The Importance of C – P – A in Math Learning

Many people (teachers, parents, and students) view manipulatives as a tool only for students who are weak or need extra help.  This, in fact, couldn’t be further from the truth.  There is a significant amount of research to support the necessity of the CPA progression for learning math – that is first concrete learning, then pictorial, then the abstract.  A very important skill to possess to think mathematically is visualization.  We need a model from which to visualize.  If I tell you to picture something you’ve never seen before, it’s far more difficult than if you’ve actually seen that object or situation. This is the purpose of manipulatives – to provide the basis for our visualization.  Many people will argue that manipulatives confuse their students.  I would suggest that students don’t have a solid understanding of a concept if manipulatives are confusing them.  This may happen with a concept like division.  Often times, students are instructed immediately at the abstract phase with things like the long division algorithm and phrases like “how many times does 3 go into 8”, drop down, arrows, subtract, etc.  For students who are able to carry out the algorithm accurately, many of them don’t know what the numbers represent and whether the division is partitive or quotative.  Do you know the difference?

As we entered double digit division in Mr. Cowan’s awesome grade 4 classroom at Viscount Montgomery, you can see how this concrete learning supported the students understanding of the situation.  Some students felt comfortable operating at the pictorial stage while others had built models with manipulatives that they could use to help them communicate their thinking both orally and in writing.  Skipping these developmental steps will lead, if you are lucky, to short term success and almost guarantees lack of retention or understanding in the long run.

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