17 December 2008

The School Mathematics Project

JJ had me look at a set of old mathematics textbooks, and I found this.

4.1 Division and repeated subtraction

We can write 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7 × 9 = 63.

(a) What is 63 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7?

(b) What is 63 ÷ 7?

(c) Explain the connection between the last two questions.

(d) If you were to work out 65 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7, what would you find? How would you give your answer?

4.2 Division of a whole number by a whole number

Example 11 (Method I)

If you were asked to work out 5489 ÷ 12 by finding out how many times you could subtract 12 from 5489, you wouldn't be very pleased!

5489
-12
5477
-12
5465
-12
5453
-12
5441
-12
5429
-12
5417

This is just the start. It would certainly take a long time. However, as you will have realized, there are quicker ways of doing this division.

(Method II)

12 )5489 Consider 5400. There are more than 400 (but less than 500) twelves in 5400. Let us subtract 400 of them all at once.
4800 (400 twelves)
689 Now consider 680. There are more than 50 (but less than 60) twelves in 680. Subtract 50 of these all at once.
600 (50 twelves)
89 Finally, we know that there are 7 twelves in 89 which if we subtract them leave us with a remainder of 5.
84 (7 twelves)
5

So we have subtracted (400 + 50 + 7) twelves and have 5 left over.

5489 ÷ 12 = 457,   remainder 5.

If we were dividing in order to find the answer to a ‘fair shares’ question, we would write

5489 ÷ 12 = 457 5/12

You will probably have recognized this method. Why?

I'll stop there. What struck me as cool about this is that it takes long division, a complex procedure which most students learn by rote, and at once (a) explains why it works (b) makes it seem simple and obvious.

The example is from SMP Book C, published 1969 by Cambridge University Press. JJ has the whole series. They seem quite good, relative to what I recall from grade school. The approach is conversational with a lot of questions. Very few paragraphs are more than a few lines long. There are exercises but no “word problems”. The books are printed in black and red ink. There are no photographs or sidebars. The subject matter is richly mathematical: very little arithmetic, which must have been a separate curriculum; but in the first few books (hard to tell but they appear to be directed at students 12-15 years old) there are chapters about things like relations, directed graphs, symmetry, counting possibilities, why a slide rule works.

The SMP stands for School Mathematics Project, a British nonprofit. They're still making mathematics textbooks.

1 comment:

Blank said...

Nice explanation. I plan to use it in class. I also usually reinforce the procedure by asking my students to work these long division worksheets out.