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# 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, exposes their structure:

The representation makes it clear that an odd number is one more than an even number and that it’s not divisible by two. Exposing these key features deepens children’s understanding, and avoids them having to remember that 3,5,7 etc. are odd. Powerful stuff.

Peer talk is perhaps one casualty of home-schooling which is a real loss. High quality dialogic mathematical talk which can occur in classrooms has great potential for deepening children’s understanding (Williams, 2008; Askew, 2012). However, there is scope for promoting this with the range of dynamic, engaging online representations which are proving central to home learning. Encouraging your children to talk to their grown-up about how they’re using the chosen website is a great starting point.

For example:

Show them these two tens frames and ask them to discuss ‘what’s the same and what’s different’ about them.

Then, ask the children to explore the interactive representation and make one change at a time which they discuss and explain. Again, what the tens frames represent can change to fit the children. For example each counter could represent 7, with the whole frame representing 70 allowing the distributive law of multiplication to be explored:

Giving children something to talk about, rather than abstract ideas, is essential for getting their thinking heard. Manipulating representations enables children to follow their own lines of enquiry which is great mathematical thinking (DfE, 2014) and often an illuminating window into their understanding and thinking.

Some non-screen time is good too - click here to see a range of games using ten frames which are great as mathematical talk tends to emerge organically when playing. Games also facilitate joint activity with the representation, which Askew (2012) deems essential to children filling them with meaning.

With the very real challenges of teaching via Zoom, the temptation is to present children with a range of jazzy models and images, and it’s so easy to do at the click of a button. For maximum impact though, take a moment to consider, what mathematics is this showing? How is it developing children’s understanding? And you may find that the one you’ve been looking for has been there all along!

References

Askew, M (2012) Transforming Primary Mathematics, Oxon, Routledge

Boaler, J, Lamar, T, Langer-Osuna, J, Suurtamm, C (2020) Supporting high-quality math teaching in uncertain times Available at: https://app.knowledgehook.com/app/ReturnToLearn/SupportingMathTeaching?utm_source=Youcubed%20Updates&utm_campaign=07d1374f78-EMAIL_CAMPAIGN_2020_08_10_06_13&utm_medium=email&utm_term=0_230e567c40-07d1374f78-133944497 (Accessed: 5th February 2021).

Department for Education (2014) The national curriculum in England: complete framework for key stages 1 to 4. Available at: https://www.gov.uk/government/publications/national-curriculum-in-england-framework-for-key-stages-1-to-4 (Accessed: 5th February 2021).

DfE/NCETM (2020) Mathematics guidance: key stages 1 and 2 Non-statutory guidance for the national curriculum in England, Available at: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/897806/Maths_guidance_KS_1_and_2.pdf (Accessed: 5th February 2021).

NCETM (2020), Representations in our primary video lessons, available at:https://www.ncetm.org.uk/features/representations-in-our-primary-video-lessons/ (accessed 2/2/21)

Williams, P (2008) Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools, available at: https://dera.ioe.ac.uk/8365/7/Williams%20Mathematics_Redacted.pdf (accessed 4th Feb. 2021)