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Problem Solving: does a context help?
By Clare Hudson (M719) on Tuesday, September 26, 2000 - 11:31 am:

I am a 4th year student teacher and doing a dissertation on problem solving. Does anyone know any recent books or web sites that will give more information on whether children find it easier to solve problems with a context rather than abstract ones? If not, do any teachers have any examples of using problems with contexts with the children?

By Michael Doré (P904) on Thursday, September 28, 2000 - 10:58 pm:

I came across a Keith Delvin book in a library a few months ago (I can't remember the title I'm afraid) which gave a good example of how a problem can seem easier if set in relevant context.

The basis of it was as follows. You have 4 cards, each with a number on one side and a letter on the other. You have to check that the following rule is obeyed in all 4 cases:

"if there is a vowel on one side there is a odd number on the other side".

The cards are on the table (obviously one side is face up and the other face down). They read:

E 3 H 6

Which cards do you need to turn over to check whether the rule is obeyed?

Answer: the first and fourth card. The first one: you do need to turn it over to check there is an odd number on the other side. The second one: it doesn't matter; if there is a vowel on the other side then there is indeed an odd number on the first side, and if not it doesn't matter. Third one: no need to check that because H is not a vowel. The fourth one: you do need to check this. If the other side has a vowel then that is illegal (as the first side should have an odd number).

Now that is actually pretty tricky to figure out. Lots of people would say you should turn over the first two and forget about the fourth (thinking that as it is even it doesn't apply). The reason most people say you should turn over the second one is that an odd number is involved, and the criterion does mention odd numbers. Perhaps people think that the converse of the criterion also has to be fulfilled but of course it doesn't. Anyway, this puzzle confuses most people apparently.

Now consider a second (and apparently different) problem. Suppose you are an inspector going round a pub, checking everyone who’s drinking is old enough. You come to a table where four people: Anna, Brian, Caroline and Dennis are sitting. You have to check that the drinking laws are not being breached.

Anna is clearly under-age, but you can't quite see what she's drinking. You can see Brian is not drinking alcohol, but you cannot tell whether he is old enough yet to legally drink alcohol. Caroline is above the required age, but it is unclear what she's drinking. Finally Dennis is drinking alcohol but you can't tell whether he is the right side of the age boundary.

Which people do you need to investigate further? Clearly Anna and Dennis. You need to make sure Anna isn't drinking alcohol (as she's underage). It doesn't matter whether Brian is old enough as he isn’t drinking alcohol. And it doesn't matter what Caroline is drinking as she is above the age limit. But you do need to investigate whether Dennis is old enough to be drinking the alcohol he is drinking.

So surely two very different problems, the second being much easier than the first.

But these problems are the same! A vowel is analogous to 'drinking alcohol'. An odd number is analogous to being 'above the age limit'. And the two problems are completely identical subsequently. Yet because the first one is more abstract, it seems harder, and apparently surveys have shown that the general public (and particularly children) are infinitely more successful with the second one. (It would be quite interesting to conduct one of these surveys.) This is good evidence that problems are much easier if put in context.

By Anonymous on Saturday, September 30, 2000 - 07:35 pm:

What a fantastic example to explain what you are saying!!
Could provide some interesting discussion with my year 12's!!

By Michael Doré (P904) on Monday, October 2, 2000 - 03:12 pm:

Glad to have helped, but full credit for this example must go to Keith Delvin!!