I will have to read your book sometime; I have glanced through some of it to get a feel of the book first, and it looks quite interesting and useful. These comments from mathematicians indicate at least some of the problems with gaps or holes in traditional arithmetic curricula. That book might prove useful for my own remedial arithmetic book I am working on. I am fed up with college remedial arithmetic books which are terrible because they reduce arithmetic to meaningless manipulations. And much of the problems college students have with arithmetic is that their past teachers and books did not explain much of the reasoning and meaning behind arithmetic so that arithmetic and mathematics in general has become for them a laundry list of meaningless rules to perform on meaningless symbols.
On 1/2/2011 at 11:36 am, Alexandre Borovik wrote:
> I collect stories about first encounters with > mathematics told me by my fellow professional > mathematicians, and this is one of them, about > multiplication as repeated addition of the same > quantity: > > %% > When I was 7 or 8 year old, ny uncle decided to teach > me to count up to a million. I do not remember up to > what number I could count at that time. Teaching was > done in an abbreviated form: ... after a thousand > goes two thousand, then three thousand, ..., ten > thousand, eleven thousand, twenty thousand, thirty > thousand, ..., one hundred thousand, two hundred > thousand, three hundred thousand, ..., one million. I > accepted that as a literal truth, and came to > conclusion that one thousand multiplied by 37 gives > one million. That is, it was assumed that one > thousand is followed by one thousand and one, but I > somehow missed that point, since it was not told to > me explicitly. But when I told to my big brother > about my discovery that one thousand multiplied by 37 > makes a million he looked at me as if I was crazy, I > was embarrassed, started to think and everything > somehow fell into its place (although I do not > remember how long it took). > %% > > You kind find more (including some discussion of > scaling) in my draft book, > http://www.maths.manchester.ac.uk/~avb/ST.pdf > > And here is another story, about division (and > scaling): > > %% > I could not understand the ``invert and multiply'' > rule > for dividing fractions. I could obey the rule, but > *why* was > multiplying by 4/3 the same as dividing by 3/4? > > My teachers could not explain, but I was used to > that. I couldn't > work it out for myself either, which was less usual. > > Finally I asked my father, who was an accountant. He > said: if you > divide everything into halves, you have twice as many > things. > Suddenly not just fractions but the whole of algebra > made sense > for the first time. > %% > > > Happy New Year -- Alexandre Borovik