The multiplicative situation.
Hurst, Chris
The relationships between three critical elements, and the
associated mathematical language, to assist students to make the
critical transition from additive to multiplicative thinking are
examined in this article by Chris Hurst.
Multiplicative thinking--a very 'big idea' of number
Multiplicative thinking is a critical stage in children's
mathematical understanding and is indeed a 'big idea' of
number (Siemon et al., 2011). Wright (2011) suggests that children often
need to reconceptualise their thinking about what is involved in
multiplication and division in order to understand the
'multiplicative situation'. This article suggests that the
level of reconceptualisation required could depend on how multiplicative
relationships are taught. Multiplication and division are often taught
as separate entities whereas in reality, the 'multiplicative
situation' is rich in connections and links. If teachers know this
and teach to it, they are more likely to help children understand this
important idea. This article considers that there are three critical
elements to be considered--an understanding of the 'multiplicative
situation', a deep understanding of multiplicative arrays, and the
notion of factors and multiples and associated language--and the myriad
connections that exist between those three elements. However, it is
important to briefly consider some rationale for moving from additive to
multiplicative thinking.
Additive to multiplicative
The three elements mentioned above are critical to the development
of multiplicative thinking. One of the elements is the multiplicative
array. Downton (2008) used the term 'composites' to refer to
an equal grouping structure that greatly assists children to think
multiplicatively. It is suggested here that this would be further
enhanced by the use of grid arrays to promote the idea of a composite or
'entity'. Figure 1 shows the progression from additive to
multiplicative thinking through the use of the array. The final right
hand grid is considered as a more powerful representation that enables
the link with area to be made.
[FIGURE 1 OMITTED]
It is essential that children move from additive to multiplicative
thinking. Devlin (2008) described these two stages of development as
preceding the third stage, or exponential thinking, and suggests that if
children are explicitly taught in the first instance that multiplication
is 'repeated addition', they are likely to remember that to
their detriment when they need to be thinking multiplicatively. He notes
an essential difference between the two ideas--"Adding numbers
tells you how many things (or parts of things) you have when you combine
collections. Multiplication is useful if you want to know the result of
scaling some quantity" (Devlin, 2008, p. 1). The idea of
multiplication as scaling rather than adding is a key aspect of thinking
that needs to be developed. 'Repeated addition' may seem
innocuous enough but it is often harder to 'unlearn' a
concept, or 're-conceptualise' as Wright (2011) suggested,
than learn it appropriately in the first place. Nunes and Bryant (1996,
p. 153) note that multiplication and division can certainly be done
through repeated addition and subtraction but they hasten to add that
"several new concepts emerge in multiplicative reasoning, which are
not needed in the understanding of additive situations". Devlin
comments that this is often what happens with beginning mathematics
instruction and working with small, positive whole numbers but it is
important to move beyond the notion of 'repeated addition' and
to see multiplication as a 'scaling' concept.
This is reinforced by Watson who noted the following:
A shift from seeing additively to seeing multiplicatively is
expected to take place during late primary or early secondary school.
Not everyone makes this shift successfully, and multiplication seen as
'repeated addition' lingers as a dominant image for many
students. This is unhelpful for learners who need to work with ratio, to
express algebraic relationships, to understand polynomials, to recognise
and use transformations and similarity, and in many other mathematical
and other contexts. (Watson, n.d.)
Given the importance of moving beyond additive to multiplicative
thinking, let us now consider the three elements mentioned in the
introduction.
One situation ... three quantities
First, Jacob & Mulligan (2014) specifically use the term
'multiplicative situation' to describe the relationship that
exists between multiplication and division. Perhaps they have chosen to
use that term in order to emphasise the link, rather than consider the
two terms individually. This is a key to how teachers should think about
and teach children about multiplicative thinking. It might seem an
insignificant point to make but it is better to teach children about ONE
situation--the multiplicative situation--not about multiplication and
its inverse, division. The multiplicative situation is about the three
key quantities--the number of equal groups, the number in each group,
and the total amount. If we know the group size and the number of
groups, we multiply. If we know the total amount and one of the other
quantities, we divide to find the one we don't know. The idea of
the multiplicative situation as one situation, is a more powerful way of
thinking than simply considering multiplication and division as the
inverse of one another.
The multiplicative array
Second, the case for the use of the multiplicative array has been
well made for some time (Jacob & Mulligan, 2014; Jacob & Willis,
2003; Kinzer & Stanford, 2014; Siemon et al., 2011; YoungLoveridge,
2005). The power of the multiplicative array lies in its ability to
focus children's attention on the three quantities involved in the
multiplicative situation at the same time (Siemon et al., 2011) and
therefore the array is critically important in developing multiplicative
thinking. Young-Loveridge (2005) specifically described the power of
arrays to develop flexible partitioning of numbers, and Jacob &
Mulligan (2014) showed the strength of the array in helping children
understand a range of multiplicative story problems. As well, Kinzer
& Stanford discussed how arrays can help children develop an
understanding of the distributive property noting that "Students
who make friends with the distributive property early on will find that
is a friend for life."(2014, p. 304). They base this claim on the
distributive property being a key element in developing an algorithm for
multi-digit multiplication, amongst other things.
Factors, multiples and other language
Third, the language associated with multiplicative thinking is
important. Specifically, the terms 'factor' and
'multiple' are vital in helping children to understand the
situation and to be able to explain why the properties of multiplication
work and why the inverse relationship with division works. Children
often learn these ideas in a procedural way and explain them in terms of
"You're just switching the numbers around" (commutative
property) or "You're just splitting up the number"
(distributive property). The group size and number of groups need to be
considered as factors and the total amount as the multiple. If both of
the factors are known, we multiply to find the total, or multiple. If
one factor and the multiple (or total) are known, we divide to find the
other factor.
Downton noted the significance of language saying that
"placing emphasis on the relationship between multiplication and
division and then language associated with both operations, before any
use of symbols and formal recording, needs to be a priority" (2008,
p. 177). This article highlights the many links and connections within
multiplicative thinking, and for these links to be understood,
discussion, reasoning, inferring, and justifying need to be at the
forefront of classroom activity. Hence, language assumes a vital role
and it needs to be supported with the extensive use of materials to
assist children to learn from a conceptual standpoint rather than learn
procedures.
Specific connections within the multiplicative situation
So, just what connections and links can be found within the
multiplicative situation? How can the explicit teaching and highlighting
of these connections enrich children's understanding?
How can these connections and ideas be shown to children?
Factor-multiple relationship
To begin with, the factor-multiple relationship needs to be shown
in two ways. (See Figure 2.)
Note the use of the terms 'scaled up or down by a factor of 3
or 5'. This is important as it highlights the essential difference
between multiplicative and additive thinking described by Devlin (2008)
ates thinking about ratio and proportion. As noted earlier, the
development of understanding of the concept of the multiplicative
situation is a complex process that takes time. The specific use of the
term 'scaling' is included here to show the connection that
exists, but its introduction in the teaching process requires caution.
In keeping with the idea of 'big idea thinking' and the
connections that are inherent, there is an opportunity here for teachers
to make such a connection explicit for their students.
That is, the idea of 'scaling' is multiplicative in
nature and essentially the same as using scales on maps. For example, on
a map with a scale of 1:100, real measurements and distances are
'scaled down' by 'a factor of 100'.
Commutative property
In terms of the properties of multiplication, the array is a
powerful representation as has already been indicated. It is widely
noted that the array can simply be rotated by ninety degrees to
represent the commutative property (e.g., Siemon, Breed & Virgona,
2006) but an even more powerful way of viewing this is to maintain the
same orientation for the array and consider the different numbers of
equal groups as shown in the second part of Figure 3.
[FIGURE 3 OMITTED]
To support children to understand the commutative property,
teachers could use a simple task like 'Building Arrays'. In
this, two 5 x 3 arrays could be cut into pieces--one grid is cut into
three groups of five and the other into five groups of three. The first
grid is then rebuilt as a vertical array as shown in Figure 3 (with the
groups of five squares being in columns) and the second grid can be
overlaid on the first grid (with the groups of three squares being in
rows). This is a very powerful representation of the commutative
property without even having to rotate the array! It should be noted
that the arrays in Figure 3 are shown as dot/grid arrays--it is
suggested that an even stronger representation would have the dots
removed.
The Australian Curriculum: Mathematics describes the development of
the 'Understanding' proficiency in terms of students
connecting ideas that are related, when students represent concepts in
various ways, and when they identify similarities between different
aspects of a concept. The statement about the proficiency of Fluency
discusses flexible use of procedures and recognizing robust ways of
answering questions (ACARA, 2013). The Building Arrays task described
above helps develop those aspects of the Understanding and Fluency
proficiencies.
Distributive property
The distributive property can be equally well shown with an array
(see Figure 4). That is 13 x 6 can be considered as (10 x 6) + (3 x 6)
or in combination with the commutative property, 6 x 13 can be
considered as (6 x 10) + 3 x 10). As discussed by Kinzer & Stanford
(2014), this use of the array can also be beneficial for helping
children learn the 'harder number facts' by splitting them, as
well as developing the algorithm for multiplying larger numbers. Also,
this use of the array is important for developing a standard algorithm
for double digit multiplication for an example like 36 x 28. The second
part of Figure 4 demonstrates this by splitting 36 into 30 and 6, and 28
into 20 and 8, to derive the four parts of the algorithm as shown by the
shaded parts of the array.
[FIGURE 4 OMITTED]
Arrays, fraction concepts and division
Fractions are notorious for the perception that they are difficult
to teach and understand. However, it is suggested that much of this
perception arises from procedural teaching and a lack of linking and
connecting of fractions to other ideas. One such idea is the division
construct for fractions which is often ignored. Again, there is some
interesting language such as the notions of 'thirding' and
'fifthing', as described by Siemon et al. (2011), and which is
a part of developing the understanding of this idea, as shown by Figures
5 and 6. The understanding that these connections exist within the
multiplication situation, and the making of those connections explicit
to children, are characteristics of strong teaching.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Arrays, fraction language, and 'times as many'
The importance of language has already been discussed, and children
need to be exposed to a range of ways of considering the multiplicative
situation and discussing it. Too often, multiplication facts are learned
in isolation and not linked to other representations. Askew (1999) used
the notion of 'free gifts' as a way of considering
'free' or additional facts that come from knowing one fact.
This notion can be applied here, as shown in Figure 8, to learn about
fraction concepts based on known facts.
[FIGURE 8 OMITTED]
Arrays and area
The concept of area is easily shown by the grid array and can be
explicitly developed as multiplication facts are learned in a conceptual
way through the use of arrays that incorporate the commutative property,
inverse relationship and associated language. Again, this is another
example of Askew's notion of 'free gifts'.
[FIGURE 9 OMITTED]
The language of 'times as many' and 'a third as big
as' is clearly important and forms an important bridge to the
understanding of ratio.
It has already been pointed out that young children are capable of
understanding multiplicative situations with sharing or splitting as the
basis.
Arrays and combination problems
Further to this, Jacob & Mulligan say that "Even young
children need to be exposed to a range of multiplication and division
problems" (2014, p. 36). Such a range of problem types that
characterise the multiplicative situation are described in various
sources (Department of Education, Western Australia, 2013; Van de Walle,
Karp & Bay-Williams, 2013). One example of problem types that can be
represented well by an array is the combination problem based on a
familiar canteen menu scenario as shown in Figure 10. The array can be a
powerful tool for developing a multiplicative view of this type of
problem rather than an additive one.
Arrays and ratios
Multiplicative thinking underpins the development of proportional
reasoning and the concept of ratio and this is associated with
Devlin's (2008) view of multiplication as a 'scaling
concept'. It is inherently linked to the notion of 'times as
many' and as such, informs the development of understanding of
percentage. Figure 11 shows how an array can be used to highlight yet
another of the rich connections within the concept of multiplicative
thinking and specifically the multiplicative situation. The concept of
ratio can be quite a difficult one to master, probably because of the
two types of ratio to be considered. Rathouz, Cengiz, Krebs &
Rubenstein (2014) make the distinction in terms of an additive
comparison of the two parts (part-to-part ratio) and a multiplicative
comparison of the one part to the total (part-to-whole ratio). The
distinction can be seen quite simply with the aid of an array, as shown
in Figure 11.
Conclusion
It is well established that multiplicative thinking is a critical
stage in the development of children's mathematical understanding
as it underpins much of the mathematics that follows.
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
Understanding the multiplicative situation and the three quantities
involved, having a flexible knowledge of the terms factor and multiple,
and knowing about the extended use of the multiplicative array, are
three key elements of multiplicative thinking. It is important to
realise that such conceptual development takes place over time and
requires a whole-school approach beginning in the early years and
extending throughout primary school. This article has attempted to show
the myriad connections and links within the idea of multiplicative
thinking as well as how aspects of it link to and inform the development
of other ideas. The power of the multiplicative array as an enabling
tool is central to children making sense of these connections and links.
Initially, teachers may not necessarily see the connections between
ideas presented in this article and the elements of the multiplicative
situation, but a major purpose of the article is to point out that these
connections do exist. The multiplicative array can indeed be a powerful
way of representing the distributive property, combination problems, and
even equivalent fractions, as has been shown here. The construct of
fraction as division is very much a part of the multiplicative
situation--when fifteen is divided by 5, it is the same as
'fifthing' the original quantity. Similarly, the concept of
division as sharing into equal groups is inextricably linked to
equivalent fractions. The connections based on the multiplicative
situation are there.
If teachers understand that and make the connections explicit to
children, they are enabling them to understand mathematics conceptually
and in a truly connected way.
References
Askew, M. (1999). Figuring out: Free gifts. Junior Education, May
1999, p. 39.
Australian Curriculum Assessment and Reporting Authority (ACARA)
(2013). Australian Curriculum: Mathematics. Retrieved from:
http://www.australiancurriculum.edu.au/ mathematics/curriculum/
Department of Education, Western Australia. (2013). First steps in
mathematics: Number. Retrieved from: http://det.wa.edu.
au/stepsresources/detcms/portal/
Devlin, K. (2008). It ain't no repeated addition. Retrieved
from: http://www.maa.org/external_archive/devlin/ devlin_06_08.html
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Nunes, T., & Bryant, P. (1996). Children doing mathematics.
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auckland.ac.nz/mathwiki/images/4/41/WATSON.doc
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Chris Hurst
Curtin University
<C.Hurst@curtm.edu.au>
Figure 2. Factor and multiple relationships.
Factor of 15 Factor of 15 Multiple of 3 & 5
Number of Number in Total of all
groups each group groups
5 x 3 = 15
3 is scaled up by a factor of 5
Multiple of 3 & 5 Factor of 15 Factor of 15
Total of all Number of Number in
groups groups each group
15 / 5 = 3
15 is scaled down by a factor of 5
