# 4 BIG IDEAS about MULTIPLICATION

Times tables are more important than just memorising sums – they can form the memorable groundwork for children’s greater mathematical understanding.

Having multiplication facts and times tables knowledge as a key part of our **‘mathematical tool box’** is hugely important because we use them constantly in everyday life.

However, instead of traditional approaches using rote learning, which cause children to perform at a significantly lower level than those taught with a focus upon the ‘big picture and connections’ (Boaler, Mathematical Mindsets) let’s look at how we can all engage children with meaning and building strong mental connections and flexibility working with how we know the brain learns best!

Here are 4💡BIG Ideas 💡 and approaches to help your children with multiplication with RELEVANCE and UNDERSTANDING:

Researchers have noted that multiplication and division are conceptually demanding due to the levels of abstraction required and the complex semantic structures they involve (Anghileri, 1989; Clark & Kamii, 1996; Greer, 1992; Steffe, 1994). Although multiplication tends to be formally introduced in school, research has shown that pre school children can model both grouping and sharing.

Nunes and Bryant (1996) suggest that the simplest form of multiplicative situations that children will meet is probably one in which there is a *one-to-many correspondence* (e.g. 1 car with 4 wheels) which relates to *ratio*, or *scale factor*, and is the basis for multiplicative, rather than additive thinking.

**SKIP COUNTING** is so often an early strategy to encourage children to count in equal groups - especially if it’s supported with concrete and pictorial representations BUT…

**BEWARE** the * ‘Mary gad a Little Lamb’ *approaches to multiplication table learning. Teaching children to reel off a list of numbers can sound like they have multiplication knowledge: ‘Let’s count in 2s – 2, 4, 6, 8, 10…’. But when we examine this more closely they are often starting in the same place every time and have essentially just learned to recite the equivalent of a nursery rhyme. Using and applying this knowledge flexibly then becomes almost impossible.

Skip counting is only one way of thinking about multiplication.

**Unitising** is simply, where many things are treated as one thing; a packet of 10 chocolate biscuits, for example. They is one packet, but at the same time they are 10. **And the thing about unitising, is that it’s efficient! **

The shift from additive to multiplicative thinking is not easy and may take considerable time to achieve as it requires a** "cognitive reorganisation on the part of the learners." **(Fosnot & Jacob, 2010)

**EQUAL GROUPING**

Learners should explore how objects can be arranged in equal groups. When describing equally grouped objects, the number of groups and the size of the groups must both be defined. Equal grouping structure is at the core of multiplicative concepts and thinking:

Equal groups can be represented with a repeated addition expression:

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

4 + 4 + 4 + 4

8 + 8

Repeated addition is only one way of thinking about multiplication.

**A shift from attending to how much is in each group (multiplicand) to how many groups (multiplier) is a critical step and leads to the recognition of the number of groups as a factor.**

Moving objects into rows and columns provides better opportunities for exploring the commutative rule than static pictures which are sometimes difficult to see in the two different ways.

For example, the array above could be read as

**efficient way to learn.**

This is a challenging concept for young learners but I believe it’s the **hallmark of real multiplicative thinking**.

The development of multiplicative thinking requires a long period of time (Clark and Kamii 1996).

While I do believe that building a conceptual understanding through multiple representations of proportional reasoning is very important, I think it should be explicitly stated that taking this approach will not necessarily speed up the learning process. Realistically, it could take longer due to the depth of knowledge we are striving for. I’m a firm believer that anything worthwhile takes time and effort. Our mathematical understandings are no exception.

If one car uses 4 tyres and we want to know how many tyres are needed for 6 cars the thinking involved is **PROPORTIONALITY. **

Often multiplication is only taught as REPEATED ADDITION and the PROPORTIONAL REASONING has been sacrificed. Don’t do that! Try this:

The transition from additive to multiplicative thinking, however, constitutes an obstacle for many children (Ehlert et al. 2013; Gaidoschik et al. 2018; Götze 2019a, b; Moser Opitz 2013; Siemon et al. 2005).

The qualitative studies of Breed (2011) and Götze (2019a, b) indicated that such language-responsive instruction might help children to develop multiplicative thinking as unitising.