The same but different

Imagine this: On the news this morning you hear a segment about the weather – the maximum temperature tonight is predicted to be -2℃ and this is 10℃ colder than last night!
Question: What was the maximum temperature last night?

Now imagine this: You step into an elevator in a tall building and travel down 10 floors to the car park. When the doors open, you see a sign on the wall saying ‘Level -2’.
Question: If the ground floor was Level 0 , on which level of the building did you enter the elevator?

Next, imagine this: You go to the cinema and spend £10 on a ticket. Later, you check your bank balance and see that it is -£2.
Question: What was your balance before you purchased the ticket?

Finally, imagine this: You sit down in your mathematics class and see the equation 𝑥-10=-2
Question: What is the value of 𝑥?

What do these scenarios have in common?

Perhaps you noticed that they all have the same numerical answer? Maybe you recognised that they are all asking the same numerical question: Which number is 10 more than -2?

We sometimes describe such problems as ‘isomorphic’ – they have the same underlying structure but different surface details. Those surface details often add context to otherwise abstract mathematical problems, and it is common for us as teachers and designers to try to include these contextual or ‘word problems’ in teaching materials. But for what purpose? I suspect the immediate answer that comes to mind is one based on ideas of numeracy or mathematical literacy – ‘because learners need to be able to apply their knowledge to solve problems in the world around them’. Another common answer might be simply ‘because these sorts of questions will come up in the exam’. A perhaps less common answer could be ‘because they help learners to understand a concept’.

But how many students would recognise that the problems posed earlier were in fact isomorphic? Or do they tackle each of these as isolated problems, never noticing the connection between them? Does it matter either way?

In the two problems below (adapted from those in Greer and Harel, 1998, p. 20), problem 2 is provided by the teacher to help a student who is struggling to solve problem 1.

1. In the diagram, 𝑎₁=𝑎₂ and 𝑏₁=𝑏₂. Find the value of 𝑎₂+𝑏₁
   
2. You and your sister had £180 altogether. Your sister gave me half of what she had and you gave me half of what you had. How much money do you have left between you?

Here the teacher is supposedly trying to support the student by providing an isomorphic problem that they might find easier to make sense of. However, Greer and Harel reported that in this case, the learner saw no connection between the two, that is until after they had constructed a solution for problem 1 (which rendered the teachers’ attempt at support somewhat unsuccessful!). And this finding is not unique; it is commonly reported (e.g. Barniol and Zavala, 2010 ; Lin and Singh, 2011) that learners simply do not spontaneously recognise isomorphic problems – that which is an obvious analogy to the teacher (an expert) often remains invisible to the learner (a novice).

I am reminded of a time when my GCSE mathematics class were exploring linear graphs in the context of Celsius and Fahrenheit temperature scales. After some time, a student loudly protested, ‘You haven’t given us enough information – we don’t know this value!’ (referring to the freezing point of water in degrees Celsius). Surprised, I prompted them to think of their science classes, to which they responded, ‘Well in science it’s 0℃. But this isn’t science, it’s maths!’

This interaction has stayed with me for many years. I had made assumptions about the ways in which students implicitly connected their knowledge, that they would automatically recognise where and how knowledge could be transferred from one setting or context to another. But they didn’t, even with something as elementary as the freezing point of water.

So, recognising sameness in context matters – that there are universal facts and knowledge that can be applied to a problem, be it in mathematics, science, geography, or in what we sometimes call ‘the real world’ (that mysterious place outside the classroom). But what about sameness in (mathematical) structure – does it matter if students don’t recognise two problems are isomorphic if they can solve each of them correctly anyway?

Here I ask you to consider the following three problems, this time concerned with combining vectors. Try to reflect on the first mental image that comes to mind in each case; you might like to note or sketch something for one before moving to the next.

These problems are also isomorphic and did you notice, the context likely influenced your mental image, your ‘sense’ of the problem and the solution path you might take?

Let’s consider this further. Problem A is the most abstract; it requires us to draw upon our knowledge of mathematical conventions and terminology and procedures associated with adding vectors, and to undertake significant mental manipulation of the two mathematical objects (the individual vector arrows) to form a sum. 

Problem B focusses on displacement and brings with it the benefits of intuition around sequential movement. Here it’s much easier to sketch the vector combination correctly and to recognise the effect of total displacement.

Problem C focusses on forces; it requires us to have a sense of what a force is and knowledge of conventions associated with free-body diagrams (most likely from physics classes). In this context, our intuition sometimes leads to initial misconceptions – for example, that the resultant vector will join the individual vectors end-to-end (completing a triangle).

Now, knowing that these problems are isomorphic, did/does any one help you to make (more) sense of another?

I am hoping that the answer is yes! As humans we are excellent at pattern spotting and building analogies and noticing, or imagining, similarities – it’s how we build connections and retrieve memories. Our ability to work flexibly with a mathematical concept is related to exactly this: our capacity to build and tap into that rich network of connected ideas, experiences and problems. But this doesn’t happen spontaneously – it is cultivated over time, an accumulation of experiences and prompts to compare and contrast situations. So, I wonder, how often do we give students time and guidance not to practise solving different, isolated problems but to understand a concept in different ways or contexts, and moreover, how often do we deliberately give students isomorphic problems presented in those different ways?

In the same way that we might support an individual student who is struggling to solve 𝑥-10=-2 by offering them a relatable, analogous scenario or way of thinking about the problem, why not incorporate explicit opportunities for all students to see, compare and even create their own isomorphic problems? Perhaps then they will begin to construct stronger, richer networks of knowledge, or ‘mental libraries’ of contexts and problem types that they can visit when confronted by a new problem or concept, seeking, not an answer, but a sense of the problem and possible paths to a solution.

Let’s finish with a challenge – how many problems can you create that are isomorphic to this?

Solve the equation 10𝑦=2

And how about problems isomorphic to this?

Solve the equation -10𝑦=2

• Barniol, P., & Zavala, G. (2010). Vector addition: Effect of the context and position of the vectors. In C. Singh, M. Sabella & S. Rebello (Eds.), 2010 Physics Education Research Conference (pp. 73–76). American Institute of Physics.
• Greer, B., & Harel, G. (1998). The role of isomorphisms in mathematical cognition. The Journal of Mathematical Behavior, 17(1), 5–24.
• Lin, S. Y., & Singh, C. (2011). Using isomorphic problems to learn introductory physics. Physical Review Special Topics—Physics Education Research, 7(2), 020104.

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