[CallMeSpooky]

This may be partially a rant, but I also have a suspicion that some of the people here will be able to provide bits of wisdom that I will find useful.

I have, lately, been frustrated a bit by my own mathematical ability or knowledge, as I started to do more serious research. I'm not bad at maths, though. In scool, for what it's worth, I've always been one of the best (except in geometry, as I absolutely lack spatial/visual thinking). I've always been able to do the problems and to see the connections between concepts, like something being just a special case of something else. And I've always been fast at learning new stuff, usually being the first to pick up how something's done. The mathematics I face now is different though.

There are two main things that give me trouble.

One is proofs. I usually just don't get them, and when I do, it takes me a long time to. Throughout my formal education, the most difficult math problem I've ever had to do was what is actually a fairly simple proof by induction. Just the typical, prove that something holds for all natural numbers. It took me many tries to do that proof properly, and even then it didn't really "click" in my mind. Same trouble happened through the rest of maths education in university. It was hard for me to do proofs, I never knew where to stop, as in where the point comes that I've actually proven something!

The other problematic thing that I especially note now is the mathematical notation. Some papers in my field are written by people with a background in maths, others not. The ones written by maths people are much harder to read. As an example, suppose the paper says something like, let F be the sum of all even positive somethings, and let G be the sum of all odd negative somethings. That's much, much easier for me than if the paper says let F be... and then spells it out in mathematical notation. It gets significantly worse when more complex processes are being described. I see, let F define... notation..., then we define a transformation of F ...notation... My brain initially skips over most of that, and working out the meaning of what it says takes me too long. I know it takes longer than it should. I will very much appreciate anybody who has tips on getting used to working with this.

Now that I am more about trying to use maths for actual problems, it's difficult to figure out where to begin. As in, which area of maths even to use. There's an intimidating amount of stuff. How do I get an intuitive grasp of what will lead me where? Maybe I need to transform this thing to a wavelet? Maybe I just apply various statistical measures to it?

Together, this has led to some frustrations for me. Such as when I'm working on something, I *know* some property holds for it, but I have no idea of how to prove it. Or when I know roughly where I want to get, but have no idea which mathematical toolkit to use to get there. I realize part of this is down to a lack of experience with doing this. It is - obviously - not the same as just doing problems in a maths class. But I also have a strong feeling that I am missing something, some gap in my knowledge that I can fill.

Really, this is sort of a disjointed rant, but I will very much welcome comments.

[roninpro]

I think that part of the problem might be your perspective on mathematics. At least from what I've observed, many math majors believe that they are "good at math" from their experience in high school and some very basic university courses, but when they try to do something more advanced, they suddenly become bad. In the end, it turns out that those people were really just good at following directions and had no understanding of what they were doing. Maybe you really need to sit down and decide if this is the case for you. When you learn a new technique, do you really understand what it means? Or do you just approach it like a recipe and apply it to made-up examples? (This probably addresses your problem with applying math to real situations, as well. Perhaps you really don't understand the content and context of what you have learned.)

This kind of leads to your issues with proofs. You need to think about what a mathematical proof is: it is a justification for a mathematical statement. Of course, there are varying degrees of "proof", ranging from a loose and informal argument to fully rigourous and very formal reasoning. Nonetheless, the goal is always the same - to convince others of a fact. Just to see where you are, how would you prove (i.e. justify; convince others) that an even number plus an even number is another even number?

Finally, I just want to mention that I share your issue with mathematical notation. I've found that a lot of mathematical literature is just a flood of symbols and formalism, to the point where it is completely unreadable. On top of that, often very little is said about what a result means or why it is important to anybody. (I've even heard that some editors are guilty of perpetuating this. An author providing more than a line or two of explanation can be flagged down.) I think that it's a problem in the field that eventually needs to be handled. Unfortunately, the only thing that I can recommend for now is to try to increase your vocabulary of mathematical notation. There's sadly no way around it.

Best of luck.

----

Afterthought:

Sorry, I forgot to mention one other thing. Geometric reasoning is one of the most important aspects of mathematics that you need to develop. These days, almost all traces of geometry have been removed from textbooks and literature, but it is always there in the background. Many theorems that have a ridiculously messy algebraic presentation can often be reduced to a simple picture and a few words. And on top of that, their proofs follow a (hidden) geometric outline.

The best mathematicians out there always have a picture in mind, and I think that it would serve you well if you could have one too.

[CallMeSpooky]

Good points. For clarity, I'm not a maths major - my degree is in computer science. And I think I have no illusions about my skill, when I say I'm decent at maths, I am comparing myself to the average student. I know I had less trouble with it than most in school and university. I realize that, if I compare myself to people actually studying math, I'm pretty crap.

I think part of my problem is the transition to not using maths in an organized way. Courses are always organized. With the vast majority of topics, I've been able to do much better than just treat them as ready-made recipes. But the meaning I understand is in the mathematical sense. Take something simple, like function derivatives. I know how to do it and I understand the mathematical significance. I understood it well enough, actually, to intuitively make some correct conclusions before they were mentioned in class, like f(x) = |x| not having a derivative at 0. The problem for me comes in tying these mathematical meanings together with other engineering or real-life concepts. Since I am doing my own research instead of doing courses, maths become a tool for the research and for understanding papers of others, but that can have much less context.

Say, I understand what a wavelet is and can apply a wavelet transform, but it doesn't occur to me that a wavelet transform is a good way of achieving lossless compression on graphical data. Or does this mean exactly that my understanding is not deep enough?

Good example there on the even + even = even proof. I know the correct proof to that one, but it didn't occur to me originally until I got a hint. In what's a typical fashion for me, I am instead tempted to construct a logically sound but non-exhaustive argument. For instance,

2 is the smallest even natural number, and 2 + 2 = 4, which is even, and also 4 + 4 or 4 + 2 are even. So when we add an even number (2) to an event number (4) that we, in turn, got as a sum of even numbers, it seems that even + even is always even.

And the problem there is that I don't myself make the leap to generalizing it, to saying, instead of "2 is even", that we can write an even number as 2x, and thus the addition of two even numbers as 2x + 2y. When I get the hint about writing those down generically, I figure out that addition of two even numbers is 2x + 2y and from there the obvious step that 2(x+y) is an even number. Would I have figured it out by myself? Probably, but after a disproportionate amount of time.

Or a more recent, real example of me failing to generalize. I was working with some normally distributed function f(x), and observed some useful properties. I saw they will hold for a function as long as f(0) > f(1) and f(0) > f(-1). Simple enough. But it took me way too long to realize that this, in fact, means it will hold for a function with a Cauchy or Laplacian distribution as well - that I'm working with a certain special case but can make a broader conclusion.

[roninpro]

	Originally Posted by CallMeSpooky To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Good points. For clarity, I'm not a maths major - my degree is in computer science. And I think I have no illusions about my skill, when I say I'm decent at maths, I am comparing myself to the average student. I know I had less trouble with it than most in school and university. I realize that, if I compare myself to people actually studying math, I'm pretty crap.

No problem - I wasn't assuming that you were a math major, but I referenced them for an example, since I deal with them the most. Some of the things that I do are directly mixed in with computer science (algorithms, complexity theory, etc..., specifically related to computations in finite fields), and the people from that department are really good. I'm sure that you're not as bad as you claim.

I think part of my problem is the transition to not using maths in an organized way. Courses are always organized. With the vast majority of topics, I've been able to do much better than just treat them as ready-made recipes. But the meaning I understand is in the mathematical sense. Take something simple, like function derivatives. I know how to do it and I understand the mathematical significance. I understood it well enough, actually, to intuitively make some correct conclusions before they were mentioned in class, like f(x) = |x| not having a derivative at 0. The problem for me comes in tying these mathematical meanings together with other engineering or real-life concepts. Since I am doing my own research instead of doing courses, maths become a tool for the research and for understanding papers of others, but that can have much less context.

This is already really good and definitely shows understanding. Having a little intuition goes a very long way.

The way I learn mathematics is actually very disorganised - I usually look through Wikipedia for something I might have heard about in passing, and one topic leads to another, ending up in a sprawl of things I have looked at. It may not be bad at this stage to just go out and explore freely. Just find a topic that is related to whatever you are doing and look around for some good commentary on it. (I know that it is exceptionally difficult to find, especially in newer results.) I think that it is surprising how much you can gain by doing this.

	Say, I understand what a wavelet is and can apply a wavelet transform, but it doesn't occur to me that a wavelet transform is a good way of achieving lossless compression on graphical data. Or does this mean exactly that my understanding is not deep enough?

This is probably an example of an ingenious observation from somebody who happened to be working on wavelets and compression at the same time. I probably wouldn't expect anybody to just make the connection out of the blue like that, so you shouldn't really compare yourself to it. However, the moral of the story is that having a broad range of mathematics that you have understood will increase your likelihood of making such a connection in your own work.

Good example there on the even + even = even proof. I know the correct proof to that one, but it didn't occur to me originally until I got a hint. In what's a typical fashion for me, I am instead tempted to construct a logically sound but non-exhaustive argument. For instance, 2 is the smallest even natural number, and 2 + 2 = 4, which is even, and also 4 + 4 or 4 + 2 are even. So when we add an even number (2) to an event number (4) that we, in turn, got as a sum of even numbers, it seems that even + even is always even. And the problem there is that I don't myself make the leap to generalizing it, to saying, instead of "2 is even", that we can write an even number as 2x, and thus the addition of two even numbers as 2x + 2y. When I get the hint about writing those down generically, I figure out that addition of two even numbers is 2x + 2y and from there the obvious step that 2(x+y) is an even number. Would I have figured it out by myself? Probably, but after a disproportionate amount of time.

The basic issue here is that you have relied on a very specific instance of even + even, which I'm sure you realise isn't a strong argument. It obviously doesn't show that it is guaranteed to work for some extreme case, say, two even numbers of astronomically large size. (The same criticism can be given for any assertion one can make. In computer science, if you want to prove that an algorithm outputs a correct result, you need to consider inputs in general. Testing that it works for one specific example doesn't help, as there could easily be another that breaks the algorithm.) To give a more encompassing argument, you need to ask yourself what you mean by an even number. There are many ways to define "even number", but maybe we can say that it is a number that has a factor of / is divisible by two. In plain English: if you take two numbers with a factor of two and add them together, you can still factor out two, so that number must be even as well. (Your algebraic argument is equivalent to this.)

So this is kind of a rough sketch of what I do when I set out to try to establish something. I need to set a definition for that thing and then use whatever tools I have to try to show it. It's something that takes a lot of practise and experience - things get easier as you work on them more frequently.

Or a more recent, real example of me failing to generalize. I was working with some normally distributed function f(x), and observed some useful properties. I saw they will hold for a function as long as f(0) > f(1) and f(0) > f(-1). Simple enough. But it took me way too long to realize that this, in fact, means it will hold for a function with a Cauchy or Laplacian distribution as well - that I'm working with a certain special case but can make a broader conclusion.

If you could post something a little more specific, somebody might be able to make some comments on your technique.

[CallMeSpooky]

Good points again. I'll also keep in mind the opportunity to ask more specific questions of the clever minds here.

And to get this off my chest, I want to post an approximate example of what I mean by some papers becoming hard to comprehend due to the abundance of notation. Suppose one paper goes like this:

We say that a function f(x) is foobaristic if f(x) = -f(-x) for all negative x and f(x) = f(-x) for all non-negative x.

And then a second paper expresses the thought without natural language almost, like this, with apologies to LaTeX notation being used here:

We define S_0 \eqiv {x | x < 0, x \in R} and S_1 \eqiv {x | x \geq 0, x \in R}. We call a function f(x) foobaristic if \forall x \in S_0 f(x) = -f(-x) \vee \forall x \in S_1 f(x) = f(-x)

Sure, the second is more formal, but I think it comes at the expense of readability. It's Not a real example, but it's representative of the two different styles I've noticed in papers. Authors with a stronger mathematical background tend, probably unsurprisingly, to gravitate towards the latter style.

I actually found a post on another forum that describes how I feel exactly:

The main problem I have with mathematical notation right now is that I can't skim it. If I am reading a document with some math notation in, I tend to just skip past it and figure out what's going on from the surrounding text. I can read it quickly and just see a bunch of apparently meaningless symbols. Or I can read it very, very slowly and carefully and figure out exactly what everything means and exactly what's going on. But there's nothing in between. Computer code I find rather easier to skim, and natural language is much easier.

That's absolutely me. With computer code, I can very quickly get the main idea of an algorithm, but this can of course be chalked up to me being very used to code. But that post captures the source of my frustration with the notation very well. If a paper uses a lot of natural language and less notation, I can first get the general idea and then go back to the maths-heavy parts for the details. But when the content is mostly described through formal notation, I can't just get the basic idea. I either understand nothing or have to go in depth and get all the details.

On a semi-related note it's ridiculous how maths - the beauty of which is largely in logic and consistence - has a fairly large amount of oddities and inconsistencies in the actual notation.

[Latro]

One advantage of set notation is that it's compact once you've defined what sets are involved, which means that if you use some sets again and again, it's easier. A simple example is intervals, but there are more subtle examples, such as common function spaces. One problem I tend to have is that in advanced material, they tend to assume a certain amount of background in the notation that you may not actually have. That is, if you're somewhat new to a field but have the background needed to understand the material from that field, you may hit a complete road block when you run into some notation you haven't seen before.

For what it's worth, computer code can sometimes be very very hard to skim, especially if it involves a lot of mutable state or an unusual recursion pattern.

[CallMeSpooky]

Oh yes, that rings true. In an unfamiliar subfield, or even in a familiar one with unusual notation, it's hard to look up what some notation actually means, if you don't know what the symbol is called. I encountered the notation A \ominus B for the symmetric difference of sets, instead of the more familiar A \Delta B, and couldn't look the \ominus symbol up directly. Not that it's an insurmountable hurdle, but it's an annoyance.

[Monte314]

Sounds to me as though you are struggling with the long-term effects of a teaching disability.

[CallMeSpooky]

Oh, my thread has been honoured by the all-knowing mathematical feline!

It may be true, but only in part. In university, my "mainstream" maths courses (linear algebra, calculus, diff. eqs.) were taught by a fantastic professor. A high-class mathematician and possibly the best teacher I've encountered, he had a knack for making people understand stuff. I think everyone who was willing to put any effort into his classes ended up getting the material. But with some areas, it wasn't so bad. Took a semester of discrete maths with possibly the worst teacher - certainly the worst maths teacher - I've encountered. His language wasn't very good, his voice very quiet and he spoke with a bad lisp, so even understanding the words was hard, and his teaching methods were bad. I'm pretty much self taught in basics of abstract algebra (possibly a waste of time, haven't needed it so far), automata theory and probability, for example.

I'd like to see an example of your lectures, given that you've assured the forum of their quality
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

[Monte314]

I am astonishing, there's no doubt about that.

Tomorrow in class I'll be deriving the notion of the Shannon Entropy from scratch; my lectures are usually recorded, so perhaps I can get a copy.


[HIDE="In the meantime, you will have to content yourself with this..."]Like almost everything here, it sucks wind.
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
[/HIDE]

[porousshield]

I've run into the same problem. I had a prof for a more advanced course in linear algebra that taught it at a masters level. His pass rate for the course is/was very low. I had a room mate in college who completely switched degrees because of that one course with that prof. I digress but my point was that the way he taught was all mathematical notation with insufficient explanation for much of it. It's like learning a second language. With that being said I read a lot of education papers that are put out and they are so dense with terms that it hinders understanding.

The other problem I think is that those papers/courses are written/taught by people who are long on understanding and short on explaining. They get it so they can't conceive how others wouldn't get it right away like they did.

[Latro]

I've seen people try to teach linear algebra from the perspective of an algebraist before. It's a beautiful way of looking at the subject, though it misses some even better geometric aspects of the subject, but it's just impossible for a first semester look, which is what was done at this case. I'm a bit surprised at having trouble in a second semester course, if only because I don't really understand what they could cover that would be so difficult, other than just covering a lot of material.

[porousshield]

Was your post in response to mine Latro?

Even if it wasn't, that particular linear algebra prof was disorganized, rambled on, and completely full of himself. He also wrote the textbook which shared the same problem. He was a completely terrible prof teaching a subject at a fast pace at a high.