Key teaching stages for developing multiplicative thinking in students

Abstract

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. However, multiplicative thinking is not clearly stated in curriculum documents of many countries including Australia. In the Australian Curriculum: Mathematics, it is implied throughout as multiplication and division with emphasis on recall of multiplication facts, factors, and multiples of a number, grouping into equal sets, and the use of mental solution strategies. In this theoretical paper, we discuss how this lack of an explicit framework for students' development of multiplicative thinking might be remedied by: (i) making curriculum and teaching to be explicit about multiplicative thinking from early primary school (F-3), and (ii) implementing three key teaching stages for developing multiplicative thinking in students throughout the middle primary (4-6) and junior secondary years (7-8).

Full text

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AMEJ 3 (1) 2021
The authors identify the lack of an explicit framework for students’ development of
multiplicative thinking in the curriculum documents of many countries—including
Australia. They present ideas as to how this might be remedied.
Mayamiko Malola
University of Melbourne, Vic.
<m.malola@unimelb.edu.au>
Key teaching stages for developing
multiplicative thinking in students
Max Stephens
University of Melbourne, Vic.
<m.stephens@unimelb.edu.au>
Duncan Symons
University of Melbourne, Vic.
<duncan.symons@unimelb.edu.au>
Abstract
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. However,
multiplicative thinking is not clearly stated in
curriculum documents of many countries including
Australia. In the Australian Curriculum: Mathematics,
it is implied throughout as multiplication and division
with emphasis on recall of multiplication facts,
factors, and multiples of a number, grouping into
equal sets, and the use of mental solution strategies.
In this theoretical paper, we discuss how this lack
of an explicit framework for students’ development
of multiplicative thinking might be remedied by:
(i) making curriculum and teaching to be explicit
about multiplicative thinking from early primary
school (F–3), and (ii) implementing three key teach-
ing stages for developing multiplicative thinking in
students throughout the middle primary (4–6) and
junior secondary years (7–8).
Background
A big idea in mathematics is a way of thinking about key
aspects of mathematics without which, students’ progress
in mathematics will seriously be impacted (Siemon, 2013).
Big ideas allow mathematics to be understood as a set of
connected ideas, other than disconnected concepts, skills,
and facts (Charles & Carmel, 2005). Multiplicative thinking
is clearly highlighted in mathematics education research
as one big idea in mathematics. For example, Hurst and
Hurrell (2016) contend that multiplicative thinking is a
‘big idea’ of mathematics that underpins much of the
mathematics learned beyond the primary school years
(F–6). However, studies have revealed student under-
performance on multiplicative thinking tasks. For instance,
in South Africa, studies of middle grades (4–6) students’
mathematical progress indicate continued use of highly
inefcient counting-based strategies when solving multi-
plication related problems (Venkat & Mathews, 2018).
In a similar study conducted in Australia with Year 8
students, Seah (2004) found that many students
demonstrated limited understanding of the multiplication
concepts, with their knowledge restricted to procedural
rather than conceptual understanding.
A different study conducted in Australia by Seah, Tocq,
and George (2005) found that Year 8 students’ achieve-
ment on multiplicative reasoning was low. Furthermore,
in two related studies, the Middle Years Numeracy
Research Project (Siemon, Virgona, & Corneille, 2001)
and Scaffolding Numeracy in the Middle Years Project
(Siemon, Breed, Dole, Izard, & Virgona, 2006) identied
low levels of multiplicative thinking contributing to poor
performance by students in mathematics. Hurst and
Hurrel (2016) also found that over half the students lacked
a conceptual understanding of multiplicative thinking,
and that many students could not condently make
connections between multiplication and division. Askew
et al., (2019) emphasise that multiplicative thinking is a
critical area within primary school mathematics and key
to creating a more secure foundation for secondary
school and tertiary mathematics.
Because multiplicative thinking is critical for students’
success in mathematics, it requires a systematic and
focused approach at different levels of schooling. We
therefore advocate a more explicit treatment of multi-
plicative thinking in curriculum and teaching resources
for teachers and to enable them to better support
students’ learning, we draw attention to three key

10 AMEJ 3 (1) 2021
Malola, Stephens & Symons
teaching stages for developing multiplicative thinking.
These three stages are; (1) Transitional stage (moving
from additive to early multiplicative strategies),
(2) Multiplicative stage (involving multiplication and
division word problems), and (3) Developing proportional
thinking stage (where students begin to use explicit pro-
portional reasoning). Teaching implications using relevant
examples for each key teaching stage will be discussed.
Multiplicative thinking
Multiplicative thinking is increasingly emerging an area
of focus for many researchers (see Askew et al., 2019:
Boylan et al., 2015, and Venkat & Mathews, 2018, in
mathematics education. This may suggest an increased
awareness among the mathematics educators of the
pivotal role multiplicative thinking plays in supporting
students’ success in mathematics from primary school
through ter tiary. Siemon, Breed, and Virgona’s (2005)
denition of multiplicative thinking has been broadly
acknowledged in literature. They explain multiplicative
thinking as:
– a capacity to work exibly and efciently with
an extended range of numbers (for example,
larger whole numbers, decimals, common
fractions, ratio, and per cent),
– an ability to recognise and solve a range of
problems involving multiplication or division
including direct and indirect proportion, and
– the means to communicate this effectively in a
variety of ways (for example, words, diagrams,
symbolic expressions, and written algorithms).
(Siemon, Breed, & Virgona, 2005, p. 2)
These characteristics allow multiplicative thinking to
be summarised as the ability to recognise where to use
multiplication and division solution strategies, being able
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. For those teachers who
have some appreciation of how multiplicative thinking
develops, the Australian Curriculum: Mathematics (ACM)
(Australian Curriculum Assessment and Reporting
Authority (ACARA), 2016) will provide many useful points
of connection. However, for teachers who may not have
a clear understanding of multiplicative thinking, the ACM
while providing many isolated “dotted points” may fall
short in providing any substantive connective tissue.
Supporting learning of key topics
To illustrate what we mean by “connective tissue”,Harel
and Confrey (1994) argue that understanding multi-
plicative thinking in terms of proportional reasoning
is fundamental to students’ learning of key topics in
mathematics such as probability, patterns, fractions,
measurement, percentages, statistical thinking, algebraic
thinking, similar gures, and rate of change. Although
multiplicative thinking is implicit in these and other key
topics in mathematics, some teachers may not appreciate
how multiplicative thinking underpins these topics in
mathematics, and accordingly may not effectively support
students’ learning progression of this concept.
Multiplicative thinking and the Australian
Curriculum: Mathematics
In the next section, we uncover where multiplicative
thinking begins in the Australian Curriculum: Mathematics
(ACM), how it progresses in the early primary years (F–3),
and the different challenges it presents to students.
Multiplicative thinking is implied as multiplication and
division throughout the ACM, but nowhere is this term
explicitly stated. Presenting multiplication and division
as separate and independent topics limits students’
and teachers’ capacity to explore these two related
operations and further limits their appreciation and use
of varied solution strategies and representations, which
according to Barmby, Harries, Higgins and Suggate (2007)
are important indicators of mathematical understanding.
According to the ACM, Year 1 students are expected
to skip count in twos, ves, and tens, understand
partitioning of numbers, group in tens, and split objects
into equal pieces and describe how the pieces are equal
(ACARA, 2016, pp. 12–13). Skip counting, grouping, and
partitioning form the basis for emerging multiplicative
thinking through repeated addition (Siemon et al, 2011).
Teachers who understand that multiplicative thinking
begins to emerge in these early stages can structure their
teaching in ways that support students’ gradual transition
from additive to multiplicative strategies.
In Year 2, multiplication is introduced to students as
repeated addition, groups, and arrays, whereas division
is introduced to students as “grouping into equal sets”
(ACARA, 2016, p. 17). We acknowledge that these
approaches to introducing multiplication and division
help to build strong foundations for developing multi-
plicative thinking. However, presenting multiplication
and division as distinct and separate topics is unlikely
to help teachers to focus on the complementarity of
these two operations which is fundamental to multiplica-
tive thinking. We contend that multiplication and division
should be taught as complementary concepts.
In Year 3, the ACM places a focus on “recall” of
multiplication facts and the use of mental strategies,
and an emphasis on factors and multiples of numbers
in subsequent years. Here are important opportunities,
for example, to link multiplication and division; otherwise,
a focus simply on “recall” may promote rote learning
other than conceptual understanding. This concern is

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Key teaching stages for developing multiplicative thinking in students title
repeated by Seah (2004) and Hurst and Hurrel (2016) who
found that many students demonstrated procedural rather
than conceptual understanding of multiplicative thinking.
Curriculum and teaching resources should be more
explicit about multiplicative thinking from the early years
(F–3) of schooling through middle (4–6) and junior second-
ary school (7–8) to effectively support conceptual progres-
sion. As such, teachers need to possess the appropriate
pedagogical content knowledge to scaffold students’
development of multiplicative thinking at each year level
throughout the primary and secondary school. In the next
section, we discuss one framework which can support
teachers’ own knowledge of multiplicative thinking.
After that we will present our position on three key teach-
ing stages for the development of multiplicative thinking.
An Australian Learning and Assessment
Framework for multiplicative thinking
The Learning and Assessment Framework (L AF) for
multiplicative thinking developed by Siemon et al. (2006)
presents a developmental progression of multiplicative
thinking from early primary, through middle to secondary
school years. The development of LAF was grounded on
the view that teachers are better able to support student
learning of multiplication and division if they understand
students’ developmental pathway and locating students’
multiplicative thinking on a learning continuum (Siemon et
al., 2006). Using a Rasch analysis of item responses to rich
assessment tasks of nearly 3200 Year 4 to 8 students, the
authors of the LAF suggest that, to develop into compe-
tent multiplicative thinkers, students progress through
nine hierarchical zones each one indicating an increasing
level of sophistication of ideas and strategies: primitive
modelling, intuitive modelling, sensing, strategy exploring,
strategy rening, strategy extending, connecting, and
reective knowing (https://www.education.vic.gov.au/
school/teachers/teachingresources/discipline/maths/
assessment/Pages/learnassess.aspx).
In Table 1 (see page 12) we present a summary of the
rst six of the above levels of developing multiplicative
thinking according to the LAF, with corresponding Year
levels of the ACM, together with three key teaching stages
that we will discuss in the following section.
Knowing that the LAF was constructed from responses
received from a large sample of students in Years 4 to 8,
as discussed earlier, we can be condent that assessment
tasks involving proportional thinking, starting with simple
proportional thinking at Level 4 in Table 1 below, are
accessible to students prior to Year 9 when proportional
thinking is rst introduced by the ACM. This we believe
is too late and believe that teachers should consider
introducing students to situations involving proportional
thinking before the end of primary school as suggested
by the LAF (Siemon et al., 2006) to effectively support
their development of multiplicative thinking.
Three key teaching stages
We advocate the teaching of multiplicative thinking as
needing to progress through three successive, yet exible
key stages. The stages are: (1) Transitional stage (from
additive to multiplicative), (2) Multiplicative stage (multi-
plication and division word problems), and (3) Fostering
proportional reasoning stage. It should be stated that
these key teaching stages are a result of our own synthesis
of literature presented in Table 1 and other sources, for
example, Singh (2012) and Askew (2018).
Teaching Stage 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,
Irwin, and McNaughton (2012), Malola, Symons and
Stephens (2020) acknowledge 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. For example,
Table 1 shows that the ACM recommends that in Years 1
and 2 students should be able to skip count, represent
multiplication as repeated addition, groups, and arrays.
This corresponds to Levels 1 to 3 of the Learning and
Assessment Framework (LAF) for multiplicative thinking
developed by (Siemon et al, 2006) and pre-multiplying
scheme and iterative multiplication by Singh (2012). To
help teachers of Years 1, 2 and 3 develop basic compe-
tencies and skills that support emerging multiplicative
thinking in students, we give several examples of teaching
activities at this transitional stage in the Figure 1 below.
While the ACM encourages students in Years 1 and 2
to solve multiplicative problems using repeated addition,
researchers such as Askew (2018), Hurst & Hurrell (2014),
Siemon et al., (2011), and Wright (2011) remind us that
repeated addition and subtraction become inefcient
with increased sophistication of problems. For example,
if 21 Year 4 students each bring to school their 3 favourite
books, how many books in total? Adding 3 + 3 +3 …….
twenty-one times is clearly inefcient. But being able to
think about twenty-one as ten plus ten and one more, 10
students bringing 30 books, allows students to represent
the total as 30 + 30 + 3 (where every 10 students bring 30
books and the 3 represents the number of books brought
by the remaining one student). The above example shows
the importance of rening additive strategies so that they
become more efcient and make a steady and smooth
transition to multiplicative thinking.

12 AMEJ 3 (1) 2021
Malola, Stephens & Symons
actions and images are analysed using strategies such as
partitioning and splitting to aid the solution. This stage
is arguably a core feature of multiplicative thinking where
students use multiplicative actions without relying on
additive ideas. Vergnaud (1983) recommended that teach-
ers should help students to view multiplication and divi-
sion as nested and inseparable ideas allowing students to
move backwards and forwards between multiplication and
division. For example, if the student knows that 17 × 6 =
102 this helps to work out the answer to 102 ÷ 6 (Hurst &
Hurrel, 2016). This approach to teaching, allows students
to appreciate understanding of reciprocal relationship
Teaching Stage 2. Multiplicative stage (understanding
and solving multiplication and division word
problems)
The multiplicative stage involves a higher level of abstrac-
tion in solving multiplicative problems without drawing
on additive thinking. With reference to Table 1, the ACM
recommends that in Year 4 students should be able to
solve word problems involving multiplication and division
and solve multiplication and division problems involving
whole numbers and decimals using mental strategies.
This corresponds to Level 4 of the LAF for multiplicative
thinking (Siemon et al, 2006). According to Singh (2012),
Our key teaching
stages LAF Levels
(Siemon et al, 2006) Australian Curriculum
(ACARA, 2016)
1. Transition from
additive to
multiplicative
thinking
Level 1. Solves simple division and multiplica-
tion problems, rely on drawing, models, and
count-all strategies, use skip count for groups
less than 5, extends simple number pattens,
and multiplicative thinking is not apparent.
Level 2. Counts large collections efciently,
keeps truck of counts but need to see all
groups, shares collections equally, counts equal
groups, recognises multiplication but cannot
follow through to solution, and some evidence
of multiplicative thinking.
Level 3. Demonstrate intuitive sense of pro-
portion, use doubling or repeated halving to
compare fractions, works with useful numbers
such as 2 and 5, and rely on count-all methods.
Year level 1. Develop condence with number
sequences to and from 100 by ones from any
starting point. Skip count by twos, ves and
tens starting from zero.
Year level 2. Recognise and represent multiplica-
tion as repeated addition, groups, and arrays.
• Recognise and represent division as grouping
into equal sets and solve simple problems using
these representations
• Apply repetition in arithmetic operations,
including multiplication as repeated addition
and division as repeated subtraction
2. Multiplicative
(multiplication
and division
word problems)
Level 4. Solves simple multiplication and
division problems involving two-digit numbers,
tends to rely on additive thinking, drawings, and
informal strategies to tackle problems involving
larger numbers, decimals and less familiar
situations, partition given number or quantity
into equal parts, and begin to work with simple
proportion.
Year level 3. Represent and solve problems
involving multiplication using efcient mental
and written strategies and appropriate digital
technologies.
Year level 4. Solve word problems by using
number sentences involving multiplication
or division where there is no remainder.
Year level 5. Solve problems involving multipli-
cation of large numbers by one- or two-digit
numbers using efcient mental, written strategies
and appropriate digital technologies.
Year level 6. Multiply decimals by whole numbers
and perform divisions by non-zero whole numbers
where the results are terminating decimals, with
and without digital technologies.
3. Proportional
reasoning
Level 5. Solves whole number proportion and
array problems systematically, solves simple
2-step problems using recognised rule,
determine all options in cartesian product
situation, partitioning, and beginning to work
with decimal numbers and percent.
Level 6. Systematically lists number of options
in cartesian product situation, developing sense
of propor tion, developing capacity to work
mentally with multiplication and division facts,
renames and compares fractions, and uses
partitioning strategies.
Note that Proportional thinking appears rst in the
ACM in Year 9. “Students solve problems involving
direct proportion. Explore the relationship
between graphs and equations corresponding to
simple rate problems (p.59)”
Table 1. Key teaching stages Versus LAF (Siemon et al, 2006) and Australian Curriculum: Mathematics (ACARA, 2016)

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Key teaching stages for developing multiplicative thinking in students
between multiplication and division. We recommend that
curriculum documents adopt a more unied approach to
multiplication and division rather than presenting multipli-
cation and division as two distinct topics. We use the sh
task in gure 2 below to demonstrate how teachers can
help students to consolidate this understanding.
Figure 2. Fish task (Adapted from Clark & Kamii, 1996)
Given that in Figure 2, Fish A measures 5cm, Fish B
is 10cm, and Fish C is 15 cm, and that the length corre-
sponds to the amount of food consumed by each sh.
Students and teachers could engage with the following
questions:
(a) If sh A consumes 2 grams of pellets, how many
pellets will sh B and C each consume?
(b) If 9 grams of pellets is required to feed sh C, how
many pellets will be needed to feed sh A and B
respectively?
For question (a), students will have to establish the
following relationships in Figure 3.
(i) Fish B is twice the length of sh A.
Fish A consumes 2 grams,
then Fish B will consume 2 x 2 grams
Fish B will consume 4 grams.
Figure 3a. A solution to question (a i)
(ii) Fish C is three times the length of sh A.
Fish A consumes 2 grams,
then Fish C will consume 3 x 2 grams
Fish C will consume 6 grams.
Figure 3b. A solution to question (a ii)
Question (b) could be approached as shown in Figure 4.
Fish C is three times Fish C consumes 9 grams
the length of Fish A
Fish A consumes
9 gr ams ÷3
3 grams
Figure 4a. A solution to question (b i)
Fish B is twice the Fish A consumes 3 grams
length of Fish A
Fish B consumes
2 x 3 grams
6 grams
Figure 4b. Question (b ii)
The solution strategies presented in the gures above
show how students can move backwards and forwards
between multiplication and division. In this way, students
can better appreciate the complementary relationship
existing between multiplication and division.
Counting all in ones
Skip counting in threes
Figure 1. Teaching activities at transitional stage.
Arrays
4 groups of 3
Part-part- Whole
Equal groups

14 AMEJ 3 (1) 2021
Malola, Stephens & Symons
problems require familiarity with simple decimal currency
ideas. Teachers may choose to simplify the exact amounts.
These sor ts of investigations allow students’ proportional
thinking to proceed in several quite exible ways.
Students could begin by adopting, or being introduced
to, different strategies involving scaling up or scaling
down. As the following examples in Figure 6 show,
scaling up and scaling down involving the twin relations
of multiplication and division as we discussed earlier.
Scaling up from
$8.55 (150 g)
$8.55 per 150 g = (300 gs ÷ 150 g)
× $8.55 = $17.10
Scaling up from
$2.85 (50 g)
$2.85 per 50 g = (300 grams ÷ 50 g)
× $2.85 = $17.10
Figure 6. Examples of scaling up.
Alternatively, students could use “scaling down” from
300 grams. Or they could use “scaling up and down”
from 150 gram. Their strategies may also vary in efciency.
Teaching proportional reasoning needs to be open to
scaling up and scaling down, and clearly builds on a range
of exible multiplicative thinking strategies developed
over the preceding years.
Being introduced to these various scaling up and scaling
down approaches helps students to nd that the 300
grams jar at $13.99 represents the best buy. Further to
using proportional reasoning, these strategies, including
those used in the examples 6 above, rely on both multi-
plication and division to solve the problem. As suggested
earlier, this could be the approach to teaching to support
the development of multiplicative thinking in students.
Summary and conclusion
The importance of multiplicative thinking in supporting
students’ learning of key topics such as probability,
patterns, fractions, measurement, percentages, statistical
thinking, algebraic thinking, similar gures, and rate of
change and success in further mathematics is well empha-
sised in literature. We have proposed three successive,
yet exible key teaching stages that help to develop
multiplicative thinking incrementally throughout the
primary and early secondary school years.
While the ACM rightly focusses on multiplication and
division, and the need to develop uency in these areas,
it is not, in our opinion, sufciently explicit about interre-
lated elements of multiplicative thinking such as devel-
oping and communicating multiple solution strategies,
applicability in diverse problem contexts, and introducing
primary students to proportional reasoning. Teachers
who do not have a clear understanding of multiplicative
thinking may experience difculties in designing teaching
in ways that scaffold students’ development of multiplica-
tive thinking across primary and secondary school.
Teaching Stage 3: Fostering proportional reasoning
Researchers such as Askew (2018) emphasise that teach-
ing multiplicative thinking with a focus on proportional
reasoning empowers students to engage successfully
with more sophisticated problems. The teaching of
multiplicative thinking with a focus on proportional
reasoning has also been supported by Harel and
Confrey (1994) as discussed earlier in this article. Their
argument further highlights the importance of multi-
plicative thinking to support further study in mathemat-
ics and to be able to apply mathematics in daily life
and across related elds of study.
Developing multiplicative thinking in terms of
proportional reasoning is not explicit in the ACM for the
primary years, and only appears in Year 9. However, as
the LAF shows, many students in the primary years have
entered a proportional reasoning stage of multiplicative
thinking, exhibiting performances at Level 5 and above
of the LAF, long before Year 9 where proportional
thinking is introduced by the ACM. For example, the
LAF says that at Level 5 students solve whole number
proportion and array problems systematically, and at
Level 6 students solve a broader range of multiplication
and division problems involving 2-digit numbers, pat-
terns and/or proportion but may not be able to explain
or justify their solution strategy (State of Victoria, 2008).
Multiplicative thinking in terms of proportional reason-
ing needs to be introduced with an appropriate teaching
emphasis in the latter primary and early junior secondary
school years to help students to build on knowledge
acquired at prior stages. The following example adapted
from Siemon et al. (2011) shows.
Coffee is available in three sizes. Which size represents the
best buy if 300-gram jar costs $13.99, then 150-gram jar costs
$8.55 and the 50-gram jar costs $2.85? (adapted from Siemon
et al., 2011).
300 g 150g 50g
Figure 5. Coffee Jars.
Posing this question gives opportunities for students
to use various (multiplicative) proportional reasoning
strategies to nd the “best buy”. We accept that these

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Key teaching stages for developing multiplicative thinking in students
It is important for both curriculum and teaching to be
explicit about the importance of fostering multiplicative
thinking from the early primary school years through to
the junior secondary years. It is a major cornerstone in the
preparation of students for further mathematical study
and for successfully utilising mathematics in their daily
lives at school and beyond.
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