I was surprised to hear that in the pre-2013 Key Stage 3 national curriculum there was only one statement about ratio, whereas in the current version there are ten statements. I was surprised because I did not remember, from my work on the development of the current curriculum, that there were so many, so I […]

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Consistent use of images supports the understanding of mathematical structures. The outstanding examples of this in mathematics have become ‘canonical’, that is part of the mathematical canon. At school level these canonical images are: number line; function graphs (thankyou Descartes); 2-dimensional combination grids (thankyou Omar Khayyam and Cayley); Venn diagrams (thankyou Cantor and Charles Dodgson). […]

Read moreI have been floating the idea of treating key mathematics ideas as handrails for teachers and students. Recently several positive comments about this perspective have floated back to me, so I am sharing the idea with you. In mathematics, teaching key ideas could be handrails. Handrails can be held, used as guides or supports, or […]

Read moreWhen Euclid wrote about ratios of lengths and areas and similarity, without algebra, theorems were dependent on spatial representations. Five diagrams, all related to each other, appear in his text in various places, so I designed a ‘Match the theorem’ task in the manner of Malcolm Swan’s tasks. There isn’t a one-to-one correspondence and you […]

Read moreFirst, 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 […]

Read moreIn 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 […]

Read moreEuclid 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 […]

Read moreNationally, 70% of Yr 7s achieved the ‘expected standard’ in KS2 Maths SATs tests. This ‘expected standard’ was decided by a panel of teachers looking at the scores from completed test papers and comparing them to the national curriculum, as well as by the testing agency. Perhaps eventually there will be more detailed information about […]

Read moreI 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. […]

Read moreI 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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