## Teaching to understand

I’ve been thinking a lot this week about understanding. There are different levels of understanding, there’s the deeper understanding (often referred to as Conceptual understand or relational understanding), then there’s a surface level understand (usually called instrumental understanding). The deeper understanding implies that someone understands why things are happening and a surface understanding implies that someone understands what to do in a given situation. This paper by Skemp (1976) is excellent on this topic.

A simple example might be if you have a student a simple equation: 3x + 1 = 10 . I’d the student is working with a surface understanding they may be able to solve it because they recognise that you “undo” the terms to find x. While a student with a deeper understanding would understand the role of the = sign and know why the inverse operations work.

As far as mathematical knowledge goes, a deeper understanding is far preferable to a surface understanding, but as far as passing a GCSE exam goes both of these levels of understanding will yield the exact same results, and as the surface understanding is quicker and easier to achieve a lot of teachers focus on this method to get their students their target grade or above. For many this is down to the pressures crated by the system of league tables and Ofsted gradings that drives a lot of the people involved in this profession. This is one of the problems with the system we currently have, and certainly suggests to me that many people have the wrong focus. Education should be about educating, not just teaching students to algorithmic answer questions that come up on a paper.

The current GCSE is set up in a way that you can get a B, or even an A, with poor algebra and trig skills if you are excellent at the statistics side of it and have memorised enough algorithms for the algebra, this means many with these grade struggle immensely when they attempt mathematics A level as they don’t have the pre requisite ability. The new GCSE spec will go some way towards combating that, but the focus on instrumental understanding to achieve the grades will mean that students are unable to make the links between topics and are unable fully grasp the follow on concepts as they haven’t fully understood the basic ones, and through no fault of their own.

This week I spoke to a head of maths from another part of the country and he was talking about how he’d improved his schools results by 25% at GCSE A*-C by introducing a scheme of work that focused on the big mark questions and a pedagogy that made the algebra easy- when he described said pedagogy it was purely algorithm based and promoted no deeper understanding. He was at a point where he could not understand why his AS results were poor, despite the cohort having strong GCSE grades. It’s a situation I’ve been in and seen before. The grades are a reflection of what questions they can answer, not what mathematical concepts they understand.

I spoke to another teacher this week from a different school in a different area of the country, and she had found that her students all got part b this C3 question wrong last year.

She had completed her question level analysis and decided that it was due to the students not being able to recall the rules involved and had set about creating mnemonics to help them. I disagree with her analysis. I think the inability to answer this question comes from only having an instrumental understanding of the concepts. If you understand that the range of a function is the “y” values it produces, that the nature of a turning point means it’s gradient is 0, and that the derivative of a function at a point gives the gradient of a tangent at that point then this is an easy 6 marks.

She went on to discuss a number of mnemonics shed come up with for C3 trigonometric identities. This seems particularly daft to me. Why would you create extra things to remember? If you have a deep understanding of the unit circle and right angled triangle trigonometry then the two main trig identities are obvious. *Sin^2 (x) + cos^2(x) =1 and sin (x)/cos (x) = tan (x) *

Once you know the reciprocal trigonometric ratios then the rest fall out from there.

I’ve seen A level students able to differentiate and integrate perfectly, but not be able to find the area under a curve or the gradient of a curve because they’ve never been told what these algorithms they are performing do.

I feel we need to move away from this, we need to encourage a deeper understanding or we are setting out students up to fail in the future.

*What are your views on this? I’d love to hear them. If also be interested to hear if this is a problem in other subjects or just maths. I’d imagine that it does occur elsewhere, but I’d love to hear confirm2 (or otherwise) of that, and specific examples of where it occurs if it does. *

**Reference**

Skemp, R. R. (1976). Relational understanding and instrumental understanding. *Mathematics Teaching* available here.

We want to know what works well? lots of the obvious stuff… excellent teaching, clear high quality presentation, using interactive programmes judiciously, repeated practice and consolidation, problem solving exercises and yes lots of work around the equals sign …specifically writing and constructing equations beginning with building linear equations based on simple “I think of a number…” sums. This is primary school practice in exemplar form and worth replicating. Ultimately cementing their understanding that “algebraic operations follow the same conventions and order as arithmetic operations” … a phrase that was big and prominent in all the Numeracy Strategy materials from aincient history -that is the early 2000s – worth revisiting and all available online if one is inclined to dig a little deeper e.g.

Content like this will enable them to move from basic inverse operations methods to robust algebraic manipulation skills so they progress from KS2 all the way to KS5… building their own House of Maths … brick by brilliant brick

I’d never heard of SOHCAHTOA until I started teaching and as an engineer I used to use trigonometry on a regular basis. I had learned the Sine and Cosine Rules by just learning them but basic trig was just deeply entrenched in my understanding and I always try to teach in such a way that they understand this. I always say to even my Further Maths students that if they can understand what they are doing they should be able to derive necessary formulae in the exam, even statistics and trig functions