Research School Network: The Criticality of Context in Counting James Gray, discusses the importance of different representations and the use of manipulatives in maths.

“Which is bigger, red or blue?” I ask my class of five year-olds. ​“Red!” comes the reply, from Marvin, who is steadfast in his answer. ​“Why?” I ask. ​“I like red. Red is always bigger.” He responds. Of course, the answer is neither correct, nor incorrect because the question is devoid of all context. According to Martin Hughes (1983), this is an important issue in mathematical teaching. Are we sure, that we always give adequate context, when asking children to explore numerically; particularly when children are performing operations? Or are they simply ​‘going through the motions’, with only a superficial understanding of the mathematics before them?

3 + 2 = 5, is not simply this. This is merely one representation. It can be ​“Alex found three yellow marbles and then two more red marbles, making a set of five all together” or some other context, which can be readily applied. Haylock and Cockburn (2006), promote the idea of developing relationships, when exploring number and this links also to Piaget’s research on developmental pathways (Halpenny and Petterson, 2013). Storying in Maths is an ideal way to enrich experience, give context and begin to lay the foundations of these important multi-mapping experiences. It allows children to see that number is all around them; part of everyday routines; part of everything. And as children have a deep connection to stories; this should be an important element of number work (EEF Early Years and KS1 Research Review).

Children also enjoy creating their own stories to express number and represent their thinking. Many tell these with enthusiasm and gusto!

Representations with manipulatives and pictorial symbols are then important developments of this concept. The greater the different amount of representations explored, the greater the possibility for more flexible thinking. According to Clements, Baroody and Sarama (2013), there is evidence to support ​“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.” The EEF research review for Early Years and Key Stage One also recommends the use of manipulatives, which can play a key role in supporting children’s exploration and learning, and according to Griffiths, Back and Gifford (2016) manipulatives and representations can be powerful tools to support children’s learning, helping them to ​“make sense of mathematical concepts [and] develop visual images.”

In our earlier example, Norah’s story focused on representations of birds. She may go on to draw images of those birds, representing them pictorially in a range of ways (Hughes, 1986, refers to this as pictographic: representing one-to-one correspondence and moving towards greater levels of symbolism) and she may begin to represent the birds using manipulatives as symbols, for example: three red cubes and two yellow cubes, eventually leading to coloured counters on a ten frame.

It is the flexibility of thinking and the willingness to accept and see different representations, within the same context, which is important here. Norah is developing the potential to go on to problem-solve with ever greater flexibility, because she can explore and represent with plasticity. It is this type of experience, which helps to pave the way towards later, more complex mathematical concepts such as unitising. Haylock and Cockburn (2006) take the stance that there should be a relationship between all types of operations. Addition should be paired with subtraction to sow the seeds of equivalence and fortify young children’s skill in reversing processes and forming generalisations. In Norah’s explorations above, she might begin to consider what happens in this context, when the yellow birds leave the trees and then later return again, and how this experience might provide a blue print to generalise when she problem-solves.

- Always teach through meaningful context.

- Use manipulatives carefully, as part of a wider scope of symbolic representation, ensuring they support language development and flexible thinking, rather than rooting thinking within one particular resource.

- Deepen children’s knowledge, skills and understanding, through multi-mapping/ schematic experiences, which allow children to create valuable links in their learning (i.e. objects can be a collective set; they can be partitioned into smaller groups; they can be added to and subtracted from; represented in many varying ways and they can be symbolised using cubes, counters and more).

- Select the context carefully, so that it is linked to children’s own experience and reasonable, particularly as children deal with larger quantities in their problem-solving. Larger amounts such as 12 + 6 = 18, do not lend themselves so readily to cakes and cookies (i.e. ​“Sally ate twelve cakes and then ate six more” becomes unrealistic! Whereas, ​“twelve sheep in a large field, with six more in a partitioned pen”, is not).

Without these varied and important preparatory experiences with context, manipulatives and representations, children can fall short of the foundations on which they can build a secure mathematical awareness.

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.

Improving Mathematics in the Early Years and Key Stage One Guidance Report: Education Endowment Foundation. 

Griffiths, R., Back, J. and Gifford, S., 2016: ​“Making Numbers: Using Manipulatives to Teach Arithmetic: Oxford University Press

Halpenny, A. and Petterson, J., 2013: ​“Introducing Piaget”: Routledge

Haylock, D. and Cockburn, A., 2006: ​“Understanding Mathematics in the Lower Primary Years”: Paul Chapman Publishing

Hughes, M., 1986: ​“Children and Number”: Basil Blackwell Ltd.

Willingham, D. T., 2017: Article in ​‘American Educator’: ​“Ask the Cognitive Scientist: Do Manipulatives Help Students Learn?” (Accessed January 2023)