when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what they mean.”
The most common complaint I hear about undergraduate students is that they aren't able to blindly manipulate symbols, and consequently get hopelessly confused when they have to deal with "unintuitive" concepts.
It's even worse than that. It's more like "I was very good at doing enough of the practice/homework problems to memorize all the possible combinations of problems that will end up on the test."
Exactly. You have to remember who the loudest complainers in a school are - not the nerds, not the losers, but the keeners. The kids who play the game of earning marks instead of learning to think.
That's who the current system panders to, because they're the ones who scream loudest when it doesn't. And they find it morally offensive if you ever ask them to solve a problem that isn't extremely similar to an example they've already seen.
And they're the ones who think they're "good at math". That's who highschool caters to because that's what standardized testing caters to and that's what the noisiest parents and students want... and teachers that have higher ideals are swimming up-stream if they want to actually challenge the kids.
From teaching a little CS and numerical computing in college and now teaching programming, I agree with you. Too many students come out of high school without the ability to manipulate mysterious symbols without any real understanding of what they mean. Some students only know how to follow scripts to solve a problem from a template. Others need a concrete understanding before engaging with the abstraction, so they can't explore the math in order to gain an understanding. Maybe that's what he was trying to get at with "following obscure steps".
High school teachers, please make your students manipulate mysterious symbols more.
This is what my exact thoughts were. That's why I struggled with Calculus, while I blew through Physics and high school 'math' with no real problems. I didn't need to care about the pedantry because it all just made sense to me. Once it stopped making sense, and it veered off into the abstract, I had to actually study to do well.
It's funny because when that article started I was totally on board with him, then we got there and I just laughed.
As far as I can figure out (from the electronic copy I got my hands on) that quote isn't in Bach's book, but from some glances it would appear some of the concepts might be?
The quote does appear in HHGTG.
[edit: I don't have a copy of the original radio scripts on hand, but from the book:
"The Guide says there is an art to flying," said Ford,
"or rather a knack. The knack lies in learning how to
throw yourself at the ground and miss." He smiled
weakly. He pointed at the knees of his trousers and
held his arms up to show the elbows. They were all
torn and worn through.
"I haven't done very well so far," he said. He stuck
out his hand. "I'm very glad to see you again, Arthur,"
he added.
-- Life, the Universe and Everything
That doesn't mean the phrase doesn't appear on any of Bach's writing, though -- but I've been unable to find it.]
Not being being able to do the symbol manipulation is a different problem. That's what you see from the students who have B's and C's in their previous math courses. The problem discussed in the blog post is what you see from the A students. It's probably more obvious in a class like analysis than in calculus.
I wonder if 'blindly manipulate' just lacks precision.
Isolating a variable is, in some sense, a blind manipulation, but so is, "I see a plus sign, that means I have to subtract". As a first thought about a problem, they represent very different understandings.
when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what they mean.”
The most common complaint I hear about undergraduate students is that they aren't able to blindly manipulate symbols, and consequently get hopelessly confused when they have to deal with "unintuitive" concepts.