I actually got to be on *two* wonderful panels for FTBCon, quite different from each other! In this one, Matt Lowry, Dan Linford, Jason Thibeault and I just chatted free-form about how we teach critical thinking in our classrooms.

Thoughts:

- Important: When I say math here, I mean high school math, not logic or model theory.
- The truth is I am incredibly conflicted about things like this (and really education as a whole). My thoughts go something like this: teaching breaks up into four categories: skills, conceptual understanding, love of math, meta.
- The first is most important if you think students should need to know those skills and be able to use them or remember them in the future. That goes for everyone when you’re talking about addition and number skills, and engineers or other applied scientists or mathematicians for the advanced stuff. Since I don’t think most of my students need this, I’m ok with saying that dropping a negative is not a big deal and doesn’t necessarily warrant points off. BUT there is of course value in, you know, getting the right answer, and maybe this focus is taking away from actually being able to do the problem, as this Atlantic piece argues.
- Conceptual understanding is important for mathematicians and anyone hoping to do complex math, but it’s also important for grasping connections and getting a sense of how math describes the world. I think it’s the coolest and most interesting part, and so it’s what I focus on. Being able to perform skills is subordinate to this, largely. If you can’t intersect lines well and smoothly, you won’t get what’s so cool about parametric equations. If you have trouble with mental math, integration is going to be a slog, and not for good reasons. I also tend to think, but maybe this is the Typical Mind Fallacy, that the better you understand something, the easier it is to perform skills, since everything flows and makes sense, rather than being a rote list where you’re racking your brain for what happens next. BUT you can focus on conceptual understanding to the detriment of skill building, which can lower confidence since students don’t even know where to start. Giving them systematic approaches (i.e. rules)
- Love of math is half selfish, half not. I think math is cool and beautiful, and I want other people to think so. I also think math is so amazingly capable of describing the world around us that it’s valuable for educated people to have an appreciation for it, and a general sense of how it works. (things move in parabolas when thrown, it has to do with earth’s gravity; if you know how many tickets to the interview sold and how much money you made, it is a solved question how many were $6 tickets and how many were $15 tickets). I *also* think that if you have to sit through school-mandated math you might as well enjoy it and have enough understanding and care to find things in it that interest you.

- I, and many other teachers, can talk all day about all the cool strategies we use and all of our lofty goals, but I certainly don’t know what the outcomes are. I don’t have data. I got a lovely card from a student last summer that said “thanks for teaching me a new way to think” but I also had students fail their final exam, possible because we did so much “sure it works in practice, but how does it work in theory” that they couldn’t actually use equations and formulae. At least, that’s my fear.

I hope to write more in 2015 about education and my teaching experience, but if you want to find some cool gifs and general ramblings, my math tumblr is here. But now I’ll turn it over: what are your thoughts on the panel or about critical thinking in education?

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Just an observation based on my experience. We had a lecture course on Fourier Series during our Metallurgy Degree course (1967-70) that was taken by a post-grad, who packed a 2 term course into one term, so he could go into the RAF to do computing. None of us understood what he had told us, so our spokesman pointed this out to one of our Metallurgy Profs. He then took that same course during the remaining term, explaining what Fourier Series meant in terms of real structures, e.g. ships’ propellor shafts, and it made sense (at least to me). So your title ‘merely real’ is highly appropriate in the practical teaching of maths. A friend who is a high school maths teacher tells me that she also finds much value in this approach.

Thanks for the comment! It can definitely help to connect things to real world events. But it can also be just as valuable to find a genuinely perplexing and interesting theoretical problem. It all depends.