Vision in Elementary Mathematics (9780486143620) by Sawyer W. W

Author:Sawyer, W. W. [Sawyer, W. W.]
Language: eng
Format: epub
ISBN: 9780486143620
Publisher: Inscribe Digital
Published: 2012-10-10T05:00:00+00:00

ADDITION

We have already noticed that arithmetic tends to give children the impression that an ‘answer’ should not involve any plus signs. This leads to a common type of mistake. If you ask children to add 2a and 3b, they are not happy with the correct answer 2a+3b. This answer admittedly is an anticlimax! Pupils tend to make such guesses as 5a or 5b or, on occasion, 5c. However, if our friends, Alf, Betty, and Charles are invited each to think of a number, there just is no way of saying more simply ‘Twice Alf’s number added to 3 times Betty’s number’. It will not (as a rule) be 5 times Alf’s number (as the answer 5a alleges), nor 5 times Betty’s (as 5b would imply) and there is certainly no reason to suppose it will be 5 times Charles’ number, unless telepathy has been taking place. One can illustrate the meaning of 2a+3b as we did in Chapter 3, with boys standing on men’s heads. The man can be supposed to be a feet high, and the boy b feet high. It is then clear that the picture for 3a+2b (the height in feet of 3 men and 2 boys) is not the same as that for 5a (the height in feet of 5 men), nor is it the same as 5b (the height in feet of 5 boys).

The difficulty here is that a question such as ‘Add a to b’ is rather like the riddle, ‘What is the difference between an elephant and a postbox?’ People rack their brains for some witty epigram contrasting the two things. Whereas in fact no such epigram is known. The riddle merely gives you a chance to insult the person when he admits that he does not know the difference between an elephant and a postbox. In the same way, when we ask children to add a and b, they start looking for some profound mathematical idea about the sum of two numbers. No such idea exists. The question is at first purely one of notation. It might prove less misleading if we asked, ‘If a stands for a number Alf is thinking, of, and b for a number Betty is thinking of, how should we write in shorthand Alf’s number added to Betty’s?’ The pupils would then probably answer a+b. We might then ask, ‘Is there any shorter way of writing it than a+b?’ and some discussion might be necessary before the pupils reached the conclusion, ‘No’.

We begin to reach something that looks more like a serious calculation, when we ask for the sum of 3x and 4x. We have already had several ways of picturing this. We can consider x feet to be the height of a man (Chapter 3), or x to be the number of stones in a bag (Chapter 4), or the number of objects in a line, with a long thin cloud preventing us from seeing their number (Chapter 5). By any of these devices we can satisfy ourselves that 3x+4x=7x.

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