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.
The danger with what you're saying now is that someone with an IQ of 90 -- someone who could be a fine contributor to society in lots of practical, necessary fields that don't require lots of abstract thinking -- could be inspired to throw away lots of time and risk lots of frustration trying to be a mathematician. We should deal with the fact that wasting education on someone who can't use it is as much a tragedy as failing to educate someone who can use it. By pretending that 'smart' just means 'trying hard', you're doing more harm than good.
Everyone carries around the absurd burdens of judgments and measurements, but they don't always mean what people think they mean.
It would be interesting to know more details, which I'm sure you can't divulge without violating someone's privacy. But it would be neat to find out if those people had other skills that correlate strongly with IQ, like the various digit-memorization/recitation tests, or reaction time. Were these people autistic? Were they taking a test in a language they didn't know too well?
I once heard IQ defined as "ability to navigate bureaucracy, getting the answers others think correct in a manner testwriters imagined, and color in the lines" (paraphrased). This may have something to do with the phenomenon here.
The odds that someone would 'get a degree, earn lots of money, stay out of jail, vote often, delay having kids, etc.' seem to have more to do with successfully conforming to certain values of society's upper-middle class.
I think I'm capable of inventive thought, but I don't particularly want to get another degree, or earn much money, or delay having kids, or vote often, and the sort of things that one has to do to stay out of jail, are, honestly, quite often absurd, and I often rail against them.
And if you start to observe, closely, just how these things are tested, you'll start to get the impression that maybe bright people will find their ways through the cracks more than anticipated. There's an enormous weight given to quick answers -- time directly influences scoring. Linear answers are expected, and alternative interpretations are docked. There's often insufficient information in the questions, or it's based on a model of the world that's wrong. Domain knowledge like mathematics or vocabulary is brought into it. Analogies are made to hone in on one relationship from the many that could exist. Scorers of essays give insufficent weight to substance and too much to form -- despite their lipservice, they are indeed swayed by big words.
Even tests like GRE physics are bad. They claim to be testing rapid physical intuition, but in practice what divides good from poor scores is prior experience with 100 simple systems and the ability to get the factors of 2 and pi right with three minutes per question.
"...seem to have more to do with successfully conforming to certain values of society's upper-middle class. I don't particularly want to get another degree, or earn much money, or delay having kids, or vote often, and the sort of things that one has to do to stay out of jail, are, honestly, quite often absurd, and I often rail against them."
My point is that, all else being equal, you could get a great job if you chose to, and that if you do illegal things, you apparently do them in such a way that you won't get caught. IQ seems to correlate with the ability to delay gratification, which itself seems to be a better predictor of success than IQ (sadly, it hasn't been tested in a rigorous, long-term way -- so it can't match the hundred or so years worth of data people have compiled on IQs).
One of the reasons that these tests emphasize time is that 'quickness' is a component of IQ. Francis Galton, the first guy to really study the subject, liked to think of things in that way -- and given that physical reaction time correlates so well with IQ, he had a point.
I agree that the essay part of standardized tests is messed up. Lots of the recent changes to tests seem to arise from political correctness. Test-makers found out that you can't design a test that has predictive value without getting politically incorrect results, like lots of men on the extremes, or lots of high-IQ Jews and Asians and low-IQ Mexicans and blacks. So they periodically adjust the scoring mechanism or the test to get bell curves closer to the same median and standard deviation (more focus on the median than the SD, since most of the people who complain about such things don't know what 'standard deviation' means). So keep that in mind when complaining about the essay, or the rebalanced scores, or the analogies (which were dropped from the SAT -- analogies happen to be more IQ-weighted than other categories of questions).
My main point is that truly excellent thought doesn't depend on the same skills that would allow scoring highly in an IQ test. In many cases such skills would inhibit it. In an IQ test it helps to rapidly adapt to the assumed constraints of the problem and come quickly to the closest, most linear answer. If you have a mind compelled to bring questions back to reality, to challenge assumptions or think of things from many different angles, you'll do, on the whole, more poorly than if you had not. But these behaviors are sometimes precisely what you'd want!
The millions of immigrants from there to here don't seem to buy your argument. Which Latin American countries do you mean?
"... truly excellent thought doesn't depend on the same skills that would allow scoring highly in an IQ test."
I happen to agree with you! One thing that's bugged me for years is that you can't design a test that distinguishes, in advance, between cleverness and original stupidity. So that means that whole lot of time-consuming activities can only be measured after the fact if at all. The difference in our views, I think, is that I would argue that IQ tests measure the kind of raw data-processing skills that are useful in any situation (the military tests people heavily, and for all tasks they've found that IQ correlates positively with results -- I think for some technician jobs, it explains about 60% of the variance in individual performance). There just aren't any studies I know of showing that people with low scores on IQ tests go on to succeed in any measurable way. It would be convenient, to say the least, if you argued that the success-deficit among low IQ people is more than compensated for by a success-surplus that happens to be impossible to measure. So I'd like to know if you can find some way to quantify your argument. It would change my thinking on a lot of subjects if I found that doing poorly on an IQ test predicted doing well at some other task.
This is contentious, but for most of the immigrants, they didn't move because the culture was more enjoyable here. Most of the Latin American immigrants had ther livelihood strip from them in two stages: one, from general industrialization pressures that have affected practically every country, forcing specialization into cash crops to compete, two, increasing competition from the USA and other countries, coupled with crop failures endemic to semi-arid areas.
The few areas in which modestly educated Latin Americans could reliably compete were in illegal cash crops (for which they had less competition in the USA), which continues to cause economic distruption and criminal activity, and manual labor. Those forced into either business might have better chances in the USA, but on the whole they're happy countries. When you're there, you can feel it.
As for success and low IQs, if I recall correctly, someone got the bright idea of testing Caltech professors while Feynman was there -- this may have been prompted after he won the nobel prize, and found out his score from highschool was 125. When they tested the professors, the scores were surprisingly low. I forget who it was, but someone got to make a big deal of his 105 score, since it was three points higher than Feynman's.
My point is that these measures can be stunningly irrelevant, and, if so, while they might have some utility for, say, selecting an undergraduate class, they are often best ignored by an individual when it comes time to decide what one is capable of.
The tester asked me, "Who discovered America?"
Ah, I thought, a trick question. "The Indians!" I said.
Pause.
"Who is generally credited with discovering America?"
But the bottom line though, is that it's none of your business to determine how one should spend ones effort and life. Nobody asked for your take on parameterizing their lifes limits. Thats what living is for. I could tell you what you should do with your life after a test or battery of tests and send you on your way. Like that idea? Even if you do, let it apply to you and you alone.
I mean, I could tell you that if you are really super-passionate and very very outraged, you can convince me to abandon my factual beliefs for some kind of feel-good claptrap. But that might lead you to make another comment of similar tone and content to the one to which I responded, leaving us both worse off. Do you see where I'm coming from?
I am in a business that most batteries of tests would predict I'd fail at, so I clearly don't follow your parody of my thought processes.
The problem with this assertion is the assumption that you have all the facts, facts meaning absolute certainty what a person is capable of given a scalar score from a test.
But that might lead you to make another comment of similar tone and content to the one to which I responded, leaving us both worse off. Do you see where I'm coming from?
I'm guessing not from a vantage point of omniscience.
I am in a business that most batteries of tests would predict I'd fail at, so I clearly don't follow your parody of my thought processes.
Not a parody, just an observation that your aggregates mean nothing to a given instantiation. And congratulations on defying what the stats say you would fail at. Hard worker, indeed.
I would appreciate it if you could explore some middle ground between omniscience and total ignorance. Tests that actually measure stuff give you data, and proudly claiming to be unswayed by data because you can imagine an exception is not rigorous.
Agreed, and likewise I would appreciate an understanding of the lack of absolute certitude of the predictive power of said data to an instantiation without giving external factors weight.
Again, my argument was not intended as a parody, just as a nudge toward the "middle ground" you speak of.
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."
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/...
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.
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?
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.
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.
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.
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...
ultimately, all learning is self-taught.
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).
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.
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 :-).
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.
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