An explanation for the use of arrays to promote the understanding of mental strategies for multiplication.
Day, Lorraine ; Hurrell, Derek
As part of Understanding proficiency, the Australian Curriculum:
Mathematics requires that "They (students) develop an understanding
of the relationship between the 'why' and the 'how'
of mathematics" (ACARA, 2014, p. 2). In this paper we would like to
address the issue of how we develop the 'why' around the
'how' of multiplication, leading to students being fluent
users of multiplication for mental and written computation.
In order to add meaning rather than relying on memorised procedures
for multiplication and division, students need to be able to think about
multiplication in a number of different ways. As early as Year 2 the
Australian Curriculum: Mathematics states that students should
"recognise and represent multiplication as repeated addition,
groups and arrays" (ACARA, 2014). In particular, arrays and regions
assist to support the shift from additive thinking ('groups
of' model) to multiplicative thinking
('factor-factor-product' model) (Siemon, 2013) and eventually
to proportional and algebraic reasoning.
When asked to create four groups/lots of three, students will often
create a model that looks something like Figure 1. While this is
correct, it is not necessarily the most helpful or even efficient
representation, if we want to move the students from additive to
multiplicative thinking. Whilst Figure 1 does show four lots of three it
can encourage students to use repeated addition which does not
"address all situations in which multiplication is helpful"
(Willis et al., 2008, p. 28). Whereas Figure 2, a region or array model,
not only shows the strategy of repeated addition it also encourages
other understandings.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
One such understanding is the identification and naming of factors
and multiples, an important and often undervalued piece of mathematical
understanding. Giving students a strong understanding of factors and
multiples through manipulating the materials to come to a shared
definition of these terms is extremely empowering. Another important
mathematical understanding that arrays can help to develop is that of
commutativity (Figure 3). As a mental and written computation strategy,
commutativity, the understanding that for multiplication (and addition),
it does not matter which order you use the numbers, the result will be
the same, is vital, and the capacity to rotate an array to show that
four lots of three gives the same total as three lots of four is
illustrative of this.
[FIGURE 3 OMITTED]
Another understanding that can be developed through the use of
arrays, and should not be underestimated, is 'number
families'. Often students are told that if you know 3 X 4 you know
the associated facts of 4 X 3, 12 / 3 and 12 / 4, and many accept this
as being the case without ever seeing why it is so. The array model
created with materials and then manipulated to discuss and illustrate
these 'number families' is a very visual and powerful model.
This same model then readily lends itself to being represented on square
grid paper, and then later still, having the associated facts
represented abstractly though numbers. This movement from the concrete
to the abstract can all happen whilst keeping the materials handy to
maintain a clear illustration of the connection between the associated
facts, and therefore, developing fluency through, and with,
understanding.
Another strength of the array model is that it can be extended into
two-digit by one-digit multiplication (Figure 4). Representing
multiplication in this way provides a meaningful illustration of
partitioning of numbers and encourages an understanding of the magnitude
of numbers in a very visual manner. It also encourages the development
of the distributive property (that is that 14 X 3 is equivalent to 10 X
3 + 4 X 3, another important mental computation strategy) and the link
between multiplication and area.
Once the model has been used to develop two-by one-digit
multiplication, it is then a reasonable step to move to a representation
of two-by two-digit multiplication (Figure 5).
Students who have not had the benefit of using an array model may
assume that 13 X 12 can be calculated by 10 X 10 + 2 X 3.
[FIGURE 4 OMITTED]
Using the array model and identifying the areas assists students to
see why this is not the case. Once again the array proves to be an
efficient construct which supports the Australian Curriculum:
Mathematics Year 3 content descriptor which encourages students to
"represent and solve problems involving multiplication using
efficient mental and written strategies ..." (ACARA, 2014).
[FIGURE 5 OMITTED]
At this point it is not unreasonable to start representing the
algorithm in a more abstract manner. By having the array representation
at hand (Figure 5), a direct comparison can be made between this
representation and a more abstract representation (Figure 6). The use of
a non-standard algorithm rather than a standard algorithm may better
serve to bridge the understanding between the eventual use of the
'abbreviated notation' employed in the standard algorithm and
to what the 'abbreviated notation' actually refers (for
example, the five in 156 refers to five-tens).
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
This model can also be used to demonstrate that it does not matter
how the numbers are partitioned--we use ten because it makes the
calculations easier, but the distributive property is not restricted to
partitions involving tens (Figure 7). This knowledge provides students
with more flexible mental computation methods.
The Australian Curriculum: Mathematics Year 5 states that students
should "solve problems involving multiplication of large numbers by
one- or two-digit numbers using efficient mental, written strategies and
appropriate digital technologies" (ACARA, 2014). This model lends
itself to further exploration of these larger numbers (Figure 8).
A standard, written algorithm is a useful tool. A standard, written
algorithm, which is constructed with understanding, is an exceptionally
powerful tool. Once the non-standard algorithm in Figure 6 has been
established, discussed and understood it can then be used to develop an
understanding of the components of the standard written algorithm
(Figure 9) if this is seen as necessary.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
Although not a focus of this article (and there is probably another
article which can be written further unpacking the move into numbers
other than whole numbers) the array model can also be utilised for
providing a compelling visual model of multiplying decimals (Figure 10)
and fractions (Figure 11). The Australian Curriculum: Mathematics
suggests Year 6 students should "multiply decimals by whole
numbers" and in Year 7 "multiply and divide fractions and
decimals using efficient written strategies and digital
technologies" (ACARA, 2014).
By forming the foundation of array or region models in the primary
setting teachers are not only providing powerful understandings to aid
mental and written computation, they are also paving the way for easier
connections to be made in secondary mathematics. In Year 7 according to
the Australian Curriculum: Mathematics, students should "apply the
associative, commutative and distributive laws to aid mental and written
computation" and "extend and apply the laws and properties of
arithmetic to algebraic terms and expressions".
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
Furthermore in Year 8 students should "extend and apply the
distributive law to the expansion of algebraic expressions" and
"factorise algebraic expressions by identifying numerical
factors" (ACARA, 2014). Another great strength of the region or
array model is that is can easily be extended into algebraic reasoning.
It may be a good idea at this stage to re-introduce a concrete model
here (such as Algebra Tiles Australia (Day, 2014)).
[FIGURE 12 OMITTED]
The model can be used for expanding quadratic functions:
[FIGURE 13 OMITTED]
The ability to 'see' how 0.5 x 0.5 is equivalent to 0.25
is such an important visualisation for students especially for those who
believe that multiplication always makes the answer larger. It also
assists with the understanding associated with division of decimals and
fractions, as the idea of the inverse relationship has already been
developed.
There are many reasons, to use an array or region model in the
teaching of multiplication and relating it seamlessly to division. The
most compelling of these reasons is to "support the shift from an
additive groups of model to a factor-factor-product model which is
needed to support fraction representation, the multiplication and
division of larger whole numbers, fractions and decimals, and
algebra" (Siemon, 2013, para. 4). We believe that the array model
is a very powerful way in which to take students to a robust
understanding of not only the 'how' of multiplication but the
'why' as well. It supports students in a way which simply
teaching the mechanics of the algorithm cannot.
To summarise the array method:
* Is visual
* Makes sense
* Supports a strong instructional practice of moving from concrete
to representational to abstract
* Encourages multiplicative thinking
* Links multiplication to area
* Demonstrates the distributive property
* Models commutativity
* Makes the transition to algebraic reasoning easier.
References
ACARA (2014). Australian curriculum: Mathematics. Retrieved from
http://www.australiancurriculum.edu.au/mathematics/ content-structure
Day, L. (2014). Algebra tiles Australia: A concrete, visual,
area-based model. WA: A-Z Type.
Siemon, D. (2013). Common misunderstandings--level 3:
Multiplicative thinking. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/pages/lvl3multi.aspx
Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. &
Treacy, K. (2004). First steps in mathematics: Number- Understand
operations, calculate, reason about number patterns. VIC: Rigby.
Lorraine Day
University of Notre Dame
<lorraine.day@nd.edu.au>
Derek Hurrell
University of Notre Dame
<derek.hurrell@nd.edu.au>
