By Korovin S. K., Il’in A. V., Fomichev V. V.

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What is truly astonishing is that subtler linguistic aspects of mathematics can be felt by children. What follows is a story from JA3 . I was 10 or 11 years old, in the final year of primary school in London. I am a native English speaker. e. ). The teacher said, “When we are doing fractions, ‘of’ means ‘multiply”’, and I thought, “No it doesn’t. ‘Of’ can’t change its meaning just because we are doing fractions. ” And in that moment I saw mathematics as a set of conventions for which this teacher at least did not have a coherent understanding.

Down the right side I made a list of the results of the additions. It was clear that this process could continue as long as I wanted and my attention went to the vertical sequence on the right. 1 2+2 3+3+3 4+4+4+4 5+5+5+5+5 6+6+6+6+6+6 7+7+7+7+7+7+7 ... 0 1 4 9 16 25 36 49 It was clear that the numbers in the sequence increased more rapidly as you went down so I formed the sequence of first differences. Of course I obtained the odd numbers. So I thought, “Is this true in general? ” I also thought, 4 RSR tells about himself: “ As an adult I obtained a PhD in mathematics, differential topology–group actions on manifolds and fibre bundles, and now am retired if mathematicians ever retire.

For the record: EHK is English, female, has a PhD in Mathematics, teaches mathematics at a highly selective secondary school. AB is Turkish, female, has a PhD in Mathematics, teaches mathematics in a research-led university. 43 44 5 Adding One by One (by reverse counting: 100, 99, 98, 97), 96 are left. But what will happen if you remove 4 objects from the bottom of the stack? This is a phenomenon well known and well studied in the mainstream pedagogical research. James Hiebert reports from a field study [667]: Fifty-five percent of the responses to the verbal problem a+b= and to the verbal problem a−b= included modeling the sets with cubes.

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