"If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom."
This advice generalizes to many subjects besides mathematics.
AFTER EDIT:
Here's what one math professor says about the delightful aspects of mathematics to learn besides calculus as it is now taught. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer Verlag, 1998):
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
Stillwell is wrong, or at least not doing his homework, when he says "Everyone concedes that [ideas of number and space] are fundamental, but they have been scandalously neglected..." Back in the early sixties, when Bourbakism was hip and new (as mathematical ideas go at least), everyone agreed that set theory and mathematical logic were fundamental, whence the debactacular attempt to begin teaching first-order logic to seventh graders.
The uncomfortable truth is that math is like a language in that it requires a student to put together a large and messy collection of terms and concepts to express any kind of thought at all. The fact that all of the thousands of entries in this vocabulary can be derived in principle from maybe a dozen axioms is what gives math its power, but nobody learns it that way, proving this or that fact as needed. Real skill comes from practice, which starts out as imitation and then generalizes into problem-solving and proof.
All the emphasis on The Fundamentals of math is really just evidence that most mathematicians are bad at teaching -- it's an attempt to impose mathematical order and rigor on the fundamentally messy and organic process of human learning.
I know John Stillwell, and I think his opinions and beliefs are being mis-quoted by being taken out of context. I think he is right that ideas of number and space are fundamental, and I think he's right that they are being neglected, which is a scandal. I agree that "New Math" was a complete debacle, but I think it's not relevant to this question.
I agree that math is like a language, but I disagree that it requires the student to put together a large and messy collection of terms and concepts. That may be the way you view it, but it's not the way I view it, it's not the way I teach it, and I get pretty good results in masterclasses, and more generic classes, by helping the development of the web of inter-related ideas, without large numbers of apparently unmotivated definitions.
In short, I think Stillwell is right, and I think you're missing his points because you appear not to have read his original works.
Stillwell is wrong, or at least not doing his homework,
Have you read his book Numbers and Geometry, or any of Stillwell's other books?
I actually think he takes the approach you advocate, but you haven't given me quite enough examples of the approach you think best for me to be entirely sure.
(Disclaimer - I am most likely biased since I suffered through twelve years of US public school math, often racking up low grades from boredom even while distinguishing myself in math competitions)
I agree that the US system teaches mathematics in too formulaic and granular a way. The way I see it, a good way to divide math students into two categories is to look at those who will use math in their future careers as scientists, engineers, etc, and those who will not and are only studying math to "see the world from another angle".
In either case, a spitting-back-of-formulas approach is counterproductive: Scientists and engineers need to (or at least would do very well to) understand how the mathematics they use works. On the other hand, those who will not use math in the future have no real need to pack their brains full of formulaic tidbits; they will get the most value through pursuing a problem-solving approach that generalizes well to other areas of life.
In short, no matter what the class's level, mathematics definitely needs to be taught from a more problem-solving, integrated perspective. "Pre-med math" that consists mainly of memorization and regurgitation does nobody any good.
The main problem, I imagine, would be recruiting teachers that can actually teach mathematics in this way. That raises a whole different question, though - why do we have "teachers" and not mathematicians teaching math; why do we have "teachers" and not historians teaching history? It seems that much of what education school teaches is how to keep order and run a classroom of bored and antsy students, but wouldn't it be more effective to simply teach interesting material and keep the kids engaged?
I think your division into two camps is missing the largest cohort of math students by far: those who are taking math because it is required as a checkbox item on a "before you get your degree <or diploma>, you must have passed at least this much math classes".
"Will this be on the test?" is a far more common query in my experience than "Could you show me some additional math so that I may better understand the universe and world that I live in?"
I suspect the foundations of my mathematical problem-solving skills were laid when my parents signed me up for extracurricular math classes with teachers from the local Russian community, when I was in elementary and middle school. I started on competition-type problem solving with my middle school math team coach, an extraordinary teacher contrasted with the largely incompetent ones in the curricular program. However, the majority of the techniques I applied to competitions, and I suspect the majority of my overall problem-solving skills, I accumulated through self-study, not a small part of which was from the Art of Problem Solving books from the website linked in the original post. If those were used as textbooks and the teachers knew (and felt) the material, then I think the mathematical education in this country would get a significant boost. If some of the material turned out to be too hard for those less talented at mathematics, the approach would still be the one to use in simpler textbooks.
The student in the highest class should go to something that isn't a class. Real mathematicians, after a while, don't spend much of their time taking classes. Instead they collaborate with other mathematicians to discover new mathematics.
Stillwell knows what he's talking about. In his fantastic book "Mathematics: A History" he makes the point that calculus is taught entirely backwards from its historical development, and he says that approach ruins most undergrads in truly learning calculus. In line with Abel's dictum to "Study the masters, and not the pupils", it's obvious that sweeping the why and how of calcuclus under the rug, and instead presenting a polished and abstract end-product is a disaster as far as learning goes. As natural tinkerers and hackers, surely we can appreciate how much more you can learn by peaking behind the curtain, taking apart the machine and trying to put it back together again.
Sure we can thank Cauchy and Weierstrauss for working out the logical kinks and putting centuries of ungrounded assumptions on a proof-based foundation, but the FTC is a hell of a lot easier to understand on an intuitive level if you started with series, derived integrals and then saw how derivatives came out of that, rather than the other way around.
The real problem with mathematics education is that they are teaching to the least-common denominator on a mass-scale, adopting the attitude that 99% of students won't need to really know how it works under the hood. If students can just grind through the problem sets, it's good enough, so the entire curriculum, text books, and TA lectures are tailored to that. Unfortunately, this approach has infected even the pure math programs--I think it's silly to teach the group who will specialize in mathematics in the same way say biology and economics students are taught caluclus.
This was interesting, and speaks to my own experience.
In high school, I did the local quals for the international math, physics and chemistry olympiads. I did quite well on the physics and chemistry ones and went to the international versions as part of the Swedish teams. On the math one, though, I plain sucked. In post-processing, I've realized that what happened was exactly what this article talks about. The math questions were questions where I just went "Huh? I have no idea how to even approach this," because my math education had largely only prepared me for solving specific kinds of problems. I'd never been exposed to things like proving theorems or other kinds of creative thinking.
The first math competition I entered was an interesting experience. There were so many completely foreign concepts like sigma notation that precluded me from even trying to answer the 'tricky part' of the questions. Luckily, the proofs came out on a later competition, after some good math competition coaching by the quizmaster.
I read this article in high school, and it permanently changed my math life. Without it, I would continue to think I know derivatives because I learned twenty formulas and did forty exercises (which in fact every idiot could do).
I'll allow myself to give some advice for those who are interested in problem-solving, but have no experience. If you have some, then this should be well known.
* Keep problem / exercise ratio as high as possible. This is impossible with many calculus books; find a book with hard problems. An "exercise" is something which checks your understanding of definitions and theorems; a "problem" is something which exercises your skill and forces to think. Do exercises if the theory is unclear. If a task starts with "using mathematical induction prove that..." then it is an exercise. A problem forces you to think how to do it.
* Doing differentiation exercises will give you some speed, but after five-twenty minutes your brain will stop thinking and start to rot. Healthy mathematics - just like programming - hates doing the same thing again. Of course you have to learn some algorithms, but this is a tip of the iceberg.
* Always take 20 minutes (some say more) on a problem, unless you think it is ill-posed; giving up early is stupid. If you think the problem is impossible, try proving it. Think about some way of solving, reject it quickly if you made a thinko; if you sense "this might work" go deeper. Use paper.
To a smart person, after learning about it, calculus is simple and self-evident. The rate of change of something is the slope, and to find it at a point you take two points arbitrarily close together. The rate of change of the area under a curve IS the curve. That's calculus, and the details of the rest follow.
Why not give a bright young person those insights and let them play?
They never teach the really interesting and cool parts about calculus. How many people know that the rate of change of volume of a sphere _is_ the surface area, for instance? It makes wonderful sense.
I still vividly remember an "Aha!" moment in AP Calc in high-school where a problem was stated as (paraphrasing very roughly):
Suppose a spherical snowball melts according to the formula:
dV/dt = K * pi * r^2
What is the shape of r plotted vs t for a snowball with initial radius of R1?
It was astounding to me at first that it was linear, and then when I thought it through and realized that one might naturally expect it to be linear (in real life), and that the change in volume per unit time to accomplish that MUST be proportional to the surface area just snapped into focus all at once. It's so vivid, I can recall what the room looked like, where that problem was on the page, and what desk I was sitting at 23 years ago. This from someone who can't remember what I went upstairs to get just 23 seconds earlier.
Both parents were high school math teachers, and they lamented even back then that the focus of high school education was much more about teaching the minimums to pass the exams and frankly, babysitting students than it was instilling a lifelong love of learning or deep understanding of the subject matter.
Famous quote from a math teacher at my high-school (neither of my parents :)): "The square root of two is an irrational number because it says so on page 169."
the standard curriculum is not designed for the top students it’s not a question of learning calculus that's easy the question is how to keep up with your potential.
If you understand calculus and diff EQ you can do 95% of the useful math for most walks of life. But getting to that point at 16 is not much more useful than getting there at 20. What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math.
PS: High level math completions in the US are also focused on a wider range of math skills. With a little effort you can start to step out of the "normal mold" and compete at that level.
> What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math.
If you learn the basic ideas of calculus when you're 12, this is not going to dissuade you from learning about topology and number theory too. Prime numbers pretty much sell themselves.
There are a lot of useful tricks for finding formulas for the rate of change and its inverse, though. You could probably teach them in a more socratic, leading-the-student way than is traditional.
One of the best classes I took in college was Abstract Algebra, and I only took it because my math-major roommate kept telling me that I needed to "learn more math." My favorite part was coming to the realization that linear algebra was not just a set of tools for solving particular problems, but rather an instance of a much more general mathematical structure with a beauty and simplicity of its own.
Was my favorite subject as well, precisely for those reasons. Not sure how to teach the beauty of mathematics to those who are uninitiated though. I think it really should come in at the elementary school level, when you can show really cool things about numbers like Fibonacci sequence or binary system or simple cryptography, and make mathematics seem magical and cool (and not wait until high school with these concepts, when it's too late to convince most non-predisposed folks of the poetic beauty of mathematics; you can absolutely teach these concepts earlier).
First day of college: "prove that 0 times x is equal to 0". That got rid of everyone who wasn't really into maths :-)
That exercise was literally the first question, on the first hour of back to school, after 5 minutes of a welcome speech.
I can't even figure out the proof anymore, although it really can't be complicated given the ridiculously small amount of tools you are given (basically x + -x is 0, and (a+b) times c equals a times c + b times c. You are given nothing else.
by commutativity of multiplication, distributivity, the existence of a neutral element 0 for addition, and the commutativity of multiplication once again. Thus, we have that
x*0 + x*0 = x*0.
For clarity, define
A = x*0.
Then we have that
A + A = A
A + A + (-A) = A + (-A), by the existence of additive inverses.
A + (A + (-A)) = (A + (-A)), by associativity of addition
A + 0 = 0, by the definition of additive inverses
A = 0, by the definiton of the neutral element 0 of addition
I read a book called "My Job Went to India" (it's a tongue-in-cheek tutorial of how to stay competitive in the computer industry) and one of the advice was:
Always be the worst on your team.
The idea is, of course, not to suck but to put yourself on a team with people that are more awesome than you, because that's how you grow. People usually do a double-take when I quote this, though. :-)
On every team, everyone except one is worse than someone else. On every team, someone is the worst.
People work with those worse than them all the time, as those more experienced share their acquired wisdom with the less experienced ones. (Not to mention that you don't have to be the worst at everything. Ideally the places are switched in other realms of expertise.)
For you personally, it means reaching outside of your comfort zone and doing something you're not sure you can do.
You're unwarrantedly assuming a total order on team member qualities, even though in your fourth sentence you point out that in a vector model of skills that assumption is invalid. It's not the only one.
That is what I thought. But not how it is playing out when applying for jobs that put me as the worst one (from my perspective anyway). Perhaps other factors are at play.
Very fascinating article. I fell into the Calculus trap myself, and I admit that it definitely would have been more interesting to have avoided it.
As it was I got an A in Calculus, breezed through the course as an early college course student.
How much better it would have been to have enjoyed the alternate possibility presented:
"Going from ‘top student in my algebra class’ to ‘average student in my city’s math club’ is a huge step forward in your educational prospects. The student in the math club is going to grow by great leaps, led and encouraged by other students."
I lost interest in maths in high school when I realized that the problems we were solving were specifically designed to be solvable with the techniques we were taught. Their generality was specious. This was at an expensive private school, in the so-called advanced maths stream...
I much preferred programming, because things made sense there. You could work things out. Even the instruction set for the Z-80 processing I was coding for was more consistent and general than the maths I was taught. The compound instructions were obviously compound, and - and this is the best part - the aspects that I didn't understand enriched me when I finally did understand them (I realized they had machine code support for structs, before I'd heard of structs - I 'invented' structs because my code needed the concept).
Today, after bachelor, masters and not-yet-submitted doctorate in Computer Science, the maths I see in papers still seems technique-based rather than about the truth. I use set-builder notation, and grammar notation (regular expression and CFG syntax), but only because it is logically consistent, not because it's "mathematics".
It seems to me that there isn't any way to learn the true mathematics - the mathematics that is our heritage as human beings - except by the good fortune of finding someone who does understand it, or long and personal study, using historical sources, because these are the only ones that show the point of the techniques - why they are useful.
There must be good text books out there, but the ones I've seen either dive into obscure notation, or use inadequate metaphors. Disclaimer: The only maths I took at uni were taught by the engineering dept. Maybe maths taught by the maths department might have helped me. Actually, my very best comp sci lecturer was from the maths dept - he took care to make it logical.
'Calculus brought a change in focus from "details and things" to "arguments and proofs",'
If your class has "arguments and proofs" in it, then you were not in the type of calculus course this essay is about; you were in the type the author was advocating. I assume this is different than the calculus you got in high school, or you'd have mentioned; that is the type of calculus class he was talking about.
I learned calculus in high school from Howard Anton's textbook, which presented some kind of informal proof for everything it taught. It might have been different if I'd taken a class in it instead.
There's another more fundamental issue with education at the high school level in the US: scheduling. Usually high school students go through 6 periods of 50 minutes each daily, often with one of them being a study period alternating with physical education.
Putting the students truly curious about math, who usually make up a small percent of the students in the college bound math classes, and aligning them with a teacher capable of keeping them engaged is possible. The problem lies in having them attend their other subjects at a level appropriate to them. Keep in mind that each member of this subset of students may not be at the same level as their peers in other subjects.
If these curious math students are not segregated out, which often happens, then a teacher has 20-30 good math students with a few being curious enough to be elite. Keeping the two distinct groups of that class engaged requires distinct lesson plans for each group. The goals of each day's lesson plan need to be met within no more than a 40 minute window - I'm being generous that taking attendance and assigning home work take up no more than 10 minutes of class time.
I admit it's a worthwhile goal to have the curious math students more engaged. However, I don't see it being possible unless a more fundamental restructuring of high school education occurs. Perhaps have it structured more like a university schedule in which students take 3 subjects daily with each class being 1-1.5 hours.
From what my high school math teacher told me, it's because the modern U.S. curriculum was last drastically revamped in wartime to pump out future engineers and physicists, which is why there is so much emphasis on trigonometry and calculus rather than sets, logic, statistics, or discrete math.
Are you talking about the school algebra that for most students precedes differential and integral calculus, or are you talking about the abstract algebra that in most curricula follows calculus as an upper-division undergraduate course?
I don't buy that at all. That attitude is precisely what makes people think they can't "get it". As if they lack some sort of natural aptitude or genetic predisposition towards numbers.
I don't know about genetic stuff, I'm just trying to make a relative statement between calculus and number theory.
Who knows what causes it, but there are a lot of smart 15 year olds who are less good at math than they are at most things. If those people really work their asses off at calculus, it works better than working their asses off at number theory. That's because calculus contains more memorization and less problem-solving.
"There is a widespread notion that 'ability' counts for more in learning mathematics than does effort. This is a doubtful notion, at best, and it has at least two bad consequences: it provides a recipe for failure and an excuse for failure when new ideas do not come easily."
I also didn't really enjoy Calculus in high school, only learning to love it in hindsight, when I took Real Analysis in college. Then again, I was always interested in theoretical mathematics (poetry, abstraction, & all that) rather than practical (yucky numbers & approximations :); so maybe pragmatists enjoy Calculus as taught?
I'm convinced that young people should be learning descriptive statistics and move on to inferential statistics online like in: http://davidmlane.com/hyperstat/index.html it's easy now to learn from online data sets and you can collect more information through social media (vs. using physical paper questionnaires.) I learned these kinds of statistics when I was 12 years old - I wish someone told me what I could do multivariate analyses ...
This advice generalizes to many subjects besides mathematics.
AFTER EDIT:
Here's what one math professor says about the delightful aspects of mathematics to learn besides calculus as it is now taught. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer Verlag, 1998):
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
http://www.amazon.com/gp/product/0387982892/