## Thoughts on the understanding paradox and introducing trigonometry

Recently I read a blog entitled “The understanding paradox” (William, 2017) which discussed the idea of maths teaching and put forward the idea that actually, it is better to bypass understanding when first teaching a topic and then fill that understanding in later. This was then applied to the teaching of right angled triangle trigonometry in an example that I found confusing to say the least.

The author, Rufus William, suggested that when teaching trig for the first time we should be solely teaching procedurally using SOHCAHTOA as a mnemonic, but then went on to say we shouldn’t be discussing ratio or similarity and how that links until later on. This confused me as the mnemonic SOHCAHTOA is designed to help you remember the trig ratios. I.e. Sine is the ratio of the opposite side over the hypotenuse. Just by teaching that you ARE teaching the trig ratios and purely by the fact that you are teaching the students that this will work for all right angled triangles you are telling the students that the ratios are the same for any triangle with the same angle no matter what the length of the sides are. THIS IS THE VERY DEFINITION OF SIMILAR TRIANGLES.

This perplexed me a lot and I spent a lot of time thinking about it and asking the author to elaborate on what he meant. The only way I can fathom to teach this without reference to ratio and similarity would be to say: ” “SOHCAHTOA” it gives you 3 triangles. Label the sides circle them to see which triangle you use. Put numbers in, cover the missing one, its either a divide or a times”. To me this seems like a backwards way to go about things. It feels like you are teaching them unnecessary procedures to avoid discussing the underlying concepts of trigonometry, and it doesn’t really make sense to me.

I find that by the time students reach right angled triangle trigonometry they have already met the concept of similarity, I like to use this a way in to discussing the topic and to show that ratio of two sides that are the same in relation to an angle will be the same for all similar triangles. Students will have always encountered simplifying fractions before they meet trig and as such can see why this is. This is when I specifically discuss the sine, cosine and tangent ratios and introduce the procedural manner in which they can solve the problems, although I do avoid the dreaded formula triangles (for many reasons which I have blogged about here). I will show them some common mnemonics, and SOHCAHTOA is one of them. I’m not a fan of mnemonics personally, I’ve never found them that useful except for musical ones, but I know a lot of people do.

Rufus does make some salient points in his post about teachers who refuse to allow students to memorise things and the dangers this will have on learning. Although I’m not entirely sure that they exist, and if they do I certainly don’t think there are many of them. I’ve certainly never met any.

He also suggests that students cannot have a full understanding of the ins and outs of trigonometry when they first meet it. I would very much agree with him in that respect, I know many people who have taught trigonometry for decades and still don’t, but I don’t think that means we have to bypass all information.

**Reference List:**

Cavadino, S.R. 2014. Formula Triangles. 12th October. *Cavmaths. *[online] accessed 14th July 2017. available: https://cavmaths.wordpress.com/2014/10/12/formula-triangles/

Cavadino, S.R. 2016. Catchy Mnemonics. 16th September. *Cavmaths. *[Online] accessed 14th July. Available: https://cavmaths.wordpress.com/2016/09/16/catchy-mnemonics/

William, R. 2017. The understanding paradox. 7th July. *No easy answers. *[online] accessed 14th July 2017. available: https://noeasyanswerseducation.wordpress.com/2017/07/07/the-understanding-paradox/

Interesting blog, thank you.

It’s not that teachers refusing to allow students to memorise things, it’s that some teachers have an idealistic aversion to SOHCAHTOA, so the students don’t get a chance to learn it. I don’t believe you can deny this is the case, as I said, one maths teacher told me I should retrain as a PE teacher because I advocated teaching it.

For clarity, I am arguing for teaching SOHCAHTOA purely procedurally: label the sides, choose the relationship, use the formula correctly.

Once, this is engrained in the students, then I talk about ratio and similarity. I know this is unusual to hear if you have not come across the idea before but I have given it a lot of consideration, I am informed about teaching, and I am open to admit that I could be wrong!

That clarifies a tad. I don’t see a massive distinction between the two methods, and I’m not sure how it is detrimental to talk ratio and similarity before.

I’m sorry to hear you received that sort of a comment. I’m not sure I understand the position that person was taking to be honest. I’m not sure why there would be an aversion to the mnemonic as it’s just a way of learning the ratios (or “relationships” if you prefer!). As I said, I’ve not encountered this but I would love to discus with them and find out their reasoning.

I don’t think they’d realise it was ratio and similar triangles unless it was explicitly pointed out to them. I think it clutters their mind if you teach them understanding first. I often teach procedure first, then, after they can do it, understanding.

I didn’t “understand” trigonometry until after I’d done my Maths degree and had been teaching for a few years. I don’t know if it was never explained to me, or I wasn’t listening or what, but, I was doing something unrelated and it just suddenly flashed into my brain. I think that understanding often doesn’t occur in lessons, it comes later, when the unconscious mind has time to work on it at leisure.

I can see your argument, but I can’t see how it would clutter the mind to see a link between opp/hyp being the same for similar triangles. Perhaps it could be n issue for foundation students who now need to be able to use it.