Mathematics education uncovered and recovered

Fake maths of mnemonics


Although the mnemonic SOHCAHTOA is not mentioned in the curriculum, nor is it referred to in exam mark schemes, it is hard to find a Maths GCSE textbook or revision guide without SOHCAHTOA. It seems that almost every teacher happily uses it.

The abbreviation is intended to help students to memorise the meaning of sine, cosine and tangent of an acute angle in a right-angled triangle as ratios of the sides of the triangle. Its purpose is similar to that of a baby walker: help to make first steps in trigonometry. However, unlike with children’s walkers, the SOHCAHTOA walking frame becomes indispensable and effectively paralyses their progress.

The problem is that for most students SOHCAHTOA appears out of the blue. There is no discussion about the similarity of right-angled triangles with equal acute angles, and the convenience of having the ratios of the sides packed into values of trigonometric functions is not recognised. It is not surprising then that, having been introduced to the magic mnemonic as a replacement for understanding, students are stuck with it forever.

Knowing only as much trigonometry as SOHCAHTOA can hold might be enough for practical purposes of answering GCSE questions, but the lack of understanding is not only regrettable as such, but becomes a real hindrance if students continue with Maths and Physics at A level.

Some teachers lead their students even further away from proper understanding by offering them ridiculously meaningless rules for memorising the values of trigonometric functions of some common angles. Using these rules, a student trying to recall the value of  \( \mathrm{\tan {45} ^{\circ}}\) will associate it with finger-counting rather than thinking about equal sides in an isosceles right-angled triangle.

I assume that SOHCAHTOA was a result of a well-meaning attempt to simplify school mathematics and help struggling students. But its overuse made it a surrogate for real mathematics. SOHCAHTOA, as well as FOIL, “Speed triangle” and many other “tricks” became shallow substitutes for the rich mathematical content, and the quasi-maths they produce is not really worth learning.

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