First, apologies to all for having been off the blog for such a long time. All the things happened at the same time. But things have settled down and I have time to write again! [Nothing bad happened, but suddenly everything got insanely busy.]
Anyway, “The Best Question To Ask Always”? Seems a little bold, eh? Probably, but apparently even if we’re taught not to judge a book by its cover, it seems that’s what people do. Flashy headlines for all those clicks.
Ok on to reality. It look me a long while to realize how to truly learn Mathematics (and of course, any mathematician will tell you, they don’t know all of Mathematics). As a kid, I thought that learning Mathematics was all about computation. Times Tables! (teaser: blog post on that is in the works!)
As a teenager, I thought it was about symbolic manipulation, though admittedly I didn’t know the phrase “symbolic manipulation” at the time. But looking back, I know that’s how I thought about Math. Could I perform complex algebraic maneuvers in my head? Could I isolate \(x\) in large complex equations?
As an undergraduate studying Electrical Engineering, I had shed the “silly” notion about “fast computation” and had morphed it in favor of highly precise computation as a consequence of heroic symbolic manipulation. Taylor series were all the rage and all error could be tied back to the limits of instrumentation.
As a graduate student, I was convinced that these three prior understandings were nonsensical. I adopted a more abstract approach, focusing on concepts and theory to help generalize the topics I was taught all my life in fractured and piecemeal ways. I strove to connect the missing dots and was always met with a great sense of satisfaction when I was able to really get at the heart of a concept and how it could be viewed from so many different angles. In other words, I loved reading a diverse set of proofs for the same theorem.
So, as I grew up continuing to learn Mathematics, I would approach learning the subject based on what I thought it was about. Computation? I learned to challenge myself with ever larger multi-digit arithmetic. Symbolic manipulation? I learned to challenge myself with increasingly more complex formulae and equations. As an undergrad, I kid you not, I tried to find Taylor series of every expression I could get my hands on! I tried to solve every integral and differential equation that I came across. But I shied away from Topology and Analysis. I took Topology, but that wasn’t my math, that was a “non-practical” math, at least from that world view.
As a graduate student, ah, it was all about “what is it that I’m trying to prove, what do I have at my disposal, and what do I need to complete the proof?”. Far more abstract and by most measures, far more mathematical.
All of this, I tell you, was good training. I had by accident built sound fundamental mechanics that allowed me to continue to move forward towards greater levels of abstraction. All good stuff.
Now, a decade after finishing my PhD and working as a mathematician both in industry and in academia, I’ve had a long time to think about learning mathematics far removed from the pressures of having to learning mathematics. I still have to learn mathematics for various reasons. But the grade pressures aren’t there, so the method of learning has changed.
In the last decade, I came to realize at least one thing that had escaped me about learning Mathematics. As we are taught any subject we are always battling against failure, ie, incorrect answers. And wait! Before you roll your eyes and leave, this isn’t about some populist edu-nonsense about “embracing failure” or “failing fast”, etc. What I mean here is that we are taught to approach how we learn as a refinement process through mistakes ultimately getting to correct answers (not necessarily numerical, but also theoretical and abstract). This isn’t a bad process. Self-improvement is great!
But the best question to ask one’s self regardless of subject topic? For now, I’ve landed on this, “What mistake will I make?” Give this some thought if you haven’t already. I have found this to be an extremely powerful question.
Teaching arithmetic? Teach it however you do. Let the students practice a bit on their own. Then ask them to ask themselves “What mistake will I make?”. Teaching fractions? Algebra? Geometry? Calculus? Analysis? Topology? Monte Carlo Methods? Programming? Piano? Soccer?
Anything you’re learning. After you’ve taken a crack at it a few times, ask yourself “What mistake will I make?”. The wonderful thing about this question is it forces us to stop and reflect. You may be surprised at how self-aware your students are when they are asked to answer this one question and how much easier it will be for you to teach.
Bonus teaching question: What will you do when a students says that they will make no mistakes?