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I am a math professor at a large state university in the US. A comment and a question: > Mathematicians teach math to do more math, which in turn is used to to

I am a math professor at a large state university in the US. A comment and a question: > Mathematicians teach math to do more math, which in turn is used to to even more math. That is true of some mathematicians, and of some subjects. But I think most mathematicians teach and learn math because they think it's cool, not because it's useful -- either to learn more math, or for the "real world". I might also add that I always appreciate it when teachers take a very long view of things. For example I have trained in the martial arts, and most teachers take the perspective that you want to master the art completely, rather than just learn one or two things. Some people don't like this, but I deeply appreciate it. > The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit." As a math instructor I am always looking for ways to improve, but I'm having trouble extracting useful criticism from your remarks. "They throw it at you and expect you to deal with it" seems to be true of challenging lessons in anything. I would cheerfully welcome general advice and suggestions from you or any other HN readers. (But please keep in mind that both my interests and my expertise skew heavily theoretical!)

worren 15y ago
"They throw it at you..." I've the same gripe. Maybe, I can elucidate the parent's point. I can appreciate the mathematician's fancy for reducing numerical manipulation to an indivisible, atomic precision. Creating poetic abstractions is dear to many in applied sciences. Where it sticks in the craw, where it stops making sense, is where you have this beautiful construct and it's purpose is left a mystery. If it's a piece of art, fine. It's welcome to hang on the wall over there next to the others, but I'm not going to remember it in much detail without some context. If there isn't a concrete use, if there isn't a purpose, the idea won't make it out of short-term memory. "But it's the foundation for all of these other beautiful edifices of logic that we'll cover in the next chapter!" It's like being told, "This is red paint. This is green paint. This is burnt umber paint...", without ever being shown a Picasso, Renoir, or da Vinci. If you ground the presentation with messy, imprecise, real world numbers, the logical purity of the subject will not be lost or sullied, the esteem for it's power only more appreciated. Pull the pin, show us what it can do! You may not be a physicist or engineer, and that their respective topics will be covered in their respective classes. But those same physicists and engineers probably aren't mathematicians. So what business did they have teaching me math... better than a math teacher ever did?
luke_s 15y ago
Slowpoke's comment really resonated with me, so perhaps I can provide some concrete examples. 8 years ago, I graduated from my Software Engineering degree at Swinburne University in Australia. As part of my degree I had to sit through 3 years of engineering maths. In our first programming tutorial we were taught how to write a program to display “Hello world”. It was immediately obvious to me why we might want to write a program that displays messages on the screen and what sort of problems this would be useful for solving. In our first maths tutorial we were taught how to add, subtract and multiply complex numbers. To this day, I still do not understand why I would need to do that. Where do my complex numbers come from? What do they represent? When I add them together, what does the answer mean? The only problems that I can apply my maths education to, look exactly like those on the tests: what is (-3.5 + 2i) + (12 + 5i) ? It is possible that in my job I am besieged daily with problems that I could use complex numbers to solve. But if so, I am totally incapable of recognizing them! Fourier series were a particularly egregious example – the subject started, when the lecturer came in and wrote up 3 boards of dense maths, and said something along the lines of “… and this is the formal derivation of a fourier series!” It was as if somebody had tried to teach programming by explaining the algorithm a complier uses for translating source code into machine code. Then expecting the students to just figure out how to write actual useful programs, all by themselves! I believe this is what Slowpoke was talking about when he said: “The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit." The way we were taught fourier series particularly hurt. Years later, I found out by myself, that the things are actually incredibly useful. As it turns out, that there are these things called fast fourier transforms, that programs use all the damn time, to do fantastic stuff! So, to give a very concrete example of how maths education in universities could be improved: If only the first lecture on fourier series had instead explained what they are used for, and why they were so important that we were going to spend 5 weeks on them. Then perhaps I would have had a much better understanding of what they are and how to use one. It helps so much, to be able to think as the lecture is writing boards filled with formulas: “Shit! Now I can use this to do X!” Thanks for taking the time to listen. It’s such a shame that maths is being taught this way, because mathamatics is both so very useful and very important… Edit: Actually I reflection I sat through 3 years of maths, not 4
slowpoke 15y ago
>But I think most mathematicians teach and learn math because they think it's cool, not because it's useful That's exactly the point I'm making. That's math for math's sake - which is perfectly okay if you're actually studying math, because in that case it's what you signed up for in the first place. The problem is if you study something else (for example, Computer Science, or Physics), and are thrown into the exact same math lectures, with no one telling you how all that fancy math is actually useful for anything but doing even fancier math. Only after the mentioned experiences at my internship, where I had the chance to work and talk with people who actually do applied math did I see the point of it. What I think is that mathematicians - and I don't mean this as an accusation - are often unable to understand that while math for math's sake works fine for themselves, it's not really the best approach for people who want to actually apply math to their own field of science.
bpyne 15y ago
Here are a few observations based on my undergraduate days as a math major. Engineers, mathematicians, and anyone in the sciences together took a 3 term calculus sequence and ordinary differential equations. After ODE in our second year, disciplines started to split apart. The calculus courses were taught by math professors and their TA's. Calculus was presented through exercises with a small tribute to proofs. Right here is I think where things went wrong. The engineers really didn't care about proofs and, as important, the exercises were learning by rote. At no time were engineers given anything practical to do with calculus. They would have been better served by exercises that led to building something: a model bridge, a model circuit, anything that makes the use of calculus more concrete. The mathematicians were concerned with why calculus works but weren't given a good grounding in proofs. Consequently, I was not prepared for the sudden jump in proof maturity when I took real analysis during my fifth term. Abstract algebra equally boggled my mind the following term. Rather than study calculus with engineers, I would have been better served by learning calculus through historical development with emphasis on the reasoning process the great mathematicians went through in developing calculus. Engineers and mathematicians had diametrically opposed purposes for learning math yet were forced through the same curriculum that was a compromise between the two disciplines. Most likely the compromise was due to resources, but it did nothing to make math more interesting to either student engineers or mathematicians. As an aside, my math adviser asked me about continuing to graduate school. I replied that I had no confidence in my ability to understand higher math. His reply, "Oh, as an undergraduate we largely leave you to figure things out for yourself. Once you're a graduate student we start showing you how math really works." This is not a good policy for attracting potential mathematicians. I too have been practicing martial arts for close to 20 years. You see the same difference in purpose there as well. There are people who practice martial arts and then there are martial artists. Martial arts schools have the luxury of letting the 2 groups grow apart over time. Universities have 4 years to get the 2 groups developed for different purposes. They need to become more aware of the time limitation.

worren 15y ago

"They throw it at you..." I've the same gripe. Maybe, I can elucidate the parent's point. I can appreciate the mathematician's fancy for reducing numerical manipulation to an indivisible, atomic precision. Creating poetic abstractions is dear to many in applied sciences. Where it sticks in the craw, where it stops making sense, is where you have this beautiful construct and it's purpose is left a mystery. If it's a piece of art, fine. It's welcome to hang on the wall over there next to the others, but I'm not going to remember it in much detail without some context. If there isn't a concrete use, if there isn't a purpose, the idea won't make it out of short-term memory. "But it's the foundation for all of these other beautiful edifices of logic that we'll cover in the next chapter!" It's like being told, "This is red paint. This is green paint. This is burnt umber paint...", without ever being shown a Picasso, Renoir, or da Vinci. If you ground the presentation with messy, imprecise, real world numbers, the logical purity of the subject will not be lost or sullied, the esteem for it's power only more appreciated. Pull the pin, show us what it can do! You may not be a physicist or engineer, and that their respective topics will be covered in their respective classes. But those same physicists and engineers probably aren't mathematicians. So what business did they have teaching me math... better than a math teacher ever did?

luke_s 15y ago

Slowpoke's comment really resonated with me, so perhaps I can provide some concrete examples. 8 years ago, I graduated from my Software Engineering degree at Swinburne University in Australia. As part of my degree I had to sit through 3 years of engineering maths. In our first programming tutorial we were taught how to write a program to display “Hello world”. It was immediately obvious to me why we might want to write a program that displays messages on the screen and what sort of problems this would be useful for solving. In our first maths tutorial we were taught how to add, subtract and multiply complex numbers. To this day, I still do not understand why I would need to do that. Where do my complex numbers come from? What do they represent? When I add them together, what does the answer mean? The only problems that I can apply my maths education to, look exactly like those on the tests: what is (-3.5 + 2i) + (12 + 5i) ? It is possible that in my job I am besieged daily with problems that I could use complex numbers to solve. But if so, I am totally incapable of recognizing them! Fourier series were a particularly egregious example – the subject started, when the lecturer came in and wrote up 3 boards of dense maths, and said something along the lines of “… and this is the formal derivation of a fourier series!” It was as if somebody had tried to teach programming by explaining the algorithm a complier uses for translating source code into machine code. Then expecting the students to just figure out how to write actual useful programs, all by themselves! I believe this is what Slowpoke was talking about when he said: “The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit." The way we were taught fourier series particularly hurt. Years later, I found out by myself, that the things are actually incredibly useful. As it turns out, that there are these things called fast fourier transforms, that programs use all the damn time, to do fantastic stuff! So, to give a very concrete example of how maths education in universities could be improved: If only the first lecture on fourier series had instead explained what they are used for, and why they were so important that we were going to spend 5 weeks on them. Then perhaps I would have had a much better understanding of what they are and how to use one. It helps so much, to be able to think as the lecture is writing boards filled with formulas: “Shit! Now I can use this to do X!” Thanks for taking the time to listen. It’s such a shame that maths is being taught this way, because mathamatics is both so very useful and very important… Edit: Actually I reflection I sat through 3 years of maths, not 4

slowpoke 15y ago

>But I think most mathematicians teach and learn math because they think it's cool, not because it's useful That's exactly the point I'm making. That's math for math's sake - which is perfectly okay if you're actually studying math, because in that case it's what you signed up for in the first place. The problem is if you study something else (for example, Computer Science, or Physics), and are thrown into the exact same math lectures, with no one telling you how all that fancy math is actually useful for anything but doing even fancier math. Only after the mentioned experiences at my internship, where I had the chance to work and talk with people who actually do applied math did I see the point of it. What I think is that mathematicians - and I don't mean this as an accusation - are often unable to understand that while math for math's sake works fine for themselves, it's not really the best approach for people who want to actually apply math to their own field of science.

bpyne 15y ago

Here are a few observations based on my undergraduate days as a math major. Engineers, mathematicians, and anyone in the sciences together took a 3 term calculus sequence and ordinary differential equations. After ODE in our second year, disciplines started to split apart. The calculus courses were taught by math professors and their TA's. Calculus was presented through exercises with a small tribute to proofs. Right here is I think where things went wrong. The engineers really didn't care about proofs and, as important, the exercises were learning by rote. At no time were engineers given anything practical to do with calculus. They would have been better served by exercises that led to building something: a model bridge, a model circuit, anything that makes the use of calculus more concrete. The mathematicians were concerned with why calculus works but weren't given a good grounding in proofs. Consequently, I was not prepared for the sudden jump in proof maturity when I took real analysis during my fifth term. Abstract algebra equally boggled my mind the following term. Rather than study calculus with engineers, I would have been better served by learning calculus through historical development with emphasis on the reasoning process the great mathematicians went through in developing calculus. Engineers and mathematicians had diametrically opposed purposes for learning math yet were forced through the same curriculum that was a compromise between the two disciplines. Most likely the compromise was due to resources, but it did nothing to make math more interesting to either student engineers or mathematicians. As an aside, my math adviser asked me about continuing to graduate school. I replied that I had no confidence in my ability to understand higher math. His reply, "Oh, as an undergraduate we largely leave you to figure things out for yourself. Once you're a graduate student we start showing you how math really works." This is not a good policy for attracting potential mathematicians. I too have been practicing martial arts for close to 20 years. You see the same difference in purpose there as well. There are people who practice martial arts and then there are martial artists. Martial arts schools have the luxury of letting the 2 groups grow apart over time. Universities have 4 years to get the 2 groups developed for different purposes. They need to become more aware of the time limitation.