TOAD Talk: Dr. Kenny Nguyen

At my alma mater, faculty and students began the year by talking about the ideas and people who have influenced their thinking, and so I’d like to talk about two hugely important mentors in my life. In grad school, I was lucky enough to study with Jere Confrey who herself studied with Ernst Von Glasersfeld or, as we affectionately called him, Vongie. Vongie was called the father of radical constructivism and, although I only met him once, through Jere and through his writing, I became a radical constructivist. We could spend an entire year deconstructing radical constructivism, but I'm going to give you the crash course. 

First, a question: how have you come to know what you know? How have you come to accept some ideas, but reject others? Consider this broadly. This could be evolution, or quadratics, or the best way to start a fire on an EDT, or who you trust in this community. 

Vongie believed radical constructivism rests on two principles. The first is that knowledge is not passively received: it is something we actively build. Jere argued that many educators stop right there and call that constructivism, while others misapply it badly, assuming that students will discover math on their own, without good tasks and good guidance. Vongie did himself no favors by calling those educators “trivial constructivists,” angering pretty much everybody. The second principle, the radical piece, is usually misunderstood or rejected. Vongie writes, “the function of cognition is adaptive and serves the subject's organization of the experiential world, not the discovery of an objective ontological reality.” 

Okay: we have to unpack that. What he means is that we begin with us, the subject or knower, trying to make sense of the world. We begin with our prior knowledge: the ideas, the actions, and the schemes that have been viable for us in the past, or at least that we haven’t yet found a contradiction with. But that knowledge is sticky and stubborn. You will not reevaluate it until it bumps up against failure. Failure that is so spectacular, that it creates cognitive dissonance and forces you to change. Some of you know this one already. You have a scheme called “I do my best thinking at 2 AM, so start the paper the night before it's due.” It’s worked for many of you. I’ve even heard some of you brag about it: “I started this paper last night and still got an A minus. I rule this school!” Nothing I say is going to change your mind. But I have bad news for you. There is most likely going to come a day when this won’t work, and only then will you revise your scheme. Not because someone told you to. But because reality pushed back. 

Vongie argued that all knowledge consists of invariants (the things you believe) which you create and then maintain or adapt in the face of changing experience. And we do this not to discover some true reality “out there,” but to make sense of our own experiences and our place in the world. Or, as Jere would say: there is no out there there. Radical constructivism places you, the observer, into the picture. It forces you to confront, with humility, that you can always be wrong because you never have the whole picture. And with that humility comes an obligation: to be clearer, to be more explicit, to provide evidence, and to surrender the right to stand on the false authority of declaring your view as the truth. 

Okay: So what? Before radical constructivism, I subscribed to the idea that there was one ideal understanding of every mathematical concept, and that my job was to transfer that perfect understanding into your heads. Vongie points out the fallacy: that there are many keys that can fit a lock, and so knowledge is more like devising a key that works rather than making a mirror image of the lock. Which means your way of understanding something does not have to look like mine to be right. 

Let's do some math! 

One of the most challenging definitions for beginning math majors is that of an open set. You may have been told that the set of all numbers between 0 and 1 is an open set. But here is the formal definition, which I found daunting even as I was teaching it: “A set A is said to be open if for every x in A, there exists a delta greater than zero such that the delta-neighborhood of x is completely contained in A.” I'm sure you will find it shocking that my students learned nothing when I explained this by simply repeating the definition more loudly. 

 

But Vongie made me think about how other keys, such as analogy or metaphor, could help. So I went to work with my students and we rephrased the definition in terms of nudging: an open set is a set where every point can be nudged some small distance in every direction and still stay in the set. So in your mind’s eye, can you picture a number line with all the numbers between 0 and 1? Take any one of them. Isn’t it possible to move it a little to the left and a little to the right and still be between 0 and 1? Even at .999: can’t you nudge that point by .0001 units either way and still not leave the set? Great. We all agree: the set of all numbers between 0 and 1 is an open set! 

Editor's note: the following section included a back-and-forth exchange with members of the student body at Assembly.

So now I need some audience participation. [Pick a random member of the audience.]: What's your name? Excellent. Here is my question: is the set of all triangles in the plane an open set? 

Do you all see why this question is unfair? Because a minute ago, we were considering numbers on a line, and now I’m asking about triangles in a plane. But let’s change the question the way my former student Paige rephrased it, which unlocked it for the rest of her class: given any triangle, can you nudge the vertices in some way so that the shape remains a triangle? 

Yes! Therefore … the set of all triangles in the plane … is an open set! Congratulations! You have just answered one of the most abstract questions in beginning set theory. You're all mathematical geniuses. 

One of the great honors of my life is that I got to work with Jere on an article honoring Vongie on the anniversary of his death and so, in pretentious academic fashion, I'll conclude by citing Jere and myself: 

So my wish for all of you is: find your purpose in life, be humble enough to reexamine your thinking, remember that always, somewhere, someone is fighting for you, and go set the world on fire. 

Dr. Kenny Nguyen is the Chair of the Mathematics Department at Thacher.