Should I Become a Mathematician?

[SuperiorPrep]

Thanks for this discussion and forum - my cousin is studying physics in the UK and I am definitely going to have to refer him!

 

[Mépris]

What are your (mathwonk, Dave, homeomorphic and all the guys here) thoughts on exam technique and exams in general, for math?

Do you care a lot about getting an A/A+ or would you not mind getting a B+, simply because you haven't done 50+ problems (drilling/grinding can be fun but it can get tedious...) and thus couldn't finish within the required time? I had an exam today (stats) and lost three marks (out of fifty) because of a timing issue.
The instant my exam was taken from me, which was about a minute after "pens down", I figured out how to solve one of the problems. I also lost another 4-5 marks, mostly because of time issues. Now there's also potential mistakes in the things I thought I did correct!

At any rate, I don't think I would mind (once I enroll in a college) if I were to get Bs. As far as I'm concerned, as long as I have at least a 3.5 GPA, I'm satisfied. That is of course, I am not at a school/majoring within a department that is known for leniency!

 

[Mépris]

I found some (seemingly legal; in case they aren't, say the word and I'll remove them!) links on a webpage of the Asian Institute of Technology.
http://kr.cs.ait.ac.th/~radok/math/mat/startall.htm

Included, is the calculus text by Courant.

 

[dkotschessaa]

There is conflicting information out there. A lot of people tell you give you this sort of line that in "real life" (whatever that is) "grades don't really matter." But the fact is that there are a lot of opportunities and programs that will exclude you if you do not get high marks. At least that's what I'm seeing.

I believe grades don't really fairly reflect my understanding, because I am not a good test taker. I'm sure lots of people feel this way. I think my strengths will eventually come out in project oriented work and research, which I love doing. But in order to get into those programs you are often expected to have high marks. So it's a bit of a conflict.

I make every attempt to do a lot of problems and get As on everything. I'm usually a bit upset or annoyed when I don't, because I have pretty high expectations at this point in my life. But I get over it fairly quickly and put it into context. I ended up with a C+ in Calc III this semester and was not happy about it, because I felt I had a very pedantic professor who tests so short (about 7 questions) that it was really impossible to get As or Bs on them. That C might actually exclude me from some things I wanted to get into later. Hopefully my ability to connect with others and network and my strong work ethic and maturity will make up for some of that. Otherwise I have had no lower than a B in any math or physics class and no lower than an A in any non-math class.

I am being recognized by some professors at my school for my sheer enthusiasm and dedication to the subject. They aren't asking about my grades.

It's also been said though that "they" (people that care about your grades) do take the nature of your classes into consideration. A "B" in Analysis is probably as good or better than an "A" in Calc I.

Curious what Mathwonk thinks about this too, especially in regards to the conflict I have mentioned above.

-Dave K

 

[drummer - em]

hi good day pips can i ask your opinion about my thoughts..i am 30 yrs old and I've been interested in math lately, i have learned math accidentally..when one of my co teachers ask me to substitute to her math class for two weeks, so i decided to somehow read and study intermediate algebra.. and from now on I've enjoyed doing it..i enjoy reding and solving math books even I am alone...and i have decided to study my second degree in mechanical eng. can i still be a mathematician? does my age not a hindrance to become an engineer or i just need to stop coz i am too old for that dream? how can i improved in math? I've been dreaming math everyday..thanks hope to hear from you guys...Godbless you all

 

[dh363]

Whew! I just finished reading this entire thread! In my spare time over the past couple of weeks of course. Thank you all for being so helpful.

 

[grendle7]

Does anyone know how to subscribe to a thread, without having to post a reply?

 

[jbunniii]

Does anyone know how to subscribe to a thread, without having to post a reply?

Look under "thread tools" in the blue bar right above the first post on any page of the thread.

 

[homeomorphic]

Do you care a lot about getting an A/A+ or would you not mind getting a B+, simply because you haven't done 50+ problems (drilling/grinding can be fun but it can get tedious...) and thus couldn't finish within the required time? I had an exam today (stats) and lost three marks (out of fifty) because of a timing issue.
The instant my exam was taken from me, which was about a minute after "pens down", I figured out how to solve one of the problems. I also lost another 4-5 marks, mostly because of time issues. Now there's also potential mistakes in the things I thought I did correct!

Generally, I care much more about whether I am learning a lot than whether I get a good grade. As Mark Twain said, "Never let your schooling interfere with your education."

However, I will say that I do like to over-learn things. It helps with long-term retention. If you learn it really well, so that it is at your finger-tips, it will probably help with the test. So, I wouldn't worry overly about the grade itself, but it could be that you could benefit from learning things a little bit more thoroughly than you think is necessary.

 

[mathwonk]

I love and agree with the previous answer. However, when I needed some good grades to get a fellowship or some such, I tried to nail one. it isn't that hard. you learn everything presented, and then you practice old tests given by the same prof and then you study further from outside sources and you are likely to get an A. I did this this as a senior and got into grad school with a fellowship.

I.e. indeed learning is somewhat peripherally related to grades, but some things are given based on grades, so you need to know how to learn and also how to get grants.

I.e. don't complain that grants are given in a way that ignores knowledge, just do both, acquire knowledge, and survival skills.

 

[dkotschessaa]

So well said. I guess that even if there isn't always a direct correlation between grades and knowledge, grades are most of what they have to go on when evaluating for certain programs.

I'm sure you've heard this before, but my grades suffer from "dumb mistakes." I don't know how to stop making them, and I don't know if they are something that is eventually going to be ironed out or if I have to find another way to fix this. I really do take my time with everything, but they still seem to crop up.

-Dave K

 

[homeomorphic]

I'm sure you've heard this before, but my grades suffer from "dumb mistakes." I don't know how to stop making them, and I don't know if they are something that is eventually going to be ironed out or if I have to find another way to fix this. I really do take my time with everything, but they still seem to crop up.

I am a master at making dumb mistakes. That's part of why I did so much better when I got past high school math and lower division math. In the long run, it doesn't matter that much, as long as the mistakes are inadvertent ones. In "real world" situations (including research), you can check your work 20 times if you want to get it right.

 

[homeomorphic]

From childhood I was passionate about mathematics but I noticed I can not afford to become a mathematician.

Anyone can afford to be a mathematician to some extent. In America, all you have to do is do really well in high school and you can get a scholarship. Then, in grad school, you usually get paid. Even if you don't go to college, you can still teach yourself quite a bit on your own.

 

[mathwonk]

you might try becoming a mathematician who spends more time with her family. you could start a trend.

 

[dkotschessaa]

you might try becoming a mathematician who spends more time with her family. you could start a trend.

I recently started getting invited to gatherings with our math department, and it was funny to start finding out how many of the professors were married to each other. I had no idea, because most of the women kept their last names. So, I guess that's one way!

-Dave K

 

[mathwonk]

there are at least 5 couples in our department such that both spouses are either professors or instructors.

 

[Mépris]

Good to know. That was my intended course of action. (go outside the "syllabus" if I feel like it but then when there's exams, I focus on those)

A lot of what motivated my initial question was that I had some ~12 exams within the span of 3-4 weeks and they were all exams that are much in the vein of the usual standardised testing...

Does anyone here have any experience with the Jerry Shurman (at Reed College) notes on single variable calculus? I'm currently checking out Apostol and Spivak using the free previews available on Google Books and Amazon, before choosing which of the two to buy. Shurman says that he learned from them, Courant and Rudin.

Mathwonk, I read on another post that you used Sternberg and Loomis after Spivak back in the day. What do you think about this course compared to the modern alternatives - Apostol's second volume, I guess? Would one be correct in assuming that the current MATH 55 course at Harvard assumes (equivalent?) knowledge of both that book and Spivak?

 

[Mariogs379]

@mathwonk,

Bit of a specific question but I thought you might be a good source of advice. Here's my background/question:

Went to ivy undergrad, did some math and was planning on majoring in it but, long story short, family circumstances intervened and I had to spend significant time away from campus/not doing school-work. So I did philosophy but have taken the following classes:

Calc II (A)
Calc III (A)
Linear Alg. (B+)
ODE's (A)
Decision Theory (pass)
Intro to Logic (A-)

Anyway, I did some mathy finance stuff for a year or so but realized it wasn't for me. I'm now going to take classes at Columbia in their post-bac program but wanted to get your advice on how best to approach this.

They have two terms so I'm taking Real Analysis I in the first term and, depending on how that goes, Real Analysis II in the second term. I'm planning on taking classes in the fall semester as a non-degree student and was thinking of taking:

Abstract Algebra
Probability
(some type of non-euclidean geometry)

Anyway, here are my questions:

1) What do you think of my tentative course selection above?

2) How much do you think talent matters as far as being able to hack it if I ended up wanting to do grad school in math?

3) I'm also having a hard time figuring out whether math is a fit for me. By that, I just mean that I really like math, I'm reading Rudin / Herstein in my free time, but I've spoken with other kids from undergrad and it's clear that they're several cuts above both ability and interest-wise. Any thoughts on how to figure this out?

Thanks in advance for your help, much appreciated,
Mariogs

Thought I'd update. This 6 week real analysis class covers the first 6 chapters of Rudin. I'm finding the homework hard but we have a midterm on Monday; he showed us the one from last year and it looks *relatively* easy (definitely compared to the HW). Anyway, thinking I'm going to take RA II, and some classes in the fall, decide about applying to grad school the following year.

In short, material's harder than I appreciated but also much more interesting. I think I'll enjoy it even more once I get more comfortable with some of the concepts (I feel like I spend a lot of time trying to understand Rudin's language/terminology/general technical writing even when he's conveying a *relatively* basic idea. A good example is his def. of convergence; easy now, but was a bit confusing at first. Tho I think once I'm able to get the ideas more easily, it'll be even more rewarding.

Thoughts?

 

[eliya]

Thought I'd update. This 6 week real analysis class covers the first 6 chapters of Rudin. I'm finding the homework hard but we have a midterm on Monday; he showed us the one from last year and it looks *relatively* easy (definitely compared to the HW). Anyway, thinking I'm going to take RA II, and some classes in the fall, decide about applying to grad school the following year.

In short, material's harder than I appreciated but also much more interesting. I think I'll enjoy it even more once I get more comfortable with some of the concepts (I feel like I spend a lot of time trying to understand Rudin's language/terminology/general technical writing even when he's conveying a *relatively* basic idea. A good example is his def. of convergence; easy now, but was a bit confusing at first. Tho I think once I'm able to get the ideas more easily, it'll be even more rewarding.

Thoughts?

Wowza. Six chapters of Rudin in six weeks? How many times do you meet every week?

I'm not sure what you meant by ``thoughts?", I'll take it that you ask how to understand the material quickly. I don't think there's a tired and true method to expedite one's understanding other than practice in time. I'll also add that if you manage to understand the ideas in Rudin in 6 weeks, then you're doing fine. Also, this stuff takes a lot of time to understand. With that being said, try the following:

Write definitions, proofs, concepts, whatever you see fit really, in your own words. By explaining the ideas to yourself, you'll start figuring out how you understand things, and how to approach them. So next time you read a definitions or a proof, you'll be faster.

Get a few more books from your library. Sometimes Rudin is terse, and sometimes those proofs are hard. Other authors expand on the material more than Rudin. It'll be worth it to look some stuff up in those books. I recommend Charles Chapman Pugh's Real Mathematical Analysis. It has the same breadth and depth as Ruding, although sometimes the author does things with less generality.

Read about some of this stuff on Wikipedia. I tried to avoid Wikipedia for a long time, because I was afraid that I'll read an entry that was edited by some crank. All entries I've encountered were nicely written, explained the ideas in depth, and have a nice way of tying things together (how one theorem relates to another, why it's important, generalizations, etc.)

Good luck!

Especially if your first course in upper level math is with Analysis from Rudin. Rudin isn't a bad book, and in fact I like it quite a bit, however, it's a little hard for beginners

In fact, I think that practice and time will help you understand things more quickly

 

[mathwonk]

in my opinion loomis and sternberg is a show offy book (my book is harder than yours) and the two volumes of apostol or the two volumes of spivak, or of courant, are much better.

 

[grendle7]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

---

Does anyone have experience with the math departments at these colleges:
- Berea College
- Carleton College
- Reed College
- UChicago
- Colorado College
-Grinnell College
- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

Just to let you know, it's MUCH harder to get into Berea, than Harvard. G'luck! Out of the "foreign students" pool of accepted students, only 30 aspiring applicants can be chosen, out of thousands. I'd still apply, if I were you. Just cross your fingers for good outcomes, from crazy probabilities. They usually prefer to accept "brilliant" foreign applicants who are living under crisis conditions, really deserve going to college, and/or won't ever have a chance at it; like that talented math-wiz living in Homs, Syria right now.

Either way, it's a great liberal arts school. In my opinion, you could get a great mathematics education there because it seems that their mathematics students graduate with a broad knowledge in mathematics, ranging from pure mathematics, applied mathematics, and statistics/probability; which is ideal, I think. Check out their http://www.berea.edu/cataloghandbook/dpc/mat-c.aspx! The only problem is, though, that they don't offer much variety in mathematics courses :b

And, have you considered, the best one of them all for math (in general), the University of Waterloo? It's in a town close to Toronto, Canada. I'd go there, if I didn't mind getting into debt; "Lulz."

By the way, unless you want to be chocking in debt after you graduate, then go to Colorado College! I'm infatuated with their block plan and great academic programs; and the MAGNIFICENT LOCATION; but it's totally not worth graduating with $130,000+ in debt.

Lol

 

[Mépris]

Just to let you know, it's MUCH harder to get into Berea, than Harvard. G'luck! Out of the "foreign students" pool of accepted students, only 30 aspiring applicants can be chosen, out of thousands.

Coincidence I came back to see this post. I read it before it was edited.

I think my grades may actually be just good enough to get me into Waterloo but it's really not worth the money...that I don't have. I don't know much about Colorado; it looked nice and has financial aid on offer, but it's very limited, as with most liberal arts colleges. I probably won't apply there. There's also the issue of limited coursework but few math/physics majors mean that one can try get some "independent study" thing going on. It doesn't mean grad-level courses, though.

Yeah, I read that about Berea. It's definitely going to be competitive but I believe it's free to apply, so I might as well give it a shot. There's also a list of those "free to apply to" colleges, somewhere on CollegeConfidential. It's easy to find - in case you can't find it, let me know and I'll try dig it up.

Another thing about liberal arts colleges is that bar a few (Amherst and Williams, being one of those), there just isn't much money to give to international students, which makes the competition even fiercer. It makes more sense to apply to larger colleges. Casting too wide a net is also not a very good idea. Too many essays, too much money on application fees, etc but some people can manage that just fine. ;)

This looks interesting:
http://en.wikibooks.org/wiki/Ring_Theory/Properties_of_rings

in my opinion loomis and sternberg is a show offy book (my book is harder than yours) and the two volumes of apostol or the two volumes of spivak, or of courant, are much better.

It's the post below, on another thread, that made me ask the question. I had also, per chance, stumbled upon the book, which is available for free on Sternberg's website.

In spite of its "show offy" nature, is the book any good? As for Spivak, are you referring to "Calculus on Manifolds" or is there another text which comes after "Calculus"?

In the old days, the progression was roughly: rigorous one variable (Spivak) calculus, Abstract algebra (Birkhoff and Maclane), rigorous advanced calculus (Loomis and Sternberg), introductory real and complex analysis via metric spaces as in Mackey's complex analysis book, general analysis as in Royden, (big) Rudin, or Halmos and Ahlfors, algebra as in Lang, and algebraic topology as in Spanier. Then you specialize.

 

[mathwonk]

It depends on your definition of "good". I have already stated that i think it is not as good as the other three I named.

Of course Loomis - Sternberg is very authoritative and correct and deep and well written. But the show offy aspect refers to very little attempt to make it accessible to anything like an average student, or to cover what is really needed by that student.

Differential calculus is done in a Banach space, possibly infinite dimensional, essentially the last case anyone will ever need. Most people will benefit far more from a careful treatment of calculus in 2 and 3 dimensions instead.

E.g. after giving all the definitions of differentiation in infinite dimensions, most applications are to finite dimensions. Even the brief discussion of calculus of variations is apparently influenced by Courant who devotes a chapter to it.

The treatment of the inverse function theorem again in Banach space is overkill, and gives little intuition that is actually needed in everyday practice. The implicit function theorem should be understood first for single valued functions of two variables.

Loomis is an abstract harmonic analyst. His own personal preference is to render everything as elegant as possible, not as useful or understandable.

But make up your own mind. These books are available in many libraries. Just because my course of lectures from Loomis left me feeling very disappointed, with little intuition, and almost deceived as to what is important in calculus, does not mean it may not help you.

If you read Loomis and Sternberg at least you will learn that a derivative is a linear map. That's a lot right there. Indeed that's about all i got from loomis, and it has been very helpful. But I recommend Fleming, Calculus of several variables more highly. Loomis used that book officially in his course, before writing his own.

If you want a very high powered book that also does things in banach space, but manages to be very useful, in my opinion, there is dieudonne's foundations of modern analysis. he also perversely adheres to a credo of making life harder for the reader by banishing all illustrations from his book. but it is good book with a lot of useful high level information not easy to find elsewhere. he explcitly states however that one should not approach his book until after mastering a more traditional course, (e.g. courant).

Another book Loomis used that I do not recommend either is the super show offy book by Steenrod Spencer and Nickerson. As one reviewer put it roughly, this book is more about the ride than the destination. However I do have all these books on my shelf, I just don't look at them all very much nor with the same pleasure.

Your last quote from me above is a historical account of life at Harvard in the 1960's, not a personal recommendation, indeed to some extent the opposite.

Spivak's second recommended book is indeed calculus on manifolds, an excellent place to learn the most basic several variable calculus topics, but very condensed. moreover he makes the proof of the general stokes theorem look very abstract and to me off putting. to understand this result, just work it out on a rectangle in the plane, as lang does in the back if his book, maybe analysis I.

now that i reflect, i am not familiar so much with sternberg's (second) half of the LS book. i only heard him lecture once and was quite impressed with his down to Earth and insightful approach. maybe that half of the book would suit me more.

but I'm not much into physics.

In my opinion you are spending more than enough time here asking for advice, i.e. "dancing around the fire", and need to get to work in the library reading some of these books.

 

[mathwonk]

well you provoked me to go look at LS and i did in fact like Sternberg's chapter 12 on integration.

This whole discussion is beginning to remind me of a friend telling me that his brother warned him off of reading a famous algebra book, so i myself also avoided it for years.

Finally I was required to read some of it and found it wonderfully clear. When I went back and asked my friend's brother he said he had never said it was bad, just "tedious". by which he seems to have meant overly detailed, just what I appreciated about it.

so please take what we have said with a grain of salt and try to get a good look at these books yourself.

Even Loomis' half of the book helped me in the section on "inifinitesimals" and his slick proof of the chain rule.

But the abstract implicit function theorem in terms of projections from a product of banach spaces, there left me wondering what Mumford even meant when he said the theorem simply says you can solve for some of the variables in terms of the others.

 

[mathwonk]

oh, also the intro to LS says plainly that apostol, spivak, and courant are suitable prerequisites for their book. if that includes both volumes of those books, i would agree.By the way, Jerry Shurman's calculus notes from Reed are to me, as an old professional, far too wordy, hence hard to get something out of in a reasonable amount of time. But a beginner might like them just for that reason, so you must be your own judge.

 

[RJinkies]

Hi sahmgeek...


Quote: 'I am seeking advice on a receiving a math degree (e.g. Master's + Secondary Ed cert) however I have very little formal math training beyond high school...

...Given that I would need to start from scratch, I wondered if taking the basics at a community college (Cal 123, Linear Algebra Abstract Algebra, Finite Math, and ODE) and, of course, doing very well...

...stay home with my 2 very young children and would most likely need to go back to school part-time. I might have some time during the day to work on math, but most of my free time would occur in the evening, 8pm and later. I am concerned that this isn't enough free time to really study this subject...

...I am not very concerned about my intellectual capabilities, but with my time constraints perhaps this is an unrealistic goal given the rigorous nature of math. I do, however, like the idea of studying math for it's own sake, even if the end result is purely for personal gain...

------
------

Well, I would think that there shouldn't be any problem for self-study, if you feel that you can put in the time to read the textbooks and do all of the problems.

With the rule of thumb for an Ivy League school, it's about 48 hours of class work a week with a full schedule. And you do that for 12-15 weeks to complete one semester or half of the textbook. One can figure out how to pace yourself pretty okay on your own. As long as you know what the good textbooks and supplementary textbooks [1-3 texts - old and new] should be, and the books fit best with your learning style.

It sounds like you could actually learn the subject better and on your own terms, setting your own schedule as long as you're motivated to get the most out of the textbook by reading all the pages and doing all the problems 95% of the time.

------

As for goals, that can change semester by semester as you master one more notch in the textbook ladder, and your interests may change, and perhaps your direction... If you want to take some courses later, by all means, but I'd probably do most of the work on my own, but it all depends how much time to spend with the family, and how quickly you want to zoom up the ladder of checking off the courses that you got a lettergrade in.


Quite a while back, people could teach high school math where i was with at least a minor in math and maybe a major in education or something else... [like a major in physics and a minor in math and some education courses]

-----

But it sounds like you want to do a BA/BSc in Math and well just know what the basics are, and then add the stuff that interests you to your liking. As you finish off one textbook and then go to the next tier, you get to choose your own path pretty much.

a. getting your Calculus I II III ... and IV [aka Vector Calculus]
b. Taking you Analysis courses and thinking of them as one stream of at least four semesters to like Real Analysis - say from RG Binmore/Apostol/Rudin/Hartle/Strogatz/Royden...
c. Linear Algebra - and up
d. Differential Equations - and up into PDE and Non Linear Dynamics/Chaos
e. Complex Analysis [Applied if you're for Physics, Pure for just math, or maybe both]

You can always figure out if you want to go into [most people might only do 5 courses worth [20%] of these...

f. Geometry - like Coxeter's book
g. Number Theory
h. Mathematical Logic and Set Theory
i. Abstract Algebra [helpful with Analysis to get into Topology]
j. Topology - Munkes and Guilleman as the main two books
k. Probability
l. Differential Geometry and Tensor Calculus - like Synge's book [what you'd want after Vector and for say Wheeler's Gravitation]
m. Mathematical Physics stuff [like if you took Symon and then Goldstein in physics] and then wanted to go into the mathematical side of LaGrange and Hamilton
n. Fluid Mechanics [if you're more physicist/engineering curious]
o. Continuum Mechanics [if you're more physicist/engineering curious]
-----

I tend to think of Grad School as basically what textbooks did you find cool in Third and Fourth Year, and sometimes the rest of those books are your grad school classes [like Royden] or the supplementary reading in those texts...

Me i would choose a Mathematical Physics like route where you can get the best of the Applied and the best of the Pure, i don't think people think of things as Pure Math anymore like Hardy...


I think of it as, spend the 400 hours on each textbook, master things on your own with completing the reading of one chapter as your self-mastery, and then doing all the problems in that book, as the proof of your self-mastery.

That way you don't get hung up on midterms and finals, you see the ladder of math or science as a bite filled chunk as a single chapter, accomplished usually in a week with maybe 20 hours of effort, getting to that goal of the last chapter and 400 hours clocked on your mental library card for your own textbook, using your own dining room table as your own little uni.

 

[RJinkies]

Hi PrinceRhaegar


Quote: my second semester of college as a mechanical engineering major, but I'm thinking about switching to math. The reasons are simple; recently I've found that I'm better at math than any other subject (especially physics...

...I just think math is cooler than any other subject I've seen so far. The reason I'm really hesitant to do so is because firstly, I have no idea what I'd do with my degree after I graduate, and secondly, and this may seem a bit shallow, I know that I'll likely be making more money as an engineer...

...In a perfect world I'd major in math and get a job as an engineer (or at least in an engineering company). This is because I love math and I feel like I'd get a TON of satisfaction out of doing useful stuff for the world while also doing what I love...


-------

Honestly, it sounds like the ideal path is to do both, and just take that extra one or two years for your B.Sc and do a double major

There are people out there that sound a lot like you and they do things like get a Mechanical Engineering degree and double it with a Physics Degree...

if you really wish to slow it down, and you got zero problems with the textbooks, you can almost accomplish it all, and think of it as engineering as a hobby and math as a hobby, and then think of the engineering stuff as your income...

-------

Some bizarre and brilliant souls in 5-6 years end up with a satisfying thing of doing four Bachelor degrees. [maybe 6-7 for ordinary mortals with the same goal]

a. Mechanical/Aerodynamic Engineering
b. Physics Degree
c. Math Degree
d. Electronics Engineering Degree

since there is considerable overlap and his future goals worked out that he used most all of it in his career... though he wasnt as deep as some that just took one and only one path...

But you can be 65%-80% fluent in two courses with a Double Degree.


so there is a LOT you can accomplish with an extra 1.5 years of your life, that these sorts of things are possible.

The Hardest thing is knowing how to self-study and how much effort to put into things, and not fearing failing or exams anymore... the second dilemma is what really makes you happy, and a career may or may not conflict, if you just put some extra time into things.

but sure if you go up a ladder in academia you do tend to end up stuck there, where people who get a physics degree who almost wanted to go to grad school in pure math, and they find they *had* to pick one or the other, but if it's a hobby, or circumstances are right, you can sometimes slide into both worlds... all depends how happy you are, and you like the results..

 

[mathwonk]

or perhaps, just start with one small significant goal, like learning calculus, and do it well. then go further.

 

[RJinkies]

mathwonk - 'perhaps, just start with one small significant goal, like learning calculus, and do it well. then go further...'


Agreed!


I just think it's good to know that the most important thing are probably the two streams

a. Calculus I II III IV - where you know Vector Calculus
b. some Foundations in Analysis that takes you by the hand up to Real Analysis.. [RG Binmore's three books/Hartle/Strogatz - which are all friendlier to start off with than Rudin]


which boils down to a promise that you'll get through 1-2 calculus/Vector books and two Analysis texts...


you could think of Linear and Abstract Algebra as stuff that gets 'analysis' heavy so (b) gets to be important...

and if you're doing (a) you get to see how it all works in the real world with differential equations.

---------------

but it does boil down to, how does one start off... and that's usually with a good algebra text. You suggested one of the classics of the 50s Welchons and Krickenberger [about 1953], and there was also the Dolciani books [about 1964] which was probably the only non-experimental text to come out of that Yale SMSG Special Math Studies Group...

Munem's algebra books in the 80s seemed like a easier path than dolciani also...

And well, i still think that the two old classics from the 1910s and 1930s still work out pretty damn good. Syl Thompson [Calculus made Easy], and JE Thompson [Calculus made Simple]. And we got Mathwonk and Martin Gardner to recommend the first [though Gardner's edition sounded totally unnecessary], and Feynman to recommend the second Thompson.

As for calculus, i think just realizing that mastering one chapter almost perfectly, is better than rushing through the book with 65% comprehension. And most any of the texts from the 1920s to the 1950s i think are great since like the older algebra books, are truly meant to be read front to back, without too many frills of abstraction/formalism/the new math]. And one can always rush to Courant/Spivak/Apostol after the easy books... for the 'deep stuff'.

And that's one of the deepest things i got from Nathan Parke III, about self-study that you go from vigour to rigor. Read the baby book on calculus, and then read the elegant book on calculus, or physics or anything else scientifical...

It's hard for lots of people to appreciate Hardy or Rudin, or Apostol without some gentle breaking in...

--------

and knowing one chapter deeply, by just spending enough hours on it, reading it, and rereading it, and doing all the problems, i think shakes people off from thinking about teachers, exams, and the course as a whole as one hurdle.

If you make just one chapter in physics or math your hurdle, and you take 10-20 hours jumping it, you don't need to worry about falling down, much later down the path...


I just that unis would still offer physics and math right from the basic nuts and bolts more often, rather than expecting people to learn it all perfectly and then some in high school.

 

[dimasalang]

i want to be a mathematician


but I am not very good in abstraction and analysis




is there a magic ingredient to be very good in math?

 

[chiro]

i want to be a mathematician

but I am not very good in abstraction and analysis

is there a magic ingredient to be very good in math?

Hey dimasalang.

The key ingredients are persistance, and thinking about continuously. If you do these and do what you can to learn and later teach what you have learned, you will surprise yourself.

 

[homeomorphic]

i want to be a mathematician


but I am not very good in abstraction and analysis




is there a magic ingredient to be very good in math?

I think the magic ingredient is the ability to conceptualize (although you might say that conceptualizing is just one possible style, and people have different styles). To ask why and find an intuitive answer whenever possible. For the best possible example of this in action, the book that allowed me to take this to the next level when I was an undergraduate was Visual Complex Analysis. I would say reading that book was one of the keys to my success (the other being a few years of trying to understand electrical engineering as deeply as possible). I read it before I took real analysis and then breezed through the class with probably over a 100% with some extra credit when everyone else in the class was struggling (although, I dropped all my engineering classes that semester to switch to math, so I also had a lot of time to spend on it).

Another tip is to go to office hours and talk to professors. I I didn't need too much help, but by talking to my professors a bit when I was stuck, I got to know my professors on a more personal level, which is helpful for getting recommendation letters for grad school. Anyway, generally speaking, one on one conversations are much, much better for communication purposes than lectures (and often written material) are, so it's always good to take advantage of that. Because I'm not very sociable, I think that still holds me back. I'm not that good at picking peoples brains and getting more of their intuition, which is often easier to find by talking to mathematicians in person, one on one.

 

[homeomorphic]

Also, I forgot to mention, if you need to get used to doing proofs, it's good to warm up by studying something like naive set theory, where the proofs are easier (and in set theory, you also get to learn some foundational concepts). For example, Halmos wrote a book on that.

Another route might be to try to study the foundations of geometry, which is also a bit easier than analysis, but is proof-based. I don't know a good book for that. I took a class like that which just used lecture notes.

 

[AnTiFreeze3]

Is it normal to have to spend a relatively long time understanding proofs in a math book? I'm currently going through Basic Mathematics by Serge Lang (recommended to me by MicroMass), and I've noticed throughout the book that, besides the exercises, I'm spending the majority of my time rereading and going over proofs.

Is understanding proofs just a skill that you develop over time, or would it be beneficial for me to pick up a math book that is solely made for better understanding proofs? If so, what would be a good book that would help me out with understanding proofs?

EDIT: Mathwonk, if you do see this, maybe it would be a good idea to update your original post including MM as an active mathematician on this forum, seeing as that's true. When people first join this forum, and possibly read this thread, then they are reading your statements from 2006, not present day, and would be missing out on the information that we have the fine mathematic mind of MicroMass who is also capable of helping out around here.

 

[RJinkies]

Probably the only 'magic' is knowing just how much time to spend on reading and re-reading a chapter, and really grappling with each puzzle you face. You just realize that in most cases, spending a lot more time to know each example inside out, and tackling all the problems.

For me, i find just making a 'single chapter' your goal and putting in something like 20 hours into it, rather than 2-3 hours on things, works for me.

-----

I like to think of semester of math [usually half a textbook], not as one course, or consisting of 4 big exams for marks...

but as single chapters... or sub chapters...

so that half a math text is like 70 mini courses, with no exam, and no teacher.

-----
So one textbook to me is like 30 weeks of reading, with 400 hours of my time to
a. read it
b. keep re-reading it
c. studying all the examples, inside out [or getting out the schaums outline if needed etc]
d. doing ALL the problems
e. if i feel uncertain, do the problem again, or try it another way [or three], and don't surrender so easily

sometimes, you can do this with a textbook and 2-3 supplementary texts, but you really need to watch out about 'order'...

but that can be a bit brutal in a classroom situation or a demanding schedule... but if you eat sleep and breathe something 2-4 hours a day, six days a week, it's amazing what one can do in 3 or 4 months.

if you're less ambitious, finishing 'one chapter' is magic...
and if you got the stamina, go for the next one...

usually you can finish the book.

----
for me the real magic is figuring out what the best textbook for me is, and the 2-3 supplementary textbooks with it are.

I just care about the next rung on the ladder, i don't think about running up the ladder quickly, and i don't think about the next few floors either...