Having a vision of mathematics, of what is possible in the way of appreciating and comprehending mathematics (distinct from simply gaining facility with arithmetic), is surely an important part of teaching people to think mathematically. One important aspect of mathematical vision is being aware of ubiquitous themes, and one theme that has been impressing itself […]

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Consider x + y = 5 How do you know what this will look like plotted on equal aspect axes? How do your students know? Some possible ways include: using a graphical calculator display rearranging the equation and plotting three points (including one to ‘make sure’) using one intercept and the gradient of –1 recognising […]

Read moreOne of my favourite lessons on graphs is when I use the floor of my classroom as the x–y plane. I fix an origin in the middle of the floor (in order to make sure negative coordinates are included). I use masking tape to represent the axes along the floor, with masking tape dashes marked […]

Read moreSuppose you want students to gain facility in factorising the difference of two squares, and make use of this when expanding a product of the sum and difference of two quantities. A direct approach might be to invite learners to expand products of sums and differences of quantities, varying the complexity and format of those […]

Read moreSo much of the secondary and A-level mathematics curricula can be approached as if procedures are the way into mathematics, and all that is required to do well is to learn the procedures and spot where and when to apply them. By contrast, people who use mathematics work the opposite way round – curiosity about […]

Read moreThe fast-approaching Christmas break is often a period of revision for students – some may be preparing for mock exams and others taking time to recap topics covered in their first term. At this point, I find myself asking what it is all for, and how revision might be transformed into learning for the longer […]

Read moreEven now I remember how odd it felt when, at A-level, we had to find partial fractions. Instead of solving for x, we had to solve for the numerators of the partial fractions. I don’t remember it ever being explained that we were finding coefficients, the parameters of an equivalent way to write the given […]

Read moreIsn’t it amazing how quickly and easily adolescents can learn the words of a song, and yet they struggle to remember mathematical formulae? Memory in mathematics is quite peculiar. In common with learning the words of a song, once you get started, it can be surprising how much comes back, but, as with songs, it […]

Read moreEveryone is immersed in algebra because algebra is the expression and manipulation of generality. Whenever a generality is present, algebra is present too; wherever a generality is manipulated or varied, algebra is taking place. Whenever you purchase something, you are immersed in algebra. As a customer, you are interested in the final price, whereas the […]

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 […]

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