Ask HN: Where did you top out in math classes?

https://news.ycombinator.com/item?id=206304

by andrewparker • 18 years ago

26 74 18 years ago

news.ycombinator.com

Pretty much everyone (except perhaps tenured professors of mathematics) hits that point in their math training when they realize "I'm just not smart enough to get this." What point in your math career did you hit a brick wall?

LPTS• 18 years ago

For me the brick wall was social, not related to the math, and I hit it in geometry class in 10th grade. Our teacher was an idiot, and had trouble solving the more difficult proofs at the end of the homework. Day after day I would go up to the board and write down the solution to the problem he couldn't solve. Then, I would get an F for not turning in my homework, which I rightly considered to be a waste of my time, and the people who copied down my answers would get an A's. That was when I decided I had enough formal mathematical education. I taught myself the math to do AP chemistry in high school and tested out of pre calculus in 2000 in college. I am currently teaching myself calculus from Michael Spivak's Calculus. You couldn't get me back in a math class, but I love learning math.

nostrademons• 18 years ago

Long division. 1st grade.

I never really topped out, because I always figure that when I don't understand something, I'm not smart enough to get it yet, but that doesn't mean I'll never get it.

This pattern started, as I said, in 1st grade with long division. My dad had been trying to teach me math early, and I whizzed through addition, subtraction, and multiplication, but I just couldn't understand long division. My mom (who always took a dim view of acceleration) said "Just let him learn it in school with the other kids." So that's what we did, and when 3rd grade rolled around and we did long division in class, I got it right away.

I did similar things with algebra (dad first tried to teach me it in 2nd grade, didn't get it then, but I started rederiving it on my own in 6th grade and my teachers figured it was time to get me an algebra textbook) and logarithms (which I first tried in 8th grade, but didn't understand for 4 full years...that was my block through all of high school).

As for how far my formal mathematical training has gone - I aced up through vector calculus in college, and also took discrete math late in college and aced it. Also took Functions of a Complex Variable and Mathematical Logic, but got lost around halfway through each of them. Passed, but not really competent in them.

eru4• 18 years ago

Most of us have forgotten how to do long division on paper - I guess. It is quite a nice exercised (if a bit geekish) to rediscover it on your own.

ctkrohn• 18 years ago

When taking 2nd level honors linear algebra, we were given a takehome exam. I spent ~40 hrs on it and got just under the median grade. I was able to pull out a decent grade in that class, and I graduated with a degree in math, but now I know I'm not smart enough to hack it as a professor.

Not that I'd want to, anyway. Math is a great subject, but it's not what I'd want to do with my life.

cperciva• 18 years ago

Algebraic number theory, when I was a graduate student in Oxford.

Well, sort of -- by that point I was pretty firmly on the CS side of the fence, and was just sitting in on the number theory classes out of interest. I probably could have grokked class field theory and L-functions if I had taken the time, but I was busy and it wasn't my research area...

g00dn3ss3• 18 years ago

Recent evidence tends to indicate that 'not smart enough' is probably a myth. Almost everything can be attributed to exposure and effort at some point rather than some innate smartness.

A lot of advanced math takes some serious concentration to understand. For some non-practical aspects, I found that I lacked the motivation rather than ability to understand it . One particular class where I seemed to hit my tolerance was a theoretical linear algebra class. I could understand the practical applications of most of the topics but some of the theory seemed just out of reach. The book was extremely dry and I think the professor may have been taking lessons from Ben Stein.

Give me a private tutor, a theoretical linear algebra for dummies book, and a pending disaster for which this is the solution, and I bet the outcome would be a little different.

aneesh• 18 years ago

Very true. I really enjoy math, but after abstract algebra and real analysis, I slowly lost the motivation to go deeper into pure math. Interestingly the most enjoyable areas of math for me are those arguably most applicable to CS: graph theory and number theory.

flashgordon• 18 years ago

Totally agree. Case in point. Up until highschool I was very good at math. Mainly because I would actively seek out books and try to solve problems where ever I could find them. As soon as I started Uni, my interest dwindled (was replaced by love for all things CS). People who I had dominated back in highschool were now getting a good 10-20% higher than me. All in practise I think.

manvsmachine2• 18 years ago

Haven't gotten to that point yet, but I can definitely attest to the significance of good materials and enthusiastic teaching. Because I started out as a engineering major, I had taken the entire Calc sequence up through Diff. Eq's, but I ended up having to take Prob/Stats and Discrete when I Switched to CS. I was sick of taking math classes at that time, but my Discrete prof was awesome and motivated me to see how far I could go with it. I went into fall semester never wanting to take another math class again; now I'm taking more so I can get a minor in it. I'll be taking combinatorics in the fall and topology(?) the following spring to close it out.

bkovitz• 18 years ago

Yup, motivation is the blocker.

I have repeatedly gotten tired of math when it seemed like meaningless puzzle-solving. Why do I really care, for example, how many nonabelian groups of order 36 exist? When math seems to me like it's providing a vocabulary for framing and answering deep questions about the world, or making it computationally feasible to find the stamp of causal influence in data or design real things that are impossible to make without out, then the motivation is there.

This might just be an artifact of the way math is often taught. For example, real analysis is often taught as meaningless proof-finding (e.g. proving things that seem either pointless or obvious). But there's a fantastic book, _A Radical Approach to Real Analysis_ by David Bressoud, that teaches the exact same subject matter as the fruit of deep, pressing, and non-obvious questions that stirred debate among mathematicians for around 100 years.

manvsmachine2• 18 years ago

Many branches of math start out as being "meaningless". Number theory was once widely considered meaningless until it was given importance through cryptography. The current head of Microsoft's Cambridge research location literally said to Bill Gates as she was being hired that her work was completely unrelated to computer science; her work was in combinatorial optimization. I think that may be a part of the problem: not enough math professors teach their subjects with any regards to what they are used for. Sometimes, they don't even know themselves.

doubleplus• 18 years ago

Would you happen to have any links or book references to this "recent evidence"? While most of success comes from effort and discipline, I've always been under the impression that only the naturally gifted (who also have discipline and drive) can/have achieve/achieved certain things. The dwarf down the street from me will never be a professional football player, and someone with a 100 IQ will never be an astronaut. I've always felt the you-can-always-achieve-whatever-you-want-no-matter-what-if-you-want-it-bad-enough approach to be more than a little cliched and misguided. If recent scientific evidence has disproved this stance of mine, I'm open to changing it.

elai• 18 years ago

To be honest, I believe that all intelligence comes from innate motivations and practice. You suddenly become 10x more "intelligent" when you actually give a shit or get some innate joy or obsession relieved when you do it. If that is something that is constant within you, you become more and more practiced and a virtuious cycle results. Nothing is too hard to explain, it's just mathematicians are really horrible at communicating.

g00dn3ss3• 18 years ago

Here is a pop-sci version of some recent research:

http://www.sciam.com/article.cfm?id=the-secret-to-raising-sm...

However, this isn't particularly surprising. Research on "grandmasters" in many activities, like chess or pilots, usually shows they've put in 25K hours of practice in their field.

I'm certainly not in favor of saying anyone can do anything. Especially in physical activity. There are definitely physiological advantages.

The same is also true in mental activities to some extent. A kid with down syndrome probably isn't going to become a math professor. However, I think the point is that most normally developed folks probably haven't hit a ceiling in their math aptitude. They just stopped caring or putting in the effort.

mlinsey• 18 years ago

I don't think this is true - or at minimum, higher math/theory certainly comes easier to some people than others, and there probably comes a level of intelligence where you were progressing slowly enough that it really wouldn't make a good career option for you.

I hit my brick wall in theoretical classes when I felt like I would understand everything I had been taught up to that point, but to actually construct a proof to solve certain problems I had to come up with some flash of insight which just wouldn't come to me. I simply wouldn't know where to start.

I don't think that this is uncommon. When many people write about famous mathematicians and how they solve extremely difficult problems, it usually happens that they have a certain intuition or insight which didn't really follow logically from the problem as stated up until that point. Perhaps anybody could put enough effort into these problems that these insights would come to them as well, but I think this is doubtful considering just how brilliant these insights are.

Put another way, when you reach a certain level of math it starts requiring a large degree of a particular kind of creativity, which simply not everyone has.

g00dn3ss3• 18 years ago

Actually, there was a recent article on YC news about the large number of discoveries that were made independently at almost the same time.

Our made-for-tv histories usually show a lone scientist toiling away for years, making huge discoveries. But in reality, most discoveries are incremental. The stories we are taught about these discoveries are usually mythologized.

It is true that there is some level of creativity needed. But it's probably way more common than our histories would lead you to believe. The recurring theme you will hear from scientists is that "chance favors the prepared mind."

edw519• 18 years ago

I'm still asymptotically approaching the "top out limit point".

voidfiles• 18 years ago

Last class I attempted was trig, and failed completly. I enjoy math, but have given up untill after I finish collage.

rms2• 18 years ago

Calculus III... I think I could have got it, but it's tough material and I had a terrible professor, so I didn't really learn any Calculus III. I passed and did much better in my final math course, a one semester differential equations/linear algebra hybrid course.

rms2• 18 years ago

Also, my first semester I took one class of "Honors" calculus which was a proof based differential+integral Calc course. I dropped that really fast in favor of Calc II. So I also hit a wall at proof based calculus. Most of the people who actually took the class didn't get it either and resorted to memorizing all the proofs needed to derive calculus.

tjr• 18 years ago

I went to Cornell College, which schedules classes on a "block plan", one class at a time for a month each. My first semester, I took four consecutive courses in calculus. By the fourth one, I was extremely burned out on math and barely squeaked by.

I ended up taking a couple more math classes before graduating (discrete math and linear algebra), have continued to study math on my own, and have recently been contemplating a master's degree in math, just because I want to learn more.

All of that is to say, I'm not sure that the math brick wall is constant, but perhaps sometimes you need to take a break to allow your mind to digest what you've learned so far.

jfarmer2• 18 years ago

I have yet to find it, but Algebraic Topology took me a long, long time to get.

mnemonicsloth3• 18 years ago

My first thought on reading this was "dude, you stole my answer."

After thinking about it for a while, though, I think General Topology is worse. I still don't get it. Or, what's worse, maybe I do get it, and just don't care enough about the subject to register that fact.

davo112• 18 years ago

I got it I think and still open up the books every now and then, but it was about topology and model theory that I started to question its applicability. I remember asking my lecturer this (I could have phrased it a little better in hindsight, I sort of burst out during a lecture 'whats the point of this' The lecturer got a little upset :-)).

Transfinite numbers are always my favorite example of something way out there, different sorts of infinities and which infinity is bigger than the other, the models are beautiful, but I couldn't spend my life arguing about them. I think for most people maths starts losing its relevance a lot earlier, it justs takes some longer to admit it, because it's prettiness is so seductive :-).

rtf• 18 years ago

I don't know, how about every subject?

Multiplication. Long division. Algebra. Geometry. Trig. Calc.

I was never very motivated to study math. The problem was, my older brother was very into it(and now is a math grad student, ever-so-slowly getting his thesis together). This set a model that I could not hope to emulate, but it only meant my mom pushed me more, talked to the school to get me into the advanced/accelerated classes I didn't really want to take. She probably would have done some of that without my brother around, but not as much.

This led me down the "please the parents" line of study, which naturally meant some surreptitious, embarrassed attempts at cheating. This only made me feel worse, of course.

In college I started into computer science, thinking that I at least liked the programming. But integral calc sunk me for good, and in a particularly bad quarter that was my low point, I tried taking linear algebra as well as a repeat of calculus, thinking that perhaps the extra pressure would do something good.

Of course not. I dropped linear algebra and failed calc again. After that, I decided to declare in economics, restarted calculus with the "ez-for-econ-majors" series and sailed through those courses with a solid B average. I struggled through, but passed on the first try, the two intermediate econ courses which started introducing serious mathematical modelling. The remainder of the major was electives, and not difficult ones.

I never knew, until after that whole period of my life was over with, exactly what was holding me back. Now I'm pretty sure that it's about motivation and dedication. My brother is fairly normal but can get interested enough in math problems to sacrifice his well-being. The genius researchers of the field sacrifice well-being regularly, without really knowing it, and are typically slightly unhinged socially.

As for myself, I tend to run away from a challenging math problem. So, even if I'm forced to tackle it, it will probably take me 10 times as long to solve as it would my brother(not even factoring in his years of experience now). Once I overcome those hurdles particular to a new category of problem I am fine, but I have to take considerable effort to do so.

Summing that difference up over a long-term period like that of a college course, the best students can zoom far ahead because of this motivation factor, even if they aren't necessarily the _smartest_. Indeed, many math students reach the upper-division levels on memorization alone and get stuck from there, as proofs take on more and more importance. That's a major failing of current math education in the United States - overdependence on rote techniques. (The former Soviet educational system, OTOH, had probably some of the strongest math education, and much of it has been translated to English - pick up a book from that period and you will probably see a small and dense text that introduces high-level concepts in great, if unforgiving, detail. Very different from the thick drill+practice textbooks I'm used to.)

My conclusion: many academic fields can accommodate a half-hearted practice. Math is not one of them. And our society doesn't respect that difference, shoving it under the rug as "I'm just not good at math."

plinkplonk• 18 years ago

"The former Soviet educational system, OTOH, had probably some of the strongest math education, and much of it has been translated to English - pick up a book from that period and you will probably see a small and dense text that introduces high-level concepts in great, if unforgiving, detail. Very different from the thick drill+practice textbooks I'm used to."

Could you provide a url for some of these? Should I look up any specific publishers? Thanks in advance!

gunderson• 18 years ago

I second that.

This is one of the best ones:

http://books.google.com/books?id=ikMAzFXpFOsC&printsec=f...

btw if anyone wants to do a collaborative chapter by chapter self-study, just send me a private message. I have had it on my shelf for a while but haven't taken the time to do a few pages a day as intended.

Shortnick• 18 years ago

Not pure maths, rather applied, but one would have to mention the Landau course as sharing those characteristics.

http://www.amazon.com/Landaus-Course-in-Theoretical-Physics/...

epi0Bauqu• 18 years ago

Didn't. Just lost interest.

mattmaroon• 18 years ago

Same. Realized I wasn't going to go through life as either a mathematician or a physicist, so didn't pursue beyond Calc 2.

a-priori• 18 years ago

My most advanced math course is currently Calculus II (integral calculus and so on). However, I don't believe it's my "wall", and I'm planning on taking courses on differential equations and linear algebra whenever I have some time.

streblo• 18 years ago

Multivariable Calc and Linear Algebra kicked my ass. I didn't really do much beyond that.

vegai• 18 years ago

I never bothered to invest much to the math classes. The benefit didn't seem so large.

Of course, now I see how completely stupid that line of thought is. The time I saved by not attending classes or demonstrations was indeed not well spent.

rw• 18 years ago

Set theory & mathematical logic.

Learning math is all about spending time with the material. I find that upper-level math is easier than lower-level topics. Less grunt work, more pondering.

mnemonicsloth3• 18 years ago

Foundations proofs are the worst. I can still remember proving something like "For every left-paren, there is a matching right-paren." Ugh.

mnemonicsloth3• 18 years ago

With one notable exception, math seems very easy to me. I open the book, write down verbatim anything labeled Definition, Theorem, or Lemma, and trace through anything labeled Proof. There's nothing particularly hard about it -- you just have to make sure you actually do every step listed inside your head.

It is a little slower than reading for pleasure, but with practice, I'd say only by about half.

I think the primary reason I do well, though, is that I take adderall for ADHD. Stimulants make it trivially easy to maintain the necessary level of focus, but whenever I forget to take them before a lecture or study session, I'm gnashing my teeth and tearing my hair out by the end.

muriithi• 18 years ago

Do you take adderal because you suffer from ADHD or you take it get focused?

jfarmer2• 18 years ago

As with most subjects there are levels of understanding.

One level is the ability to understand what other people are saying or doing. Another is the ability to do it yourself.

Once you get to higher level classes, at least where I went to school, the reading is relatively light but the work is relatively hard.

I don't think most people had trouble understanding the proofs -- they had trouble applying the lessons in a novel way to exercises and problems they'd never seen before.

loowee• 18 years ago

Geometry, Logic, Trigonometry

strlen• 18 years ago

My undergraduate degree is in Computer Science, which at the time I attended the university was (if one was doing their degree in the College of Arts and Sciences vs. School of Engineering) in the math department. In the end I ended up going further than I would have naturally (just by the nature of department requirements).

First response to this question would be "what kind of math"? In terms of continuous math, I've topped out at Differential Equations (much like any other CS Major). In terms of discrete mathematics I topped out at abstract algebra (the first class that made me sigh with relief upon seeing actual numbers) and combinatorics. I felt that I could certainly go on further in the discrete field (I particularly regret not taking number theory - other than what I've learned in cryptography courses - and graph theory).

I could have received a Math major, but that would have required taking the analysis series (real and complex), which I felt would go beyond my level of abilities (particularly since I was aiming for an early graduation and wanted to take as much of classes that I felt would interest me more).

ekzept• 18 years ago

there's learning on an academic track timeframe, and there's learning. there are lots of courses i took which i did not do well gradewise, but i learned a lot, enough to go back and re-learn the material at my own pace and better.

ultimately, all learning is self-taught.

Tichy2• 18 years ago

Seems to me all or most human understanding of maths is poor. Most things, even if we can prove them, we can not really understand them.

I wonder if there is a different way to understand maths out there.

nazgulnarsil2• 18 years ago

you're probably a victim of rote learning math classes. learning math for real (getting it) is not a linear process. lots of aha moments where lots of concepts fall into place suddenly and then you move on to more advanced things.

that said, I would consider everything below calculus to be merely clever ways of tallying stones. algebra can be tricky at times (abstract algebra sucks) but it is really not that interesting in of itself (yes you can solve a lot of practical problems with it but the questions aren't generally interesting, so the answers aren't that interesting). you do need to have it down pat in order to "get" calculus however. And if you "get" at least the basic concept of calculus down a lot of things fall in to place.

OTOH I could be totally biased and/or wrong, but I personally didn't really see how math could describe complex systems in a way that let me get those complex systems until calc. Some really mind blowing aha moments.

Tichy2• 18 years ago

I did understand a lot of maths, I just mean that I think the way we do maths often is not really understanding. If we have proved something with a hundred step proof, I might be able to verify every single step of the proof and conclude that it is correct. But we might not really have a feel for why it is correct. Can't really think of a good example right now :-/ I mean usually we jump through quite a few hoops to prove something - maybe there are other ways of looking at it that would make more sense.

One example might be the autistic people who just know if a number is prime or not. Not sure if they are really doing anything special, or just calculating really fast, though.

davo112• 18 years ago

funny, I always thought it was the other way around, calculus was just a clever calculating trick, but give me a good algebra, then we can really do something :-)

akd• 18 years ago

My first math class at MIT. :-\

jamiequint• 18 years ago

Vector Calc, because I stopped caring. Somehow Diff. Eq. held my interest but vector didn't. Calculating the area of partial 3D oval objects for the sake of doing it was painful.

eru4• 18 years ago

Diff. Eq.: Difference Equation or Differential Equation?

cousin_it• 18 years ago

Got my masters in math, then kinda lost interest, but I'm pretty sure that everything is understandable with a little effort. When I get genuinely interested, no effort is required at all.

pageman• 18 years ago

did he mean "tap out"? as in UFC "tap out"?

can't get enough of Math (although I barely passed Linear Algebra and Numerical Analysis). Maybe it's the challenge. I shall try more of it when I take up a Masters in CS. :P

maxklein• 18 years ago

Math is pointless if you are not planning to apply it at some point in your later career. Learning algebraic topology when you are studying digital circuits is like learning literary analysis - it may make you feel smart, but it's a damn waste of time.

thorax• 18 years ago

I don't know if I topped-out, but I did lose interest. Calc III felt hard with not very much useful information (my comp sci advisers basically gave me grief for taking it). Linear Algebra sparked my interest again, but only because it started out wicked hard and then (just like with Calc years before), it just "clicked" and suddenly it was a lot easier/interesting. I didn't pursue further because after I graduated I preferred diving into new compsci (and engineering) concepts rather than focusing in math areas.

nothingHappens• 18 years ago

I very nearly hit that point taking Differential Equations my last semester, but thanks to persistence and a very helpful prof, I got over it. Not sure where I'd have ended up if I kept going, but I've always regretted not taking more abstract/"modern" algebra

DaniFong5• 18 years ago

I'm not sure I ever had that reaction. I explored a few topics in different lectures in grad school which were pretty obtuse: ergodic theory, for example. I didn't get it, but I assumed it was because the topic wasn't suited to lectures, and after learning of the basic direction they were headed in and setting out myself, I found I could get pretty far.

I've seen others give up on some topic rather than the method they used to try to understand it. It's the equivalent of someone saying "I can't get math" because they're spending all of their time remembering formulae, or someone saying "I'm no good at programming" while spending their time reciting bubble sort line by line in preparation for the exam. Giving up on higher mathematics is like that. Almost always, there's some path to real understanding -- you just have to find it.

geebee• 18 years ago

In retrospect, I topped out in my upper division coursework for the math major. I still got B's and even some A's in those courses (Real Analysis, Abstract Algebra), but I was reduced to studying solution sets rather than mastering the material creatively. This was when I got more interested in applications (optimization, scientific programming, etc).

I figured I'd pick a grad program in Industrial Engineering so that I could get away from math and more into applications. Unfortunately, I picked Berkeley. A graduate intro to optimization was pretty much a series of proofs about convex sets. Same deal for stochastic processes (it might be different with different professors, though). BTW, this isn't meant as a knock on Berkeley - this theory focus is a big part of why it's such a powerhouse program. But it probably wasn't the right choice for me. I've heard Stanford is more flexible (I've also heard the grass is greener, so who really knows).

At this point, I topped out in grades as well as understanding, because I didn't have the motivation to continue. I tanked, and squeaked by with a MS. I actually missed problems that I would have gotten right as an undergraduate. I was thinking "I've had enough of proofs, I just don't want to do this anymore."

In a way, I got a lot out of the experience. My interest in academics waned, and I pretty much just started hacking with like-minded students. In other words, I finally, belatedly arrived at the process of becoming self-educated, which is really the only kind that sticks, right?

eru4• 18 years ago

Perhaps differential equations for me. But that's more a question of motivation. I like optimization more and look forwand to having more lectures.

jimbokun• 18 years ago

I was required to take two semesters of Statistics as an undergrad. I didn't think it was especially difficult, but I didn't see applications for the things I was interested in (mainly Computational Linguistics at the time).

A few years later, pretty much all Computation Linguistics/Natural Language Processing research revolved around Machine Learning approaches, which is largely Probability and Statistics applied in various ways. I really kicked myself for not taking Statistics more seriously at the time, and not learning more.

I was away from this field for about a decade, but have been working with researchers in this field again in my current job and have been taking classes to catch up. In addition to Probability and Statistics, I've been learning Linear Algebra as fast as I can and want to take a course on Optimization.

So, I suppose the moral of the story is: you never know when that particular branch of Mathematics that doesn't seem at all relevant to you might suddenly turn out to be very relevant.

Ask HN: Where did you top out in math classes?

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