Math With Bad Drawings had a nice post with pedagogical thoughts on the tendency of students to commute two functions that don’t commute:
The author’s proposed remedies:
- Teach the distributive law more carefully. Draw pictures. Work examples. Talk about “bags.” Make sure they understand the meaning behind this symbolism.
- Teach function notation much more carefully. Give them the chance to practice it. Think like Dan Meyer and seek activities that create the intellectual need for function notation.
- Keep stamping out the “everything is linear” error when it crops up. Like the common cold, it’ll probably never be entirely eradicated, but good mathematical hygiene should reduce its prevalence.
I agree with all three points. Concerning the third point, here’s an earlier post of mine concerning these kinds of mistakes (and others), with a one-liner I’ll use to try to get students to remember not to make these kinds of mistakes:
- I wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”
- If that fails, then I’ll cite Finding Nemo, trying to minimize frustration by keeping the mood light.
- And if that fails, I’ll cite The Princess Bride. One of the most common student mistakes with logarithms is thinking that
When I first started my career, I referred to this as the Third Classic Blunder. The first classic blunder, of course, is getting into a major land war in Asia. The second classic blunder is getting into a battle of wits with a Sicilian when death is on the line. And the third classic blunder is thinking that somehow simplfies as .
Sadly, as the years pass, fewer and fewer students immediately get the cultural reference. On the bright side, it’s also an opportunity to introduce a new generation to one of the great cinematic masterpieces of all time.