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

26 点作者 andrewparker将近 17 年前
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?

32 条评论

g00dn3ss将近 17 年前
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.<p>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.<p>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.
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rtf将近 17 年前
I don't know, how about every subject?<p>Multiplication. Long division. Algebra. Geometry. Trig. Calc.<p>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.<p>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.<p>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.<p>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.<p>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.<p>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.<p>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.)<p>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."
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DaniFong将近 17 年前
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.<p>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将近 17 年前
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).<p>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).<p>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."<p>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?
nostrademons将近 17 年前
Long division. 1st grade.<p>I never really topped out, because I always figure that when I don't understand something, I'm not smart enough to get it <i>yet</i>, but that doesn't mean I'll never get it.<p>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.<p>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).<p>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.
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mnemonicsloth将近 17 年前
With one notable exception, math seems very easy to me. I open the book, write down verbatim anything labeled <i>Definition</i>, <i>Theorem</i>, or <i>Lemma</i>, and trace through anything labeled <i>Proof</i>. There's nothing particularly hard about it -- you just have to make sure you actually <i>do</i> every step listed inside your head.<p>It is a little slower than reading for pleasure, but with practice, I'd say only by about half.<p>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.
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tjr将近 17 年前
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.<p>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.<p>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.
ctkrohn将近 17 年前
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.<p>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将近 17 年前
Algebraic number theory, when I was a graduate student in Oxford.<p>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...
epi0Bauqu将近 17 年前
Didn't. Just lost interest.
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ekzept将近 17 年前
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.<p>ultimately, all learning is self-taught.
strlen将近 17 年前
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).<p>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).<p>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).
jimbokun将近 17 年前
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).<p>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.<p>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.<p>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.
jfarmer将近 17 年前
I have yet to find it, but Algebraic Topology took me a long, long time to get.
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rms将近 17 年前
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.
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vegai将近 17 年前
I never bothered to invest much to the math classes. The benefit didn't seem so large.<p>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.
a-priori将近 17 年前
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.
rw将近 17 年前
Set theory &#38; mathematical logic.<p>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.
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thorax将近 17 年前
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.
voidfiles将近 17 年前
Last class I attempted was trig, and failed completly. I enjoy math, but have given up untill after I finish collage.
edw519将近 17 年前
I'm still asymptotically approaching the "top out limit point".
nothingHappens将近 17 年前
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
Tichy将近 17 年前
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.<p>I wonder if there is a different way to understand maths out there.
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cousin_it将近 17 年前
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.
jamiequint将近 17 年前
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.
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eru将近 17 年前
Perhaps differential equations for me. But that's more a question of motivation. I like optimization more and look forwand to having more lectures.
streblo将近 17 年前
Multivariable Calc and Linear Algebra kicked my ass. I didn't really do much beyond that.
maxklein将近 17 年前
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.
akd将近 17 年前
My first math class at MIT. :-\
loowee将近 17 年前
Geometry, Logic, Trigonometry
LPTS将近 17 年前
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.
pageman将近 17 年前
did he mean "tap out"? as in UFC "tap out"?<p>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