Multiplicative reasoning

In November we considered the progression through multiplication and division.

In this article we are going to follow this up by briefly exploring the idea of multiplicative reasoning which often doesn’t get a mention in primary mathematics. It is a very important concept because a lot of what we need to teach in the national curriculum is based on the expectation that children can reason in a multiplicative way. It enables children to develop a deeper understanding of multiplicative structures such as fractions.

'It has been suggested by research that many pupils (and adults) fail to move on from additive structures and this can lead to many misconceptions and errors in subsequent mathematical study.' 
NCETM July 2013

Many children will, for example, say that to make 2 into 10 you add 8 or to make 5 into 10 you add 5. This is additive thinking. If they were thinking multiplicatively, they would multiply 2 by 5 or 5 by 2 to make 10. Purely thinking in an additive way, as research suggests, can put a ceiling on children’s learning. In the national curriculum, various topics need an understanding of multiplicative reasoning, for example, multiplication, division, scaling, area, ratio and proportion. It is a useful concept to teach the children as it also helps to build mental calculation strategies and develop reasoning. Multiplicative reasoning is essentially a recognition and use of grouping in the underlying pattern and structure of our number system.

It contributes to an understanding of place value and makes it possible for children to see different kinds of relationships between numbers. When we ask children to count in steps of different sizes we are helping them to make the transition from additive to multiplicative reasoning because this shows the structure and efficiency of counting in groups and highlights sequences and patterns. When we do this with our children it is important to model with manipulatives, such as towers of interlocking cubes, and visual representations, such as shapes on the whiteboard, so that these orders are learned with meaning and not simply as a rote counting method, which is often what happens and results in the children not truly understanding the patterns and sequences they are saying.

When children initially think about grouping they are often introduced to the idea of halving and doubling which they seem to be able to do quite easily.

Halving is an early stage of multiplicative reasoning where they ‘split’ groups equally, for example halving six:

The inverse nature of halving and doubling enables the children to see that if they double three they will get back to six. Teaching multiplication through repeated addition is commonly done and this is one of the structures of multiplication. However, if the children only think of, for example, 6 x 3 as 6 + 6 + 6, they will find it difficult to deal with calculations such as ½ x ¼ later on. The connections between repeated addition and multiplication and repeated subtraction and division are somewhat limiting if we are to help them build a strong understanding of multiplicative reasoning. To support children in their understanding of multiplicative reasoning we need to ensure that they have access to effective models. Difficulties can arise, as in all areas of mathematics, if they are simply expected to memorise formulae or procedures without any conceptual grounding.

The two most common models of multiplicative reasoning are creating like groups and the use of area. Linking grouping with scaling is also helpful. I have used this with all children from year 2. We can use grouping in a scaling model to work out any number of facts. For example, if one apple costs 12p, we can use doubling to work out the cost of two, four and eight and then make up combinations of these to work out the cost of any number: 1: 12p 2: 24p 4: 48p 8: 96p 3: 12p + 24p = 36p 6: double 36p or add 24p and 48p 10: multiply 12p by 10 12: 48p + 96p….. and so on. Young children could explore, for example, beans in a cup. If there are two beans in a cup, they can work out how many will be in two cups, three cups and so on, by simple doubling and addition. This can link well to counting in twos. Arrays are an excellent way to develop multiplicative reasoning because they link multiplication and division and allow for the explorations of commutativity. They also give the children their first introduction to area.

It is often better to begin teaching multiplicative reasoning through a problem, such as: There are two cats in a basket and four baskets, how many cats? They could set this out as an array or develop a bar model to help them find the solution.

For example: There are 30 children in the class. The class is going on a trip in mini buses. If each bus holds six children, how many buses are needed?

Before starting problems such as these, ensure that the children are able to make use of manipulatives such as cubes or counters, so that they can physically move them into groups. Another topic requiring multiplicative reasoning, as mentioned above, is area. The children are usually introduced to this concept within a measures topic when in fact the links to multiplication and division are very strong and possibly provide the best context in which to explore area. This might be worth considering when you next plan these areas of mathematics. The model for this can be developed in a similar way to the grouping model discussed above. You could ask the children to draw a rectangle of 2cm by 1cm on centimetre square paper. They could then count the squares and record 2cm x 1cm = 2cm2. They could then find the areas of other rectangles up to 10cm by 10cm. Ask them to view these as arrays and discuss how they could use multiplication to find their areas. You could repeat this by giving the children an area and asking them to use their knowledge of factors and division to find the dimensions. The array model can also be used to develop the idea of multiplicative growth. Using arrays, children can make conjectures about what will happen if they make, for example, a three by four array twice the size or half the size. This article has really only scratched the surface of multiplicative reasoning but we hope it has given you some food for thought in this area.