Linear vs. Non-Linear ideas in teaching math

Three tweets over the last two days have had me thinking about linear and non-linear ideas that arise when students are learning math.

In the order I saw them:

(1) A linear approach to understanding fractional exponents:

(2) A student mistaking a non-linear idea in trig for a linear one:

(3) A second example of mistaking a non-linear process (functions / logarithms) for a linear one:

Something in the example in the first tweet left me uneasy and I couldn’t quite put my finger on what it was. Seeing the next two tweets, though, helped clear the fog – at least a little.

The first example takes a non-linear process – a geometric series – and uses it to illustrate how to understand a linear process – adding exponents. BUT, it is maybe a little surprising, especially to students, that the linear / non-linear relationship works so nicely in this situation. As the next two examples from students show, a non-linear idea doesn’t always simplify so easily. I worry that the first example subtly plants the idea that everything is linear and could lead to the type of misconceptions illustrated in the 2nd two tweets.

Still trying to think through all of this, though.

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