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This report, on one of the three research projects to receive the British Curriculum Forum’s Curriculum Investigation Grant for 2018–2019, investigates how variation theory might be applied to the use of manipulatives (that is, any objects that can be moved
and handled by learners) to support understanding of early number.

Variation theory was developed by Ference Marton, has now become dominant
in current mathematics discussions. It draws attention to underlying relationships in mathematics by focussing on the careful design and sequencing of mathematical tasks, including the use of multiple representations of a mathematical concept in order to draw out what it is and what it is not.  

The authors, Ruth Trundley and Helen J. Williams, aimed to increase awareness of current narratives of variation theory and ascertain how applicable these narratives might be to younger learners of mathematics. Little work has yet been done to explore the application of variation theory to teaching and learning mathematics in the earlier years.

The project involved 12 children aged 5–6, and focussed on the move from ‘counting-all’ to ‘counting-on’ (the ability to establish the new quantity in a group in which the amount
has been increased, without needing to recount the original group), which is known to be critical for numerical understanding yet difficult to reliably establish. The project explored which manipulatives might be effective in highlighting the essential features of this mathematical idea.

The report identifies two key sub-skills and understandings that appear to make a significant contribution to children’s ability to count-on:

• understanding cardinality and abbreviating the augend (the original quantity)
• keeping track of both location in the number system and of the addend (the number being counted-on) using objects.

This research makes a substantial contribution to the existing literature on counting-on , and has important implications for the teaching of counting-on to young children.