The Calculus Trap (2005)
artofproblemsolving.com
artofproblemsolving.com
(In my view, solving integrals is a search problem, and search problems are for computers. I want to understand the concepts and the algorithms, not be a glorified calculator.)
I am glad I followed the curriculum as well, though. In the end, you'll need to take calculus (et al) at some point, and taking it as soon as possible meant that it could help form my mind earlier, and allowed me to progress beyond even sooner. Specifically, I followed the sequence: differential calculus, integral calculus, multivariate calculus, differential equations, and linear algebra (and then I transferred to Berkeley's math program). That's a lot for a young mind to soak in, and the type of student these classes select for make an excellent social environment as well (inevitably, the type that likes learning).
If I were to make a recommendation between following the curriculum, and pursuing extra-curriculars, I'd say: do both.
I thought most high schools taught calculus. Both my and my wife's did. Why is this 15 year old going to a local community college or university for that?
http://www.maa.org/the-changing-face-of-calculus-first-semes... agrees, and points out that 3x more people take AP Calculus now than when I was in high school. Which is about the same time that the author went to school.
(FWIW, I do realize that my high school was unusual, in that it also offered linear algebra, differential equations, and modern algebra. There were enough of us who had taken calculus and were interested in additional classes. It was a community college course taught at the high school by a high school teacher.)
"I met nearly all of them through activities or employment that selected for thinkers. In school, these activities were (and still are in most schools) extracurricular programs, not curricular ones."
Nope, don't recognize that in my experience. The class with the most "thinkers" was probably AP European History. I can't think of an extracurricular which matched it. (AP American History, AP English, and AP Calculus were pretty close.)
How much of this essay is the author's hypothesis, and how much of it is based on actual research?
In sum: as far as my experience goes, educational standards are set up specifically to frustrate and impede fast learners.
That being said, I think UGRADs spend waaaaaay too much time learning calculus: Calc I, Calc II, multivariable, vector calculus, etc. That's like 4 semesters of calculus! This signals to the student that calculus is somehow a big deal, when in fact it is not: it's just calculation techniques. I think this should be cut-down to 2 semesters: Calc I+II together, then multivariable+vector calc. Just learn it quickly and get it over with[2].
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[1] Gilbert Strang saying we should learn more LA and less Calc. http://www-math.mit.edu/~gs/papers/essay.pdf
[2] My short book on Calculus and Mechanics http://minireference.com/launch40 (disclaimer: self-plug, but on topic)
I would say that calculus is the first "real math" that students study which is so complicated that they can't really learn the first principles without years of effort. That's why classes teach the most immediately useful parts instead of covering the details. Someone who only studies those basics might even come out thinking that calculus is just a bunch of techniques, without grasping how gorgeous the underlying concepts and abstractions are.
Take geometry as something I think you regard as "real math." How come geometry classes never prove that it's impossible to trisect the angle? Or double the cube? It turns out that compass and straightedge can only produce quadratic constructions, and Wantzel showed that these problems require cubics. This shows just how limited geometry really is, which is partially why students can feel that they have a handle on the entire topic. (Also, few if any secondary schools cover non-Euclidean geometries, or Euclid's book X as it explores incommensurable magnitudes.)
I would also say that trigonometry is the first applied math course that students learn, not calculus.
Let's go to calculus. Take for example the chain rule, D(g∘f)(c) = Dg(f(c))∘Df(c). This is one of those rules everyone learns in Calculus I. Can you prove it? I once could. Here's the start of the proof from my text book:
"The hypothesis implies the c is an interior point of the domain of h = g∘f. (Why?) Let e>0 and d(e, f) be as in Definition 39.2. It follows from Lemma 39.5 that there exists strictly positive numbers g, J such that if |x-c|<=g then f(x) is an element of B and |f(x)-f(c)|<=K |x-c|. For simplicity, we write L_f = D f(c) and L_g = D h(b). By Theorem 21.3 there is a constant M such that |L_s(u)|<=M|u| for all u in R^q. If |x-c|<infimum(g, (1/K) ..."
And so on for another 10 lines of the book.
Do you really want to subject all calculus students to that level of detail? I don't. Who other than a mathematician needs to learn about Lebesgue integration and measure theory, which are even more advanced topics in calculus?
BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. Also, calculus books do have some proofs. For example, every chapter I looked at from http://ocw.mit.edu/resources/res-18-001-calculus-online-text... contains a few proofs.
So did mine, which is biting me on the ass now that I'm studying for my Machine Learning exam in graduate school. This course expected matrix calculus.
I'm counting at least eight classes that were arguably Calculus in my undergrad program:
* Calc I (basic differentiation/antidifferentiation) * Calc II (integration techniques/applications, series/sequences) * Calc III (vector calc) * Careful/Proven Calc (Real Analysis I) * Careful/Proven Calc II (Real Analysis II) * Differential Equations * Partial Differential Equations
Five of which I took (unsurprising for a math major).
It's a lot, and I could definitely wish there was more linear algebra (the intro LA course was a significant part of what convinced me to look at the major, and I took the grad level matrix analysis class as one of my electives because there wasn't anything else). At the same time, I can see why it's all there.
[0]: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...
http://www.ted.com/talks/arthur_benjamin_s_formula_for_chang...
However, my personal experience was the exact opposite. Calculus classes, along with the accompanying physics-with-calculus classes, were what transformed my concept of math from a game you play with symbols into a powerful way to describe the way the world works. Once I realized that equations like d=1/2at^2 just fell out of applying calculus to the idea of change in position over time, everything suddenly made sense.
[1] A double entendre. A mine field is a set with the standard field operations, which belongs only to the educator.
High school students would be much better served in life studying statistics instead. Far more important decisions in life and public policy are informed by statistics than by calculus.
There was a time when I thought more breadth - number and set theory, proofs, and so on - would be better in place of calculus. And while I think it would be an improvement on teaching calculus, it would not be nearly as useful for most students as a better understanding of statistics.
[1] http://steve-yegge.blogspot.com/2006/03/math-for-programmers...
This article is lacking examples. What exact case makes the standard curriculum bad? What are the good examples of the alternatives?
The article mentions, "more importantly, the gifted, interested student should be exposed to mathematics outside the core curriculum, because the standard curriculum is not designed for the top students."
The whole site that the article comes from serves as an example of mathematics teaching that goes deeper and connects topics together better than the standard curriculum in United States schools. Other authors have written on the same topic. 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."
Personally, I hated calculus. It turned me off of math for a long time -- there was too much emphasis on memorizing heuristics for solving problems, like integral tricks and trigonometric identities. Worse, calculus was used as the canonical example of "college-level math", so it seemed that further math courses would just be about memorizing more and more problem solving tricks.
This second comment to be posted expresses what many students miss out on if their secondary school curriculum rushes to get to calculus as early as possible without also being designed to help them understand mathematics as well as possible. That's what the submitted article is about.
If I were to make a recommendation between following the curriculum, and pursuing extra-curriculars, I'd say: do both.
Yes, the both-and approach is helpful. That's what the article says when it says "Developing a broader understanding of mathematics and problem solving forms a foundation upon which knowledge of advanced mathematical and scientific concepts can be built. Curricular classes do not prepare students for the leap from the usual ‘one step and done’ problems to multi-step, multi-discipline problems they will face later on. That transition is smoothed by exposing students to complex problems in simpler areas of study, such as basic number theory or geometry, rather than giving them their first taste of complicated arguments when they’re learning a more advanced subject like group theory or the calculus of complex variables."
Lockhart's Lament [0] comes to mind.
[0]: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...
Lockhart's Lament is indeed also a response to an era (different from the era I grew up in) when many high school students are rushed into a calculus class before reaching a profound understanding of fundamental mathematics.
"you’re in ninth grade and you’ve already taken nearly all the math classes your school offers" I thought most high schools taught calculus. Both my and my wife's did. Why is this 15 year old going to a local community college or university for that?
The author is indeed writing for a particular audience (which, as you correctly point out, is growing in size) of young people who have blazed through the United States mathematics courses that are now typical at ages once thought unimaginable. My late dad took his calculus course in the late 1940s as a second-year college student. I had just seven high school classmates in the mid-1970s who took calculus in high school at all. Most students in my generation who took calculus at all took it as a first-year university course. My oldest son began a formal course in calculus at eighth-grade age, through an accelerated local program that was founded in the 1980s. My second son is taking AP calculus BC as high school junior (eleventh grader). People are rushing into calculus much more rapidly than ever before in the United States, but often lack "profound understanding of fundamental mathematics (PUFM)" before starting the calculus course. A link that furthered my process of pondering how students might learn mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma.
http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf
Another review of that excellent book by mathematician Roger Howe
http://www.ams.org/notices/199908/rev-howe.pdf
is also food for thought. In some countries, elementary mathematics is not considered "easy" mathematics, but rather fundamental mathematics, which must be understood in full context to build a foundation for later mathematical study.
However, my personal experience was the exact opposite. Calculus classes, along with the accompanying physics-with-calculus classes, were what transformed my concept of math from a game you play with symbols into a powerful way to describe the way the world works.
There are definitely a lot of students who enjoy a calculus course for that experience. That seems to be a form of enjoyment that especially comes to students who have had time to learn about other topics on the way to learning calculus. Russian mathematical instruction tries harder than instruction in the United States to bring in examples from physical science at all ages, so that the mathematics that explains physics is taught to students who have a decent background in physics.
Your example has a 15 year old going to community college or university in order to go to a calculus course. You named it the "calculus trap", which includes as a negative the social problem of having a 15 year old in a class full of 19 year olds. This implies that it's unlikely that the high school offers calculus.
My observation is that more and more high schools offer calculus in the high school, so the 15 year old you described, who was taking college courses to learn calculus, is now more likely to be a 15 year old at high school taking courses with 16 year olds (like your second son).
In that case, the severity of the trap is lessened, no? If only because the age gap is so much less.
I'm not saying that you are wrong about how math knowledge should be developed. I point out only that the arguments from your hypothetical case feel a bit out of date.
Than again, suppose the 15 year old does take calculus at high school, then takes a tertiary education class at age 16. What class might that be? That's about the time the standard college curriculum branches away from calculus, to include algebra, differential equations, or discrete math.
In that case, it's not really a "calculus" trap, no? :)
Also, I read the reviews of KTEM. Cross-cultural observational comparisons are often enticing, but it's hard to draw firm conclusions from them. Had you read, say, a comparison with the Finnish model then perhaps you might have drawn different conclusions?
My hypothesis, btw, is that the US is entirely too car dependent. Extracurricular activities like a city math club would be much easier if teens had ready access to transportation independent of their parents.
Many people that are very advanced in math get bored. Topics like number theory can reintroduce the fun and wonder in math that has been beaten out by the system. Bringing "Wow, how can that possibly be?" back into mathematics should be a goal in and of itself. (This isn't meant to degrade the "master the basics to master the complex" argument either)