Use manipulatives and representations to develop understanding
Improving Mathematics in the Early Years and Key Stage 1
Manipulatives and representations can be powerful tools for supporting young children to engage with mathematical ideas.
Education Endowment Foundation
Education Endowment Foundation
What are manipulatives and representations?
A manipulative is an object that children or practitioners can interact with and move to represent mathematical ideas. Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). could include everyday objects such as pine cones, buttons, and small toys as well as resources like interlocking cubes, Cuisenaire rods, Dienes blocks, and building blocks.
A ‘representation’ refers to a particular form in which mathematics is presented.1 Representations include informal drawings, mathematical symbols, and more formal diagrams, such as a number line or graph.
Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and representations can be powerful tools for supporting young children to engage with ideas across many areas of mathematics. They can help children make sense of mathematical concepts, develop visual images, increase engagement and enjoyment, help practitioners see what children understand and provide a bridge to abstract thinking.2 Children benefit from practical, first hand experiences of moving and interacting with manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to develop mathematical ideas.
It is important that children have opportunities to engage in both free and structured play with manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods).. However, practitioners must help children to understand the links between the manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). or representations and the mathematical ideas they represent.3 As children’s understanding of mathematical ideas develops, practitioners should encourage children to use pictures, symbols and more abstract diagrams to represent and communicate ideas and concepts.
There is some evidence that physical whole-body movement and gestures may support the learning of mathematics, for example, moving along a physical number line,4 or jumping and clapping while counting. Practitioners should encourage children’s use of fingers, which can be important manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). for children.5 Fingers can be useful for supporting
counting and later on for counting in groups.
What does effective practice look like?
The evidence suggests some key considerations:
- Ensure that children understand the links between the manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and the mathematical ideas they represent.3 Children need support in linking a manipulative with the mathematical ideas it represents. For example, a child may be confident using Dienes blocks to add but be unable to connect this to a written addition. This requires practitioners to explicitly help children to link the materials (and the actions performed on or with them) to the mathematics of the situation. This should enable children to develop related mathematical images, representations, and symbols.
- Ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. Practitioners should consider carefully how the manipulative will be used to build on existing understanding, and help develop increasingly sophisticated approaches and ideas.6
- Encourage children to represent problems in their own way.7 Practitioners should support children to become familiar with a repertoire of strategies to use to represent mathematical ideas, including their fingers, drawings, and marks such as tallies and arrows. Children should be free to invent and explore their own representations to record their thinking and communicate their understanding.
- Be aware that young children can be distracted by the surface features of a novelty manipulative—this can take away from the intended learning aim. Using a given manipulative regularly, or introducing it through play to gain familiarity can be beneficial.2
- Use manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and representations to encourage discussion about mathematics.2 Children can work in pairs and small groups using manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to solve problems and to encourage questions about other children’s strategies and reasoning. This can prompt the sharing and comparison of different approaches. Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). can also be used by children to communicate what they know.
There is evidence of the importance of showing children different representations of number and then helping them to make connections between them in order to support a fuller understanding.8 For example, understanding that the numeral ‘3’, a three on a dice face, three cubes, and a three-step on a number line all represent aspects of ‘three’.
It is likely that practitioners’ understanding of mathematical concepts needs to be strong in order to use manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and representations effectively, and this may need to be a focus of CPD for some practitioners.9 Settings should also plan their use of manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and representations to ensure a consistent approach.
Box 8: Using manipulatives to explore ‘one more than’
A Reception practitioner had recently watched ‘Numberblocks’ (CBeebies) with the children. In this particular episode (season 3 episode 28), the number 15 is represented with interlocking cubes in a staircase pattern.
The practitioner wanted to further explore the ‘one more than’ relationship between counting numbers with the children. In order to do this children need to know the cardinal value of the numbers (that the last number counted represents the overall amount in the group) and realise that adding one to any number produces the next counting number.
The practitioner showed the children how to build ‘staircases’, making each ‘stair’ by matching the previous one, then adding one. The practitioner encouraged the children to use large bricks to make staircases, which were easy for them to handle, and encouraged physical movement of the manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods)..
In order to support the children to understand the link between the pattern and the ‘one more than’ relationship, the practitioner started to make a spectacular giant ‘staircase’ in the playground with some cable spools. This also presented the same pattern in a different context, helping children to recognise the structure in a different orientation and with circular objects.
The practitioner modelled the process of making the staircase, saying, ‘How many do I put next?’ Sometimes he did it wrong deliberately and matched the previous stair. When a child said, ‘You need another one’, he said ‘Oh, yes, it’s got to be one more than the one before’, modelling the language to describe the relationship. When it was finished he asked, ‘What do you notice?’ One child said, ‘It’s a pattern. It goes up one, one, one.’ One child counted the stair columns, saying ‘one, two, three, four, five’. Another child standing at the side said they could also count ‘five, four, three, two, one’, pointing to the rows. This showed that children were becoming aware of different features of the pattern.
The practitioner later made a staircase with sticks of interlocking cubes wearing number ‘hats’ to explicitly show the continuing pattern with numerals.
In creating and playing with physical and symbolic representations of the staircase, and discussing ‘wrong‘ or muddled examples, the children were repeatedly meeting the same pattern and becoming familiar with ideas such as the inverse relationship between ‘one more’ and ‘one less’ and of hierarchical inclusion—that each successive number is equal to the previous number plus one.
Box 9: Using manipulatives to explore bridging through ten
A Year 1 teacher wanted to teach children to use their number bonds to add numbers by bridging through ten (adding two numbers whose total is greater than ten by counting through to ten then adding the remainder). Using the example of 7 + 5, she chose to use double sided counters and ten-frames and demonstrated putting seven red counters on the frame, then adding five, yellow side up. This allowed the children to focus on splitting the five to into three (to make ten) and two more, with the ten-frame clearly showing the resulting 12 as ten and two. For children who found it difficult to keep the counters in place on the ten-frames she provided chunky ‘jewel’ counters and ice cube trays in tens.
The teacher then modelled this in abstract form alongside the ten-frame.
This helped children to represent this concept abstractly and make the explicit link between the manipulative and the idea it represents. The children then played a game in pairs, using a dice numbered five to ten to decide the first number, then adding five.
Box 10: Using manipulatives to explore making rectangles
A Year 2 teacher wanted to introduce children to the multiplicative composition of numbers, for example that 8 can be expressed as two fours and four twos, and to increase familiarity with some key multiplication facts.
He challenged them to make as many different rectangles as they could with 20 cubes. This enabled them to see that 20 could be made with two lots of ten or four lots of five. He then asked them how they knew they had found all the arrays for 20 and to check. Most children worked randomly, trying different numbers in rows. One child checked all the numbers from 1 and then discovered the 1 x 20 array. The teacher chose multilink cubes because these could be joined to form arrays, which could be turned round to clearly show, for instance, that two rows of ten are equivalent to ten rows of two. This demonstrates the commutativity of multiplication (that the order of multiplying two numbers does not affect the result).
He asked the children to describe the five by four array and to say what they noticed when they turned the array round. One child said, ‘I can see five and five and five and five, and when I turn it round I can see four and four and four and four and four’. Other children said things like, ‘It’s five rows of four this way and four lots of five the other way’ and ‘It’s five fours and four fives’. This showed the teacher that the first child was thinking additively, whereas children who could talk about ‘lots of’, ‘rows’ or ‘four fives’ were thinking multiplicatively. When he asked them to record all the ways they had found, this similarly showed their understanding, as some wrote 5 + 5 + 5 + 5 while others wrote 5 x 4. In this activity the manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). supported the children to articulate their reasoning and helped the teacher to identify children beginning to work multiplicatively.
Activity adapted from ‘Making Numbers: Using Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to Teach Arithmetic’, Oxford University Press.
For more examples of using manipulatives:
Griffiths, R., Gifford, S., and Back, J. (2016). Making Numbers: Using Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to Teach Arithmetic. Oxford University Press.
The Nuffield Foundation commissioned a literature review on the use of manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). for teaching arithmetic to children. This made recommendations for practitioners, which can be found at:
Making Numbers was a book produced as part of this project to provide exemplification of using manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods)..
References
1. National Centre for Excellence in the Teaching of Mathematics (2014). ‘Curriculum Glossary’. https://primarysite-prod-sorte...
2. Griffiths, R., Back, J. and Gifford, S. (2016) Making Numbers: Using Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to Teach Arithmetic, Oxford: Oxford University Press.
3. Carbonneau, K. J., Marley, S. and Selig, J. P. (2013) ‘A Meta-Analysis of the Efficacy of Teaching Mathematics with Concrete Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods).’, Journal of Educational Psychology, 105 (2), pp. 380–400. https://doi.org/10.1037/a0031084
4. Sung, W., Ahn, J. and Black, J. B. (2017) ‘Introducing Computational Thinking to Young Learners: Practicing Computational Perspectives through Embodiment in Mathematics Education’, Technology, Knowledge and Learning, 22 (3), pp. 443–463. https://doi.org/10.1007/s10758-017-9328-x
5. Deans for Impact (2019) ‘The Science of Early Learning: How Young Children Develop Agency, Numeracy, and Literacy’, Austin, TX: Deans for Impact. https://deansforimpact.org/wp-content/uploads/2017/01/The_Science_of_Early_Learning.pdf
6. Cross, C. T., Woods , T. A. and Schweingruber, H. (2009) Mathematics Learning in Early Childhood: Paths Towards Excellence and Equity, Washington DC: National Academies Press. https://doi.org/10.17226/12519
7. Thomas, N. D., Mulligan, J. T. and Goldin, G. A. (2002) ‘Children’s Representation and Structural Development of the Counting Sequence 1–100’, Journal of Mathematical Behavior, 21, pp. 117–133. https://doi.org/10.1016/s0732-3123(02)00106-2
8. Clements, D., Baroody, A. J. and Sarama, J. (2013) ‘Background Research on Early Mathematics’, background research for the National Governor’s Association (NGA) Center Project on Early Mathematics. https://morgridge.du.edu/marsi...
9. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusman, U., Krauss, S., Neubrand, S. and Tsai, Y-M. (2010) ‘Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress’, American Educational Research Journal, 47 (1) pp. 133–180. https://doi:10.3102/0002831209345157
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