My Answers to “Six Questions Math Educators Need to Answer Honestly”

[Questions can be found here.]

1. What is mathematics?

Don’t know. Pass.

2. Where should we learn mathematics?

Don’t know what “should” means here. Pass.

3. Should mathematics be a compulsory subject?

Question: What would the immediate effect of changing of making math non-compulsory tomorrow would be?

Answer: Worse teaching.

We’ve had decades (centuries?) to learn a bit about how to teach what we teach. I’m not saying it’s easy to teach systems of inequalities well; I’m saying that if I’m going to teach systems of inequalities there are dozens of resources I can turn to, curricula that have thought carefully about how to put the pieces of the course together to reach many students.

Have you ever taught a new course or an elective? It’s hard, isn’t it, not knowing how to go about the work. A world where we radically change the curriculum or where we eliminate mandatory classes is that world. It’s not like number theory and graph theory is easier to teach than polynomial division. Yeah, number theory is really cool, but it’s hard to turn cool things into learning. That’s the whole art of teaching.

So go ahead, make math non-compulsory. Honestly, I don’t care, what we ask kids to study in school is mostly arbitrary anyway. In the 1920s it was unclear whether math would be a compulsory subject in high schools — NCTM was created, in part, to protect math compulsory math offerings. If they’d lost that fight, who knows what hoops we’d be asking students to jump through before graduating. Mandatory home ec? Mandatory workshop? What reason do we have to think that these mandatory offerings would be better taught?

And if we make most of the curriculum non-mandatory? That’s like college, and if you don’t like how k-12 is taught then, wow, get excited for higher ed.

The fact is that there are tremendous pedagogical benefits from having a lot of people teaching and learning the same thing. School requirements are arbitrary, but there are benefits to keeping things the way they are.

4. How should we learn mathematics?

Too broad, pass.

5. Who should learn mathematics?

Don’t know what “should” means here, pass.

6. What is the purpose of mathematics education?

Tempted to pass, but I think there’s something interesting to think about.

The purpose of math education depends on the purpose of math, and the purpose of math has varied across history, across cultures, across math departments, varied so much in the time of Fermat that there wasn’t even a single thing called “mathematics.” From The Mathematical Career of Pierre de Fermat:

With due allowance for exception, one may sort those practitioners into six broad categories: the classical geometers, the cossist algebraists, the applied mathematicians, the mystics, the artists and artisans, and the analysts. Although only one or two of these categories constitutes what one might call a “school” of mathematics, and although the work of many individuals falls into several categories, nonetheless each category distinguishes itself from the others by characteristic attitudes toward the nature and purpose of mathematics, its problems and methods of solution. Each category has a distinctive style, and the different styles often conflict in essential ways.

When people talk about the “values of math” or the “purpose of math education,” know that the question is ill-formed. Math education emerges from the competing needs of students, parents, communities, our government. It’s inherently contradictory, and given that the conflicts are genuine it probably should be sort of a mess. Teaching is not hampered by these conflicts, teaching is the job of making sense of these conflicting needs. That’s the job.

Math in 2018 is used for joy, pleasure, getting rich, pursuing justice, attacking foreign governments, passing tests, teaching, and dozens of other purposes too. When educators say “we have to get back to the values of math” that’s a lateral move; that’s no clearer. Math has as many purposes as math education.

“Math education is a very young child of mathematics,” Singh writes, a completely incorrect statement. One thing I learned from Learning Modern Algebra is that pedagogy is a major source of mathematical innovation. Part of why you’d need a way to find Pythagorean triples is to help you come up with nice problems for your students.

Singh calls math education “a spoiled and rotten child” of mathematics, as opposed to mathematics, whose purposes are clear and pure:

Mathematics is not up for debate. It is what it is, and it has been tattooed in so many civilizations and cultures for many millennia. Its purpose and mandates, might run parallel to math education at times, but in reality, they operate on a higher plane of truth, justice, beauty, play and love 

Historically speaking, this is nonsense. Math changes, math has different purposes. Math is contradictory, math education is contradictory. Math is beautiful and ugly, so is math education. In math education we’re not the spoiled child of math — we’re the grown sibling, sometimes in touch, sometimes not.

So I’m entirely against Singh’s idea that we can look to the values of math and measure math education against them. In sum: Stop pooping on math education and educators. The end.

When I say people, I mean mathematicians

On the wave equation and the beginnings of Fourier analysis:

“Very soon after that, people began…when I say people, I mean mathematicians.”

I only know Rudin from his analysis books, but come on, man! What, are you worried that someone will think mere people would be interested in this?

Another not great moment comes a few minutes later:

“D’Almbert raised a rather strange objection to that. This is a curious objection to my mind, because these were sophisticated people, and why d’Almbert couldn’t see…is hard to believe.”

It’s a lecture on history of mathematics. Rudin warns us at the start that mathematical history is understudied, and that mathematicians aren’t exactly qualified for it. I’m glad he had that awareness, because history starts right here, at that moment of disbelief. The historian has the tools and penchant to turn that disbelief into curiosity.

To be fair, Rudin offers an answer…

“Part of the problem was that the concepts were not well-defined.”

…though this is just another way of saying that they d’Almbert didn’t know modern math. But what was he thinking? Why did the objection seem natural to him? Why did it make sense? (There’s something sort of similar between some of the questions the math teacher and the math historian have to deal with.)