Math education: It’s not about numbers, it’s about learning how to think

This is something I've been pondering quite a bit recently. It is my firm belief that mathematical skill and general numeracy are actually a small subset of abstract thought. Am I wrong in thinking that school math is the closest to deliberate training in abstract reasoning that one would find in public education?

Abstract reasoning, intuition, and creativity, to me, represent the underpinnings of software engineering, and really, most engineering and science, but are taught more by osmosis along side the unintuitive often boring mechanics of subjects. The difference between a good engineer of any sort and one that 'just knows the formulas' is the ability to fluently manipulate and reason with symbols and effects that don't necessarily have any relation or simple metaphor in the tangible world. And taking it further, creativity and intuition beyond dull calculation are the crucial art behind choosing the right hypothesis to investigate. Essentially, learning to 'see' in this non-spacial space of relations. When I'm doing system engineering work, I don't think in terms of X Gb/s throughput and Y FLOPS... (until later at least) but in my mind I have a model of the information and data structures clicking and buzzing, like watching the gears of a clock, and I sort of visualize working with this, playing with changes. It wouldn't surprise me of most knowledge workers arrive have similar mental models of their own. But what I have observed is that people who have trouble with mathematics or coding aren't primed at all to 'see' abstractions in their minds eye. This skill takes years to cultivate, but, it seems that its cultivation is left entirely to chance by orthodox STEM education.

I was just thinking that this sort of thing could be approached a lot more deliberately and could yield very broad positive results in STEM teaching.

I teach math at a community college. I've tried many times to teach my courses in such a way that understanding the concepts and thinking were the goals. Perhaps I'm jaded by the failures I encountered but students do not want to think. They want to see a set of problem types that need to be mimicked.

In our lowest level course we teach beginning algebra. Almost everyone has an intuition that 2x + 3x should be 5x. It's very difficult to get them to understand that there is a rule for this that makes sense. And that it is the application of this rule that allows you to conclude that 2x + 3x is 5x. Furthermore, and here is the difficulty, that same rule is why 3x + a x is (3+a)x.

I believe that for most people mathematics is just brainwashing via familiarity. Most people end up understanding math by collecting knowledge about problem types, tricks, and becoming situationally aware. Very few people actually discover a problem type on their own. Very few people are willing, or have been trained to be willing, to really contemplate a new problem type or situation.

Math education in its practice has nothing to do with learning how to think. At least in my experience and as I understand what it means to learn how to think.

> The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right?

It is not, strictly, an improvement on the concept. It's the same concept expressed two ways.

But you cannot symbolically manipulate that prose text, where as you can stick that integral statement into an equation and manipulate it there along with the other expressions of the equation. The ability to perform those manipulations is an improvement.

Is this a US thing? Why would people still think that math is about numbers? Math is about patterns, which got drilled into us by our teachers in primary school. I really don't understand how US education system can fuck up so badly on fundamental subject like math.

As a counterpoint to that, I flat-out despised abstract Math until I got to college precisely because I found it useless. At best I knew I was probably going into engineering in college and I'd been told "engineers use calculus". That was my sole motivation to learn anything about abstract math, and it was pretty bad. It didn't help that the "examples" given in class were mundane toy problems (ex: "how long will this arbitrary tank take to empty?" Well gee I don't know and I don't care. Is this the last tank of water on the island or something?)

Now physics on the other hand, chemistry, computer science, any time I could apply math to a real-world situation got me motivated. I liked Math that could solve actual existent problems, and I learned said math better as a result. The first time I learned about projectile motion I literally started picturing vectors everywhere for the rest of the day.

It also didn't stop me from learning abstraction later in college. In basic signal analysis we learned about the Fourier transform, and then we learned about the Laplace transform and how it was simply a more general Fourier transform (or the reverse if you prefer). Mastering Fourier first gave the Laplace a relatable mental context, it made the abstractions more real.

By contrast I had a Math professor once who tried to teach us theoretical induction and work his way down to examples. I had no idea what the fuck he was talking about until he hit the examples, then everything clicked all the way back up.

Maybe that's just how I'm wired, but I would argue for more and more relevant examples. There are tons of historical uses of math that people would find interesting. Talk about the math used to build the Hoover dam, or telegraphs, fly satellites, or make medicines, or decode Nazi transmissions. And talk about it in real terms with character, if you must simplify for the sake of time then say that's what you're doing. Make people realize they're learning something that has uses beyond Math class or the rarified ivory towers of Math PhDs. That by learning the material they're learning how to do meaningful things. The original, wildly popular SimCity games were basically raw spreadsheets and formulae, but people learned those spreadsheets and formulae because the games had character.

I know to a lot of Math people the abstractions are the "real" part, but most people I know simply don't process the world that way.