# Scaffolding numeracy in the middle years: Multiplicative thinking

Multiplicative thinking is indicated by a capacity to work flexibly with the concepts, strategies and representations of multiplication (and division) as they occur in a wide range of contexts.

## Key characteristics

Multiplicative thinking is characterised by:

• a capacity to work flexibly and efficiently with an extended range of numbers (for example, larger whole numbers, decimals, common fractions, ratio and percent)
• an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion
• the means to communicate this effectively in a variety of ways (for example, words, diagrams, symbolic expressions and written algorithms).

## Example

A student developing multiplicative thinking will move from a problem such as:

 3 bags of sweets, 8 sweets in each bag. How many sweets altogether?

to problems such as:

 Julie bought a dress in an end-of-season sale for \$49.35. The original price was covered by a 30% off sticker but the sign on the rack said, “Now an additional 15% off already reduced prices”. How could she work out how much she had saved? What percentage of the original cost did she end up paying?

## Multiplicative situations

Students need to construct and coordinate three aspects of multiplicative situations when they are developing multiplicative thinking:

• groups of equal size
• the number of groups, and
• the total amount.

Students then need to move from the models and representations that work for whole numbers to more general ideas accommodating rational numbers and algebra.

These more general ideas include:

• ratio
• proportion
• multiplicative comparison
• multiplication of measures, and
• the use of intensive quantities.

This is a complex process and may take many years to achieve.

## From modelling to abstracting

Multiplicative thinking begins to develop as each aspect (groups of equal size, the number of groups, and the total amount) is abstracted and can be dealt with as a mental object.

ModelledAbstracted

Equal groups

Makes all and counts all

Trusts the count and sees equal groups as a composite unit

Number of groups

Sees in terms of each group, counts all groups. For example:

1 group, 2 groups, 3 groups, ...

Can deal with number of groups in terms of part-part-whole understanding. For example, 6 groups is:

• 3 groups and 3 groups, or
• 5 groups and 1 group.

Total

Arrived at by counting all and skip counting

Total is seen as a composite of composites. For example, 18 is seen as :

• 2 nines
• 9 twos
• 3 sixes
• 6 threes.

Working with concrete models and representations is important in the early stages. However, a key aspect in moving from modelling to abstracting appears to be the capacity to:

• work with mental images
• use strategies that don’t rely on physical objects.

## Transition during middle years

The move from additive to multiplicative thinking is not always smooth. It can be a real and persistent barrier to mathematical progress for students in the middle years of schooling.

Compared with the relatively short time needed to develop additive thinking, the introduction and exploration of ideas to support multiplication may take many years. According to some researchers, multiplicative thinking may not be fully understood by students until they are well into their teen years.

## Supporting the student

One way that we can move forward is to slow down and give students time to appreciate the complexities involved. This does not mean ‘dumb down’ but ‘think through’ problem situations by asking:

• What do they mean?
• How can they be represented?
• How can we use what we know?
• Which strategies are better and why?

As teachers, we need a deeper understanding of what makes multiplication difficult and how we can scaffold more appropriate strategies.

Resources available through the Scaffolding Numeracy in the Middle Years (SNMY) project aim to provide further understanding about the transition from additive to multiplicative thinking.