Why dodecahedrons hate CPA

    Published: 31 August 2016

    Rachel Rayner is a Primary Mathematics Adviser for Herts for Learning

    For a blog about the CPA approach click here.

    Yes, teachers do label their fixed ability groups by shapes…still. Yes, pupils do end up in the circles group from the age of five and in some cases in the teacher’s head, younger.  And yes, it damages.  We are all by now familiar with the work of Carol Dweck and the idea of fixed and growth mindsets.  But in maths at least this fixed ability grouping or setting persists in Primary, despite the evidence that it can be detrimental to those pupils designated ‘circles’ or ‘triangles’.  

    Much of this has come from the culture of mathematics that is insidious in the UK. This is how we were taught, and therefore this is the way we teach.  We tend to think of mathematics as a performance, the ability to calculate quickly and accurately has been traditionally prized above all else in our classrooms.  Even when we ourselves as educators think of mathematics success more holistically, messages still pervade from home and the media that appear to reinforce the notion that people are born either good or bad at mathematics.  And let’s face it those planning forms with the 3 boxes for differentiation have caused us to fill in boxes and see it as a place to write tiered activities for each ability rather than seeing differentiation as a way of supporting and deepening.  We have a blog on the subject of differentiation here.

    I have been struck this year whenever I work with teachers and schools on improving subject knowledge, particularly with the aim of developing teaching towards mastery, how thrilled they are when a long accepted rule of mathematics suddenly becomes clearer. BODMAS for example, when we work with teachers on the idea of distribution instead, using Cuisenaire, becomes abundantly accessible.  Liping Ma’s excellent research about the differences between teacher subject knowledge in the US and China (Knowing and Teaching Elementary Mathematics), highlights the fact that because Chinese teachers were themselves taught conceptually, they have a deep understanding of the subject themselves and teach with conceptual awareness.   Against this she found that the US teachers, who, by and large, were not taught conceptually, were more likely to struggle to teach concepts fully.  I really admire the teachers we are working with who are endeavouring to teach and learn themselves alongside the pupils, what feels like ‘new’ kind of mathematics.  Several of these schools have transformed the way mathematics is taught and learnt from ‘Do as I do on the board’,  to guided carefully planned exploration by different means of the same concept by pupils with great success including through de-setting their classes and implementing the CPA approach.  In particular they cite that the groups formerly known as circles and triangles have benefited from this the most, mathematics is available and understandable and they are successful now.  It is no longer elite and the abstract is illustrated.

    This has been an exciting mind shift for many teachers, who are evangelical about CPA. Coming soon on the HfL website are two case studies which document the journey of two schools towards a mastery approach.  These will be found here.


    But…oh what to do with the group formerly known as dodecahedrons.  So far in their maths lessons (especially by the time they are in KS2) they have been perceived as the crème de la crème, confident in their prowess. They get pages of examples correct and are always first to answer and are competitive. Now they are being asked to work with others who might slow them down and horror of horrors, they have to explain and represent their mathematical thinking.  They have to explore concepts using manipulatives and now just one strategy isn’t enough, they have to use several.  This is all rather time consuming and trying if you are a dodecahedron.   And to top it all, now mathematical behaviours other than speed and good memory are being valued.  Yes I know it’s good for them, I agree, very often this is their challenge.  But perhaps we should begin to acknowledge how scary this is for some of them and why.  These are my own observations of pupils working and anecdotal accounts from leaders who have introduced CPA and mixed ability flexible groupings over the last academic year.  For our Year 1 pupils engaged, there seems very little evidence of this.  Whether this is an age related problem or whether this will diminish as pupils get used to working in this way remains to be seen.

    Let’s face it that great working memory and the speed of calculation is a great skill to have.   We should value it and not diminish it.  But early on in children’s experience we really must value a range of effective mathematical behaviours and make sure they do to.  The use of manipulatives and representation should be the norm from pupils’ earliest experiences.  Until then we have to look after our dodecahedrons and ensure that they don’t switch off from a subject they have previously felt successful at, plan challenge that appeals to them, some opportunities to record in ways that appeal to them and the occasional chance to work together and with you.

    Finally, whilst I haven’t any experience of fixed ability groupings working well, please can we at least desist from naming them after shapes or anything that obviously points to the ability we place on pupils if we are using them? They know and would you want to be labelled a circle?


    Ma, Liping (2010). Knowing and Teaching Elementary Mathematics. Abingdon, Routledge.

    DWECK, C. S. (2006). Mindset: the new psychology of success. New York, Random House.

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