How does one truly and fully learn Mathematics?

Post Reply New Topic

That's quite true for many fields such as teaching English or working in restaurant. For any kind of scientific pursuit, algebra alone won't cut it.

For those in fields such as physics, I would say that you will probably be better off if you learn the math solidly first. If you don't know the math you need in a course, it is much harder because you have to learn the physics and the math. If you already know the math, the physics is far less difficult.

Anyway, for physics and engineering I think that calculus, differential equations, vector analysis, and matrices are more important. In some areas of physics, some abstract algebra is very useful, too.

For many other scientific fields, probability and statistics is probably more important.

As for math courses, I have one observation looking back on my math courses both as an undergraduate and a graduate student -- when you start taking the courses it isn't always the case that you have any idea why the subject of the course is important. In any math course, endeavor to find out from the start why you are taking it -- that is, the purpose of learning the material in the course.

When I first took topology as an undergraduate, I had no idea why I was taking it other than it was required. I took it from one of Moore's former students using the "Texas Method" or "Moore Method" and had no textbook of any kind. It was also my first course in which we really had to come up with proofs for theorems and I wasn't prepared for that.

In the Texas Method, you start with a list of definitions and theorems and start proving the theorems. You don't study how other people proved the theorems -- you do it yourself. Whenever someone has a theorem, they go to the board and present the proof to the rest of the class. Any errors or anything you overlooked that you can't handle immediately, you fix the problem and present them the next class. If nobody has a proof to present, then the prof lectures about one or more topics in topology. With no text book as a reference, you had better pay attention to what he says because if you miss it, then you won't get it. Also, it is considered cheating to read up on the proof elsewhere. There are no tests -- the entire grade is based on your class participation in proving theorems.

My point was simply that algebra teaches basic logic skills.

I took classes beyond algebra. I took calculus, differential equations, Fourier Series, group theory, proof classes, logic classes, etc blah blah, blah. All very interesting but ask me to prove some basic theorem from one of those courses years later and I would be left either scratches my ears for hours on end or consulting a text book to give me the answer. There's a reason we learn theorems from famous past mathematicians of the past. It's because they were ahead of their times and very smart. Things that seemed brilliant to past mathematics are now standard knowledge because it is taught to us. And I suppose this proves both of our points.

I think this is excellent advice. I found it much easier to learn math when I had to use it for physics.


This sounds brutal and horrible. I would never want to be in a class like that! If you miss, then you won't get it seems like a method designed to screw people who may have medical problems or a lapse of concentration.

Seems like some weird macho right of passage thing--what purpose does this serve, other than making people who are suited to it feel superior?

It takes 8 to 10 years of hard work to make advanced mathematics effortless.

ruveyn

For most of the brightest students, it's a great way to study the subject. Anyone who excels in such a class has a right to feel superior in their math abilities. For those who learn best from a textbook, a regular class is obviously better.

One thing that you could do to get a feeling for the Texas Method is to select a math subject of choice that you are ready to tackle but preferably have not already taken. Have a friend with a good math background go through the first two or three chapters and carefully list each definition and theorem. Something like an outline. You might also wish to include the exercises as well.

Your job would be to go through the pages of definitions and theorems and prove every theorem on your own as you go without reference to any other texts or other outside sources. As you prove the theorems, present them to your friend to look for any holes in your proofs.

If you had two or three other math friends interesting in learning the material, then you could do this together and meet weekly to present your proofs of the theorems to each other.

I guess that's a good thing, if feeling superior is an important outcome of the class.
I suspect there may be more to it than that, though.

I found this description, which is slightly different:

Carefully selected sequence seems like an important detail. But the source of this description is a "second generation" participant, taught by former Texans, but not Moore himself.
http://www.jiblm.org/mahavier/pages/PDF ... aching.pdf

It sounds like the selection and sequencing of the problems set for the student to solve was an important part of the method.

Regarding textbooks, the thing which struck me as problematic was the idea that, "you had better pay attention to what he says because if you miss it, then you won't get it." This might mean that those who did well were "superior" at math, but it may also be that they were just luck--lucky not to get the flu, say, and miss a key lecture with no alternative means of learning that material.

Not an important outcome, but a natural outcome in many cases.

The Texas Method is not used to make students feel superior, but to give them superior understanding of the material.

Remember that not all mathematicians are created equal. Any efforts to try to force equality upon mathematicians would only limit the abilities of the best mathematicians while doing nothing to help those not so mathematically endowed.



Yes. In the method I gave above that could provide someone with a flavor of the method, there would be far too much to do to progress far during the course of a semester. One would want to select theorems (not problems) that lead one boldly forward.



That is a good point. In my classes using the Texas Method, it was rare to ever see anyone miss a class, but it did happen on occasion. On those cases where someone was absent, if we had time for a lecture after going through whatever new theorems had been proved, it seemed to me that the prof would usually try not to get too far ahead of them. And typically, we might have only had time for a lecture once a week or so since the lecture was only if there were no proofs of theorems to be presented.

If you missed class but the entire class was spent on presenting the theorems, it was easy enough for the student to ask whoever presented theorems for a copy of their proof.

That makes sense. It sounds like a good teacher making sure the students actually completely understand the material, kind of an ideal situation.

In a way, all the different branches of math are like different tools. What you want to be able to do is, when faced with a question you need answered, is to be able to select and apply the right tool for the problem. This is a different from the usual classroom setting, where you are told what tool to use. Also, there is also the expectation that the problem is has a relatively easy solution--you don't get problems that take 10 hours of hard work.

For me, it just came down to practice--I'd need the answer to a real life problem and I'd use the math I learned to get an answer--then I build something to see whether it made sense or not.

That would depend on the level of the classes. I've spent far longer than 10 hours of hard work on occasional problems.

I remember one problem in Celestial Mechanics that the text omitted the proof. I spent a lot more than 10 hours over a period of two weeks to figure it out. Once I had the solution, it only took one or two pages to write down the proof.