There is actually a whole field of research on mathematics education. I'm sure this 28 year old essay is nice, but probably the most up to date and comprehensive guide to evidence-based teaching of college math is this free Instructional Practices Guide from the MAA: https://www.maa.org/programs-and-communities/curriculum%20re...
For K-12 math education, I haven't been keeping as close an eye, but there are several books and resources, such as work by Jo Boaler: https://www.youcubed.org/resource/books/
Math education research is not math research: it's education research, which is among the worst among social science, which is among the worst of all sciences in evidentiary practices. There are almost no valid RCT's in educational research by the standards of non-social science researchers. My own impression is that folks who publish on this sort of thing and not mathematics, like education and do not like mathematics per se.
I've read Boaler's stuff. I agree mostly in form with RJ Milgram's attack on her claims, which basically accuse her of statistical incompetence.
It is possible to be a great research mathematician and to be an awful mathematics teacher. It is not possible to teach great research mathematics without being a great research mathematician. I also do not believe it possible to inculcate in younger students an enduring love of mathematics without having oneself an enduring love of mathematics.
> It is not possible to teach great research mathematics without being a great research mathematician.
I'm not sure it makes any sense to talk about "teaching research mathematics". If it's research mathematics (as in, discovering new math), it should not and cannot be done in a classroom setting.
With that in mind, it is certainly possible to be a good math teacher while being a poor or mediocre researcher. Their students are probably not be prepared for an academic math career, but they can walk away with a solid understanding of established mathematics nonetheless.
I mean, if we are talking about teaching elementary school, then you don't need to be world class researcher to get them ready for next level. You need to like math and, imo, you need to like problem solving, you need to understand math instead of having it memorize etc.
However, you definitely don't need to be actual researcher. I would even argue that it will be more beneficial to study psychology, child development, child behavior etc then spending time doing serious math research.
I'm not talking about elementary school, I'm talking about masters or bachelor's level advanced mathematics that has nonetheless been established for decades or centuries.
All I'm saying is that teaching and researching are different skill sets. One does not prepare you for the other except indirectly, nor is one a prerequisite for the other. One could just as easily say some pedagogy is required to communicate original research. It's true to an extent, but you don't need to be a excellent teacher to be an excellent researcher, or vice versa.
Math teaching until upper years of undergrad is "The Technical History of Mathematics", which really could be taught very well by teachers yoinked by a time machine from the 1800s...
My favorite math teacher was a linguist by training, but had a knack for getting across the intuition behind Linear Algebra. Some of my mathematics researcher acquaintances mentioned that in teaching their own children mathematics, they focus on clarity of thinking and language...
Here's a relevant RPF quote:
"Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning."
― Richard Feynman
> which is among the worst of all sciences in evidentiary practices
Linguistics is worse: It's quite literally based on hearsay.
Edit: I think that's relevant. Perhaps mathematical skill is diametrically opposed to linguistic faculty.
Of course that's nonsense, but it's an observation inferred from the completely separate treatment of these subjects -- and other stereotypes. At least they are seperated to such an effect that being good at one of these is not perceived as prerequisite for the other. If it wasn't for informatics I would never heard the word "logic" in school. And even then, CS is very much focused on number crunching.
In fact, being good at one is often purported as apology a pro pos being bad at the other. I'm not sure why such a schism exists.
This is a nice article. I teach mathematics at a community college and the last section of article resonated with me. What can we do about education of mathematics to improve it?
Years ago it dawned on me that most of the students taking a class of mine needed the class so that they could take physics. It’s been a long time since I took physics and I don’t really know what math problems students in physics classes have. I went to the physics department and asked for a list of types of mathematical mistakes that are common. Give me the list and I’ll make sure I emphasize this in my classes. No response.
I’ve since learned that what my students really need to know to be successful in physics really isn’t covered in course that is a prerequisite. I’m not really sure why I teach the topics I’m required to cover. They mostly are to give my students enough knowledge to take calculus and almost none of them end up taking calculus. I think things are done the way they are because it’s always been done this way.
>>> I went to the physics department and asked for a list of types of mathematical mistakes that are common.
A couple of ideas. First, ask the students. Second, see if you can join in the grading of physics exams, as an observer.
20 years ago, I spent a semester as an adjunct, teaching electrodynamics in the electrical engineering department at a big university. The same semester, I also taught two sections of the freshman college algebra course.
I'm going to make some educated guesses here. The math that's used in physics isn't really high level, but because you're using math en route to something else, you have to be fluent enough that it's not an obstacle to understanding the physics concepts. So for me, just being able to crank through algebraic manipulation using odd symbols and getting it right were important.
My EE students used a handful of basic differential equations that were actually given short shrift in the math curriculum, namely simple linear equations that are conveniently solved using complex exponentials. An example would be any kind of wave phenomena. Being able to bounce back and forth between time space and frequency space got my EE students hung up.
I agree with you about the college math topics. As I understand it, certain topics have to be covered in order to meet the accreditation requirements, which makes it hard to develop new curriculum at the lowest levels where it's the most needed.
I think that for students who aren't going to be STEM majors, I'd rather spend the same course time getting them fluent with Excel. That's how they're going to do math anyway.
That's the Bill Thurston! He's the guy that practically invented low-dimensional topology. He sewed up the entire field of foliation theory, and showed that most knots are hyperbolic. He won the Field's Medal ... he's someone to listen to!
HFS! I just watched the Numberphile video literally today with those two incredible wire memory, oscilloscope display calculators from the 60s. What a phenomenal piece of engineering! Cliff needs his own channel. I could listen to him talk all day.
I was actually expecting this when I clicked on the post. I don't think that Lockhart's criticisms are unique to mathematics, I think it's an endemic problem with the educational system.
Music is more or less taught as Lockhart describes in in the musicians nightmare. Art education isn't quite that dire, but it's close.
If you can't use a standardised test to evaluate somebody in a subject, it's of no value to the educational system.
I completely agree (though it is some years since I was at school).
Still, as Lockhart points out: at least art and music are recognised as arts. Surely even the ropiest music and art educations at least admit that these arts are:
1. ends in themselves, because
2. they are beautiful.
I believe these opinions are entirely absent from most mathematics education.
The following quote from the paper in my opinion applies to coding interview tests:
"The competitions reinforce the notion
that either you ‘have good math genes’, or you do not. They put an emphasis on being quick, at the expense of being deep and thoughtful. They emphasize questions
which are puzzles with some hidden trick, rather than more realistic problems where a systematic and persistent approach is important"
Decades ago when I was in school, East Germany's "polytechnical" 10 year elementary school they sent me to "Mathe-Olympiaden" (at least county-level math contests) and to after-school math courses because they thought I had talent. Maybe I did, but when I saw the tasks they gave us, and when I saw how the others solved it I simply stopped caring and just wasted time. We got a nice food bag with sweets and the day off school, that was good enough for me.
For example, loooong before we ever heard anything about this mathematical field, and this was way before the Internet so unless somebody tells you about something you have no chance of having heard about it as a young kid, we got a combinatorics question. Everybody in the room started writing down every single combination - which is why I knew what they were doing, neat columns and rows filling several pages that could only have been for this particular problem, and I had almost started doing the same. I thought it was WAY beyond silly that I should spend my time on this problem. I knew how to solve it in this simple fashion, there was no chance to come up with the real solution on the spot, so I switched to "annoyed" mode and gave up.
The after-hours math club wasn't much better, nothing deep or substantial, just some riddles.
TODAY I'd just go to Khan Academy, edX, Coursera and ignore my teachers. The Internet is great. Oh and cat pictures.
For K-12 math education, I haven't been keeping as close an eye, but there are several books and resources, such as work by Jo Boaler: https://www.youcubed.org/resource/books/