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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.
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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.
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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-
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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
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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.
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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
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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.
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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
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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
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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
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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)
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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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
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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,
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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
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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
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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).
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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
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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)
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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
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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
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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
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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,
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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,
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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
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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
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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)
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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
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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.
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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)
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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).
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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.
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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
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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,
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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
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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
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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)
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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,
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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
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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.
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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:
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(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
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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.
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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.
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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.
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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).
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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).
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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
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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.
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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
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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).
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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).
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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.
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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
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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.
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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
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