Ask HN: Where did you top out in math classes?
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?
72 comments
[ 2.9 ms ] story [ 122 ms ] threadI 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.
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.
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...
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.
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.
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.
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.
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."
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.
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.
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 :-).
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."
Could you provide a url for some of these? Should I look up any specific publishers? Thanks in advance!
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.
http://www.amazon.com/Landaus-Course-in-Theoretical-Physics/...
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.
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.
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.
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.
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).
ultimately, all learning is self-taught.
I wonder if there is a different way to understand maths out there.
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.
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.
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