Representations are used in lessons to expose the mathematical structure being taught. They are not new – we can probably all remember using counters when we were children. The history of mathematics uncovers many examples of objects, pictures or symbols used in early maths to represent concepts. Nowadays it’s not unusual to find tens frames, Dienes blocks, Cuisenaire® rods, bar models and other representations in frequent use in primary classrooms.

In recent years, teaching for mastery has encouraged the use of representations to deepen understanding, and encouraged teachers to question their purpose, and which representation to use and when.

In this piece, we bust some myths about representations and encourage you to watch them being used in a teaching for mastery context. Can you see what each representation used in the video below exposes about the mathematical structure of multiples of two?

Secondary maths teachers may find Using mathematical representations at KS3 helpful.

Myth: Manipulatives and pictures are for younger children and low prior-attainers

Representations are useful for all learners, whatever their age. Research mathematicians often use representations to explain their thinking. Teaching for mastery suggests that representations should be used throughout primary and secondary school to promote a deep understanding of mathematical structure. Once learners have a deep understanding of the maths being represented, the aim is to work with the maths without recourse to the representation, though they will often continue to work with visuals in their mind’s eye.

Secondary maths teachers may find Using mathematical representations at KS3 helpful.

Myth: Representations help children to do calculations

Objects can assist children in performing calculations – for example, a child might use three groups of five counters to then count all the counters to find the product 15. However, using the representations in this way can encourage a child to become dependent on them. Teaching for mastery encourages the use of representations to demonstrate the structure (e.g. three groups of five counters). The child’s understanding of the structure is then built on to teach efficient calculation methods.

Myth: Children should always be able to choose their own representations

When teaching for mastery, the teacher will often have a clear idea of the particular structure s/he is trying to illuminate and will choose the most appropriate representations for doing this. Children given free rein with manipulatives may use them to do, rather than see, the maths.

Myth: A particular concept calls for a particular representation (e.g. fractions as pizzas)

Teaching for mastery uses multiple representations of a concept to expose different aspects of the structure.

Watch the video lessons below and see how the teacher uses many different representations to build the concept of units of two. You might like to see if you can identify the purpose of each representation, before you check in the dropdowns:

The subsequent video in the sequence goes on to introduce other representations to deepen children’s understanding of two-ness:

The representations, now familiar from the lessons, can then be used to enable counting on in twos from a multiple of two and beginning to touch on the two times table.

Teachers may want to use the slides in these videos to teach their own lessons. These are available, along with teacher guides, on the Key Stage 1, Multiplication 1 page.

However, our Primary Team warns against teaching individual lessons in isolation.
‘Each lesson is part of a carefully planned sequence’ says the NCETM Director for Primary, Debbie Morgan. ‘The learning builds over the sequence of lessons’.

The lesson sequences were collaboratively planned by groups of Mastery Specialists and members of the NCETM Primary Team for the purpose of home learning, during the first lockdown in 2020.

You can watch all 17 lessons in the Multiplication 1 sequence or find other lesson sequences for all year groups.