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Mathematicians are chronically lost and confused

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Re: Mathematicians are chronically lost and confused

#31

Post by graycat »

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

Group theory is a standard topic in a college course, say, at the junior level, in 'abstract algebra'. So, the other topics typically are rings, fields, vector spaces, maybe the construction of the rationals and reals. The course might also toss in a little on number theory.

Anyone with a ugrad pure math major is supposed to know what a group is and the early, standard theorems.

Group theory was used by E. Wigner for the quantum mechanics of molecular spectroscopy so that at times some chemistry students want to know some of group theory and group representations.

My ugrad honors paper was on group theory.

I published a paper showing how a group of measure preserving transformations could lead to a statistical hypothesis test useful for 'zero-day' monitoring for anomalies in server farms and networks.

Group theory pops up occasionally. Get a good, standard text in abstract algebra or two or three and spend a few evenings. If you get to Sylow's theorem, you are likely deep enough for starters. Group theory is very clean, polished stuff and can be fun. Go for it.

Re: Mathematicians are chronically lost and confused

#33

Post by dalke »

"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something."

This is true for all research.

And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study.

This is why students who are "good at X", whether it be math, German, sports, or programming, may become frustrated when they find out that "good at researching X" is a very different matter.

Re: Mathematicians are chronically lost and confused

#34

Post by mathattack »

It's strange to hear mathematics described more as a search for art and structure than computation. Unfortunately most of my math education was on the computational/applied side. I'm only getting into number theory and the more esoteric math later in life for fun. As a parent I think we can't let the school system destroy our kids love of math through too much rote learning. We have to make it fun for them. (Same with music btw)

Re: Mathematicians are chronically lost and confused

#35

Post by japhyr »

acjohnson55 wrote: post #28

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety…

as the math truck barrels on ahead

I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now.

What most of my students think: "I could never do math, I fucking hate it, and I might drop out because I will never finish my math credits. I can't do math because it's stupid and meaningless and I will never get it."

What really happened to get people off track?

- Some just didn't follow one topic in some early grade, nothing else made sense after that, and no teacher was prepared to get them back on track.

- Parents split up, student couldn't focus in school for 6 months, they got off track.

- Parent/ sibling/ significant person passed away when student was young, couldn't focus for 6 months-2 years, no way to get back on track.

Any number of other external events happen, and it is perfectly reasonable for students to get off track in math.

a systematic way to deal with the discontinuities when they strike, especially that first time

Exactly. I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track. Helping a kid master some mechanics does a little to get them back on track, but diagnosing misunderstandings takes more math expertise than most elementary teachers have.

I could go on forever; thank you for putting some of these issues so clearly in focus.

Re: Mathematicians are chronically lost and confused

#36

Post by nly »

coldpie wrote: post #25
cs702 wrote: post #22

Earlier quoted context omitted.

Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The sa…

Just as a specific example, I had this experience with Bayes' Theorem " rel="nofollow">http://en.wikipedia.org/wiki/Bayes%27_theorem> . As an informal paper for my computer security class, we used Bayes' theorem to implement aimbot detection in a simple FPS. It sounds like a big, complicated theorem with a special name that some genius had to come up with and has complicated notation involving probabilities and logic…

It's the same with the crypto we rely on. There are some quite complex mathematical underpinnings for the primitives, but once understand their properties all the common protocols are actually incredibly trivial. DSA and various other iconic crypto protocols are just linear algebra using these new mathematical primitives. The complexity is about the same as using hash functions to build hash tables. Making protocols secure is actually mostly an implementation challenge (as it is with hash tables), and keeping systems secure, pragmatically, is about key management and social challenges.

I was incredibly proud when I noticed, just through algebraic manipulation, without reading it anywhere first, that you should be able to extract the public key used to create an an ECDSA signature. Schnorr signatures don't have this property. This is kind of sad in a way becaue it's trivial, but you have no know how the primitive functions and the difference between a field and a group.

Re: Mathematicians are chronically lost and confused

#37

Post by graycat »

nbouscal wrote: post #27
graycat wrote: post #13

No, the OP is giving bad advice. Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the In…

You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.

No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated in error, out of place, use the Internet, etc.

To do just the "opposite" of his straw man is not good -- for solid foundational material, Halmos, Rudin, Royden, etc., the exercises are darned important. Right the Rudin exercises where have to consider uncountability are not so good. The Royden exercises on upper and lower semi-continuity are a lot of work for a little curiosity but likely won't see again. The Fleming exercise on every bounded linear functional on a intersection of finitely many closed half spaces achieves a maximum value is mis places. Etc. The abstract algebra book I had had an exercise where the student had to reinvent Sylow's theorem; a student wrote the author and got back a letter that the purpose of the exercise was to see if a student could reinvent Sylow's theorem -- bummer, misplaced exercise.

I'm correct.

Re: Mathematicians are chronically lost and confused

#38

Post by jmclean »

acjohnson55 wrote: post #28

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety…

This hits close to home for me. I was one of those kids who was 'good at math' throughout pretty much all of school. I tested out of Calc I thanks to AP classes and began university in Calc II.

It didn't go well. Somehow this thing that had seemed so natural and intuitive now just seemed totally incomprehensible, mainly because I really just didn't understand the level the abstraction was at or something. I did poorly and it really shook my confidence. I changed majors and wound up a designer instead.

It's not a surprise to me that I've worked my way back into a field with deep math roots, though I think I would have found it much sooner if it hadn't been for that initial roadblock.

It gave me a new appreciation for what many of my classmates were struggling with in the early math that I breezed through in grade school. It's very difficult to see what the concepts you're learning are building towards if you don't have a sense of the bigger picture.

Re: Mathematicians are chronically lost and confused

#39

Post by bpyne »

I wish this post was around when I finished my undergraduate degree in Mathematics. I would have taken my adviser's advice to go to grad school. At the time, I remember telling him that I feel like a barely made it through the program. Apparently I wasn't alone. Amazing the difference 25 years and the internet makes.

Re: Mathematicians are chronically lost and confused

#40

Post by hessenwolf »

pcmonk wrote: post #30
nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating…

I regularly explain my PhD thesis on napkins. The explanation involves twins, one with a broken nose; fish; GPS satellites; and a tractor driving in a field.

Somehow this holds a crowd better than non-linear 72-dimensional space, and isometric and rigidity matrices.

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