Does your school's math curriculum satisfy you?

[Dr. Seafood]

Does your school's math curriculum "satisfy" you?

How much rigor is in your math courses? My school has a distinct math faculty (our math program is through the math faculty, not through sciences) with a variety of math majors: combinatorics, statistics, pure math, applied math, computation, etc. Even with such a distinguished name, the math classes are jokes. They're example- and algorithm-based. We're usually provided with a general problem and then an algorithm for solving it without much rigor. Proofs aren't on the exams at all.

That said, there are advanced level courses of all first and second year core math classes. These include: Calculus I, II, III, Intro to algebra, Linear Algebra I, II, Intro to combinatorics, Intro to probability, Intro to statistics. After these, the pure math classes are pretty proof heavy -- real analysis, rings/fields/groups, geometry, graph theory, etc. I'm currently taking calculus III, linear algebra II and intro to combinatorics at the advanced level and it's pretty proof-heavy. Calculus III is epsilon-delta all over the place, lin alg II is pretty much a rings and polynomials course, and combinatorics is about bijections and power series. In calculus we've already discussed open and closed sets, compactness, uniform continuity.

The standard versions of these courses are completely different. Calculus III is about how to find partial derivatives of any function, and the idea of "limit" is reduced to algebraic tricks. Linear algebra II is an expert course on row-reduction -- they don't even talk about rings or fields at all. Combinatorics is just using binomial coefficients to count how many ways there are to arrange socks. It's ridiculous in my opinion.

My question is ... is this "normal", by which I mean, are your school's math classes like this too? Or does your linear algebra class begin with a rigorous identification of fields and vector spaces? The first month of my calculus I class was devoted to sequences and sequential definitions of continuity. I don't think this is typical for intro calculus classes. We talked about finite fields in my first year algebra class. Is your school like this too?

 

[homeomorphic]

No, that's not normal. Most classes are the cook-book kind. Most students just wouldn't survive if you gave them heavy-duty stuff.

I personally don't even think it's desirable to do deltas and epsilons if it is your first time learning calculus. If you took it in high school already, then by all means.

Rings and fields are probably overkill for starting out with linear algebra, but I didn't like the course I took. It over-did the row-reduction and vector spaces and linear transformations only showed up towards the end. Maybe it's not a bad idea to start with a more coordinate-based approach, just for motivation, but I think vector spaces and linear transformations should quickly take over.

I wish I would have gotten an earlier start on more theoretical stuff, though. I was an engineering major who converted to math. I'm not a fan of mathematical macho. Nothing should be made more difficult than it needs to be, and haste can make waste when if too much is forced down the students' throats too fast. Simplest case should come first. You shouldn't blast the students with abstract generalities right from the start.

However, I am equally against purely cook-book stuff and doing math like a trained monkey on the other hand. But many low-level students will scream bloody murder if you ask them to think too hard.

So, I don't know what the solution is.

 

[micromass]

I won't answer whether yout situation is normal or not. Many classes seem to be dumbed down a lot: avoid those classes!

My freshman year at the university was already very rigorous. We immediately started off with metric spaces, uniform continuity, open/closed/compact, etc. Linear algebra did vector spaces, inner products, fields, etc.

I think that such a rigorous approach is very satisfying to me. It introduces you to what math really is. Cook-book classes are useless and annoying.

Take the rigorous classes and don't bother with the easy stuff.

 

[Dr. Seafood]

Sometimes I feel like if I didn't ever have the gentle intro to calculus or linear algebra in high school, beginning in an abstract setting would be pretty tough. That might be part of the reason why calculus is (most often) presented before progression into real analysis. My first discussion on linear algebra, in high school, was about "direction and magnitude". We only used it to describe the geometry of three-dimensional Euclidean space, and even then, we did so in such an elementary sense. Little did I know that my second year lin alg course would discuss minimal polynomials, algebraically closed fields, and quotient rings in (relative) depth. But perhaps I wouldn't understand these concepts as well if they were presented like this at first.

For example, I'm taking (what seems to be) a pretty rigorous intro to graph theory course right now. Maybe it's just the choice of topic, but it's just "definition, definition, definition, result. This is a path, this is a connected component, a graph is bipartite if and only if it contains no odd cycles." If I had taken a gentle graph theory course, seeing this stuff would be no problem. But I guess that's school for you.

 

[micromass]

Sometimes I feel like if I didn't ever have the gentle intro to calculus or linear algebra in high school, beginning in an abstract setting would be pretty tough. That might be part of the reason why calculus is (most often) presented before progression into real analysis. My first discussion on linear algebra, in high school, was about "direction and magnitude". We only used it to describe the geometry of three-dimensional Euclidean space, and even then, we did so in such an elementary sense. Little did I know that my second year lin alg course would discuss minimal polynomials, algebraically closed fields, and quotient rings in (relative) depth. But perhaps I wouldn't understand these concepts as well if they were presented like this at first.

For example, I'm taking (what seems to be) a pretty rigorous intro to graph theory course right now. Maybe it's just the choice of topic, but it's just "definition, definition, definition, result. This is a path, this is a connected component, a graph is bipartite if and only if it contains no odd cycles." If I had taken a gentle graph theory course, seeing this stuff would be no problem. But I guess that's school for you.

I understand that. But reading a mathematics text requires some skill. I'd even say that undergraduate math is built for acquiring the skill of reading mathematics text (while the real mathematics is presented in grad school). The skill actually consists of finding out the intuition for yourself. So instead of the text developing each concept, you are expected to do this yourself.

For example, when you read the definition of "connected component", then you are expected to not just read it, but to process it mentally. You have to ask yourself the question: what do I intuitively expect a connected component to be?? Does this definition satisfy my intuition?? Can I come up with some basic examples/counterexamples? How do I check this definition in an example? Can I come up with some basic lemmas concerning the definition?

Reading mathematics texts is NOT easy at all. It's actually fairly difficult. It's very normal to spend a couple of hours at one sentence (although it can be discouraging).

 

[Dr. Seafood]

The problem is that these new ideas are presented in the lecture. They're in the book too though. The theorem is drawn immediately after the definition is made, giving me no time to get intuition for a new concept and so I end up just scribbling down everything so I can read it later. It makes me feel like I'm incompetent for this material. Or maybe I just have a bad lecturer. But like I said ... I suppose that's just what studying is and this is what school's about.

 

[micromass]

The problem is that these new ideas are presented in the lecture. They're in the book too though. The theorem is drawn immediately after the definition is made, giving me no time to get intuition for a new concept and so I end up just scribbling down everything so I can read it later. It makes me feel like I'm incompetent for this material. Or maybe I just have a bad lecturer. But like I said ... I suppose that's just what studying is and this is what school's about.

It's not you being incompetent for this. Reading math texts is pretty hard for almost everybody. Even field medal winners can struggle if they encounter a new concept!

Math texts are indeed structured as
Definition -> theorem -> theorem -> theorem -> Definition -> ...

There are rarely examples. The thing is that you need to make these examples yourself. Whenever you encounter a new definition: immediately make some examples about it. Or at least try to find out what it means and why we introduce it.

Math books only care about being rigorous and correct. This can be quite a shock to students. The Definition -> theorem style is very discouraging. But it betters with time. After a while, you will be so used to it that you find it hard to read a normal text because it contains to much BS.

Another very good thing to do is to look up stuff on wikipedia. Say that you encounter the concept of bipartite graph, then go to http://en.wikipedia.org/wiki/Bipartite_graph and read some of the stuff on there. Wikipedia can be very good to understand the intuition and to understand the results that use the definition.

I know it can be tough. And I remember very well that I've been very discouraged and angry while reading my previous textbooks. But you DO get used to it. The thing you have to learn though is that you can't read a math book pasive. You need to actively be engaged in the material. This is the only way to master it.

 

[homeomorphic]

I take issue with the way math books are. That's what I mean about mathematical macho. I am quite used to reading typical math books, but I never moved on to a stage where I actually like them to be that way. The intuition is often, not just hidden, but not even there, in many cases. It's not that you are expected to come up with it yourself. You are just expected not to have the intuition or motivation. It's just not effective.

I struggled so hard trying to read Kassel's quantum groups, but then, once I found the same material presented in a more reasonable way on Baez's website, it became trivial, and I soaked it up like a sponge. It's not that I didn't understand Kassel. It was just that there was no conceptual glue to hold everything together and make it meaningful. I am not bad at coming up with my own intuition, but I was not quite up to the task, in this case. Just having examples is not enough. The rigid, formal approach just resulted in a lot of unneccesarily wasted time. It's just plain silly, at times, the way mathematicians insist on being so formal all the time, sometimes in general, but particularly when it comes to their writing.

Also, I agree with what Feynman said about physics. Someone commented about how she appreciated all the hard work that physicists do to figure everything out. He said something like, "no, we just do it for the fun of it."

Of course, it is pretty hard work. But I am in it for the fun of it. Although, you could say I'm a professional mathematician, I just do it for entertainment, and I think things should be done accordingly.

The thing is, the subject is ridiculously hard, no matter how you slice it. Therefore, everything should be done to make it as easy as possible to learn everything.

On the bright side, there are some good books out there. They may be completely rigorous and use definition, theorem, proof style, but they don't bury the intuition. Somehow, it still comes through.

Some difficult books are pretty good. For example, Vladimir Arnold's books are still difficult, but they have a lot of insight in them.

 

[homeomorphic]

Often, it is simply impossible for any normal human being to really get the intuition out of a math book, no matter how hard they try because the intuition comes from things that the books don't even hint at. They just aren't always written in a way that allows that. Sometimes, the authors just proceed formally and hide their intuition, and sometimes, it seems to me, they probably just don't have the intuition themselves. I suspect even if you were Riemann, you would be poorly served by many books and would be unable to arrive at a really deep, intuitive understanding of the subject, without maybe looking at other books or expending a colossal amount of effort over an extended period of time, which would extend well beyond the subject matter treated in the book in question.

I think a lot of the motivation for functional analysis, for me, comes from things like quantum mechanics, real analysis, PDE, Fourier series, calculus of variations, integral equations. To some extent, I don't really believe in studying it too separately from all those other subjects. Functional analysis books may mention those things casually, but I'm not aware of one that actually motivates the subject, starting from other subjects. Maybe that's part of the problem. Putting subjects in a box, isolated from other ones.

 

[kramer733]

First month of calculus: Ok kids.. today we're going to learn epsilon delta proofs
First month of linear algebra: Ok kids.. today we're going to learn about fields and modular arithmetic.

First month of discrete math: ok kids we're going to learn proof techniques..

yeah it satisfies me. At first i was frustrated and now I'm thankful. I'm still a **** ton behind but carleton university's math program is prolly far ahead of most universities. I really would like to join the army but if they're only willing to pay for courses at royal military college of Canada for me, i'd rather pay out of my own pocket and go to carleton university.