Thanks for the interesting comments. Taking the top-level comments in order of posting, I read
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 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.
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.
> My second son is taking AP calculus BC as high school junior (eleventh grader).
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.
First - you are taking on a very worth cause. I think you are missing a very important point above and beyond skill building, though.
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)
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.