No one wants students who don’t *understand* the meaning of their subject; we don’t want our students to merely regurgitate facts devoid of context, or for them to know how to answer questions in an exam yet have no idea what these things mean outside of an exam hall. And yet, on the path to understanding it is unavoidable that our students will often have to learn things that perhaps they feel they don’t fully understand, and will have to memorise things devoid of context. This is what I call The Understanding Paradox.

My view is that teachers attempting to bypass the memorisation and rote learning part of teaching in order to ‘teach for understanding’ can have disastrous consequences for students.

I want to illustrate my point by discussing trigonometry. You might remember it from school, and if you do, you probably remember SOHCAHTOA, the mnemonic device used by pretty much everyone in the UK to learn it. If you didn’t take Maths beyond GCSE or O-level then you might wonder what on Earth it was all about and I know many people feel that they have zero understanding of it.

I have heard it argued, including on my PGCE, that the idea that people who learned SOHCAHTOA reporting this lack of understanding of trigonometry makes the convincing argument that we shouldn’t teach it. Indeed, in a recent conversation on Twitter, a maths teacher said to me that anyone teaching SOHCAHTOA ‘should retrain to be a PE teacher’. There was a certain amount of opprobrium aimed at me for suggesting that it was a good method, and I empathise with that view as I used to feel similarly, mainly under the influence of my PGCE.

This is a faulty line of thinking though, using erroneous logic. Just because people who learned the method feel they don’t understand the topic, it doesn’t follow that we not teaching the method and teaching it in a different way will lead to understanding. The mnemonic has been used historically to teach the topic because it is a good method for students to learn trigonometry. This is backed up by research (although this wasn’t relayed to me on my PGCE). Learning it doesn’t hinder people from having a good understanding of trigonometry, what stops people is that they learn this and nothing else about the topic. It’s clear to me that in Y9 every teacher in the UK should embed the knowledge of SOHCAHTOA in the minds of their students, so that in Y10 they can expand on this knowledge and teach them more about trigonometry. This would avoid the situation that I often hear described by maths teachers

I try to teach them trigonometry properly through similarity and the unit circle and then those students who don’t get it, well I teach them SOHCAHTOA in the Y11 Feb half-term.

Here, by denying students the chance to learn about trigonometry in a straightforward, simple way, teachers risk embedding misconceptions about the topic. When it is realised in Y11 they don’t actually know much about how to answer questions about the topic, they then have to rectify this by masking the misconceptions with the straightforward method they could have taught everyone in the first place.

**What all this comes down to is the word understanding**

My contention is that there is no objective measurement of whether someone understands something. People use the word as a subjective, post hoc judgement that arises once a critical mass of knowledge is accumulated about a topic. This critical mass will be different for different people. Hence, teachers can’t aim at getting students to understand a topic. What they can do is ensure students know as much as possible about each topic. This is a curriculum and sequencing issue. For example, in trigonometry, teachers should first ensure students are proficient with SOHCAHTOA and then teach them more about the topic, such as how it relates to similarity, and the wave functions.

This might not feel right, but it is the right way to teach. Lets not deny our students the best ways to learn just because it doesn’t seem to fit with our ideals of how we would like them to learn.