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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I want to offer an example of a mathematical exploration which is likely to enrich learners’ appreciation of an apparently unrelated aspect of graphs of functions. Imagine the graph of . Is there a point on the curve at which the tangent passes through the origin? One thing that emerged when I used this with […]

Read moreSometimes we are unsure how well a topic has been received or understood by the class. We don’t want to wait until they can’t do their homework, which you don’t find out about until they come into your next lesson, to do something about this. Collecting a quick snapshot from every student about their level […]

Read moreHaving 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 […]

Read moreConsider 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 […]

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