# Representations

January’s blog post unpicked the Ready to Progress document, identifying one key element as the coherency of mathematical representations. February’s blog picks up this theme, and considers it within the imposed home-schooling situation which we find ourselves in where the use of virtual representations has boomed. There are a plethora of online sites…so which ones to choose and why?

Representations can support children’s mathematical understanding in a number of ways. First and foremost, their potential lies in their capacity to reveal the structures of mathematics. Let’s take the tens frame as our example, and an ubiquitous aim of supporting children to make efficient choices about calculation strategies. The associative law of addition states that sum will remain equivalent, regardless of the way in which the addends are grouped. By using a tens frame to show this, children can be supported to make positive choices about efficient calculation strategies.

For example, this set of tens frames could represent 7+6+4 or 0.7+0.6+0.4, depending upon the children you’re working with. In either case, the representation highlights that totalling the second and third addends is a sensible starting point. By applying the associative law and using known number bond facts, an efficient strategy is employed.

Doing so avoids bridging through one/ten, and leaves a simple calculation of 7+10/0.7+1. Would this have been apparent using counters or number frames? Making choices about the most appropriate representation is a key consideration for teachers to make (NCETM, 2020) so that the representation enables children to *see *the maths, rather than *do *the maths. As the DfE/NCETM (2020) states, representations can be used to help pupils make connections between known facts and related calculations rather than provide a crutch for calculating. Over time, the aim is for the representation itself not to be required, but instead to have become a tool for thinking with (Askew, 2012).

Let’s now consider odd and even numbers. Giving children tens frames showing them, such as these, expose