Mistakes That Knowledgeable People Make When Teaching

1) Confusing "conceptually simple" with "notationally compact", and 2) jumping to the most general method right away.

It’s tempting to confuse “conceptually simple” with “notationally compact.”

Just because something can be stated in fewer words, or with fewer symbols, or in a more general form, doesn’t mean that it’s easier for learners to understand.

If you want to make something easy for learners to understand, you need to start with the conceptually simplest / most explicit form, where there’s nothing to “unpack” in the notation, and then gradually generalize / compress the notation afterwards.

Another mistake that highly knowledgeable people sometimes make when teaching is jumping to the most general method right away.

Reminds me of a time back when I was teaching, when another teacher wanted to introduce students to differential equations using the terminology “kernel of a linear operator,” building up the definitions of “linear operator” and “kernel” instead of going through some concrete computational examples.

You can imagine how that went… ;)

The “kernel of a linear operator” idea definitely simplifies things in hindsight once you already have a zoo of examples to compress into this idea, but if you don’t have that zoo to begin with, then it’s pretty hard to understand.