Primary school teachers’ knowledge for teaching multiplicative thinking

Abstract

Multiplicative thinking is important for students’ learning of key topics in mathematics such as algebra, geometry, measurement, fractions, statistics, and probability. This study employed an embedded mixed-method approach to investigate primary school teachers’ pedagogical content knowledge for developing multiplicative thinking in students. The study participants (n = 62) were primary school teachers in Australia at different levels of teaching experience, from preservice teachers to novice, experienced, and expert teachers. Teachers completed a carefully designed questionnaire, and a model of Teacher Capacity was the framework for instrument design and data analysis. The investigation in this research focused on three key teaching stages for developing multiplicative thinking: transitional, multiplicative, and proportional reasoning. Estimated marginal means, pairwise comparison, and regression analysis statistical tests were conducted using SPSS 2.0. The results highlight specific areas requiring attention in teacher professional development and preparation programs to enhance teachers’ capacity to effectively support students’ learning and development around multiplicative thinking. For example, there is a need to enhance teachers’ capacity for multiplicative thinking during transitional teaching and the need to emphasize the development of teachers’ knowledge of students and the design of instruction.

Full text

Journal on Mathematics Education
Volume 15, No. 4, 2024, pp. 1175-1196
http://doi.org/10.22342/jme.v15i4.pp1175-1196
Primary school teachers’ knowledge for teaching
multiplicative thinking
Mayamiko Malola1,* , Wee Tiong Seah2
1Faculty of Arts and Education, Charles Sturt University, Wagga Wagga, Australia
2Faculty of Education, University of Melbourne, Melbourne, Australia
*Correspondence: mmalola@csu.edu.au
Received: 20 August 2024 | Revised: 17 October 2024 | Accepted: 27 October 2024 | Published Online: 2 November 2024
© The Authors 2024
Abstract
Multiplicative thinking is important for students’ learning of key topics in mathematics such as algebra, geometry,
measurement, fractions, statistics, and probability. This study employed an embedded mixed-method approach to
investigate primary school teachers’ pedagogical content knowledge for developing multiplicative thinking in
students. The study participants (n = 62) were primary school teachers in Australia at different levels of teaching
experience, from preservice teachers to novice, experienced, and expert teachers. Teachers completed a carefully
designed questionnaire, and a model of Teacher Capacity was the framework for instrument design and data
analysis. The investigation in this research focused on three key teaching stages for developing multiplicative
thinking: transitional, multiplicative, and proportional reasoning. Estimated marginal means, pairwise comparison,
and regression analysis statistical tests were conducted using SPSS 2.0. The results highlight specific areas
requiring attention in teacher professional development and preparation programs to enhance teachers’ capacity
to effectively support students’ learning and development around multiplicative thinking. For example, there is a
need to enhance teachers’ capacity for multiplicative thinking during transitional teaching and the need to
emphasize the development of teachers’ knowledge of students and the design of instruction.
Keywords: Multiplicative Thinking, Pedagogical Content Knowledge, Teacher Capacity, Teaching Experience
How to Cite: Malola, M., & Seah, W. T. (2024). Primary school teachers’ knowledge for teaching multiplicative
thinking. Journal on Mathematics Education, 15(4), 1175–1196. https://doi.org/10.22342/jme.v15i4.pp1175-1196
This paper presents research that explored primary school teachers’ knowledge for developing the
concept of multiplicative thinking in students. The importance of multiplicative thinking in supporting
students’ learning of key topics and success in further mathematics is widely and clearly stated in
mathematics education literature (e.g., Siemon, 2013; Askew et al., 2019). It forms the basis for
understanding proportions, patterns, fractions, measurement, rates, percentages, statistical thinking, the
development of algebraic thinking and understanding the complex issues in society (Askew et al., 2019).
However, there is evidence of low student performance in this area (Siemon, 2013). Multiplicative thinking
underpins students’ learning of STEM disciplines which is currently an area of focus for many
governments globally (Siemon et al., 2018).Teachers’ pedagogical content knowledge (PCK) is critical in
determining students’ learning attainment (Fennema & Franke, 1992). However, research is needed into
teachers’ PCK for developing multiplicative thinking in students. Past research efforts have largely been
directed towards understanding students’ conceptual development of multiplicative thinking and
development of both diagnostic and teaching materials to support student development in this area,

1176 Malola & Seah
ignoring those aspects to do with teachers. Fennema and Franke's (1992) argument about the role of
teachers’ PCK in determining student attainment substantiates the need for a balance of research into
understanding students’ thinking, understanding, development, and difficulties with multiplicative
problems and the understanding of teachers’ PCK for developing multiplicative thinking in students.
Furthermore, Downton et al. (2019) argued that given the complexities and importance of students’
multiplicative thinking, effort should be devoted towards promoting teachers’ PCK for developing
students’ multiplicative thinking. This focus on inquiry into teachers’ knowledge for multiplicative thinking
relates to a study conducted by Matitaputty et al. (2024) who investigated teachers specialised knowledge
for teaching permutation and combination through Teacher professional education programs. While some
teachers demonstrated pround knowledge of teaching permutations and combinations, the study also
identified the need for teacher professional education programs to enhance teachers’ instructional
strategies on the topic.
Multiplicative thinking represents learners’ mental adaptive processing of multiplication and
division concepts by using different methods and approaches in various mathematical problems (Singh,
2012). It allows learners’ capacity to successfully grapple with mathematical problems across
mathematics topics that require understanding and application of multiplicative ideas. Multiplicative
thinking is characterised by the following:
1. a capacity to work flexibly and efficiently with extended range of numbers—for instance, whole
numbers, decimals, fractions, and percentages.
2. the ability to recognise and solve a range of problems involving multiplication or division, including
direct and indirect proportion.
3. the means to communicate this effectively in a variety of ways—for example, using words,
diagrams, symbolic expressions, and written algorithms.
(Siemon, 2013, p. 41)
These characteristics allow multiplicative thinking to be summarised as the ability to recognise
where to use multiplication, division, and proportional reasoning solution strategies, the ability to
communicate and justify the solution strategies, and the capacity to solve problems requiring knowledge
of multiplication and division in a broad range of contexts using different strategies. In this study,
multiplicative thinking was conceptualised beyond the three characteristics suggested by Siemon et al.
(2006). Multiplicative thinking further entails understanding how the knowledge of multiplication, division,
and proportional reasoning is connected and applicable to key topics in mathematics. Understanding
these characteristics is vital to inform not only the design of curriculum, but also the design of teaching.
While the concept of multiplicative thinking is broad and used in many topics in mathematics, such as
fractions, probability, trigonometry, patterns, and statistics, this study looks at multiplicative thinking by
starting from its basic level of multiplication and division. This position aligns with Hurst and Hurrell (2016)
who conceded that the concept of multiplicative thinking is not simple to teach and learn, but
understanding the concept from multiplication and division offers an excellent starting point. We should
make it clear, however, that Siemon et al. (2006) conceptualisation of multiplicative thinking remained
especially useful throughout this study.
Studies have pointed to student underperformance in multiplicative thinking. In South Africa,
evidence from research of middle primary students’ mathematical progress indicates continued reliance
on counting-based strategies when solving multiplicative problems (Venkat & Mathews, 2019). A study
conducted in Australia by Seah and Booker (2005) found that Year 8 students’ achievement on

Primary school teachers’ knowledge for teaching multiplicative thinking 1177
multiplicative reasoning tasks was low. Furthermore, projects by Siemon et al. (2001) and Siemon et al.
(2006) between 2001 and 2006 identified low levels of multiplicative thinking in students, contributing to
students’ low performance in mathematics. Downton et al. (2019) argued that this low performance can
be attributed to teachers’ PCK for developing students’ multiplicative thinking.
There is substantial research demonstrating the depth and breadth of inquiry into students’
understanding, thinking, performance, and developmental hierarchy, as well as ways to support students’
development of multiplicative thinking. This claim coheres with Sowder et al. (1998) who maintained that
the primary and high school mathematics related to multiplicative structures has undergone scrutiny over
the past decade and that researchers have identified the types of reasoning and difficulties students have
with concepts of multiplicative thinking. In Australia, the development of the Learning and Assessment
Framework (LAF) from the SNMYP (2003–2006) project is evidence of efforts to improve students’
learning of multiplicative thinking. The Reframing Mathematical Futures (RMF) project (Day & Hurrell,
2015), which used SNMYP materials with the aim of improving student performance in multiplicative
thinking and proportional reasoning in Years 7 to 10, provides evidence of substantial efforts to improve
students’ attainment in multiplicative thinking.
Research is needed into teachers’ PCK for developing multiplicative thinking in students. This
study aimed to investigate teachers’ PCK for developing multiplicative thinking in students using the
Teacher Capacity model (Zhang & Stephens, 2013), with particular attention to three key teaching stages
that form the foundation to understanding key topics in mathematics: transitional stage (from additive to
multiplicative), multiplicative stage (multiplication and division word problems), and proportional
reasoning stage (Malola et al, 2021).
This study sought to answer the following two research questions:
1. Research Question 1. To what extent can teachers orchestrate student learning of multiplicative
thinking through the three key teaching stages of transition from additive to multiplicative thinking,
multiplicative (multiplication and division of word problems), and proportional reasoning?
2. Research Question 2. How does primary school teachers’ PCK for multiplicative thinking compare
across the four components of the Teacher Capacity model (mathematical knowledge, curriculum
knowledge, knowledge of students’ thinking, and design of instruction)?
METHODS
Study Design
Considering the COVID-19 pandemic environment in which this study was conducted, the study used an
embedded mixed-methods research design. The questionnaire data were used for both quantitative and
qualitative analysis. “The Embedded Design mixes the different data sets at the design level, with one
type of data being embedded within a methodology framed by the other data type” (Creswell et al., 2003,
p. 67). A representation of an embedded mixed-methods research design is shown in Figure 1.
Figure 1. An embedded Mixed-Methods Design (Cresswell et al., 2003, p.68)

1178 Malola & Seah
The study reported here used a single questionnaire to collect both quantitative and qualitative
data. This study was largely quantitative, and the qualitative data played a supportive secondary role to
the quantitative data. This was emphasised by Creswell et al. (2003) who explained that in an embedded
mixed-methods research design, the secondary data should play a supplemental role to the primary data.
This paper reports largely on quantitative data.
Framework
This study used the model of Teacher Capacity (Zhang & Stephens, 2013) as the framework for
questionnaire design and data analysis to explore teachers’ PCK for developing multiplicative thinking in
students across the three key teaching stages. The Teacher Capacity model shown in Figure 2 suggests
that teacher capacity for teaching mathematics comprises the following four elements: knowledge of
mathematics, knowledge of curriculum, knowledge of students’ thinking, and design of instruction (p.
489).
Figure 2. The model of teacher capacity
While this model recognises teachers’ own attitudes, beliefs, values, and dispositions to act in the
teaching of mathematics as external factors influencing teacher capacity, the current study focused on
the four components of teacher capacity for teaching mathematics. Table 1 provides a summary of these
four elements of the model and how they were understood, interpreted in terms of multiplicative thinking,
and used in this study.
Table 1. Summary of components of the teacher capacity model
Mathematical knowledge
Mathematical language, multiplicative ideas, the
effectiveness of solution strategies, student
correct and incorrect responses, variety of
teaching/solution strategies, connection to key
Curriculum knowledge
Identifying multiplicative thinking in curriculum,
knowing where multiplicative thinking begins and
aspects of multiplicative thinking suitable for each
year level, knowing where multiplicative thinking
knowledge is applicable in curriculum, and

Primary school teachers’ knowledge for teaching multiplicative thinking 1179
topics, knowledge of student future challenges
with the use of inefficient strategies.
knowing how multiplicative thinking is connected
to other key topics in mathematics.
Knowledge of students
Knowledge of where students “should be,”
supporting individual students, knowing when to
move students forward, knowledge of potential
challenges and successes, identifying and
correcting errors.
Design of instruction
Lesson design, addressing misconceptions,
adherence to curriculum standards, variety of
resources solution strategies, and choice of
teaching.
Questionnaire Development and Validation
A carefully designed questionnaire was used to collect data for this study. In developing the questionnaire
items for this study, the four elements of the Teacher Capacity model (Zhang & Stephens, 2013) as
discussed informed the structure of questionnaire items according to the mathematical problem
presented at each of the three key teaching stages. The items were designed to investigate teachers’
PCK in each of the four elements of the Teacher Capacity model that collectively constitute PCK for
developing multiplicative thinking in students. The questionnaire had three parts with each part focussing
on one of the three key teaching stages (transitional, multiplicative, and proportional reasoning). The
questionnaire was reviewed and validated by experts in mathematics education. Experts in science
education with special interest on teachers’ pedagogical content knowledge for science also participated
in the expert validation of the questionnaire. Pilot testing was also conducted involving three primary
school teachers before the questionnaire was used in the main study reported here. We note that a similar
approach to questionnaire design was employed by Pincheira and Alsinan (2024) to assess teachers’
mathematical knowledge for teaching algebra. The questionnaire was useful in revealing areas that
required attention in teacher education programs.
Part 1: Transitional Stage
The transitional stage involves a gradual move from reliance on counting of all to emerging multiplicative
thinking. Researchers such as Siemon et al. (2011), Ell et al. (2004) and Malola et al. (2020) have pointed
out that additive thinking in terms of counting strategies such as counting on, skip counting, counting in
groups, and breaking down and building numbers (part–part–whole) form the foundations for developing
multiplicative thinking and will appear as children move away from dependence entirely on additive
strategies. To understand teachers’ basic competencies and skills to support emerging multiplicative
thinking in students, the following question suitable for the transitional stage was presented to teachers:
A new theme park has opened in Melbourne, and you and five other friends are going
there for your birthday party. Tickets cost $43 each. How much will it cost altogether?
A sample of a student’s response referred to as Student C in this study was included in the teacher
questionnaire and is shown in Figure 3.

1180 Malola & Seah
Figure 3. Solution strategy used by student C
This question was presented and discussed at a webinar titled Australian Teachers Using Japanese
Lesson Study: A Structured Problem-Solving Lesson on Multiplicative Thinking organised by the
Australian Association of Mathematics Teachers in 2020. It encompasses the key features of
multiplicative thinking at the transitional teaching stage and adequately provided understanding of
teachers’ PCK for supporting students at this stage of developing multiplicative thinking. Seven
questionnaire items were developed from this question addressing all four components of the Teacher
Capacity model (Zhang & Stephens, 2013): mathematical knowledge, curriculum knowledge, knowledge
of students’ thinking, and design of instruction. Examples of questionnaire items that were developed
include: Can you identify other additive strategies for solving the above problem which are more efficient
than used by student C? What are the present and future challenges that students are likely to encounter,
if they continue to rely on additive strategies to solve this problem? How will you support students moving
forward from relying on additive strategies to using more efficient (multiplicative) strategies? Please
provide as many as possible.
Questionnaire Part 2. Multiplicative Stage
Multiplicative stage involves students solving word problems involving multiplication and division without
relying on additive strategies (Malola et al., 2021). This stage is a core feature of multiplicative thinking,
where students use multiplicative actions without relying on additive ideas. The Piagetian Fish Task
adapted from Clark and Kamii (1996) in Figure 4 was used to explore teachers’ PCK for supporting
students through this stage of developing multiplicative thinking.
Figure 4. The fish task
Four questionnaire items were developed from this question focusing on teachers’ PCK for
developing students’ multiplicative thinking at the multiplicative teaching stage addressing the four

Primary school teachers’ knowledge for teaching multiplicative thinking 1181
elements of the teacher capacity model: knowledge of mathematics, knowledge of curriculum, knowledge
of students’ thinking, and design of instruction. Examples of questionnaire items that were developed and
used include: How would you expect a student in Year 5 to solve this problem? What challenges are
students likely to experience and how would you support them? What content in the curriculum relates to
this problem?
Questionnaire Part 3. Proportional Reasoning Stage
Researchers such as Askew (2018), Harel and Confrey (1994), as well as Siemon et al. (2011) emphasise
that teaching multiplicative thinking with a focus on proportional reasoning empowers students to engage
successfully with more sophisticated problems. The question in Figure 5 was adapted from Siemon et al.
(2011) to explore teachers’ PCK for supporting students at this stage of developing multiplicative thinking.
Figure 5. Coffee jar
Two questionnaire items from this question addressing two components of the teacher capacity
model: mathematical knowledge and knowledge of students’ thinking, with emphasis on mathematical
knowledge. These questionnaire items are: How could you solve this problem? Suggest one or more
strategies. some students in Year 6 may find this a difficult question. Please identify one or more
difficulties. How could you help them?
Participants and Data Collection
This study sought participation from Australian primary school teachers at different levels of teaching
experience. Grouping teachers based on teaching experience resulted in four categories of teachers:
expert teachers (8+ years of teaching experience), experienced teachers (4–7 years of teaching
experience), novice teachers (1–3 years of teaching experience), and preservice teachers (those
studying towards a teaching qualification). For this specific study, the preservice teacher participants
were Master of Teaching (MTeach) students from one institution of higher learning in the Australian state
of Victoria. These students already held a first degree in other fields of study at the time they enrolled for
the MTeach degree.
Data collection was conducted using an online questionnaire through professional teacher
networks. The online questionnaire was developed using a survey software tool called Qualtrics

1182 Malola & Seah
(https://www.qualtrics.com/). Initially, the study targeted an equal number of participants per category of
teachers (25 teachers in each category). However, due to several COVID19-related limitations
encountered in the conduct of the study, equal sample sizes across teacher categories were not
achieved.
Table 2 shows that 33 expert teachers participated in the study, of which 25 completed the whole
questionnaire, and eight completed Parts 1 and 2 comprising biographical information and questions
related to the transitional teaching stage of developing multiplicative thinking.
Table 2. Distribution of participants according to teaching experience
Teacher category
Complete
Incomplete
Total
Expert
25
8
33
Experienced
8
5
13
Novice
5
0
5
Preservice
8
3
11
Total
46
16
62
Table 2 further shows that 13 experienced teachers participated in the study. Of this number, eight
completed the whole questionnaire and five completed Parts 1 and 2. It also shows that all five novice
teachers who participated in the study completed the whole questionnaire. Eleven preservice teachers
participated in the study, and eight of these completed all sections of the questionnaire while three
completed Parts 1 and 2. These numbers bring the total number of participants in the study to 62,
comprising of 16 incomplete responses and 46 complete responses.
Data Processing and Analysis
The data analysis was conducted in two phases. Phase 1 focused on individual teaching stages:
transitional teaching stage (n = 62), multiplicative stage (n = 46), and proportional reasoning stage (n =
46). Phase 2 focused on teachers’ performance across the four elements of the Teacher Capacity model
(n = 46). To explore the different teacher groups’ (preservice, novice, experienced, and expert) PCK for
multiplicative thinking, estimated marginal means were calculated and pairwise comparison analysis was
conducted at each key teaching stage. Estimated marginal means (EMMs) and the pairwise comparison
of the EMMs of teacher scores at the transitional teaching stage of developing multiplicative thinking in
students were calculated using SPSS. The EMMs give an estimate of the means based on a statistical
model other than the observed data as in the case of the descriptive mean (van Dooren, 2020). It gives
a predicted mean of the dependent variable (in this case, teachers’ scores at the transitional teaching
stage) considering any adjustments to the independent variables in the model (Coxe et al., 2009).
A pairwise comparison analysis compares the estimated mean differences in scores for paired
groups of independent variables (Hinton et al., 2014). In this study, the independent variables were the
various categories of teachers (preservice, novice, experienced, and expert). It also shows the standard
error of the estimated mean differences and whether the estimated mean difference is significant or not.
In this study, the pairwise comparison was useful to demonstrate any estimated mean differences in
scores at the transitional teaching stage between groups of teachers according to teaching experience—
for example, the estimated mean difference between experienced and expert teachers or between
preservice and experienced teachers.

Primary school teachers’ knowledge for teaching multiplicative thinking 1183
RESULTS AND DISCUSSION
Estimated Marginal Means and Pairwise Comparison
The quantitative data analysis considered 46 complete cases across all three key teaching stages, while
the qualitative data analysis at the transitional key teaching stage considered 62 cases, with 46 complete
and 16 incomplete cases. The reader is reminded of the practical limitation of this study in obtaining equal
sample sizes across the four categories of teachers despite the concerted efforts to achieve equal sample
sizes due to challenges associated with COVID-19.
Estimated Marginal Means at Transitional Teaching Stage
The results in Table 3 show that expert teachers had the highest EMM of 17.1 with a standard error of
0.57, and upper and lower bounds of 17.8 and 13.7, respectively, at a 95% confidence interval. The
transitional teaching stage part of the questionnaire had a total score of 22 points.
Table 3. Estimated marginal means at the transitional teaching stage according to the teacher capacity model
Experience
Mean
Standard
Error
95% Confidence Interval
Lower Bound
Upper Bound
Preservice
15.8
1.0
13.7
17.8
Novice
14.4
1.3
11.8
16.9
Experienced
16.1
1.0
14.1
18.2
Expert
17.1
.57
16.0
18.3
This means that if the study is duplicated with the same sample size of expert teachers, we are
confident that 95% of them will score between 13.7 and 17.8 of 22 points with a standard error of 0.57.
Hinton et al. (2014) stated that a small value of standard error indicates less variability in the predicted
means, and a larger value of standard error indicates higher variability in the predicted means, if the study
is duplicated. For the expert teachers, a standard error of 0.57 is small, and this means there will be less
variability in the EMMs if the study is repeated with a distinct set of participants.
Table 3 further shows that experienced teachers had the second-highest EMM of 16.1 with a
standard error of 1.0 and lower and upper bounds of 14.1 and 18.2, respectively, at a 95% confidence
interval. Preservice teachers had an EMM of 15.8 with a standard error of 1.0, and 16.0 and 18.2 lower
and upper bounds, respectively, at a 95% confidence interval. The novice teachers scored the lowest
with an EMM of 14.4 with a standard error of 1.3, and 11.8 and 17.0 lower and upper bounds, respectively,
at a 95% confidence interval.
Pairwise Comparisons of Estimated Marginal Means at the Transitional Stage
A pairwise analysis was conducted to establish any estimated mean differences in scores between
groups of teachers at the transitional teaching stage for developing multiplicative thinking in students.
Table 4 shows that there were significant differences (p >0.05) in EMMs between Experienced and
Preservice teachers and between Expert and Novice teachers.

1184 Malola & Seah
Table 4. Pairwise comparison of teacher scores at the transitional teaching stage
Mean Difference
Standard
Error
Sig.a
95% Confidence
Interval for a. ...
(I)
Experience
(J)
Experience
(I–J)
Lower Bound
Preservice
Novice
1.350
1.623
.410
−1.924
Experienced
−.375
1.423
.793
−3.247
Expert
−1.370
1.156
.243
−3.703
Novice
Preservice
−1.350
1.623
.410
−4.624
Experienced
−1.725
1.623
.294
−4.999
Expert
−2.720
1.394
.058
−5.534
Experienced
Preservice
.375
1.423
.793
−2.497
Novice
1.725
1.623
.294
−1.549
Expert
−.995
1.156
.394
−3.328
Expert
Preservice
1.370
1.156
.243
−.963
Novice
2.720
1.394
.058
−.094
Experienced
.995
1.156
.394
−1.338
* The mean difference is significant at the 0.05 level.
Further to the significant differences in the EMMs between groups of teachers, Table 4 allows for
groups of teachers to be ordered based on the EMMs from the highest to lowest. The expert teachers
scored the highest, followed by experienced teachers, then preservice, then novice teachers who scored
the lowest.
Estimated Marginal Means at the Multiplicative Stage
The EMMs of scores for different groups of teachers based on their teaching experience are illustrated
in Table 5. The questionnaire had a total score of 11 points at the multiplicative stage.
Table 5. Estimated marginal means at the multiplicative teaching stage
Experience
Mean
Standard
Error
95% Confidence Interval
Lower Bound
Upper Bound
Preservice
10.1
.544
9.027
11.223
Novice
8.2
.688
6.811
9.589
Experienced
5.8
.544
4.652
6.848
Expert
7.5
.308
6.899
8.141
Table 5 shows that at the multiplicative teaching stage, the preservice teachers had the highest
EMM of 10.1, with a standard error of 0.544. This means that if the study was to be repeated with the
same number of preservice teacher participants who shared the same characteristics as those who
participated in this study, it is highly likely that the EMM will be the same. The novice teachers were
second highest with an EMM of 8.2 and a standard error of 0.688. The expert teachers followed with an
EMM of 7.5 and a standard error of 0.308. Experienced teachers had the lowest EMM of 5.8, with a

Primary school teachers’ knowledge for teaching multiplicative thinking 1185
standard error of 0.544. A pairwise comparison analysis was conducted to establish any differences in
the EMM scores between the groups of teachers at the multiplicative stage of developing multiplicative
thinking in students.
Pairwise Comparisons of Estimated Marginal Means at the Multiplicative Stage
Table 6 shows that the mean difference of the EMMs was significant between the preservice teachers
and novice teachers (p = 0.034). However, there was a strong difference in the EMM between preservice
teachers and expert teachers, and between preservice teachers and experienced teachers (p < .001).
This table further shows that the mean difference was significant between the experienced and expert
teachers (p = 0.007), and between the novice and experienced teachers (p = 0.008). These significant
results are circled only once in Table 6.
Table 6. Pairwise comparison of teacher scores at the multiplicative teaching stage
Mean difference
Standard
error
Sig.b
95% confidence
interval for b. …
(I)
Experience
(J)
Experience
(I–J)
Lower Bound
Preservice
Novice
1.925*
.877
.034
.155
Experienced
4.375*
.769
< .001
2.823
Expert
2.605*
.625
< .001
1.344
Novice
Preservice
−1.925*
.877
.034
−3.695
Experienced
2.450*
.877
.008
.680
Expert
.680
.754
.372
−.841
Experienced
Preservice
−4.375*
.769
< .001
−5.927
Novice
−2.450*
.877
.008
−4.220
Expert
−1.770*
.625
.007
−3.031
Expert
Preservice
−2.605*
.625
< .001
−3.866
Novice
−.680
.754
.372
−2.201
Experienced
1.770*
.625
.007
.509
* The mean difference is significant at the 0.05 level.
These results interpreted simultaneously with the results in Table 6 mean that, at the multiplicative
stage of developing multiplicative thinking in students, the preservice and novice teachers performed
better than the expert and experienced teachers despite the significant mean difference in scores
between the expert and experienced teachers. It is important to state that the questionnaire items at the
multiplicative stage emphasised more the knowledge of students’ thinking than the other three
components of the teacher capacity model.
Estimated Marginal Means at the Proportional Reasoning Stage
The EMMs of teachers’ scores at the proportional reasoning stage were calculated out of 8 total points.
Table 7 shows that the preservice teachers had the highest EMM score of 6.1 with a standard error of

1186 Malola & Seah
0.436. Novice teachers were the second high-scoring group with an EMM of 5.8 and a standard error of
0.552. The expert teachers’ group was second from the low-scoring group with an EMM of 5.2 and a
standard error of 0.247. Of the four groups of teachers, the experienced teachers’ group had the lowest
EMM of 4.9 with a standard error of 0.436. The standard errors of EMMs for each group of teachers were
nearly 0.5 or less. This means that it is highly likely to obtain the same EMM scores at the proportional
reasoning stage if the study were to be replicated using the same sample sizes. It is evident from the
results that the preservice and novice teachers performed better than the experienced and expert
teachers.
Table 7. Estimated marginal means at the proportional reasoning teaching stage
Experience
Estimated
Marginal
Means
Standard
Error
95% Confidence Interval
Lower Bound
Upper Bound
Preservice
6.1
.436
5.245
7.005
Novice
5.8
.552
4.687
6.913
Experienced
4.9
.436
3.995
5.755
Expert
5.2
.247
4.662
5.658
It is important to highlight that the questionnaire items related to the proportional reasoning
teaching stage gave more emphasis to teachers’ own knowledge of mathematics than the other three
components of the teacher capacity model (curriculum knowledge, knowledge of students’ thinking, and
design of instruction). To establish whether the differences in the EMMs between groups of teachers
were significant or not, a pairwise comparison analysis was conducted.
Pairwise Comparisons of Estimated Marginal Means at the Proportional
Reasoning Stage
Table 8 shows that the mean difference of the EMMs was significant only between the preservice
teachers and experienced teachers (p = 0.049). This means that at the proportional reasoning stage,
preservice teachers performed much better than experienced teachers. However, the mean difference of
the EMMs was not significant between the rest of the groups of teachers (p > 0.05).

Primary school teachers’ knowledge for teaching multiplicative thinking 1187
Table 8. Pairwise comparison of teacher scores at the proportional reasoning stage
Mean Difference
Standard
Error
Sig.b
95% Confidence
Interval for b. …
(I)
Experience
(J) Experience
(I–J)
Lower Bound
Preservice
Novice
.325
.703
.646
−1.094
Experienced
1.250*
.617
.049
.005
Expert
.965
.501
.061
−.046
Novice
Preservice
-.325
.703
.646
−1.744
Experienced
.925
.703
.196
−.494
Expert
.640
.604
.296
−.580
Experienced
Preservice
−1.250*
.617
.049
−2.495
Novice
−.925
.703
.196
−2.344
Expert
−.285
.501
.573
−1.296
Expert
Preservice
−.965
.501
.061
−1.976
Novice
−.640
.604
.296
−1.860
Experienced
.285
.501
.573
−.726
* The mean difference is significant at the 0.05 level.
Summary of Teacher Performance at Each Teaching Stage
This section summarises the EMM scores of individual groups of teachers (preservice, novice,
experienced, and expert) at each key teaching stage (transitional, multiplicative, and proportional
reasoning) for developing multiplicative thinking in students. The EMM scores are presented in Figure 6.
Figure 6. Summary of the estimated marginal means
Figure 6 shows that at the transitional stage, expert teachers had the highest EMM score of 78%.
This group was followed by experienced teachers with an EMM of 74%. Preservice teachers came third

1188 Malola & Seah
with an EMM of 71%. Novice teachers had the lowest EMM (66%) compared to the other three groups of
teachers. Looking at the figure and with a focus on the in-service teachers (novice, experienced, and
expert) only, we can see that at the transitional teaching stage, teachers’ PCK of multiplicative thinking
increased with experience. This may suggest that expert and experienced teachers are better able to
support students’ transition from additive to multiplicative thinking than are novice and preservice
teachers. The questionnaire items at the transitional teaching stage emphasised the knowledge of
curriculum and design of instruction. The results may also suggest that the expert and experienced
teachers have a better knowledge of curriculum and design of instruction to support student transition.
Figure 6 shows that at the multiplicative and proportional reasoning stages, the preservice and
novice teachers on average scored higher than the expert and experienced teachers. It should be stated
that the questionnaire items at the multiplicative stage emphasised the knowledge of students’ thinking
more than the other three components of the teacher capacity model, while the items related to the
proportional reasoning teaching stage emphasised more the knowledge of mathematics. This predictive
model of analysis (EMM) and the results suggest that preservice and novice teachers have a higher
knowledge of mathematics and of students’ thinking than do expert and experienced teachers. This may
also mean that preservice and novice teachers are better able to support students’ development of
multiplicative thinking at the multiplicative and proportional teaching stages.
Mean Scores according to Components of Teacher Capacity Model
Table 9 shows the mean scores on the questionnaire of the four groups of teachers across the four
components of the teacher capacity model (Zhang & Stephens, 2013): mathematical knowledge,
curriculum knowledge, knowledge of students’ thinking, and design of instruction.
Table 9. Mean scores of teacher groups across the four components of the teacher capacity model
Experience
MK Percentage
CK Percentage
KS Percentage
DI Percentage
Preservice
M
91.3
71.0
76.4
74.1
N
8
8
8
8
SD
8.3
21.4
9.5
7.5
Novice
M
92.0
57.8
61.8
65.6
N
5
5
5
5
SD
4.5
20.2
27.8
22.0
Experienced
M
81.3
76.6
46.5
66.0
N
8
8
8
8
SD
17.3
9.2
12.7
20.0
Expert
M
86.4
80.6
59.5
66.7
N
25
25
25
25
SD
13.2
12.8
13.8
19.2
Total
M
87.0
75.8
60.4
67.7
N
46
46
46
46
SD
12.8
16.2
17.0
17.9
Note, MK=mathematical knowledge; CK= curriculum knowledge; SK= knowledge of students’ thinking;
DI = design of instruction; M = mean score; N = number of participants; SD = standard deviation.

Primary school teachers’ knowledge for teaching multiplicative thinking 1189
Table 9 shows that the mean score on mathematical knowledge for all the teachers (N = 46) was
87.0%, with a standard deviation of 12.8. The mean score on curriculum knowledge (N = 46) was 75.8%
with a standard deviation of 16.2. For knowledge of students’ thinking (N = 46), the mean score was
60.4% with a standard deviation of 17.0. We can also see from the table that the mean score on design
of instruction was 67.7% with a standard deviation of 17.9. It is evident from this table that all teachers
performed well on mathematical knowledge, followed by curriculum knowledge, then design of instruction,
and last, knowledge of students’ thinking. It should further be noted from these results that there was a
slight variation in teachers’ scores on mathematical knowledge (SD = 12.8) than the other three
components of the teacher capacity model (SD > 16.2). This suggests that there were no demonstrated
differences in teachers’ mathematical knowledge for multiplicative thinking irrespective of teaching
experience.
Regression Analysis
Regression analysis was conducted to investigate which of the four components of the teacher capacity
model (mathematical knowledge, curriculum knowledge, knowledge of students’ thinking, and design of
instruction) contributed the most to the primary school teachers’ PCK for developing multiplicative thinking
in students. Mills and Gay (2016) explained that stepwise linear regression typically starts with a model
that includes all the independent variables and then removes one at a time the variables that do not
contribute as strongly to the dependent variable. This way, we have the variables ranked according to
their contribution level to the final score. The results in Table 10 present the output of four regression
models conducted to establish the component(s) of the teacher capacity model (Zhang & Stephens,
2013) that contributed most significantly to the final score.
Table 10. Stepwise linear regression analysis showing beta coefficients
Unstandardised
Coefficients
Standardised
Coefficients
Beta
t
Sig.a
Model
B
Standard
Error
1
(Constant)
40.399
4.015
10.062
< .001
DI percentage
.469
.057
.777
8.176
< .001
2
(Constant)
26.221
3.181
8.242
< .001
DI percentage
.397
.038
.658
10.338
< .001
KS Percentage
.315
.040
.497
7.808
< .001
3
(Constant)
15.071
2.706
5.570
< .001
DI percentage
.328
.028
.544
11.644
< .001
KS percentage
.309
.028
.488
11.148
< .001
CK percentage
.213
.030
.319
6.998
< .001
4
(Constant)
–.129
1.154
−.111
.912
DI percentage
.262
.010
.433
27.490
< .001
KS percentage
.271
.009
.428
30.296
< .001
CK percentage
.219
.010
.328
22.777
< .001
MK percentage
.248
.013
.294
19.540
< .001

1190 Malola & Seah
Note. DI = design of instruction; KS = knowledge of students’ thinking; CK = curriculum knowledge; MK
= mathematical knowledge.
a Dependent variable: overall score.
Model 4, as shown in Table 10, is the first stage of stepwise linear regression and includes all four
independent variables (mathematical knowledge, curriculum knowledge, knowledge of students’ thinking,
and design of instruction). We can see that the standardised beta coefficient was lowest for mathematical
knowledge (beta = 0.294). The standardised beta coefficient measures the effect size of each
independent variable on the dependent variable (Muijs, 2011). The results mean that, for all the
independent variables combined, mathematical knowledge contributed the least to variability in teachers’
PCK for developing multiplicative thinking in students, and this was eliminated from the model.
The elimination of mathematical knowledge from Model 4 resulted in Model 3, which is the second
stage of stepwise linear regression, and comprised curriculum knowledge, knowledge of students’
thinking, and design of instruction. We can also see that of these three independent variables, curriculum
knowledge had the least (beta = 0.319) contribution to the overall score—that is, teachers’ PCK for
developing multiplicative thinking in students. The elimination of curriculum knowledge from Model 3
resulted in Model 2, which is the third stage of stepwise linear regression and comprised knowledge of
students’ thinking and design of instruction. Of these two, knowledge of students’ thinking was found to
contribute less (beta = 0.497) to variability in teachers’ PCK for developing multiplicative thinking in
students. The elimination of knowledge of students’ thinking from Model 2 resulted in Model 1, the final
stage of stepwise linear regression with only design of instruction (beta = 0.777).
These results suggest that design of instruction is the most significant predictor and determinant
of teachers’ PCK for developing multiplicative thinking in students. This is followed by knowledge of
students’ thinking, then knowledge of curriculum. The results also reveal that mathematical knowledge is
the least powerful predictor or determinant of teachers’ PCK for developing multiplicative thinking in
students. However, it should be stated that, from the results in this table, the contribution of each element
of the teacher capacity model to the overall teachers’ PCK for developing multiplicative thinking in
students was statistically significant (p < 0.001) at a significance level of 0.01.
Textual qualitative analysis was performed to give further understanding of quantitative results.
This analysis focused on the depth, quality, and conciseness of responses. As discussed, this study was
largely quantitative, and little qualitative results will be provided in this paper. A sample of this analysis in
reference to Item 1.2 on the questionnaire is shown in Table 11. Item 1.2 was chosen as a sample
because of its uniqueness as it probes teachers’ knowledge of the importance of multiplicative thinking
to the learning of other topics in mathematics and responses from a few teachers are selected.
Table 11. Summary of teachers’ responses to Item 1.2.
Teacher ID
Responses to Item 1.2:
What are the present and future challenges that students are likely to encounter if
they continue to rely on additive strategies to solve this problem?
Teacher 8
Expert
In K-6 additive strategies restrict mental computation, fluency, and flexibility. It
restricts thinking in looking at fractions as divisions and decimals and limits
abilities in measurement. So many challenges but mostly using inefficient
strategies impacts negatively on a student’s self-perception as a mathematician.

Primary school teachers’ knowledge for teaching multiplicative thinking 1191
Teacher ID
Responses to Item 1.2:
What are the present and future challenges that students are likely to encounter if
they continue to rely on additive strategies to solve this problem?
Teacher 4
Expert
When the size of number gets bigger, students will experience work overload.
Repeated addition will not work for decimals or fractions such as 1.33 x 2.41.
Teacher 6
Expert
Decline of fluency when trying to adapt that strategy to more complex problems.
A lack of transference of multiplicative skill to other areas such as calculating area.
Teacher 55
Preservice
Inefficient problem solving, inability to work with large numbers, struggle with
division.
Teacher 2
Expert
Inefficient time management, not using (hopefully) known skip counting/times
tables knowledge.
Teacher 52
Preservice
Takes longer, as numbers grow bigger, more likely to make mistakes. Not a
suitable strategy for multiplying two double-digit sets.
Teacher 37
Experienced
Repeated addition when working with tens of hundreds becomes very inefficient.
Teacher 48
Novice
Numbers get too big, so they’d have to add too many times.
Table 11 presents a summary of teachers’ responses to Item 1.2. This item was chosen as a
sample only to demonstrate how qualitative analysis was conducted. These responses are arranged in
order from the strongest response at the top to the weakest response at the bottom. On the one hand,
we can see that most teachers were concerned about student inefficiency in solving the problem when
larger numbers are involved and that this was a concern of all teachers from preservice to expert
teachers. On the other hand, while the first four responses demonstrated awareness of the connection
between multiplicative thinking and other topics in mathematics, the first two responses are more specific
in that they mention the mathematics topics. These two responses came from the two expert teachers
and supports the quantitative findings that expert teachers are more capable of supporting students at
the transitional stage of developing multiplicative thinking.
Discussion
This study showed that teaching experience is important for effectively teaching some complex concepts
in mathematics. The complexity of transitioning students from additive to multiplicative thinking and its
pedagogical challenges is well emphasised in mathematics education literature (Clark & Kamii, 1996;
Malola et al., 2020; Siemon et al., 2005). In this study, we saw expert and experienced teachers
demonstrating high pedagogical competency at the transitional teaching stage (students transitioning
from additive to multiplicative thinking). At this teaching stage, teachers support the students in moving
forward from heavy reliance on additive strategies such as repeated addition and skip counting to solve
multiplicative problems to using more efficient emerging multiplicative strategies such as partitioning,
arrays, and area model. In this study, expert and experienced teachers demonstrated higher pedagogical
competence in supporting students through this process. Their capacity to effectively support students’
transition from additive to multiplicative thinking can be attributed to more professional development
opportunities, engagement with peers and curriculum resources, and several opportunities for self-
reflection on the lessons taught. This study recommends that expert and experienced teachers be
allocated for teaching in the early years’ classes (Foundation to Year 3) in primary school. However, it

1192 Malola & Seah
should be stated that some novice and preservice teachers demonstrated higher PCK compared to some
experienced and expert teachers.
The teacher capacity model has four components (mathematical knowledge, curriculum
knowledge, knowledge of students’ thinking, and design of instruction) that constitute MKT. While all
these components were found to be important contributors to teachers’ PCK for multiplicative thinking,
this study further revealed that design of instruction and knowledge of students’ thinking are powerful
predictors of teachers’ PCK for multiplicative thinking. A similar finding appeared in Zhang and Stephens’s
(2013) study of teacher capacity. This finding suggests that explanation teachers with limited knowledge
about students’ challenges with multiplicative thinking may not design effective instruction to support
students’ development of multiplicative thinking. Teachers need to know their students’ conceptions and
misconceptions around the concept of multiplicative thinking to effectively support their learning through
carefully designed and implemented lessons.
The lower performance of teachers on the design of instruction and knowledge of students’ thinking
components of the teacher capacity model used in this study suggests the need for teacher professional
learning programs to focus on promoting teachers’ knowledge of these two components. It has been
demonstrated in this study that each of the four components of the teacher capacity model (mathematical
knowledge, curriculum knowledge, knowledge of students’ thinking, and design of instruction) is an
essential component of teachers’ PCK for multiplicative thinking. However, teachers’ knowledge of
students’ thinking, and design of instruction around multiplicative thinking were shown in this study to be
limited. Hill and Chin (2018) argue that teachers’ knowledge of students’ thinking is important to effective
instruction. This position corroborates with Malola et al. (2020) who maintain that teachers who are aware
of their students are more likely to design and stear instruction towards effective student learning.
PCK for multiplicative thinking was evident in many preservice teachers participating in the study.
Some preservice teachers demonstrated higher PCK than some expert, experienced, and novice
teachers; these were preservice teachers from one institution emphasising multiplicative thinking in its
teacher preparation program. This draws attention to the need for initial teacher preparation programs in
all institutions across Australia and beyond to emphasise multiplicative thinking in their teaching. A similar
recommendation was made by Pincheira and Alsinan (2024) in their study that assessed preservice
teachers’ PCK for algebra. We should mention it that the preservice teachers in this tended to display
more limited curriculum knowledge and mathematical vocabulary.
CONCLUSION
The importance of multiplicative thinking in supporting students’ learning of key topics and success in
further mathematics is widely and clearly stated in mathematics education literature. Several studies have
been conducted to assess students’ performance in multiplicative thinking, and many resources have
been developed to support teaching and learning in this area. Empirical evidence points to the continued
low performance of students in multiplicative thinking. While teachers’ PCK is assumed to be critical in
determining students’ learning, little explicit attention has been paid by prior studies to teachers’ PCK for
multiplicative thinking.
This study used the teacher capacity model as the framework for questionnaire design and data
collection to explore primary school teachers’ PCK for multiplicative thinking. Teachers’ PCK for
multiplicative thinking was explored across the three key teaching stages: transitional, multiplicative, and
proportional reasoning. The results highlight limited knowledge of the connection between multiplicative

Primary school teachers’ knowledge for teaching multiplicative thinking 1193
thinking and other key topics in mathematics among primary school teachers. Understanding these
connections is a powerful aspect of multiplicative thinking that supports students’ learning and success
in further mathematics. The results also indicate high PCK for multiplicative thinking at the transitional
teaching stage among most expert and experienced teachers. Furthermore, the results point to a high
PCK for multiplicative thinking at the multiplicative and proportional reasoning stages among many novice
and preservice teachers.
The results further revealed that all four components of the teacher capacity model (mathematical
knowledge, curriculum knowledge, knowledge of students’ thinking, and design of instruction) significantly
contribute to teachers’ PCK for multiplicative thinking. However, the results showed design of instruction
to be the most critical determinant of teacher PCK for multiplicative thinking, followed by knowledge of
students’ thinking, then curriculum knowledge, and last, mathematical knowledge. High curriculum
knowledge for multiplicative thinking was more evident among the expert and experienced teachers than
among the novice and preservice teachers. There was no significant difference in mathematical
knowledge among the four groups of teachers (expert, experienced, novice, and preservice).
While the existing body of literature on multiplicative thinking emphasises assessing students’
performance in this area and the development of teaching and learning resources to support both
students and teachers, the current research adds new knowledge around multiplicative thinking with an
emphasis on teachers’ PCK to effectively support students’ learning. While existing research and
literature on multiplicative thinking underscore students continued low performance without inquiring
about teachers’ capacity to support students’ understanding of multiplicative thinking, the current
research opens a new area of focus that will potentially direct research on multiplicative thinking to identify
the kind of support teachers require to be empowered to teach multiplicative thinking effectively. The role
of teachers’ PCK in determining students’ learning has been discussed throughout this article.
The limitation of this study is that sample sizes among the four groups of teachers with distinct
experience levels were unequal. This was due to the self-selected-sample nature of the study and the
COVID-19 conditions discussed in some points above. The difficulty associated with the uneven sample
sizes is the generalisation of the study results across and within the four teacher groups (expert,
experienced, novice, and expert). However, the quantitative tools and methods used to analyse the data
allowed room to control the associated generalisation limitations.
Another limitation was that the qualitative data, while valuable under better circumstances, could
have been probed and further elaborated had there been an opportunity to interview teachers. This was
impossible because there was an embargo on contacting teachers and schools directly during the
COVID-19 pandemic lockdown period during the years in which this study was undertaken (2020–2022).
Further research on teacher PCK for multiplicative thinking should interrogate the connections
between teachers’ PCK for multiplicative thinking and their students’ performance. Further investigation
of qualitative factors of teacher PCK for multiplicative thinking, such as the number of workshops attended
by teachers, breadth of teaching experience, and the location of schools, is highly recommended.
Specifically, the content of the workshops and duration, number of years taught at each year level, and
location of schools versus the location of teacher residence would be worthy of further inquiry. The
findings also suggest that future research investigates how much multiplicative thinking is emphasised in
initial teacher preparation programs across institutions. Future research should explore the link between
teacher PCK for mathematics including multiplicative thinking and students’ performance in large national
studies such as TIMMS and NAPLAN and explore connections between students’ multiplicative thinking
and their learning in other subjects such as financial literacy and STEM subjects. This study suggests

1194 Malola & Seah
that further research should consider exploring teachers’ PCK for multiplicative thinking at an international
scale.
Acknowledgments
The authors acknowledge funding support from Charles Sturt University to publish this research.
Declarations
Author Contribution
:
MM: Preparation, creation and/or presentation of the published work,
specifically writing the initial draft, including further editing, and revising the
article based on reviewers’ recommendations.
WTS: Preparation, creation and/or presentation of the published work by
those from the original research group, specifically critical review,
supervision of research project (PhD), and revising the article based on
reviewers’ recommendations.
Funding Statement
:
This research was funded by Melbourne Research Scholarships as a PhD
project.
Conflict of Interest
:
The authors declare no conflict of interest.
REFERENCES
Askew, M. (2018). Multiplicative reasoning: Teaching primary pupils in ways that focus on functional
relations. The Curriculum Journal, 29(3), 406–423.
https://doi.org/10.1080/09585176.2018.1433545
Askew, M., Mathews, C., Takane, T., Venkat, H., Ramsingh, V., & Roberts, N. (2019). Multiplicative
reasoning: An intervention’s impact on Foundation Phase learners’ understanding. South African
Journal of Childhood Education, 9(1), 1–10. https://hdl.handle.net/10520/EJC-1579679fa7
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in Grades 1–5. Journal
for Research in Mathematics Education, 27(1), 41–51.
https://doi.org/10.5951/jresematheduc.27.1.0041
Coxe, S., West, S. G., & Aiken, L. S. (2009). The analysis of count data: A gentle introduction to Poisson
regression and its alternatives. Journal of Personality Assessment, 91(2), 121–136.
https://doi.org/10.1080/00223890802634175
Creswell, J. W., Plano Clark, V. L., Gutmann, M. L., & Hanson, W. E. (2003). Advanced mixed methods
research designs. In A. Tashakkori, & C. Teddlie (Eds.), Handbook of mixed methods in social and
behavioral research (pp. 209–240). Sage. http://rszarf.ips.uw.edu.pl/ewalps/teksty/creswell.pdf
Day, L., & Hurrell, D. (2015). An explanation for the use of arrays to promote the understanding of mental
strategies for multiplication. Australian Primary Mathematics Classroom, 20(1), 20–23.
https://search.informit.org/doi/abs/10.3316/informit.059089282275483
Downton, A., Giumelli, K., McHugh, B., Roosen, T., Meredith, N., Caleta, G., King, M., Salkeld, K., &
Stenning, P. (2019, June 30–July 4). The impact of whole school professional learning on students’
multiplicative thinking [Paper presentation].

Primary school teachers’ knowledge for teaching multiplicative thinking 1195
Ell, F., Irwin, K., & McNaughton, S. (2004). Two pathways to multiplicative thinking. In I. Put, R. Faragher,
& M. McLean (Eds.), Mathematics education for the third millennium, towards 2010. Proceedings
of the 27th annual conference of the Mathematics Education Research Group of Australasia (pp.
199–206).
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=5784e0b84b51230d72e1e0b6
dd4f71e0e5de7e53
Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws
(Ed.), Handbook of research on mathematics teaching and learning: A project of the National
Council of Teachers of Mathematics (pp. 147–164). Macmillan.
Harel, G., & Confrey, J. (1994). Development of multiplicative reasoning in the learning of mathematics.
The Suny Press.
Hill, H. C., & Chin, M. (2018). Connections between teachers’ knowledge of students, instruction, and
achievement outcomes. American Educational Research Journal, 55(5), 1076-1112.
https://doi.org/10.3102/000283121876961
Hinton, P., McMurray, I., & Brownlow, C. (2014). SPSS explained. Routledge.
Hurst, C., & Hurrell, D. (2016). Investigating children’s multiplicative thinking: Implications for teaching.
European Journal of STEM Education, 1(3), 1-11. https://dx.doi.org/10.20897/lectito.201656
Malola, M., Stephens, M., & Symons, D. (2021). Key teaching stages for developing multiplicative thinking
in students. Australian Mathematics Education Journal, 3(1), 9–15.
https://search.informit.org/doi/abs/10.3316/informit.759368128817474
Malola, M., Symons, D., & Stephens, M. (2020). Supporting students’ transition from additive to
multiplicative thinking: A complex pedagogical challenge. Australian Primary Mathematics
Classroom, 25(2), 31–36.
https://search.informit.org/doi/abs/10.3316/INFORMIT.284655920537973
Matitaputty. C., Nusantara, T., Hidayanto, E., & Sukoriyanto. (2024). How mathematics teachers' special
knowledge changing: A case study in the Professional Teacher Education program. Journal on
Mathematics Education, 15(2), 545-574. http://doi.org/10.22342/jme.v15i2.pp545-574
Mills, G. E., & Gay, L. R. (2016). Educational research: Competencies for analysis and applications (11th
ed.). Pearson.
Muijs, D. (2011). Leadership and organisational performance: from research to prescription?.
International Journal of Educational Management, 25(1), 45-60.
https://doi.org/10.1108/09513541111100116
Pincheira, N., & Alsina, Á. (2024). Assessing knowledge to teach early algebra from the Mathematical
Knowledge for Teaching (MKT) perspective: A support tool for primary school teachers. Journal
on Mathematics Education, 15(2), 639-660. http://doi.org/10.22342/jme.v15i2.pp639-660
Seah, R., & Booker, G. (2005). Lack of numeration and multiplication conceptual knowledge in middle
school students: A barrier to the development of high school mathematics. Stimulating the ‘Action’
as Participants in Participatory Research, 3, 86–98. http://hdl.handle.net/10072/2510
Siemon, D. (2013). Launching mathematical futures: The key role of multiplicative thinking. Mathematics:
Launching Futures, 36–52.

1196 Malola & Seah
https://albert.aamt.edu.au/content/download/38258/547092/file/maths_launching_futures.pdf#pa
ge=42
Siemon, D., Banks, N., & Prasad, S. (2018). Multiplicative thinking a STEM foundation. In T. Barkatsas,
N. Carr, & G. Cooper (Eds.), STEM education: An emerging field of inquiry (pp. 74–100). Sense
Publications. https://doi.org/10.1163/9789004391413_006
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching Mathematics:
Foundations to the middle years. Oxford University Press.
Siemon, D., Breed, M., & Virgona, J. (2005). From additive to multiplicative thinking: The big challenge
of the middle years. Proceedings of the 42nd Conference of the Mathematical Association of
Victoria, Bundoora, Australia.
Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding numeracy in the middle years:
Project findings, materials and resources: Final report. RMIT University.
Siemon, D., Virgona, J., & Corneille, K. (2001). The middle years numeracy research project: 5–9. Final
report. Department of Education, Employment and Training, Victoria, Catholic Education
Commission of Victoria and Association of Independent Schools of Victoria. RMIT University.
Singh, P. (2012). Multiplicative thinking and learning. Encyclopedia of the Sciences of Learning, 2389–
2392.
Sowder, J., Armstrong, B., Lamon, S., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers
to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education,
1(2), 127–155. https://doi.org/10.1023/A:1009980419975
van Dooren, W. (2020). The wonderful world of marginal means. https://jasp-stats.org/2020/04/14/the-
wonderful-world-of-marginal-means/
Venkat, H., & Mathews, C. (2019). Improving multiplicative reasoning in a context of low performance.
ZDM, 51(1), 95–108. https://link.springer.com/article/10.1007/s11858-018-0969-6
Zhang, Q., & Stephens, M. (2013). Utilising a construct of teacher capacity to examine national curriculum
reform in mathematics. Mathematics Education Research Journal, 25(4), 481–502.
10.1007/s13394-013-0072-9