What is ‘variation’ in early years mathematics?
As we near the end of our year-long project into what effects five- and six-year-olds’ ability to count-on when adding, here we reflect upon our journey. What led us to this topic was the recent prevalence of variation theory (Marton & Tsui, 2005) in narratives of mathematics education and our concern about the relevance of many of these discussions to key stage 1 teaching (four-to-seven-year-olds).
Our notes reveal that in the summer of 2018 we had been mulling over the relevance of variation theory to younger learners, particularly in terms of how we were seeing it being construed in younger classrooms – that is, as exercises, worksheets or textbook examples. We discussed several ideas for how to refocus the central tenet of variation theory as the drawing of learners’ attention to underlying mathematical relationships and thus reflection on teacher design and sequencing of tasks. All of our ideas focussed on the choice and use of manipulatives, which we know to advance learning opportunities by both modelling and representing mathematical ideas and by helping children make sense of problems they are solving (Griffiths, Back, & Gifford, 2016).
Two initial questions we posed were the following.
- Why might we be using this particular representation in mathematics?
- Do the children know what it is they are supposed to notice mathematically when using this resource?
We planned to work in both year 1 and reception classrooms, exploring variation in terms of important aspects of young children’s building understandings of number. Our funding was sufficient for us to focus on small groups of learners in year 1, the first year of the English national curriculum, and we chose those children who were assessed as being on the cusp of moving from ‘counting-all’ to ‘counting-on’. This move was pinpointed in research as underlying all counting systems and thus critical in children’s mathematical development (Nunes & Bryant, 2009) and, moreover, as something that children under the age of seven find difficult (Bobis et al, 2005; Young-Loveridge, 2011; Thompson, 2013). Our experience bore both out. Furthermore, in our experience it is in year 1 where there exists particular pressure to imitate what is expected of older students, and we were hearing of many year 1 teachers being required to engage in mathematics tasks with their five- and six-year-olds that were in fact appropriate for key stage 2 students and above.
The manipulatives we chose to use to explore variation theory in relation to counting-on are those currently in common use in English classrooms.
Each of these would draw learners’ attention to underlying mathematical relationships in subtly different ways. We were interested in investigating if any of these manipulatives were of particular use in teaching children to count-on.
‘Our research identified a number of necessary sub-skills that children need to be able to accomplish in order to count-on efficiently.’
Our research resulted in us identifying a number of necessary sub-skills that children need to be able to accomplish in order to count-on efficiently. These were related to children’s developing understanding of cardinality, and matched what we read in the literature. What we had not appreciated was that there might be a previously unconsidered and additional facet of cardinality to be taken into account when working on counting-on. This additional aspect was recognising a number written as a label on a tub containing a number of items, as abbreviating the number of items contained within the set, in the same way that students recognise the number in a set through subitising, iconic representation or composition.
Reflecting on our journey, two things strike us. Firstly, it was important to keep the focus of the project tight in order to make it manageable (although we both gave far more hours to this than we planned). Secondly, we both came to the project as experienced mathematics educators and researchers. What we observed surprised us and supports the claim that all research, however small-scale, contributes to every individual’s development as well as something to the overall developing mathematics education picture.
In the event, our initial questions changed little and guided our study. We believe that with younger learners, variation is not about graded examples or worksheet questions, but the much broader drawing of learners’ attention to underlying mathematical relationships.
To summarise, our findings support international research findings that counting-on is a complex process for children of this age, and that careful selection of manipulatives can support this understanding, alongside awareness of the aforementioned sub-skills that are important to work on in building such understanding.
Research that further explores the following questions would be useful.
- Is counting-on something we should not be expecting a year 1 child to master?
- How might the choice and use of manipulatives aid its development?
Helen J Williams and Ruth Trundley’s project is one of three winners of the British Curriculum Forum’s Curriculum Investigation Grant for 2018/2019. The grant is awarded biennially to support research led by schools and colleges with a focus on curriculum inquiry and investigation – click here for more information.
Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, R., Young-Loveridge, J., & Gould, P. (2005). Supporting teachers in the development of young children’s mathematical thinking: Three large scale cases. Mathematics Education Research Journal, 16(3), 27–57.
Griffiths, R., Back, J., & Gifford, S. (2016). Making Numbers: Using manipulatives to teach arithmetic. Oxford: Oxford University Press
Marton, F. & Tsui, A. (2005). Classroom Discourse and the Space for Learning, Mahwah, NJ: Erlbaum
Nunes, T. & Bryant, P. (2009). Key Understandings in Mathematics Learning. Paper 2: Understanding whole numbers. London: Nuffield Foundation
Thompson, I. (2013). Counting: Part 1 – From ‘counting all’ to ‘counting on’. Unpublished paper.
Young-Loveridge, J. (2011). Assessing the mathematical thinking of young children in New Zealand: The initial school years. Early Child Development and Care, 181(2), 267–276.