Functions that commute (Part 2)

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:

1. Teach the distributive law more carefully. Draw pictures. Work examples. Talk about “bags.” Make sure they understand the meaning behind this symbolism.
2. 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.
3. 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.