Should I Become a Mathematician?

[tgt]

unfortunately, partly for the reasons you mention, some people at a US university tend to simplify the evaluation of your scientific work, and often reduce it simply to: "how much grant money did you bring in?", which should be almost irrelevant.

a number of years ago we had a famous mathematician interview with our administration, and he was asked if he had any current grants, since none were visible on his vita. He responded indignantly, "No self respecting mathematician would list his GRANT MONEY on his vita!"

I assure you those days are long gone.

Quiet unfortunate but it's happening as you say. One thing I often ask myself is if you can't beat them, join them. In other words, why not use your brains to make the most amount of money possible like in financial services? Have you considered such an option? Having witnessed the current situation in academia, do you think it's a worthwhile pursuit for the younger generation? Or does academia still have a decent, uncorrupted, anti money grabbing future?

 

[mathwonk]

well money is very helpful, but not sufficient. there is a dilemma, as one cannot be happy without enough money to pay bills, have healthcare, etc,...

but one has to do what one enjoys, and what one feels good about doing. when i am discussing math, i am a happy man, at least temporarily.

so do what you love primarily, but save your money, or invest it wisely.

 

[mathwonk]

it is probably best to work consistently. i am trying now, even in the midst of my teaching, to set aside at least an hour a day for research thoughts. that's enough to seed them, and then my mind takes over and pursues the themes many more hours in the day and night.

 

[PhysicalAnomaly]

I was forced to take the maths unit that is below my level because it is a prerequisite for later classes even though I've already covered all the material because of a technicallity. I've gotten into the habit of getting my hands on final year maths assignments from my friends and doing them. I've been finding them quite easy so far (been helping my friends in fact) but am worried that by the time I do those units, I'll be caught in the same situation as I'm in now - with all the material covered years before. Should I desist? What will I do to keep me occupied when I reach the final year units?

 

[mathwonk]

i am puzzled that you find it difficult to be challenged by math when math is so hard. have you read my recommended books?

 

[Gib Z]

I was forced to take the maths unit that is below my level because it is a prerequisite for later classes even though I've already covered all the material because of a technicallity. I've gotten into the habit of getting my hands on final year maths assignments from my friends and doing them. I've been finding them quite easy so far (been helping my friends in fact) but am worried that by the time I do those units, I'll be caught in the same situation as I'm in now - with all the material covered years before. Should I desist? What will I do to keep me occupied when I reach the final year units?

Personally, and not trying to brag, I am quite a few years ahead of my class mates (who are reviewing the Sine rule at the moment) and yet, it has never bothered me once. It doesn't matter if you have covered that work before - continue ahead on your own, and only do the set homework from those classes for some good revision all year round to make sure you don't fail your test and don't forget your basics. Nothing wrong with already knowing the material, just go ahead.

PS. Sorry to hijack this a bit mathwonk :( Just a personal view

 

[mathwonk]

happy to have your input.

 

[bubbles]

My professor is teaching introduction to linear algebra by copying definitions from the book onto the board and does not explain them. I am trying to do independent study for that class since both the textbook and the professor are bad. The professor said to "unlearn" geometry since algebra is not about geometry and told us to think algebraically. He teaches linear algebra from a computational/applied perspective (since that is his specialty), but does nothing but copy proofs and definitions onto the board and told us to memorize them. Is that good? I have always seen math geometrically as well as algebraically when possible. I am having trouble with my linear algebra course right now. Can you give me some advice on how to really learn linear algebra?

Also, how strong of a background do I need in linear algebra to take more advanced math courses (Linear Algebra II, Abstract Algebra, ...etc.)? How should I prepare?

 

[PhysicalAnomaly]

First of all, I'd like to note that I'm not trying to brag or anything. I'm just quite desperate to not go over stuff that I've already learned a few times over. For example, my lecturer is teaching us the binomial distribution as if it were something new. I learned that 4 years ago and have learned it or used it every year since! I feel pretty guilty about not paying attention in the lectures but neither can I bring myself to listen...

I'm breezing through Spivak. A lot of the exercises are at the A levels further maths level. I am working on Munkres and that's fun to read. But I'm worried that if I finish that in my first semester of the first year and then tackle other books at that level like Rudin and Dummit, I'd be bored in my 3rd year classes. In australia, it all seems to be pretty laidback, unlike the uk system. If I'm able to do the 3rd year assignments now, how bored will I be in the 3rd year?

PS I've not been neglecting my unit's work or anything. Been doing all the exercises and assignments like a good boy...

PPS The cause of all this is probably switcing to the australian system after A levels further maths. Going from learning linear algebra and groups to stuff that was learned years ago isn't very enjoyable.

 

[Gib Z]

Did you just completely ignore my post? I do happen to be in a similar position to you.

You have two choices:

Don't go learn ahead. That way you won't go over things you've learned before, maybe pay a bit more attention in class. Not further yourself, not actually achieve anything. Just slow yourself down for a stupid reason.

Or, Learn ahead. That way, you DO go over things you've learned before, which is a GOOD thing. When you learn ahead by yourself, you don't always pick every skill up at that one time. Many teachers have their own unique skills that they pass onto students, and going over the work again you'll always learn something new, even if its something small.

More concisely: Either learn ahead, and actually do something worth while, or just stay with your class and be an average student.

And yes, I know the Australian System is a bit slow compared to the UK, but that's still no excuse.

 

[kurt.physics]

First of all, I'd like to note that I'm not trying to brag or anything. I'm just quite desperate to not go over stuff that I've already learned a few times over. For example, my lecturer is teaching us the binomial distribution as if it were something new. I learned that 4 years ago and have learned it or used it every year since! I feel pretty guilty about not paying attention in the lectures but neither can I bring myself to listen...

I'm breezing through Spivak. A lot of the exercises are at the A levels further maths level. I am working on Munkres and that's fun to read. But I'm worried that if I finish that in my first semester of the first year and then tackle other books at that level like Rudin and Dummit, I'd be bored in my 3rd year classes. In australia, it all seems to be pretty laidback, unlike the uk system. If I'm able to do the 3rd year assignments now, how bored will I be in the 3rd year?

PS I've not been neglecting my unit's work or anything. Been doing all the exercises and assignments like a good boy...

PPS The cause of all this is probably switcing to the australian system after A levels further maths. Going from learning linear algebra and groups to stuff that was learned years ago isn't very enjoyable.

Why don't you accelerate if you feel that confident that your so good? I know that the university of Sydney has an accelerated program but that was just for students who was in one of the three science Olympiads (Physics, Biology and Chemistry). But i would imagine that if you feel confident then you can take some of the course tests (should be on the internet) and if you score quite well, i would suggest you see the dean or physics head or what ever.

 

[mathwonk]

i guess after you solve the riemann hypothesis and the ABC conjecture, youll really be bored.

If you will go back and read a few of the recommended books in this thread, you'll find enough to interest anyone for life.

and i question whether you are really breezing through spivak unless you are not doing the problems. please try all the problems and then see how breezy it is.

 

[Gib Z]

Why don't you accelerate if you feel that confident that your so good? I know that the university of Sydney has an accelerated program but that was just for students who was in one of the three science Olympiads (Physics, Biology and Chemistry). But i would imagine that if you feel confident then you can take some of the course tests (should be on the internet) and if you score quite well, i would suggest you see the dean or physics head or what ever.

I went for work experience at USYD's math department and specifically asked if they had an acceleration program and they told me no! :( Why would they offer such a program to those who are in a science Olympiad program?

 

[tgt]

Just wondering how you like to tell others (when asked what you do) that you are a maths professor. What are their reactions? I find that the general public are impressed enough by a maths student. I just like to know what the reaction is when they hear you are a maths professor. Do you find that you are very respected being a maths professor?

 

[mathwonk]

I don't have a huge social life, so do not often tell it.
oh yes, and because i try to minimize the shock, i usually tell people i am a "math teacher".
this does not impress too many of them, since i do not use the "professor" title, unless pressed.

in fact when i started posting here, i declined to say i was a math professor for a long time, until quizzed about it.
up until then lots of people argued with me over my statements about math, and it bugged me that afterwards my opinions on math received more weight than they had before.

i.e. i started out believing that anonymous correct answers to math questions would impress people just by being correct, but eventually found that more people think i must know something about math because i am a professor, than think i am a professor because i know something.

 

[tgt]

I don't have a huge social life, so do not often tell it.

oh yes, and because i try to minimize the shock, i usually tell people i am a "math teacher".

this does not impress too many of them, since i do not use the "professor" title, unless pressed.in fact when i started posting here, i declined to say i was a math professor for a long time, until quizzed about it.

up until then lots of people argued with me over my statements about math, and it bugged me that afterwards my opinions on math

received more weight than they had before.

i.e. i started out believing that anonymous correct answers to math questions would impress people just by being correct,

but eventually found that more people think i must know something about math because i am a professor,

than think i am a professor because i know something.

By stating that you are a maths teacher, it could mean a primary school maths teacher so that would lessen the effect drammatically. It's funny because the average Joe might think more highly of a uni maths student then you, a 'maths teacher'.

Do you get treated really well when they do finally find out that you are a maths professor?

But I tend to be like you and don't like to show off too much. Maybe all mathematicians are like that?

 

[mathwonk]

well i have become more modest as I got older. maybe i realize that i have good reason to be modest.

Physical anomaly, I apologize for teasing you. You are in a position of needing guidance. Your ability is a blessing. There are many good books you can enjoy and be challenged by.

best wishes.

 

[mathwonk]

yes math professors do enjoy a certain fear/respect from many strangers. it does not last much past getting to know you though. then you get the treatment your personality commands or fails to command. i.e. people assume you are smart until you open your mouth too often.

 

[RufusDawes]

Just curious if any of you are planning to become a high school mathematics teacher ?

 

[mathwonk]

i would like to become one after i retire from university teaching, but i don't know if i can get hired, and I may not have the stamina to deal with teenagers.

 

[eastside00_99]

Hey mathwonk, I need some advice about the qualifying exams. I am going to CUNY, and there are 6 areas (out of which one choose three) that offer qualifying exams. You have two years to complete the exams, they are given three times a year, and you can only fail one exam twice before having to pick a different area. Here are the six areas:

1) Real Analysis
2) Complex Analysis
3) Algebra
4) Differential Geometry
5) Topology (starting with general topology)
6) Logic

At my undergraduate school, I took the graduate qualifying sequences in Algebra, Algebraic Topology, and Real Analysis. I also took half the qualifying sequence in Differential Topology. I have not had ANY complex analysis or Logic. So, basically, the way I see it, I have three options:

1) Take the sequences that would most prepare me for specializing in Algebraic Geometry (which is quite a big subject I know). As I see it, if I follow this plan, this would mean take the following sequences: Algebra, Topology, Differential Geometry.

2) Take the sequences that I know the least about as I probably have enough (not sure how to qualify that word enough though) knowledge of the above three areas to specialize in Algebraic Geometry. This would mean take: Logic, Complex Analysis, and Differential Geometry.

3) Just take the exams that I know the most about without necessarily taking the corresponding classes. For instance, I could study all summer for real analysis and algebra, and before the semester starts, take the exams. I would be using one of my chances, but the good thing is that you are not kicked out for failing an exam twice--you just have to choose a different sequence.

I think I may attempt the real analysis exam at the end of the summer regardless of which plan I take. Also, I need to talk to a few people about this. I know a lot of students who find out what exams are the easiest and then take those. But, I don't want to do that. Of course, I would not punish myself by taking the hardest exam just because it is hard and no body passes. I am open to advice which ones would you recommend?

 

[asdfggfdsa]

Hi, mathwonk:

I've been thinking about becoming an algebraist (after completing a course in Galois theory) - are there any texts which you would consider classics in algebra?

asdfg

 

[tgt]

When you lecture at uni, do you need to look at your notes once in a while to keep track or can you walk in without any notes teaching a full lecture without referring to any notes?

 

[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 will 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 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.

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, I am 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 that's 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 unsolved 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 homework to reinforce it.

 

[Cloud]

Applied Mathematics?

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?