Today marks the 11th anniversary of my father’s death. Appropriately, I’m reading a book by Ken Bain called What the Best College Teachers Do. It’s a years long, systematic study of the philosophies and practices of those college teachers that you might have been lucky to have: those who were so challenging, inspiring, and brilliant as educators that their students didn’t just do well on exams… they reconsidered their whole life path. My dad taught at the Evergreen State College for 23 years, and was regarded, rightly, in my opinion, as being one of those rare, life-changing professors. Reading the results of Bain’s years of research, I see my dad on virtually every page.
So what do the best teachers do? The answer takes a whole book, but some things jump out at me. Bain talks about how the best teachers understand their students as having mental models that (i) are resistant to change and (ii) are often faulty. For example, researchers found that some students who aced an into physics course still conceived of the world in pre-Newtonian terms. When they asked the students to make predictions about physical systems and then demonstrated that their predictions were wrong, the students concluded that there was something unusual about just this particular demonstration, and that their (faulty, Aristotelian) understanding of the world was just fine, thank you. Eric Mazur ran into that problem, as he discusses here very briefly, and here more extensively.
In my dad’s book Teaching with Your Mouth Shut, he describes grappling with this very issue. The idea is not to pour the correct knowledge into your students (Bain remarks that the best teachers have all realized that this doesn’t work)–it’s to force various models into a productive conflict (and get the student to care about the outcome). The Greeks called it aporia, the impasse you reach where two or more ideas that felt solid are suddenly in deadly and unresolvable dissent. This moment is the pivotal point in deep education. Here’s how one of Bain’s teachers describes it:
It’s sort of Socratic… You begin with a puzzle–you get somebody puzzled, and tied in knots, and mixed up.
If you want a clue when these types of impasses were reached in the field itself, take a look at when people got furious. The Pythagoreans believed–reasonably I’d say–that you could describe any relationship as a relationship between whole numbers. Their philosophy crumbled when it became clear that the relationship between the diagonal and the side of a square was not comparable to the relationship between any two whole numbers. Apparently, this bothered some of them. A lot. Proclus describes the fallout:
And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves.
Some existential angst working itself out there.
But the point is, we’re all walking around with huge blind spots and faulty models for how the world works.
Here are two other quick mathematical examples:
- Parallel lines. Euclid assumed they exist, but would have preferred to prove it.
Centuries later, it became clear that it couldn’t be proved because it didn’t have to be true. In (1), we have “flat” space… what we’re used to. But if space is curved, as in (2) and (3), you either get NO parallels or infinitely many. This was seen as insanity at the time. I think it’s pretty counterintuitive for us today too.
- Infinite Sets. Infinity is infinity, right? But if all infinite sets are the same size, how can the even numbers fit inside the whole numbers? Two different intuitions about what “same” or “equal” mean come into conflict here. Cantor (and Galileo, to be fair) was able to resolve this by defining equality in a way that worked for finite and infinite sets, based on matching things up in a one to one fashion. But the outcome was infinite sets of different sizes!
There are plenty more examples, but those two jump to mind as causing major crises in the field. And just as ontogeny recapitulates phylogeny, so does the individuals grappling with these concepts mirror the historic struggle within the field at those pivotal moments of discovery.
But where was I? I was thinking of my father…