Discussion about this post

User's avatar
Jacob's avatar

Emotions are powerful and immediate, so if you can harness an emotion -- any emotion -- and link it to mathematical insight, you create a visceral experience for the student: the initial insight is immediate ("ughh that looks horrible!"; "pffft, that one would be so easy!") and they can marshal reasons for justifying that insight *after the fact*; thus creating the sense of having intuition or natural insight before "learning" anything.

Now every student has some expertise. Now the stage is set: the student's feelings towards the easiest and hardest problems are *different* (e.g. confident about A but intimidated by X) but clearly of the *same* type. Suddenly there is a natural motivation to discover how the two are related ("can we transform X so that it would be more like A?"). I'll often follow this up by asking students if they could think of ways we could make A appear more like X, without affecting the solution; now they're thinking like a teacher.

Implicitly, it also reinforces an important lesson in learning: feelings *aren't* facts. You might be intimidated by X, it might look impossible but now you see it's on the same spectrum as A, which is not intimidating, so you know not be afraid of a new problem simply because it looks hard.

This is one of my favourite approaches to use. It is especially effective with intimidating content *because* of the strong emotional response. In some variations, I also emphasize to my students that this really is how mathematicians work: we look at a problem and notice what we don't like about it ("hmm, this would be a lot easier if we didn't have this part") and this emotional response can often be trusted to guide us towards an appropriate approach. Students learn that mathematical intuition and common intuition are not fundamentally different -- and who doesn't feel great about being "a natural"?! It feels wonderful knowing you can trust yourself.

Another variation on this approach is to ask "if you could change one thing about this problem, what would change?"; however, it's more appropriate at a later stage, when students have already developed a sense of the diversity of possible problems.

Expand full comment
Mardalee Burwitz's avatar

Giving students the opportunity to choose the "least difficult" of problems, offering student agency - giving them a choice and an entry into their learning. However, it also provides those who are confident to be looking for the most challenging to solve if they want. I recall advanced students who would jump to looking for the MOST difficult to answer, looking for that challenge, too. Ms. Esmende also includes the discussion between students giving voice, and then, I'm sure, she would use routines for students to explain the reasoning and continue with the lesson from there.

Expand full comment
13 more comments...

No posts