Research School Network: Mathematical Manipulatives – Why and how? Deputy Director of Norfolk Research School – Rachael Wilson

They live in most classrooms, and when used purposefully and routinely, mathematical manipulatives can have a significant positive impact on children’s long-term learning through the development of a robust mathematical schema. Yet in many schools the habitual and planned use of manipulatives is not always happening, and is dependent on the confidence, capacity, and subject knowledge of individual teachers.

There are numerous beliefs and assumptions about the use of manipulatives in daily maths lessons. These range from the idea that they are predominantly for younger children or only used as a ​“catch up” to plug gaps in prior understanding as children move through school, to the idea that they are a distraction or become a ​“crutch” which then becomes detrimental to learning in the first place. Their use is not always accounted for in schemes of work or in initial teacher training beyond basic application, so there are multiple, very real barriers to using them effectively.

However as with all resources, the impact of manipulatives on learning is entirely dependent on why, how and when they are chosen and used. This blog aims to explore some of these ideas in more detail, to look at what the evidence says about the use of manipulatives, and to offer a few suggestions as to how to use them effectively across schools in order to obtain the most benefit from them.

Why use manipulatives?

In both their Improving Mathematics in the Early Years and Key Stage 1 and Improving Mathematics in Key Stages 2 and 3 Guidance Reports, both based on extensive research and literature reviews, the EEF mentions the significance of manipulatives as powerful vehicles for supporting pupils to engage with mathematical ideas and concepts.

Manipulatives are capable of not only conveying understanding in a multidimensional way, such as developing a ​“sense” of the dimensions and mass of 3D shapes or weights, but chosen wisely and explained explicitly, concrete representations are an excellent way of demonstrating the connectedness of different mathematical ideas. Examples include bead strings and Base 10, which can be used to explore fundamental mathematical concepts such as proportionality- as well as additive and multiplicative reasoning.

They support the strengthening and building of the complex mathematical schema which children develop as they encounter more concepts and explore their interconnectedness. They link to pictorial representations such as number lines, part whole or area models, and can be used to build new knowledge onto established knowledge through evolutions of their use through school.

Manipulatives can also support the acquisition of foundational concepts such as one: one correspondence, number bonds and multiplication facts, as well as a sense of the ​“size” of numbers that abstract digits alone cannot do without significant development of schema first.

They can support children in making the jump from specialising (looking at individual cases) to generalising (identifying and applying a mathematical rule). For example, using a bead string with one hundred beads to represent a ​“whole” – what if it represented ten? Or three? What would each bead be worth then?

So what? How should we use manipulatives effectively?

As with all tools and vehicles, manipulatives are only effective when used with very careful consideration. The EEF Guidance Report makes the following recommendations-

1. There needs to be a clear rationale for the selected manipulative – why and how will it be used by teachers, teaching assistants or pupils?

2. Pupils should be explicitly taught the link between the manipulatives and the concepts being taught.

3. Manipulatives should be temporary and used as a removable scaffold.

4. Manipulatives can be used at all ages – the measure should be strength of conceptual understanding, rather than age or developmental stage.

What could this look like?

In a classroom….

…. the use of manipulatives is carefully planned for before the lesson.

.…there is a clear rationale for using specific manipulatives

…. children are not left to choose them unless there has been preparation for this and it is part of the intended learning

…… routines are in place around their use.

…. teachers explicitly link their use to the learning and are assessing their impact.

.… concrete resources are used alongside pictorial and abstract ones – children are explicitly invited to compare and link different representations and explore the differences and similarities between them

Across a school….

…. the use of key manipulatives is planned for and monitored.

…. there may be a policy that supports staff, particularly new teachers, with which manipulatives to use, when and how.

…. routines are established across the school that build as children progress. E.g. use of bead strings for decimals and percentages in Upper Key Stage 2/​Key Stage 3 and not just counting in ones and groups lower down the school.

…. manipulatives are chosen to help demonstrate underlying mathematical structures, rather than viewed as a simple calculation aid or support for struggling children


Further Reading

Ofsted Mathematics subject review 

MC Brown, NM McNeil and AM Glenburg, ​‘Using concreteness in education: real problems, potential solutions’, in ​‘Child Development Perspectives’, Volume 3, Issue 3, 2009, pages 160 to 164. 

NCETM Article – Representations in our Primary video lessons 

EEF blog: Manipulatives and representations – Helping children to ​‘see’ the maths