Turns out I _don't_ know what I'm doing...
July 8, 2006 9:49 PM   Subscribe

"Math 51H provides a rigorous, proof-based introduction to linear algebra and differential calculus in several variables." Recommend a book to catch me up to the starting point for this course!

All through high school math classes were taught to me as "memorize equation Z, plug numbers/variables in, solve." I.E., taught to the lowest common denominator in terms of math theory ability. It was inane, and very limiting. Needless to say, school next year will not be that way.

I want to take a class that actually goes into the theory, but I'm a bit worried about just jumping in at this point, with absolutely no background in theory (although a very strong inclination towards math in terms of picking up material - I got a 4 on the AP Calc BC test, despite our school only offering AB as a class).

What I'm looking for, I guess, is a good introduction theory/proof-wise to the math up to this point, so I don't completely die in this class. I'm looking for something that's hopefully not a textbook (ie, is more accessable and readable) but still will give me a fair understanding of basic math (up through single variable calculus).
posted by devilsbrigade to Education (20 answers total) 11 users marked this as a favorite
Politely email Penn's Dr. Max Mintz and ask him if he would sell or snail mail you a printed copy of the textbook he wrote for his CIS 592 Mathematical Foundations of Computer Science class. It is an excellent text that goes into basic proof techniques as applied to number and set theory. In particular, explain in detail your situation as described above — I imagine he would be quite sympathetic.

The class and materials are not specific to linear algebra or differential calculus — and indeed are focused on number theory leading up to public key encryption — but the materials would give you the logical framework for the technique of and language used for writing mathematical proofs.
posted by Mr. Six at 9:59 PM on July 8, 2006

Some earlier threads: 1, 2, 3.
posted by Gyan at 10:21 PM on July 8, 2006

I took that class about 6 years ago (Stanford, right?). I got a 5 on the AP Calc BC test, and had a pretty firm understanding of the material, and 51H was still quite tough to keep up with. Why not take the non-honors version? And doesn't 51H require a 5 on the BC test in the first place?

I was still in 12th grade when I took the class though, so it might have been a bit easier with a year more experience under my belt. I wound up with a B and opted not to continue to 52H.

When I went to UCLA a year later, I found that about the first 2 1/2 quarters of calculus were the equivalent of one quarter of 51H. Which should give you some idea of just how advanced the Stanford program is. I bet that 51 would give you all the "theory" you want.. isn't math all theory anyway?

To your question, I don't have any book recommendations.. as I recall though, the course textbook was Apostol's Calculus Vol II. Maybe Vol I would be a good place to start?
posted by Sxyzzx at 10:24 PM on July 8, 2006

Response by poster: Gyan: Thanks, but I don't need any help learning math. I'm quite proficient. Its really the theory that I want to start into learning, and none of those thread adressed that.

Sxyzzx: Yeah, Stanford. It's a soft requirement at the moment (you're advised not to do it, but its not a prereq). I don't have to commit until this fall anyway (shopping period seems like a great thing), but I would like to explore the 'real' theory end more. I'll check out the book, thanks. All the links were dead on the page for the course, so I didn't get the chance to look at the text.
posted by devilsbrigade at 10:30 PM on July 8, 2006

Response by poster: That should be, of course, none of those threads.
posted by devilsbrigade at 10:30 PM on July 8, 2006

Response by poster: Oops, that was another class that didn't have the texts posted. The page says:

"Multivariable Mathematics" by Theodore Shifrin is required.
"Linear Algebra" by Steve Levandosky is recommended.

I was shying away from those originally because I didn't want the material the class itself covered, but what I "missed" leading up to it.
posted by devilsbrigade at 10:32 PM on July 8, 2006 [1 favorite]

devilsbrigade : "Its really the theory that I want to start into learning, and none of those thread adressed that."

From the 1st thread: "I'd like to understand the theory behind the math more than the math itself. Anyways, any help is appreciated."

The 2nd and especially 3rd threads also answer the same question.
posted by Gyan at 10:43 PM on July 8, 2006

To be concise, textbooks by Spivak & Apostol are what you seem to be looking for.
posted by Gyan at 10:47 PM on July 8, 2006

Response by poster: You win. I missed the Spivak reference (or rather, the link that had the Spivak reference), & I must have missed Apostol somewhere in the flood of "Calculus the Easy Way." I'll see if I can get ahold of one of those this summer.
posted by devilsbrigade at 10:57 PM on July 8, 2006

If this is the Math 51H at Stanford, I can offer a different type of input- I know several people who took the class and found that it was essentially only for people who really wanted to study math. The regular 50 series offered a pretty good background for people who liked math but didn't want to be math majors. Anyhow, that was my experience ~8 years ago; YMMV.
posted by JMOZ at 11:03 PM on July 8, 2006

Apostol's Calculus I and II were the standard 1st and 2nd year undergrad textbooks at Caltech, for what it's worth. Everyone took them, not just math majors...
posted by xil at 12:09 AM on July 9, 2006

For what it's worth, at any top-ranked university for mathematics, the honours math classes really are intended for the top 10% of incoming freshmen. Do not underestimate the calibre of your classmates or these courses. People that take honours courses typically have stellar grades, are contemplating future PhDs in mathematics, and have a LOT of time to devote to the course. That said, on one hand I somewhat regret not trying the harder course. On the other hand I was too busy adjusting to the social aspects of campus life to really deal with it. I found the regular stream of courses for math majors to be sufficiently theoretical.

Here's an alternative intro book for you: An Introduction to Mathematical Thinking: Algebra and Number Systems. This book is used at the University of Waterloo for first term mathematics students. It is the text for MATH 135, "A study of the basic algebraic systems of mathematics: the integers, the integers modulo n, the rational numbers, the real numbers, the complex numbers and polynomials." The purpose of the course is to teach students how to do proofs at the university level. Note that the book focuses on neither linear algebra nor calculus, but rather a theoretical approach to number systems and an introduction to the world of "pure mathematics". It is also rather dry, but the exercises are excellent. I also second Spivak.

I know you asked for "not a textbook", but I think you need a textbook. Reading a book is nearly useless. You must do the exercises.
posted by crazycanuck at 2:49 AM on July 9, 2006

Spivak is very good for the material you'll be seeing and I love his writing style. It's quirky, thought, and not everyone loves it equally, but if you can find a copy through the library or something, give it a shot.

But before that, you might also like to take a look at Liebeck's A Concise Introduction to Pure Mathematics. It introduces various proof strategies and terms from logic and gives you some of the basics of algebra, integer arithmetic, analysis, and sets in very digestible doses. You'll see a lot of well-written proofs and get acquainted with the vocabulary everybody else is likely to be taking for granted.
posted by Wolfdog at 5:29 AM on July 9, 2006

I second JMOZ about the 51H being for the select few. Not that you can't do it, but I had several friends who were really into math and got it handed to them in 51H. Jeez, freshman year was 9 years ago for me, scary.

Enjoy your time at the Farm. Take some classes outside of your major, don't overload the first quarter, and good luck. My email is in my profile for any questions.
posted by shinynewnick at 5:33 AM on July 9, 2006

Having taken what sounds like a similar course at another university, and the problem is not catching up to where the class will be taught, it's being able to keep pace with the material. The great thing about classes that do a rigorous development of calculus is that you basically start at the very begining and move rapidly forward. The problem sets will be much, much different than you are used to, but they won't assume any knowledge you won't have at the time.

The textbook suggestions here are good, and could certainly be of help, and you should definitely be aware that 90% of such a class will be future math majors, but the problem is with pace. It will feel like learning to swim by jumping in the deep end, but if you stick with it, it will be a fantastically useful course. Ask lots of questions, use office hours liberally, and plan to spend a lot of time on the course, try to get a study group together, and read the book on topics you don't understand. You'll be able to do it just fine.
posted by Schismatic at 7:43 AM on July 9, 2006

In my opinion, Apostol has occasioned far more worship than understanding; I sometimes imagine undergraduates with checked cotton short-sleeved shirts and pocket protectors scrambling around at the base of a giant vertical copy of his Calculus, in place of Kubrik's man-apes and monolith.

Spivak, on the other hand, seems to prefer letting mathematics itself, rather than an austere and forbidding presentation, inspire awe in the student.

I also think there is a regrettable tendency, even in institutions you'd think too prestigious for such nonsense, such as Stanford, to set up courses which are more like a mathematical boot-camp, or a medieval trial by fire, than an attempt to raise brilliant and impressionable young students to "a perception of the infinite."
posted by jamjam at 10:35 AM on July 9, 2006

When I went off to Harvard, my 5's in AB and BC calc in hand, I signed right up for multivariable calc, as you're planning to do. I got an A- in it, but in retrospect it was a mistake for several reasons.

First of all, it was a lot of work - as much as any two classes I took later on. It was really hard, and I was coming at it with a flawless math record in classes that were ostensibly the prereqs. A lot of the difficulty was adjusting to the dirty little secret of University life, which is that, in general, no one teaches you anything.

Second of all, no sophomore-level class in any other field required it as a prereq; they didn't expect freshmen to have taken that class.

Third of all, which may or may not apply to you: I *never* used what I learned in that class, nor in the following class, which was pretty much differential equations. It's been more than two decades now and I'm certain that multivariable calc and diff eq are never to appear in my life.

So my recommendation to you, which you didn't ask for, is that you retake whatever your school calls calc 1a (AB calc), and then take calc 1b (BC calc) the next semester. Worst case scenario: you study things you already know, you get A's that boost your GPA, and you have more time as a freshman to explore what's going on outside the classroom. Best case: you'll be well prepared to take multivariable calc as a sophomore.

College is a competition, but it's not the kind of race you're thinking it is. Your plan risks putting you at the back of a more advanced pack; however, it is better to be at the very front of the appropriate pack. In other words, your competition in 51H are going to be not only people who've made 5's on the AP BC calc test, but also foreign grad students who've taught multivariable calc to students in their own countries for decades. Pick your competition so that you can excel.
posted by ikkyu2 at 2:29 PM on July 9, 2006

Response by poster: Yeah, its looking like its not a good idea to take 51H. That's disappointing, but fair I guess. I'm not planning on being a math major, but I want to take at least a bit more. I may still pick up a few of the books from the library. Thanks for the responses, especially those dealing with experiences in this course & similar. I may be emailing some of you with more questions :)
posted by devilsbrigade at 2:59 PM on July 9, 2006

Welcome to Stanford. Freshman fall quarter usually has lots of crazy events.. you might want to take that quarter easy. It takes some time to get adjusted and to meet new friends.

If you are very interested in 51h, I would recommend "shopping the class." That basically means taking regular 51 and 51h concurrently, and deciding before the drop deadline (which is a few weeks in) which one is better for you. Sometimes you'll get the first midterm back before the drop deadline.

I took regular math 51 with an AP background and found it manageable. I did pretty well on the first few tests, and it was really a confidence booster. Good luck.
posted by alex3005 at 3:07 PM on July 9, 2006

Head to the library, and pick up any book on proofs. There are, literally, dozens. A proofs course (whatever it may be called), is fairly standard for a math degree. Thus the textbooks are everywhere. I have a degree in math -- the one we used was A Transition to Advanced Mathematics, by Smith et al.

I read another one that was cool, but I don't remember the name now (it focused on specifically the problem you mentioned, which is [to my mind] that you have been taught didly-squat about proofs in particular, or mathematical reasoning/learning in general). It was good -- it championed "active reading," which is the way any one that actually understands mathematical theory learns. The gist was that you must learn to read mathematics with a very active mind, understanding why a condition was placed in a proof, for example, and you must work out all of the things on your own as you read. Reading new math is not a passive activity. If you're not doing it with a pencil and paper (or whatever), and working it out as you go, there is very little chance you are actually learning it.

Many people are "great at math," but have fuck-all in the way of understanding mathematical theory. Those people are, basically, plug-and-chuggers. They are generally sharp folks, but couldn't develop new mathematics to save their lives. Unless given the theorem/axiom/lemma/whatever up front, they're stuck. Most of these folks could probably get out of that camp, assuming they wanted to [which doesn't make sense for many of them], if they had a good intro. to proofs book.

Your goal is to not be in that camp, assuming you want a degree in math (if your just taking this class to stroke your ego, and don't have much interest in pursuing advanced mathematics as a subject, I'd say you're wasting your time in such a course.).

What you don't necessarily need is a book on the math at hand (linear algebra or differential calculus). You'll do fine with that when you come to it. What you'll need to acquaint yourself with is

a) learning math on your own. You need to be able to sit down with a textbook, read it, understand every line, and be able to apply it. This is very hard for most folks in college. As a college student, your job is to teach yourself. The professor only facilitates. Most people not only don't know this, they also have the very hardest time teaching themselves math.

b) you need a gentle introduction to proofs. The bright folks can and do figure out simple proofs on their own. Most high school and elementary college math completely omits proofs (because students balk). As a result, very basic things about proofs are not completely understood by the bright math student starting out. You need to bone up on this stuff -- at first, it will seem really simple, maybe even an insult to your intelligence. It is not. Spending just a few weeks understanding very elementary proving techniques, learning all of the abstract terminology and rules about sets, logic, etc., will be truly invaluable to you.

Also, be careful with the linear algebra. Many folks think linear algebra == matrix manipulation, and thus underestimate it tremendously. Linear algebra requires you to think in extremely abstract ways if you do anything beyond the plug-n-chug matrix manipulations (which the course you mention will surely make you move beyond). Your first attempts at linear algebra proofs will be very awkward, and probably much harder than you expected. I think proving things in calculus/real-analysis is much easier than proving things in linear algebra.
posted by teece at 3:31 PM on July 9, 2006 [1 favorite]

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