Basic Skills vs. Conceptual Understanding: A Bogus Dichotomy in Math Education [pdf]
Comments
red_admiral
AStonesThrow
Math was always really tough for me. My mother professed a "math block" and didn't deal in numbers. My father knew everything but kept the numbers to himself. I do suppose he helped some on my elementary math problems. There was no shortage of calculators, slide rules, and other gadgets lying around our home.
I sometimes surprised people by telling them that as a computer hobbyist/IT professional, I needed to know very little about math. As a programmer it was true. Internet engineering can rely highly on crunching small numbers in terms of addresses, ports and other things, so you get the binary and hex conversions.
But by high school, I was questioning the goals of learning advanced math. I was railroaded into calculus and beyond by college, and my teachers weren't great at giving me real-life scenarios where I'd be using this in my career or daily life, so I sort of checked out. I typically confuse numbers and forget them easily (unless it's the first girl I phoned in high school) and it worked out that I never needed calculus or trig in my career or daily life, ever at all.
Returning to community college recently, I found reductions in math requirements for the IT degrees, and lots of help. There was a supportive tutoring center where they could walk me through anything. I took online classes where I could basically brute-force every quiz for 100%. And if I was inclined to cheat, there were calculus solver websites that could do everything where my HP calculator fell short.
I suppose if I were a city planner, or a farmer-businessman, or going for some MBA to run companies, I'd be interested in calculating volumes of things or predicting financial outcomes, but I'm not and I won't, so these days I'm just amusing myself with finding prime numbers, and factoring the other ones.
sherr
I think this is an important realisation and it brought to mind John Von Neumann, who said to Felix Smith: "Young man, in mathematics you don't understand things. You just get used to them."
I think I have sabotaged by own education in maths (to a degree) by wanting to "understand" the "why" of something rather than just get used to it, practice and get fluent. The understanding can wait and might not even be important.
dartos
I’m in the same boat. I’ve been programming since I was very young, but never did well in math classes.
In programming you can always dig into the “how” something works, but math is descriptive as opposed to programming being prescriptive.
I got it in my head that I never understood math bc I didn’t get how it worked and that destroyed my academic career in math for my whole life.
It wasn’t until I read the first chapter of “Mathematics for Programmers” that it finally clicked for me. The last paragraph had a quote that was like “if this didn’t quite make sense to you, maybe you spent too much time learning how to think like a computer” and it shattered my whole worldview.
Tier3r
I have a similar experience. It took me awhile to understand the main "working area" of math was the manipulation of abstract properties, not the examples of the properties themselves. If you learn math by rote, you learn to identify certain procedures to apply to certain problems of properties, which is effectively manipulating properties following a way that is already known. But you will have near zero transfer. But understanding math is about understanding what properties hold, and what happens under different operations. At a high level, math problems revolve around isolating certain abstract properties initial and derivable relevant to your question. While coding is the opposite, the main "working area" of programming is applying operations to examples. The properties themselves are usually quite basic, although seldom rigorously defined. In more advanced levels you start using math more (intuitively or rigorously), such as isolating invariants in loops.
I've also thought that math syntax is very poorly structured and places enormous cognitive load on the user. In code, you have both formal and technological methods to make reading it more efficient. Languages are rigorously defined by CFGs to produce (largely) unambiguous syntax, while LSPs and other parsers take off a load of the cognitive load in parsing, identifying and recognition. For math it is the opposite. Take how in calculus d looks like a variable but is actually an operator, the concept of which they have never learnt before. One of the elementary things calculus learners struggle with is unlearning the syntax rules they have acquired thus far and trying to unconsciously conceptualise what an operator is. That ambiguity places an unnecessary cognitive load on them.
I wonder if in the future this will change. Looking at the history of mathematics, formulas used to be described in plain language, which likewise placed excess cognitive load because of its ambiguity and unnecessary information. Take the solution to a quadratic from Al-Jabr:
"What must be the square which, when increased by ten of its own roots, amounts to 39? The solution is this: You halve the number of roots, which in the present instance yields five. This you multiply by itself; the product is 25. Add this to 39; the sum is 64. Now take the root of this which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for."
Which is really just x^2 + 10x = 39. It may be beneficial to conceive of some other way to structure the language of math so it can be a more powerful cognitive tool.
dartos
> I've also thought that math syntax is very poorly structured and places enormous cognitive load on the user.
This is a common sentiment coming from programmers turned mathematician.
Math notation is made to exactly describe ideas between mathematicians (read: other humans) not for implementors to have an easy time.
As a programmer first, I tend to agree, but I’ve spent a long time learning how to think like a computer and not like a mathematician.
It’s very reminiscent of the old hacker attitude of “the code is the documentation”
I think one of the key points comes at the very end: teaching algorithms becomes rote learning _if the teacher does not understand the underlying mathematics in depth_.
I wonder how many of the teachers who advocate for "visual" learning instead of algorithms are doing so because otherwise they'd have to admit they couldn't explain why the algorithm works themselves?