I’ve been taking Keith Devlin’s Massive Online Open Course (MOOC) “Introduction to Mathematical Thinking,” and it’s been really fun to see how my brain’s mathematical thinking skills line up with the course assignments. It’s especially fun to see the times when ideas are new to my brain and how that’s different from when they’re familiar.
For example, Assignment 1 was about reasoning from definitions, proof by counter-example, proof by induction — those are tricks of the trade I’m familiar with. I can read math problems and see how those tools are applicable, I know what to focus on when we’re proving properties from definitions, etc. I wasn’t flummoxed by stuff like what does it mean to divide 0 by 2, because I could work comfortably with the idea of 0 as being just another number that’s divisible by 2 — I was working on a higher level of abstraction.
But then again, on Assignment 2, I was flummoxed! I didn’t know what to pay attention to, and different stuff popped out at me. The assignment was about connecting natural language to formal logic (like if Statement A is “it will rain tomorrow” and Statement B is “it will be dry tomorrow” then is the plain language statement “It will either rain tomorrow or be dry all day” equivalent to the formal logic statement A v B (A “or” B)?) I got quite tangled up in the question of can it rain and be dry on the same day, and trying to nitpick the words in the sentences and find some hidden trick or meaning — because I didn’t have any idea what the basis of comparison was, or what it would mean to make a logic statement equivalent to a plain English statement. On the homework, I found myself confused and looking for “key words” to translate little parts of sentences into logic. In short, I felt like a student who struggles with word problems because they don’t know what it means to mathematize something, and so they are focused on different details than an expert would, and use different rules and tricks to do the mathematization. For an expert, it feels obvious, how to apply this heuristic, but for the student seeing it modeled, it’s not clear what details are salient to the expert.
I had an ah-ha! moment with Assignment 2, but more accurately, I had two or three ah-ha! moments. Two or three times this week I’ve struggled and asked my peers for help and had someone show me a truth table. Ah-ha! I said — to find out if two statements (whether in logic, plain English, or both) are equivalent I need to find out if they have the same truth table. And I could answer one question. But then when I became stumped again, I didn’t think of truth tables, because the question felt different!
I hope that now that I’ve reflected on truth tables and what it felt like to need one, and the (seemingly) different contexts that they helped me in, I will now have truth tables as a mathematician tool. Just like I have proof by induction, and showing sets are equivalent by proving an arbitrary element from one has to be in the other and vice versa, as mathematician tools that leap out at me.
Reflecting on my experience as a learner, I noticed that:
a) I needed to struggle with the problems repeatedly, without help, before I really cemented my learning.
b) I needed models of explicit tools from people who were smarter than me (about this kinda thing), but I couldn’t make sense of the tools and when to use them the first two times.
c) I needed to compare my ideas with other non-experts, to articulate them to myself and others.
d) I needed to persevere through enmeshment in all the wrong details, and be able to come up for air and entertain ideas about what other people saw, not just the alluring details. And be exposed to people just above my level, and their ideas, not just expert ideas.
I wonder how many of those features my classes and work with teachers have. Are they all necessary? Are they sufficient?