What do students know about functions?

First, a bit of theory about learning.  In 1980, Shlomo Vinner, an Israeli mathematics educator, coined the terms ‘concept image’ and ‘concept definition’ to highlight the difference between what students actually know about a concept and the formal meaning of the concept. David Tall, an English mathematician and educator, is credited with promulgating the term […]

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The problems of problem-solving

In my last blog I set a task, which was to find out and explain what happens in this situation: Given a set of consecutive natural numbers from 1 to 2n, choose any n of them. Arrange these n numbers in ascending order. Next to them, in one-to-one correspondence, arrange the remaining numbers in descending […]

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Euclid knew about diagrams

Euclid knew that using similar diagrams over and over again, each time looking at them in a different way, was a powerful way to see mathematics as a connected whole. Euclid used similar diagrams several times throughout his Book 2 and Book 13 to provide visual reasoning contexts for relationships between segment lengths on straight […]

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Now you see it; now you don’t

I presented some visual approaches in an earlier blog called ‘Finding Nemo’ and I am now going to elaborate one of the ideas to show the value of looking at a mathematical structure in different ways.  Each of these needs learners to ‘see’ the diagram or expression in one way, and then in another way. […]

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Eliminate the Negative

I have been thinking about negative integers and wondering how anyone I taught ever managed to understand them. Many did, but when I am asked, by KS2 and KS3 teachers, ‘Is there a good task for discovering the rules of negative numbers?’, I find it impossible to answer. There seem to be a few popular […]

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Finding Nemo and Dissection

I attended both conferences over Easter, ATM and MA.  As OUP was sponsoring the MA conference it was well attended by authors and publishing people. A major theme among secondary teachers in informal discussions at both conferences is whether and how the requirements for problem-solving at GCSE will affect teaching throughout secondary. Another major theme […]

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Anyone can reason

We all have stories like this: Mick was giving us a quote for laying a patio. Having measured the rectilinear space he almost immediately told us that he would need so many 4x4s, 4x2s, and 2x2s and that would cost us so much, plus labour at so much and VAT at so much. I can’t […]

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