

Thomas, G. (2012). How to do Your Case Study. A guide for students and

researchers. Los Angeles/London/New Dehli/Singapore/Washington DC: Sage

Publications.

Thomson, S., Wernert, E., O’Grady, E. & Rodrigues, S. (2016). TIMSS 2015: A first

look at Australia’s results. Camberwell, VIC: Australian Council for

Educational Research Ltd.

Thomson, S., Wernert, E., O’Grady, E. & Rodrigues, S. (2017). TIMSS 2015:

Reporting Australia’s results. Camberwell, VIC: Australian Council for

Educational Research Ltd.

Trochim, W. M., & Donnelly, J. P. (2008). The research methods knowledge base

(3rd ed.). Mason, OH: Atomic Dog.

Van Dooren, W., De Bock, D. & Verschaffel, L. (2010) From Addition to

Multiplication … and Back: The Development of Students’ Additive and

Multiplicative Reasoning Skills. Cognition and Instruction, 28(3), 360-381. doi:

10.1080/07370008.2010.488306.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.),

Acquisition of mathematics concepts and processes (pp. 127-174). Orlando, FL:

Academic Press, Inc.

Walshaw, M. (2012). Teacher knowledge as fundamental to effective teaching

practice. Journal of Mathematics Teacher Education, 15(3), 181-185.

Waring, T. & Wainwright, D. (2008). Issues and Challenges in the Use of Template

Analysis: Two Comparative Case Studies from the Field. The Electronic

Journal of Business Research Methods, 6(1), 85-94.

Watson, K. (2016). Laying the foundation for multiplicative thinking in Year 2.

Australian Primary Mathematics Classroom, 21(3), 16-20.

Watson, J. M., Callingham, R. A., & Donne, J. M. (2008). Proportional reasoning:

Student knowledge and teachers' pedagogical content knowledge.

In MERGA (pp. 563-571).

Wilson, S., Shulman., & Richert, A. (1987). “150 different ways of knowing”:

Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring

teachers’ thinking (pp.104-123). Eastbourne, UK: Cassel.

147

Yin, R. K. (2009). Case study research: Design and Methods (4th ed.). Thousand

Oaks, California: SAGE publications.

Young-Loveridge, J. (2005). Fostering multiplicative thinking using array-based

materials. Australian Mathematics Teacher, 61(3), 34-40.