Sunday, February 13, 2022

Teaching Math

Quillette has an article It's Time to Start Treating High School Math Like Football, which has the premise that we should focus math resources on those who want to learn it, not require everyone to study it.  He makes the analogy to football, where we produce many excellent players year after year, even though we do not require everyone to play football in order to see how good they are at it. It's not a crazy idea, and has some good reasoning behind it.  But it is wrong.

Math is not taught because some huge percentage of us are going to use the knowledge side-angle-side can be used to prove that triangles are congruent, or that taking the derivative of an equation will reveal the changes in its slope. At levels beyond the simplest algebra and geometry, it is taught because it is abstract and each step depends on the one before it. We have the student build and demonstrate their ability to handle abstraction in a fairly narrow time span of a few years, to see how far they can rise before the material is beyond them and they struggle. It is useful as a measuring stick more than as an ability that will be required for adulthood.  Therefore, when a student has reached their limit in math and they cease taking it, we ask no further of them.  They are done, we have our answer.  The practical math of percentages, fractions, and statistics we continue to keep fresh because they will actually use those things. The remainder, no.

We don't have to use those specific types of math in the order that we have devised. We could design it differently, so long as the principle of increasing abstraction were maintained.  Heck, we would even have to use math if we used some other topic using similar amounts of abstract reasoning and facility. I am willing to be convinced there is something better than finding the area under a curve, already an arbitrary choice.  I just haven't seen anyone make a case for that.  When people advocate away from teaching abstract math, they invariably choose something of much lower abstraction, sometimes a skill that is entirely unrelated.  These are often very useful skills worth having and measuring, requiring perseverance or social skills, or popular but less-useful abilities such as the ability to parrot political cliches.

But not the same skill as math.  We are attempting to discover the ability to think abstractly and this is currently the best method we have.  Verbal analogies aren't bad, but not quite so efficient at this.


james said...

We have lots of choices for what to teach: The New Mathematical Library was publishing new material not that long ago.

Most non-math/non-science folks I asked either liked geometry or algebra and found the other difficult. Both involve abstractions, but geometry seems to be more logic-based and elementary algebra more tool-based.

Grim said...

This is the function of the Deductive Systems course -- a kind of mathematical symbolic logic -- at the university I went to last. Ostensibly the course is to teach you advanced logic, but in fact it's a filter. If you can't think that abstractly, rigorously, and logically, you have no business in certain programs.

engineerlite said...

Looks like this is really a dodge of the diversity police, because certain ethnicities and sexes underperform in math, and it’s hard to fudge the numbers. There is no ethno-specific way to add. As a diversity dodge, I don’t see how this can work. Creating a society where one identifiable group doesn’t do math exacerbates the problem.

From a purely educational point of view, I think middle-high school is too early to write off all the careers and learning opportunities that will depend on math, much less civic responsibilities. Math is no less fundamental to a community than reading, and we wouldn’t consider allowing publicly educated children to opt out of reading or writing.

How long before we start having the conversation about how much economic inefficiency we’re willing to tolerate to appease the diversity crowd?

David Foster said...

It's not just about diversity-appeasement, though. There are an awful lot of adults today who studied the standard math programs in K-12 and whatever was required in their colleges, but have no real understanding of math beyond basic arithmetic.

This includes a lot of people in pretty influential decisions. As I noted in a comment at the Quillette thread:

"In the US, at in many other countries, mathematical modeling is an increasingly important factor in major policy decisions, whether those decisions are about economics, covid, climate.lots of things. What does the typical US congressman think when he hears the term ‘mathematical model’? For most of them, I’m afraid, it’s like querying the oracle of Delphi, or reciting a supposedly-powerful phrase like ‘The Gostak distims the Doshes.’

Yet most of these individuals have gone through typical-or-better US school and college curricula. They mostly probably learned some procedural stuff, and promptly forgot most of it.

There are definitely changes needed in the way math is taught (ditto for science), but this shouldn’t mean abandoning it for all but future specialists."

I suspect that most people would get a better feel for the abstract nature of a Variable by learning a simple programming language than with the standard approach to algebra...with the programming language, you can actually *do things* with the variable, it doesn't just sit there. I also suspect that most people would get a better feel for calculus by learning a little bit about how differential equations work and how they are solved *step-by-step*, ie numerically. To something said by Neptunus Lex, who Remarked that he had not done terribly well at math in high school and the first two years of college:

"It was not until my junior year at the Naval Academy, when we started to do differential equations, that the light came on. Eureka! Drop a wrench from orbit, and over time it would accelerate at a determinable pace, up until the moment when it entered the atmosphere, where friction would impede the rate of acceleration at an increasingly greater rate (based on air density, interpolated over a changing altitude) and that wrench struck someone’s head at a certain velocity, that any of this applied in the real word. By then it was too late, I was too far gone, and an opportunity was lost."

Lex was an obviously brilliant man with a great deal of intellectual curiosity, and I suspect that there are alternative approaches that would have worked much better in expanding his knowledge of math.

Assistant Village Idiot said...

Coupla things - Steve Sailer, following John Derbyshire's estimate, suggested that Algebra II was about the break point for abstract thinking. That seems about right to me. It seems that someone at Grim's college broke the code that it didn't have to be math per se for higher abstraction measurement, though it does have the added advantage of being practical for quite a ways. I have always thought that teaching more probability, statistics, and decoding graphs in high school would be useful.

As for that story about Neptunus Lex, it rings true, and reminds me of my puzzlement learning that CS Lewis was not good at math. He was tightly logical in many ways, used examples from mathe and science as illustrations, and his mother took a math degree and was a tutor. It made no sense. Later I learned that at the school he called "Belsen" in Surprised By Joy students were whipped for getting wrong answers in math, and whipped for getting correct answers as well. It could put one off math.

David Foster said...

Vannevar Bush, who would become FDR's science advisor in WWII, created in 1931 a system which could solve differential equations mechanically. Writing about a machinist who worked on the machine's construction, Bush said:

"I never consciously taught this man any part of the subject of differential equations; but in building that machine, managing it, he learned what differential equations were himself … it was interesting to discuss the subject with him because he had learned the calculus in mechanical terms — a strange approach, and yet he understood it. That is, he did not understand it in any formal sense, he understood the fundamentals; he had it under his skin."

There's a group at Marshall University which is pursuing the idea that such mechanical differential analyzers can be a useful educational tool in teaching math. See my post here:

Assistant Village Idiot said...

I once read (James Gleik's Chaos, I believe) that analog computers were used to plot the trajectories of artillery, and these gave a different understanding of both the trajectories and of computing in general. Makes sense.

David Foster said...

Yes...the mechanical differential analyzers were used in the WWII era to calculate trajectories. Took about 30 minutes per trajectory, and slightly less accurate than hand calculation (which required breaking down the trajectory into small intervals of time and repeating the calculation for each--very labor-intensive)

The first programmable electronic computer, the Army's ENIAC, was built specifically to solve this problem, though it was implemented in a manner which allowed it to also do many other things.