## February 23, 2005

### Reinvention

Back in second grade, I was dissatisfied with the algorithm we were being taught for doing subtraction. So I “invented” my own

363
– 185

5 is bigger than 3, so we subtract them in the opposite order (5–3=2) and take the tens-complement (8) of the result. As usual, we borrow from the 6 (which becomes a 5) and we repeat: 8–5=3 and take the tens-complement (7). Finally 2–1=1, so the answer is 178.

While only slightly different from the conventional algorithm, I felt this one to be an improvement because I never had to know how to subtract from numbers larger than 10 (e.g. who cares what 13–5 is?).

I haven’t thought much about this little juvenile act of rebellion until a couple of months ago, when I was going over my daughter’s 3rd grade math homework with her. She was doing similar subtraction problems. But, in keeping with the times, she was charged with explaining her methods for arriving at the answer.

Imagine my surprise when she explained her method to me. It was exactly the same “unconventional” algorithm that I had used when I was her age. It was not what the teacher had taught; she had figured it out on her own.

[Her method was the same, but her accuracy was not the greatest. So I taught her the other trick that I learned in that era: check your work by doing arithmetic modulo 9: 178=1+7+8= 7 mod 9, 185=1+8+5=5 mod 9. So 178+185=5+7=12=1+2 = 3 mod 9, which agrees with 363=3+6+3=3 mod 9.]

Now, I don’t know what this has to do with Larry Summers’ remarks on the dearth of women in the Hard Sciences (at least in this country). My personal experience echoes that of the AIP Study. An alarmingly large majority of the women who arrived at Harvard the year I did, intending to major in Physics, had decided by sophomore year to do something else. As a consequence, it was unsurprising that, by the time I started graduate school, there was only one woman in an entering class of 28. Sean Carroll takes on the thankless task of confronting Summers hypotheses with the data. I’m afraid I can’t muster the energy.

I’m much too busy trying to nurture that spark of creativity in my daughter, hoping that, a decade from now, she doesn’t face the stark choice that my classmates at Harvard/Radcliffe faced a generation ago.

Posted by distler at February 23, 2005 4:28 PM

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### Re: Reinvention

I am surprised you don’t mention another Harvard man’s approach to a similar problem.

Posted by: Robert on February 24, 2005 1:19 AM | Permalink | Reply to this

### Re: Reinvention

Like father, like daughter :)

Posted by: Srijith on February 24, 2005 3:05 AM | Permalink | PGP Sig | Reply to this

### Re: Reinvention

Fan that spark into a fire! Especially the abiity and creativity to explore alternative methods.

And thanks for the “unconventional” method. I would be interested to hear (since they now have to explain their methods) what her teacher thinks of her methods. I was rather discouraged early in my education by “educators” who accept any deviations from the text book.

Posted by: Eric on February 24, 2005 8:43 PM | Permalink | Reply to this

### Teachers

As I’ve written before, I’m generally pretty pleased with her teachers’ efforts to challenge the students to explore mathematics.

One of the things both pleased and surprised me was when I explained to her the “arithmetic modulo 9” trick for checking her answers. We did a few examples, and she was quickly convinced that it worked. But then she turned to me and said, “OK … but why does it work?”

I actually had to go through the proof for her, before she was satisfied. (Or was it bored? Hard to tell with a 9 year old.)

Posted by: Jacques Distler on February 25, 2005 5:09 PM | Permalink | PGP Sig | Reply to this

### Re: Reinvention

Hi Jacques,
When I was about 12 or 13 I remember I read a book called the “Trachtenberg Speed System of Applied Mathematics”. This is a number of clever and powerful tricks to multiply and divide huge numbers in your head, as well as many other things. Remarkably, it was developed by Jakow Trachtenberg when he was a prisoner in a concentration camp.
Focusing his mind this way must have helped him survive.
http://en.wikipedia.org/wiki/Jakow_Trachtenberg
I can’t remember how to do it now but the book is available on Amazon.

Posted by: Steve M on February 25, 2005 1:25 PM | Permalink | Reply to this