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The trouble with trying to arrive at any single definition of Mathematics is that Mathematics is different things to different people. A research level Mathematician might see it differently (finding patterns, abstraction, theory - axioms and proofs) from an Engineer who has a purely practical interest in it (cookie cutter methods and formulas). For everyday use Mathematics is a set of algorithms for doing stuff with percentages, fractions, basic arithmetic etc.

Currently what's frustrating is that we subject everyone to the same level of training. Those whose interest lie elsewhere are subjected to unnecessary and meaningless torture, while those who are really interested in Mathematics as a subject are left to their own devices to find additional training (which thankfully is easier these days than when I was a kid, though it could be made substantially easier still).

You beautifully sum up my experience with math at the university, as a CompSci student in Germany. The absolutely most frustrating thing for me, in the beginning, were the basic math lectures (which I, as a disclaimer, still haven't finished). At first, of course, I did what most students do in their first semester: bitch about the apparent uselessness of the subject.

That's about one and a half years back, now. I realized by myself that math is actually pretty useful. 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."

Personal eye openers for me were my internship (and continued assistant work) at a department of the university that deals with image processing (which is basically a boatload of applied math), articles like Wolfire's excellent "Linear Algebra for Game Developers"[1], and plunging myself into a lecture about Fuzzy Set Theory and Artificial Neural Networks (which again is applied math).

I actually like abstractions, generalizations and logic reasoning (I wouldn't study CS if I didn't). I still don't think that math is the only way to teach those, but it's arguably the largest field of science dealing with these things.

I don't really think the problem is math itself, it's the way it is taught to most people - in the math way. Mathematicians teach math to do more math, which in turn is used to to even more math. And that's fine - for a mathematician. For literally every other field of science that needs math as a tool, it's highly frustrating to be subjected to what you aptly describe as "torture".

[1]: http://blog.wolfire.com/2009/07/linear-algebra-for-game-deve...

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!)

"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?

>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.

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.

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

If you had enrolled in electrical engineering then by half way through second semester of first year you probably would have been having lots and lots of light bulb moments. Complex numbers and fourier series and transforms rapidly become like oxygen for EEs. Essential but consumed almost without noticing.

You suffered from the university's lack of bandwidth -- only so many lecturers, only so many lecture theatres, many other competing classes. They try to find efficiencies where they can -- e.g. by giving all of the engineering departments a common mathematics curriculum.

That curriculum is almost certainly driven by the needs of the EEs and ME/CivE's much more than chem and software, and I'd guess software engineering is looked at is if there's really no maths required. After all, software engineers just program and write documents, don't they? The truth is of course that there is just as much scope for maths, if not more, in a software engineering or computer science course, it's just a whole other type of maths that's needed.

Hmmm ... It is entirely possible that the EE's went straight out of that maths tutorial and into an electronics class that used complex numbers for a practical application.

To be honest all us software engineers understood that we didn't make up the majority of maths students. I would have been perfectly happy to hear about how I could have used complex numbers to design circuits or build bridges. But I guess the maths classes were designed to provide the pure theoretical foundations upon which other, more subject specific classes would build. It just so happens that for software engineering there were no subject specific classes which made use of the maths.

Thanks luke_s.

However, after reading your comment I still don't know what to do about it. As far as applications I'm familiar with... well, they are typically to other areas of math, as Slowpoke was complaining about.

Since you mentioned Fourier series, please imagine that I am about to teach an undergraduate course in the subject, and I want to follow your advice. However, I don't know jack shit about EE or any other practical application of Fourier series. What would you do in my shoes?

Hmmm, well its a very interesting question. I'm not entirely sure what I would do in your shoes, but some places start looking may be:

Why are Fourier series on the curriculum? There must be a reason why they are so important that a whole undergraduate course would be dedicated to them.

How and when were Fourier series first described? I understand Joseph Fourier made some pretty major contributions to maths. What was he trying to achieve? Did Fourier series help him do it? Why did other people pay attention, and how was the idea popularised?

Finally, if you really don't know anything about EE or any other applications for it may be a good idea for you to lean a bit more about the various fields that are built on top of yours. Having a better understanding about how your area of interest relates to others is always a good thing.

I know from a software point of view, even though they are hidden from day to day coding, Fourier series are hugely important. Fourier series allow us to compress audio and images down to sizes where they can easily be transferred across the internet [1]. Almost every single digital image you see, song you listen to and movie you watch will have had a fourier transform applied to it. Did you know that movies on netflix account now account for 32% of north american internet traffic? Without the fourier transform its safe to say the world wide web as we know it would not exist.

Furthermore, because it is actually practical to transfer movies and songs across the internet, mass piracy of media is possible. Millions of people are sharing (fourier transform compressed!) movies over sites like the pirate bay. This has lead to a backlash from established media corporations demanding stronger copyright protections. Currently a huge legislative battle is being fought, which will have a impact on such disparate areas as the future of censorship, what rights people have over the things they create and the role of money in politics.

All because of what the fourier series lets us do!

This is, to put it mildly, of some interest :-)

[1] : http://en.wikipedia.org/wiki/Discrete_cosine_transform

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