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Ask the same questions repeatedly

One the simplest techniques I use is to ask certain questions repeatedly through a course. Very few people hear an idea once and internalise it. It can take repeated exposure to absorb some facts or methods. Hence, the repetition ultimately helps understanding.

For example, we often need to show that an equality holds. A lot of students take the equality and manipulate it into something they know is true. (I show them that this is a dubious method via
1=-1\implies 1^2=(-1)^2 \implies 1=1.
The latter we know is true and hence 1=-1. In other situations the implications can be reserved and we are fine.)

A better strategy is to take one side and proceed to show, by a string of equalities (or similar), that it equals the other side. Hence when presenting an equality I don’t just say “Let’s pick this side and change it into the other side”. Instead I ask “How do we show an equality?” After repeated exposure they know the answer is “Pick the complicated side and do something with it”. Note that the question requires retrieval which is better for learning.

This initial approach is in fact too simplistic. For example, we need to ask which is the complicated side — sometimes it is not obvious and is often open to debate. The other important consideration is that sometimes the best strategy to show x=y is to show x-y=0. (Showing x\geq y and x\leq y is also an alternative.) Hence a follow up question is “Ok, so what should the complicated side be?” This is better than “Which is the complicated side?”.

Another question I use repeatedly is “What about the converse?” I have already described a competition I run regarding this. Quite often the converse is not true and so this becomes a great exercise in finding counterexamples.

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