I teach students between the ages of 16 and 19, specialising on mathematical components of a more generalised course .
I am uncertain as to the location of the inflection point between pushing critical understanding and just mechanically working through procedures to arrive at an answer in learning maths. However, it still continues to surprises me the frequency at which the admirable pride and inquisitiveness of students hinder their mathematical progress simply because they desire to understand why techniques or methods work before they are adequately able to do so. When I can I do pepper these explanations at the start with more detail at the end, but I have come to realise that continued explanations and proofs have diminishing returns.
I have read about the secret language of exam questions. Of how successful students develop an instinct for the implicit meaning behind types of questions whilst others must be explicitly taught or forever end up on the wrong side of “Describe…” vs “Explain…”. The same seems to be true of methods in maths. If I ignore the most capable students, I have noticed that the above average have taken it for granted that they will eventually understand how something works and are able to progress regardless of missing the minutia of the previous topic. Now ignoring the disruptive and lazy, those below average are often actually over trying rather than genuinely not understanding, though this is usually the most frequent reason. “Where did that 1/3 come from?” (It’s just part of the formula). Why does x to the power ½ equal √x? (It’s just a different way of writing it, or we could use index laws to show it… again)
A recent catchphrase which has mercifully been accepted by some of my students has been “Stop over thinking and just do the steps for the method the way I showed you”, sometimes shamefully contracted to “stop thinking, just do it”. It emerged in part due to my time as a tutor. Given enough time to actually talk to learners and hear their issues I noticed how often the problem was an additional step added to a simple problem simply because the student distrusted themselves. They found it too easy and added an extra step. So I would break it down and point out how they overcomplicated it. That their failure arose out of finally being capable of a skill they “know” they “suck at”. Secondly as my school students’ maths progress they needed to begin getting comfortable applying formulae immediately without always going through the derivation. “With the quotient rule just match the bits and lets move on. Look up how we figured it out in your own time.”
Some students still rail against this despite the advocacy of their peers that just doing the maths really did work and it is simple once you try. Doing is the only way to learn it. When we get to the next topic the process of convincing them to just do the question repeats again. My anecdotes about the value of failure or repetition or building up mathematical muscle memory again show diminishing returns.
Rote learning has fallen out of favour in recent decades. Vilified as a practice that produces unthinking dullards with no understanding. I will admit there is a risk as covered by Feyman recalling students able to describe the action of light through a transparent planar object but unable to talk about light entering glass. Nevertheless, any adult developing a new skill learns by rote, but perhaps does not notice it. No subject is free from this. Language and sport involve countless drills before open games and conversation. Crafts require instruction and recipes before free innovation. There has been a return to recognising rote memorisation as an incredibly essential tool and this article is just one example. I will keep using this method of “stop thinking” but be careful to lay out the true purpose. I just have to hope the students adopt enough of the full lesson into whatever strange simulacrum they build in their mind.