There is a question in a recent Foundation level GCSE paper that asks:
How would you go about this question?
I can think of three ways. I could set up algebraic expressions for the length and width. I could draw a diagram and recognise that the rectangle must consist of two congruent squares. But what I really did was think “The length of a rectangle is twice as long as the width. Oh yes like 6 and 3. Area – yes 6 times 3. No, 8 times 4. ”
It’s that process I want to write about here – the process of using particular numbers to help you make sense of a general mathematical statement. In the tradition of OU mathematics education courses and books we call this ‘specialising’.
I have been in a thoughtful email conversation with one of the OU tutors over a couple of weeks, discussing exactly what we mean by specialising. We found definitions and descriptions of specialising in the literature written for researchers and teachers and used them to find opportunities for specialising and generalising in some GCSE questions. This blog shares what we found.
(At least) two purposes for specialising
The OU module ME620 introduces specialising as part of the analytic framework ‘specialise – generalise – conjecture –verify’. This framework is a way of organising your thinking about mathematical activity. The module text says “The value of specialisation lies in its role as a technique for getting insight into a problem and so for suggesting one or two sensible approaches.”
This agrees with my approach to the GCSE question. I used the numbers 6 and 3 as examples of numbers in which one is twice the others (and the examiners report suggests some 16-year-olds also chose 6 and 3 and stopped there). With these concrete numbers in mind, I knew how I would work out the area – by multiplying them. Then 8 and 4 came to mind – aha! My method was close to ‘trial and improvement’ but I do not think it was, as I am not sure that I consciously decided that the answer for 6 by 3 was 18 and compared that with 32. But I certainly used specific numbers as a way of getting insight into a problem. That is one purpose for specialising.
The same module also suggests that specialising and generalising are complementary processes – they feed into each other. We specialise in order to generalise. That is the second purpose.
What interested me about this question is that I felt I had specialised but I noticed that there is no general rectangle. The question is about finding the one rectangle (OK you could swap length and width) that meets the given description.
What I actually did was use the numbers to make sense of the first statement in the task:
The length of a rectangle is twice as long as the width of the rectangle.
The statement needed interpretation because it named unknown quantities (the length and width) and gave a relationship between them. If I examine it carefully I can see that it could be a general statement – a statement about a relationship that holds in general for some collection of cases, all of which are referred to as “a rectangle”. But I think that I – and most GCSE students – actually thought it was just about the rectangle in this question. In fact the second statement also gave a relationship between the unknowns but indirectly (because I also have to recall the relationship between area, length and width). I think it’s even harder to read “The area of the rectangle is 32cm2” as a general statement about many rectangles since it is clearly about this one. So here, students are not generalising since they are not thinking about varying across a range of cases. Instead they have to make sense of these statements because someone else wrote them in mathematical language and decided to give the information via these clues.
So what we have are two slightly different ideas about the purpose of specialising. One emphasises getting insight into a problem – specialising for making sense; the other emphasises specialising as a starting point for seeing what is the same or different across several cases – specialising for generalising.
Is there an opportunity to generalise?
The OU modules are inspired by John Mason’s work since the 1980s. One of his sayings that is still yielding food for thought for many teachers is “A lesson without the opportunity to generalise mathematically is not a mathematics lesson”. We might ask if there is an opportunity to generalise in the original question.
To get away from the single answer, I would have to explore one of the statements (and ignore the other). To explore the first statement I could draw my 6 by 3 rectangle and my 8 by 4 rectangle I could observe a common ‘look’, generalise that any such rectangle can be divided into two equal squares and then reason about what kind of numbers its area must be. Or I could notice that the area increases as the length increases and wonder if it goes up the same amount each time. If instead I start with a rectangle with area 32 cm2 , there is a lot of scope for the length and breadth, but what if I said that the length had to be a whole number multiple of the width? There is no opportunity to generalise in the middle of your GCSE but by relaxing the constraints, this question could be the basis of an interesting mathematics lesson that takes in functional relationships between variables, spatial and geometric reasoning about area and number patterns.
Having decided that the rectangle question is mathematical, let’s look at how some others have approached these two ideas about specialising, and where these ideas fit in GCSE questions.
Specialising for making sense
Kaye Stacey collaborated with John Mason and Leone Burton in the 1980s. In her 2007 paper ‘What Is Mathematical Thinking And Why Is It Important?’ she reviews their work on specialise – generalise – conjecture –verify. She offers two characterisations. First, she introduces specialising as “trying special cases, looking at examples” (page 41). This characterisation seems to allow for specialising as making sense. This type of specialising would come in useful for students attempting this GCSE question (you might like to think where your students would get stuck):
Obviously, put all thoughts out of your head about why Nadia needs more than one identical ruler, and whether the shop has limited supplies on the shelf. This is a maths question after all!
Did you specialise to make sense? I don’t know about you but when I had worked out that Nadia had £3.80 left to buy rulers; I didn’t divide 380 by 30 (or 3.8 by 0.3) but thought she can buy ten for £3, then two more for 60p. At a high mathematical level that is the same operation, but I was using specific easy numbers (ten, two) and, instead of dividing, I was multiplying up to give me a sense of the situation. Trying out numbers like this seems a very fruitful way to approach the problem; and the mental arithmetic involved is closer to what you would do in a shop than long division.
You could say that Nadia herself is specialising to make sense of a problem, since she has apparently arbitrarily decided to buy 15 pencils as a way of starting her shopping.
However in the Nadia question there is no emphasis on looking for a general method; just the particular answer. Can it really be specialising if there is no generalising? In her second characterisation Stacey emphasises this second purpose of specialising since she states:
“specialising – generalising: learning from examples by looking for the general in the particular.” (page 46).
Specialising for generalising
Polya writes about trying special cases as a strategy for problem solving in ‘How to solve it’. There is a famous quote where he explicitly connects specialising and generalising:
A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.
Quoted in D MacHale, Comic Sections (Dublin 1993)
I am not sure that this throws any insight on what specialising and generalising actually are, but it suggests that Polya sees the two as necessarily connected. If specialising means ‘looking at examples’, as Stacy first suggested, maybe the important word is ‘example’. These are not just any cases. They are chosen with a potential generalisation in mind. There has to be some notion – however vague – of what the case is an example of, or else we are not specialising.
I’ll end with two more GCSE questions and a comment. The first question seems a great example of when to specialise and it includes both purposes. You are trying to articulate a conjecture about what happens in a general case and you need to check on the details, so why not choose some numbers for the distance and time, then vary them and see what happens:
The second question suggests that specialising is not always the appropriate focus.
In this question I could decide to put a as 10 then work out the answer 53, and a as 100 and get the answer 103. I might see a pattern that can be generalised. But this question wants the students to appreciate that arithmetic calculations can be carried out on unknowns as if they were numbers. I want my students to recognise that we can add the 5 a’s and then take one and add 4, without even knowing what number a is. The examiners report suggests that students did recognise that they were meant to write an equation using the given symbols but could not simplify or symbolise the result of adding 4 and subtracting 1.
My final comment is that I am struck by how rarely students were asked to generalise in these GCSE questions. The reason specialising for making sense has such a high profile across these questions, and that specialising for generalising has a low one, is that students are not often asked to make their own general mathematical statements. Instead they are being asked to make sense of ones that the examiner has written for them So it is the examiner who generalises and the student who specialises but only in response to the examiner’s words. The student does not complete the whole mathematical cycle.