Having a blog gives me a chance to defend myself against a number of people who took issue with a passage in Mathematics, A Very Short Introduction, where I made the tentative suggestion that an abstract approach to mathematics could sometimes be better, pedagogically speaking, than a concrete one — even at school level. This was part of a general discussion about why many people come to hate mathematics.
The example I chose was logarithms and exponentials. The traditional method of teaching them, I would suggest, is to explain what they mean and then derive their properties from this basic meaning. So, for example, to justify the rule that xa+b=xaxb one would say something like that if you have a xs followed by b xs and you multiply them all together then you are multiplying a+b xs all together.
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