... is to present a problem where known simpler techniques fail.
Cross-posted from here.
In my experience, the only way to successfully teach a more sophisticated technique is to present a problem where known simpler techniques fail.
For instance, anyone who’s taught algebra to kids will know that if you try to teach $ax+b=c$ equations by demonstrating on a problem that has a single-digit solution, like $2x-1=5,$ then a portion of the students with decent number sense will tune out because they can get by just fine using guess-and-check:
- "$x=0$ gives $-1,$ too low, go up a bit $x=2$ gives $3,$ tiny bit higher $x=3$ gives $5,$ BAM im so smart i dont even need algebra!!!"
The way to avoid this is to present a problem like $9x-13=18$ and say something like “take a moment to see if you can guess the solution to that… yeah, not so easy, huh? Okay, now you see that guess-and-check is not going to work for some of these trickier equations, so let me show you a trick that does work.” This way, the kids understand the value in the more sophisticated solution technique that you’re about to show them.
It’s the same thing at all levels of math. If you want your students to understand why a more sophisticated definition is preferable, then you have to introduce them to a situation where the simpler definition breaks down and the sophisticated definition saves the day.
(And if you can’t find such a situation, then I think that’s an indication that the technique is not appropriate to use in the class – though perhaps it could be made appropriate by expanding the curriculum to cover an area where such a situation arises.)