I’ve had an interesting time in the classroom recently, discovering more about the extent to which students can struggle with mathematical models. I was dismayed to discover how many of my students got the answer wrong to:

Simplify t^{2} + t^{2. }

Most of them had written t^{2} + t^{2} = t^{4 } instead of t^{2} + t^{2} = 2t^{2}

This was in a synoptic test, a few weeks after we had covered this topic in class. At the time, in the lessons, they were whizzing through this stuff. I had built it up slowly. pen + pen = two pens; 4 + 4 = 2×4; p + p = 2p and so, obviously, t^{2} + t^{2} = 2t^{2. }

At the time, they were all seemed secure with this. Any term + the same term = two x that term, including more complicated terms:

pt^{2}s^{3} + pt^{2}s^{3} = 2pt^{2}s^{3}

If’ we had stopped there, they would probably have been ok. But we had also gone further, exploring indices and patterns such as this:

t x t = t^{2}

t x t x t = t^{3}

t x t x t x t = t^{4}

(t x t) x (t x t) = t^{4}

t^{2} x t^{2} = t^{4}

Even though, at the time, we had practised various variations of this and students were getting questions right, evidently this was not making a shift in their mental models. That final line introduces the link between t^{2} x t^{2} and t^{4 }. In their world, largely governed by surface learning, that connection is strong enough to override their thinking around the operations + or x. It is common for them to want to remember answers rather than methods so t^{2} + t^{2} = t^{4 } beat t^{2} + t^{2} = 2t^{2}.

Perhaps it was because it was the most recent thing we’d done in algebra; perhaps it was because it feels more sophisticated to write t^{4 }, but, regardless of the reason, I learned that their mental model for the whole business was weaker than I realised. When faced with a question, garbled surface recall dominated over deeper conceptual understanding.

One girl even stubbornly challenged me. No! That’s wrong sir. t^{2} + t^{2} IS t^{4 }. The way she had remembered it instead of understanding it was so strong, she didn’t believe me. We had to go over it all again, step by step. I had to pull down her badly remembered model, fixed firmly in her head to replace it with mine. This told me that I hadn’t done enough of that. We need to mix up the questions, use more wrong answer/spot the mistake questions in future and, generally reinforce the meaning of the algebra over and over again.

Another example came from a simple introductory exercise around making formulae to solve problems. For example, writing a formula for the perimeter of a shape. The diagram given was this. Write a formula for the perimeter in terms of s.

One of my students had written P = 4 + S. This took be aback. Her explanation: there are four sides, so you add four to S to get the perimeter. She hadn’t simply replaced a times with a plus. We talked about how S was a length… 4 is a number of sides; they are numbers on different scales, doing different jobs. This seemed beyond her. It made more sense when we labelled the diagram fully:

Now she could see it was S +S + S+ S = 4 lots of S, hence 4S. So, the problem came from being able to see how the number 4 and the variable S combined. It wasn’t immediately obvious to her that multiplication was the appropriate operation. This is worrying. She’s in Year 10 – but we have to work with the students we have, helping them to develop a stronger sense of the meaning of numbers and mental models for the various operations. The difficulty is that, at this age, they often just want short-cut, surface memory techniques to get through it all and that just isn’t enough.

An example of this from learning in science has featured in the informal chats I’ve had with a group of struggling Y11s who are weeks away from their GCSEs. They are lower attaining students and, when I meet them, I check in to see if they are revising. What is the equation for photosynthesis? I get all kinds of responses – a random assortment of the words that make up the equation. *Water plus thing, what’s it, glucose, Arrow, sunlight plus carbon dioxide.*

I love ‘arrow’. The arrow is part of their surface visual memory of the correct equation. They don’t really understand it; they struggle to even say ‘photosynthesis’ out loud with any confidence and the whole thing is a jumble of misremembered words – fill the gaps in any order!

Clearly, the meaning of the equation has escaped these guys – with one exception. They all seem to know that it is chlorophyll, the green stuff, that absorbs the sunlight. This is the bit they can see with their eyes; plants are green. This has made sense to them over the years where as the reactants and products are fuzzy abstractions; words without concrete significance.

It seems clear to me that these boys would have benefitted from more work to help them commit this equation to memory; if they could say it correctly without fail because of a stronger programme of drilling and micro-testing, perhaps they’d have a greater chance of it making sense to them. Oxygen is product; plants make oxygen….securely remembered. However, I also think that the model itself needs more work. At a fundamental level, these students are not fully comfortable with the notion of molecules of different types interacting, swapping atoms around and making new compounds. The development of a mental model is a long term process. The challenge comes when students have an underdeveloped or incorrect model onto which they are trying to stick bits of facts in order to memorise them. As teachers, we need to be conscious of this process and try to understand it better. Why is it that this stuff just isn’t sinking in?

See Also: Mental Mathematical Models.

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