Reflections on introducing middle school to the concrete pictorial abstract pathway

    Published: 04 July 2016

    Louise Racher is a Primary Mathematics Adviser for Herts for Learning.

    I am pretty lucky. I get to “do” maths all day every day.  Gone are the days when I used to teach PE, history, geography, English … and so on, as well as dealing with hormonal Year 6 children … to have the permission to think about one thing only, mathematics, is an indulgence.

    I am still learning, and changing my opinion about which approaches will make the biggest difference, that will continue to change I am sure. What I do know is that the experiences I had in school were not good enough.  Rote learning, procedures with no reason, struggling to keep up, sometimes copying my neighbour to get me through the lesson, because maybe I will be OK on my own tomorrow.  I am passionate about the use of manipulatives across the Primary phase, and into the secondary phase.  So, to have the opportunity to talk to teachers in a middle school and really sell my passion was a great opportunity. 

    I entered the school with trepidation. My task: teach two one hour sessions modelling the CPA approach (Concrete-pictorial-abstract).  My first session would be with about fifty Year 5 pupils, and about eleven adults observing, the second session with a similar number of pupils, this time year 6, with some year 7 pupils thrown in for good measure. “Year 7 pupils!” I hear you cry.  Yes, indeed by Primary BEd was about to be stretched beyond its capacity!

    I was nervous, of course; any teacher who has had an observation knows that suddenly you question everything you thought you knew (even your own name at times). Added to the mix was the fact the pupils were not used to being taught together (they were cross class groupings) and the fact I had absolutely no idea of their level of achievement, personalities or names, then it was understandable that my nerves were a little on edge.

    My first session with the Year 5 pupils was really enjoyable. The initial question presented:


    The answer was not as important to me as the proof was. The very statement, prove it, initially flummoxed them, but after a couple of suggestions with me demonstrating this using an area model, then arranging it in a linear fashion, and of course modelling it with my beloved Cuisenaire they were able to try and replicate this.

    The explanations supporting their drawings and use of the number rods were really powerful, the teachers took notes. They saw the pupils who did really get it, those who could get the answer but not explain how they arrived there, and those who were demonstrating they had some misconceptions.  How different this is to setting an exercise of calculations – the ability to watch the children do maths exposes the thinking much more clearly, allowing teachers to observe and diagnose misconceptions or ability to reason.

    The pupils used Cuisenaire rods, which they had not been exposed to before without a care in the world, they represented ideas about multiplying fractions by whole numbers in really interesting ways From my view, as I talked to different groups, I could see the progress, all groups of pupils were benefitting, from those who needed that misconception correcting, to others who were deepening their understanding by explaining so well.


    I paused for breath before my next session with Year 6 (and a few year 7s). I was keen for the pupils to use the Cuisenaire to find unknown, I started gently asking them to record what they noticed about the relationships of some arrangements of the rods, they picked this up quickly.

    Having introduced the top rod to have a value they interpreted their statements easily, showing clear understanding of inverse operations and had a good range of language associated with doubling, halving, fractions and percentages. They were keen to impress each other.

    Next I presented the pupils with this, they built it using the rods, and I deliberately chose rods for the numerical values that were not proportional, as I wanted the pupils to consider the arrangement of rods here to help them compare and find the unknown. This might be controversial, and generally as a rule I would always use the proportions to indicate the values.


    Two girls interested me, as they were struggling to see the arrangement, when I rearranged them the light bulb went on.



    “Oh.” They breathed in unison. “I can ignore these rods; these are the important ones, I can work out the value now.”  I thought back to my own experience in school, I was conditioned to remember rules with little about understanding about how and why, cancelling like values was a procedural process in algebra.  For someone who was naturally inquisitive the answer to my question was often a polite version of “because I said so.” It was the connections which I saw being forged about rules which I was taught which I found interesting about this session.

    We progressed to looking at balancing equations with one unknown; we used the language of algebra and progressed to arranging the rods to find unknown values.

    The Year 7 group did appear more resistant, and the teacher did have questions, which I wanted to clarify, but didn’t have an opportunity to answer during the teaching of the session. I wondered if this was the beginning of the formulas being taught making it harder for the pupils to reason as to why we might follow certain rules.

    And so, after a brief interlude with a fire drill, where we all exited and returned the session was over. It was a whirlwind of talk and justification, playing and questions.  I certainly had some, as usual, reflecting on what I might have done differently, whether the adults observing could see the value I placed on the use of the resource to unpick those rules.

    I hope so.

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