Originally published at: How math people look at math, and why it works | Boing Boing
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It’s interesting that he picks trig identities as an example, because that’s a good illustration of how math tends to alienate people the way it is taught.
I think anyone can appreciate why the concept of a sine is useful; you can connect it to real-world questions in a hundred ways. And anyone can understand why you’d want to manipulate sines and cosines algebraically, which leads to memorising trig identities because that’s such a useful practical step. But then the nature of industrialised education, with its focus on testing, means learning those identities becomes the goal, with the underlying concepts treated as mere background. So a lot of students feel like math is all about these arcane symbolic manipulations, because the real content is only mentioned in passing.
In terms of what the guy in the video is saying, math is about a network of deductions, but the way it’s taught treats students as spectators to that process. You’re expected to learn the outcomes of those deductions, and no one really cares if you can do the reasoning yourself – but if you don’t, then your math career will top out at the limit of your ability to joylessly memorise stuff.
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You get an “Amen!”
For me, when I was a kid constantly playing with Estes model rockets, I wanted to know how high they flew, and I was amazed by the concept of measuring them with a protractor. That gave me the incentive I needed to learn trig, which turned out to be both fun and useful to solving a whole pile of problems. I still use it frequently.
When it came time to learn calculus, though, I didn’t have a personal attachment to anything fun. It was a prof saying “you will learn integral calculus this way” and “you will memorize these answers to this large set of derivative calculus problems.” The problems were stuff like “given a yardstick, measure how much fluid remains in this tank.” Problems that were useful, but were much less personally interesting or engaging. So I slogged through exactly enough calc to barely pass the exams, and then immediately reclaimed that part of my brain for storing useful stuff like song lyrics and Monty Python bits.
And that’s the trouble with the learning process. Not too many years later I was presented with a real-world work-related problem (figure out the optimal parting line of a plastics injection mould), and I at least recognized calc would solve the problem. But I had no idea what to do beyond that, and had to drag in a math nerd for help. Had there been an effective learning process, I would have been able to solve the problem alone.
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? Of course math is easier than history. The things it studies, even at the highest levels, are far easier to get get a solid handle on than the things history studies.
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There is also the administrative issue of expecting all students to “keep up” rather than instructors ensuring that the entire class is understanding the concepts. There is nothing worse in our educational system than shaming students who need more time to grasp what is being taught, and as a result, fall behind because they happen to be an outlier.
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There’s also the problem where each school teaches the basics of [fill in the blank with a math subset] differently. I attempted to transfer my C average in Calc to another school, and discovered that the new school used an entirely different approach from my previous one. I couldn’t learn the new method and keep up with classes in Calc, which resulted in my leaving engineering.
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But then the nature of industrialised education, with its focus on testing, means learning those identities becomes the goal, with the underlying concepts treated as mere background. So a lot of students feel like math is all about these arcane symbolic manipulations, because the real content is only mentioned in passing.
Agreed. I was pretty “meh” about math in high school. I was ok at it and I got average grades without too much effort, but it didn’t excite me in any way. Then I went off to study computer science at university because I liked computers. Like many comp sci students, I didn’t appreciate that computer science (as opposed to computer programming) involves a lot of math. I fell in to taking some “pure math” classes and it blew the back of my head off and I ended up leaving with a math degree. When people look at me quizzically, I always tell them that the way they teach math in high school seems almost designed to make you hate math. They brush aside the fundamental underlying concepts which are interesting and focus on the stuff you can do with a calculator, which is boring. (The standard refrain from one of my university profs when asked if we could use a calculator on an exam was “Sure, go ahead, let me know how that works out for you”).
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I was always godawful at mathematics in school. Nobody ever explained anything.
It wasn’t until after I left school that I realised some of the questions keeping me awake at night were mathematics. And I eventually found some of the keys to cracking into them. Too late to make it anything more than a curiosity, though.
Still, I think I was able to help my kids a bit with their maths homework. “Trigonometry is all about circles. Even when it’s about triangles, it’s about circles.” or “The two most important numbers are zero and one. Everything else flows from those.”
Alas, that’s still when they deign to listen to me, and the rest of the time they’re in class with their teachers drilling them on rote formulae they don’t understand, convinced that they’re bad at maths.
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Obligatory link to Lockhart’s “lament” in case anyone has not read it. It is a beautiful explanation of what’s wrong with math education. In short, what schools call “math” is really computation, and only tiny vestiges of actual mathematical thinking are taught in most schools.
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The “linked” relationship of mathematical concepts is an interesting point. However, I think 2 more relevant issues are variance in teaching style, and relevant applications. Some people learn better by visualization, others by detailed descriptions, others by formulas, etc. Classes seldom have the luxury of using multiple approaches, so they can hardly hope to reach all potential learners. If they’re not using your particular optimum, you’re SOL and struggle.
Likewise, I found that abstract concepts & formulae became much easier to understand or remember when illustrated with relevant problem-solving. Students require motivation to deal with the discomfort of mental strain; real-world solutions often offer such incentives.
To all the Gen Xers complaining about how terrible your math education was - you’re correct, we were taught “new math” and it was a bad idea. Too many of us came out of it believing “math is hard”, which isn’t inherently true.
Kids are currently being taught “everyday mathematics” which teaches kids to use multiple methods to reach the correct answer, helps kids learn pretty decent estimating skills before they leave elementary school, and is able to introduce more complicated concepts, earlier than “new math” allowed because there isn’t reliance on wrote memorization. The vocabulary frustrates me, but the method seems pretty sound.
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Well, the standards are completely different. If you tried to do history with math standards, you wouldn’t get anywhere. If you tried to do math with history standards (not history of math, which is history) … I don’t know what you would get.
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Judging by my children, the next generation’s complaints will be about having to explain the reasoning behind every minute detail and about having to learn six different ways of doing something they already understand. That’s what my eldest complains about, anyway.
I’m liking this as a 65 yr old male who received a privileged math education in the 60’s and 70’s, and helped his tail-end-millennial son deal with public school (BC, Canada) math instruction. In both cases, the teachers mattered so much. In my case I had at least 5 that were stellar, across grades 7 to 12. He had one teacher of that calibre, in one grade. The chain for him got broken, and no amount of parental encouragement could help. It got broken for me, too, but repaired repeatedly along the way by those stellar teachers. My son is an excellent analyst of problems, remarkably clear-thinking – just not equipped with some critical tools. But, having matured beautifully, and realized that the game was slightly stacked against him, and that if he needs the tools, they are well within his grasp, he’s not really at much of a disadvantage for what he wants to do with his life.
So, with all that said, are there any recent teachers on this forum who can comment on the training they received about how to teach math, vs other subjects? I know from speaking with my son’s good math teacher that he eventually largely ignored the orthodoxy of the 2000/2010’s (when he qualified/when he taught son) in favour of better results (which were definitely not “teaching to the test”). A brave stand, actually, which my teachers did not have to take.
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Yeah, I was one of those kids who breezed through the manipulations in algebra and trig, but actually had to do some of that chain thinking when it came to doing my proofs in geometry. That classed helped me more than anything until I got to calc.
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