If I want to teach students that you cannot treat different elements of the set of real numbers as the same element, why is it really bad , and utterly wrong, to teach them that you cannot treat different elements of the set of fruits as the same element?

Assistant Director (Secondary) Posts: 1879 Joined: 29/06/06

It is quite often helpful to use metaphors like this to help learners but some more closely match all aspects of the mathematical situation than others. I can see that the idea that you can't add two things of a different sort can be encapsulated neatly in the idea that 4 apples + 5 bananas doesn't equal 9 apples, 9 bananas or even 9 apple-bananas (which some students want to say) is a helpful aide-memoire or mnemonic.

However, the argument made against this 'fruit salad' algebra idea is that students can also fix on the idea that 5a means 5 things or objects rather than five multiplied by a value. So if apples are being talked about, 5a would mean something like 5 multiplied by the price of an apple or 5 multiplied by the weight of an apple and therefore represent a quantity or value.

I think that this idea is behind the classic misconception where, when asked to write a formula connecting feet and yards, for example, students will write 5f = 1y (because 5 feet = 1 yard) rather than 5y = 1f (because 5 times the number of yards = the number of feet).

However, the idea of 4 apples and 5 bananas being neither 9 apples or 9 bananas but 9 pieces of fruit does match the situation of adding two fractions quite closely and so may be a more effective metaphor in this situation. I think that this 'Adding two fractions' Gattegno audio clip from the ATM web-site conveys this quite nicely.

So maybe this question is all about the difference between a mnemonic and a metaphor. This, for me, resonates (again!) with the rich conversations we had way back somewhere in this or some other community some while ago about relational and instrumental understanding:

'I think that this idea is behind the classic misconception where, when asked to write a formula connecting feet and yards, for example, students will write 5f = 1y (because 5 feet = 1 yard) rather than 5y = 1f (because 5 times the number of yards = the number of feet).'

Apart from there being 3 feet in a yard, I have no idea why this is an objection to apples and banana algebra.

If f is the number of feet and y is the number of yards, I often will write 3f = y.

But this is nothing to do with apples and banana algebra. It is just a mistake. One that is very easy to make.

It is because I forgot to ask myself the basic question when doing these sorts of equations - which is the bigger number?

If told that there are 10 times as many students as professors, a lot of people write 10s = p. They are just translating the English into maths too literally, not because they are confused by apples and banana algebra.

If we are adding 5 vectors, 5a + 4b , translates very nicely into 5 of these things and 4 of those things.

08 September 2012 16:47 - Last edited by petegriffin on 08 September 2012 16:50

Assistant Director (Secondary) Posts: 1879 Joined: 29/06/06

"Apart from there being 3 feet in a yard"
Oops - apologies!
"If f is the number of feet and y is the number of yards, I often will write 3f = y.
But this is nothing to do with apples and banana algebra. It is just a mistake. One that is very easy to make."
Does the mistake not signify anything? When you write 3f = y you know that you are using a different type of notation from the 3a + 2b type.
Do students do the same?
If so, then that is fine and the apple and banana analogy serves as a useful mnemonic. But do you really need a mnemonic if you get what a's and b's represent in algebra?

08 September 2012 17:00 - Last edited by stevencarrwork on 08 September 2012 17:03

Posts: 680 Joined: 28/07/09

a's and b's represent elements of the set of real numbers, just like apples and bananas are elements of the set of fruits.

So apples and banana algebra capture the essence of what is actually happening - adding different elements of a set.

The mistake of writing 10s = p, is because English does not always translate immediately into maths. '10 times as many students as professors' means the number of professors is being multiplied by 10, not the number of students. The '10' is on the 'wrong' side of the equation in the English sentence :-)

The mistake of writing 10s = p has nothing to do with apples and banana algebra, and I have never seen anything other than mere assertion that it does, nor any explanation of how apples and banana algebra leads to that kind of mistake, nor any evidence that avoiding apples and banana algebra produces students who don't do that sort of mistake.

Because apples and banana algebra are about elements of a set, that makes it easy to see what 3a + 4b is when adding vectors. You have 3 of these objects and 4 of those objects and you combine them - something difficult to do if you are told that letters can never be physical objects.

'y = 3f' and 'f = 3y' can both represent true statements, but students need to understand that the two statements are about different kinds of thing.

If 'y = 3f' is a symbolic representation of the statement '1 yard equals 3 feet', then ... f represents a particular length which is called a foot y represents a particular length which is called a yard
... sof and yare not variables.

If 'f = 3y' is a symbolic representation of the statement ' the number of feet equals 3 times the number of yards', then ... f represents any number of feet y represents any number of yards
... so f and yare variables.

Students need to understand the difference between these two kinds of statement, and the implications of the difference.

08 September 2012 19:33 - Last edited by mary_pardoe on 08 September 2012 19:34

Posts: 177 Joined: 29/01/08

Stephen, I don't understand what you mean by 'apples and banana algebra capture the essence of what is actually happening'.

In the expression '3a + 4b' what can a and b possibly represent?

If students interpret a and b as symbols that are being used in the way y and f are used in y = 3f (1 yard = 3 feet), then it makes sense to interpret 3a + 4b as '3 apples plus 4 bananas', and this will 'work' for them if they are doing 'collecting like terms' exercises; 3a + 4b + 4a = 7a + 4b because they have 7 apples altogether and only 4 bananas. But what happens to this concept that they have in their minds when they see 3a = 6? Then they have to change their understanding of what a represents from 'an apple' to 'a number', and they need to be conscious of, and able to describe, this change.

15 September 2012 09:17 - Last edited by NCETM_Moderator on 17 September 2012 13:06

Posts: 19 Joined: 26/07/12

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i don't see the need to resort to a metaphor to explain a rule. you can just deliver a rule, say, 'a' and 'b' are different letters so cannot be added to one another. i'm sure their mathematics long moved away from thinking as the number 47 as being '47 elephants'. that's the nature of mathematics - it moves towards the abstract and there is an important lesson in that.