The Should I Become a Mathematician? Thread

mathwonk

eastside, it is good advice to just get the quals out of the way as quickly as possible. so i would take them in the areas i knew best, and can prepare for soonest.

it is also good advice to learn something about complex analysis, since beginning with riemann it has been a key tool in doing and understanding facts from, algebraic geometry.

mathwonk

asdfggfdsa, classic texts in algebra are listed earlier in this thread, e.g. artin's algebra, and jacobson's basic algebra.

mathwonk

i seldom look at notes. i find it better to go through a calculation without notes, since that forces me to actually see what i am doing, and then maybe the class will see too.


once i had a post class evaluation that criticized me as follows:

"this man comes to class with just a box of chalk and a sponge to erase the board,

no lesson plan at all!"

of course the lesson plan is in my head, and i have filled up many pages with calculations the night before, which there is no need to consult again in class.

usually the only time i have notes, is when i do not understand what i am presenting, but sometimes i write out and copy a complicated calculation, or at least I may copy the problem so it will be one with numbers that will come out nice.

mathwonk

i am usually not trying to present a canned set of information for people to memorize, but to show a way of thinking about the topic.

i try to show what to do first, then second, and so on,...

i am always trying to prepare people for that moment when they are alone with a problem.

i.e. where do we begin? how do we remember key formulas? how can we recover them if we forget? how can we shortcut the work in special cases?

usually this can only be done by remembering what the calculations mean.


e.g. some books teach multiple integration, and then how to compute them by repeated integration,

then they state greens theorem but say they wikll not prove it.

In fact they have already proved it, since just looking carefully at what repeated integration says, shows that it may be stated as greens theorem.

i.e. greens theorem computes a path integral as a double integral, but repeated integration computes a double integral as a moving family of single integrals, which is just a path integral around the boundary of the double integral region, i,.e. greens theorem.


even earlier, seeing that repeated integration works is just seeing that the derivative of the moving volume function, is the height function. but to see this one must know the meaning of the derivative as a limit of ratios [in this case volume/area = height] , not just know a bunch of derivative formulas.

mathwonk

the first 4-5 pages of this thread have a lot of book recommendations, but the specific cheap copies i located then are surely gone by now.

tgt

I find that a lot of younger presenters need notes. The older professors don't need them. Even if you know how to prove any theorem and do any problem, how do you keep track of the order in which you want to present the material? Or is there a natural order in your head which come to you easily?

I guess the ultimate test for your knowledge of some material is if you can present it without referring to notes?

kurt.physics

mathwonk,

I was just wondering, do you, every course (like 1901 or something), put onto the board one huge equation, theorem or something and get your class to come up with a proof by the end of the course?

That would be so wicked, im currently in Australia and in High School going to uni soon and that would be the one thing i would want to do, that is, have the professor write up a thing thats almost impossible and ask us to prove it. I think that would be good as it would motivate students, like myself, to think outside of watching lectures and doing questions. But to get first hand experience of what it is like to be a mathematician, of trying to prove something (probably related to the course) and have competition to, looking at it several different ways, probably improving their math ability.

mathwonk

kurt,

well that would be a different world from the one i inhabit. i struggle with many of my students to get them to even think about math as a process of reasoning rather than computation.

since anyone can teach strong students, the older you get and the more experienced you become as a teacher, it can happen that the more you are asked to teach weaker students, and leave the teaching of more creative ones to younger colleagues.

IN my whole life I have only had one teacher, a great inspiring graduate algebra teacher, maurice auslander, do something like what you said, but even then he only handed out very terse notes in which he had sketched the proof of a very deep result he was proud of, (all regular local rings are ufd's, 1965), and made it the goal of our semester to read and understand the proof.

as to presenting a problem and arriving at a proof of it during the semester, i proposed that once in a faculty seminar, and even there some audience members were astonished at the optimism of the idea.

bott on the other hand, at harvard, used to present hard problems in grad classes, and according to lore, once challenged a class including john milnor with an unsoklved problem that milnor actually solved as if hw.

i myself also was in a class at harvard where hironaka challenged us with a hard but preliminary version of an open problem, that was soon solved by his future phd student mark spivakovsky.

but i am usually so isolated from such students that recently when i wrote an honors calc exam, from long habit i made it too easy, and left off some thoughtful questions i later wish i had asked.


here is one i decided would be too theoretical for my undergrads, to my regret, as i would have liked to see what they did with it:


Assume f is differentiable on some interval [a,b], that f '(a) > 0, and f ' (b) < 0, but not that f ' is continuous.
i) Use the definition of derivative to prove there is some e >0 such that f(x) > f(a) for all x in the interval (a, a+e), and f(x) > f(b) for all x in the interval (b-e, b).
ii) Assuming standard theorems from diff calc, prove f '(c) = 0 for some c with a < c < b.


you see i am only asking them to understand the meaning of differentiability, and use that understanding to derive the intermediate value property for possibly discontinuous functions which are known to be derivatives of other functions. but i lost my nerve about asking even this of a group of honors level undergraduates. in hindsight however i should have done so, as they had already seen many of the more standard problems i did ask, and some of them were very creative and insightful, and i would like to have seen how they handled this slightly offbeat problem.

mathwonk

tgt, it is usually only possible to do one thing in one class, so the order of topics is not too important.

usually the order is as follows:
introduce and motivate the topic with an interesting problem.
take guesses as to how to solve it.
either run with any good ideas wherever they lead,
or at some point lead the discussion to the tool you want to present, and present it,
making it as precise as necessary.
give examples of the workings of the tool, with specific numerical computations.
give hw to reinforce it.

Cloud

Hi,

I finished with Computer Engineering and Electrical Engineering for my undergraduate degree. Thinking about pursuing MS and may be PhD if I can totally absorb into it.
But I find it difficult to choose among engineering/applied Mathematics/Physics. I roughly aim for applied mathematics for now and applying schools. Can you please advise me on this matter? Thank you in advance.

mathwonk

at my university we struggle to teach students to stop expecting us to use class to carry out model calculations for them to imitate later, and to begin to appreciate that we are there to help them understand the meaning of the calculations, and the theory behind them. the specific calculations are for them to practice at home.

at some schools, the teachers just read and explain the book in class, at others they expect the students to do this at home, and in class they show what the material is good for, and how it can be extended. the teacher at a school like harvard introduces material in class that he/she knows from their own expertise, that is not found in the books.

there is a constant struggle to increase the depth of the students' experience, without submerging their heads under more than they can absorb.

of course occasionally i have students so strong i myself cannot keep up with them, but only occasionally, (every decade or so?).

mathwonk

I cannot advise on applied math, but perhaps others will?

PowerIso

Keep in mind I am speaking from my experience as an applied math guy. It's important to realize that there is a reason why it's called applied mathematics. The goal of applied mathematics is NOT to make tools for engineers or physicists, but rather to study interesting mathematical problems that may be applied but doesn't have to be applicable.

Just look at Combinatorial analysis. It can be applied to computer science, finical analysis, stats, and many other fields, however, much of the research that goes on within the field are purely mathematical questions.

Don't get me wrong though, there are a good number of applied people who do actively solve problems that can by used by engineers and physicist. If that is what you are interested in, then when looking for a graduate school in applied mathematics, try to find one that has a research group that is more about that than the what I presented earlier.

mathwonk

I am reminded that all research is the free flow of creativity and problem solving from the individual researcher, often without any focused regard for its use.

I have often made the error of assuming that research in math education was directed towards improving classroom instruction. while some is, much is just exploration of problems and concepts about learning.

I once asked a new friend who was doing research in learning psychology when his work would find its way into the classroom, and he replied he had no interest in that, but was merely engaged in "bringing order out of chaos".

matqkks

Paul Erdos said a mathematics is like a machine which coverts coffee into theorems and proof.
Marcus in his book "Finding Moonshine" says mathematician is a pattern searcher.
Lord Kelvin asked the question, whom do you call a mathematician?
He answered a mathematician is a person who finds the integral of e^(-x^2) from infinity to minus infinity as easy as you find 2x2=4.

asdfggfdsa

Thanks for the book recommendations - I have picked up Algebra, by Lang, from the uni library.

mathwonk

well lang is good but not sufficient, as it is all theory and no examples.

i recommend you add hungerford to it.