7 ms·

Dismantling the calculus pyramid

37 points by timwiseman 17y ago

RiderOfGiraffes 17y ago
See also: http://news.ycombinator.com/item?id=680063 http://news.ycombinator.com/item?id=680063 http://news.ycombinator.com/item?id=701367 http://news.ycombinator.com/item?id=701367
- arketyp 17y ago
  Why? There are no discussions there.
  - deleted 17y ago
    [deleted]
  - RiderOfGiraffes 17y ago
    Hmm. Maybe what I really meant was: this has been submitted at least twice before, three and four months ago or so, and in neither case did it generate any discussion. I hope it does this time, because it's been shown to be interesting enough to be "discovered" by more than one participant. And I do. I've never understood the US schools emphasis on calculus, and building it up to be such a huge deal. I first did calculus aged 16 in Year 10, and it was just another part of the syllabus. One more step to understanding the way stuff works. Replacing it by another "BIG IDEA" seems not necessarily to be a Good Thing(tm). Perhaps there should be more of a plain, and less of a pyramid. I agree that more elementary stats done earlier would be a really Good Thing(tm), but replacing one pyramid with another does not seem to me to be so.
    - rgoddard 17y ago
      First there is the recognition that there is no pyramid. Since math is so highly interrelated, learning about one area almost always helps understanding in a different area. Granted some areas require a certain prerequisites, hence you will encounter peaks, but I agree aiming for only one peak does not make all that much sense since calculus is only useful in certain situations.
amichail 17y ago
Bad idea. Some consequences: * People would stop voting. * Crime would probably increase. * People would be less friendly/helpful to strangers in large cities. * People would be less willing to drive (causing chaos in cities without sufficient public transportation). * People would no longer care about environmental issues. * Entrepreneurship would decrease. * Stock trading would decrease.
- chollida1 17y ago
  Are you sure you replied to the correct story. This story is about teaching more statistics and less calculus to students as it is vastly more practical for them.
  - amichail 17y ago
    These are consequences of a greater understanding of probability and statistics.
    - crux_ 17y ago
      No, not really. You seem to be assuming that a greater understanding of probability would lead to decision-making that takes only probabilities into account and not, say, one's morality. But that's a ridiculous thing to assume.
- chipsy 17y ago
  Following the same "logic" you have: * People would follow healthier lifestyles. * People would avoid gambling. * People would avoid bad marriages. * The government would be more efficient. An understanding of statistics wouldn't realistically change all of these things. In practice, people can always be swayed by a powerful rhetoric, emotions, and cultural habit. Statistics lend more weight to arguments, but only a tiny fraction of people will go back and verify a statistic for accuracy.
  - amichail 17y ago
    Losing democracy is a big deal though. While it is true that many people vote due to social pressure, I suspect that this social pressure would decrease with a better understanding of probability and statistics. You could try mandatory voting, but people could just vote randomly to avoid wasting any time thinking about the matter.
WalkingDead 17y ago
Best advice of this century in it's class. I don't know why we are still wasting our time with calculus while I see a lot of people suffering in their daily life because of their lack of understanding in probability theories and statistics. Statistics is easier and even more fun to learn than calculus.
- kscaldef 17y ago
  Sadly, calculus is a pre-requisite for anything more than very basic probability and statistics.
  - dagw 17y ago
    True, but you don't need a lot of calculus to do a large amount of interesting statistics and probability. And the little you do need is probably better being taught as needed in response to actual problems showing up. Most people learn better when they see a clear reason for what they are being taught. Start with discrete probability, no calculus needed. Then when you move to continuous case show how integrals show up naturally in place of sums, and hey presto a clear and obvious intro calculus which far more people will understand the use for.
  - winthrowe 17y ago
    I think this is directed mostly at the people who will never take another math class after they leave high school. Speaking from a canadian perspective, I had Grade 11 Math, Grade 12 Math, and Calculus classes. about half of the work in grade 12 was directly preparatory for the calc class. Students persuing a trade/vocational school or no post secondary education would only have taken 11 and 12 math. if the half of the math class that was 'wasted' calc prep was replaced with 'probability in the real world' type material, you'd still be in the basic precalc stats area, but possibly giving something more useful to the kind of people that are not well represented at a site like this.
- bbgm 17y ago
  Because calculus is beautiful, and if you are even remotely interested in the physical sciences, absolutely necessary. I can't even imagine a high school curriculum without calculus. It's what made some of the decisions I took about grad school possible.
tokenadult 17y ago
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/ http://www.amazon.com/gp/product/0387982892/ See also http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf with comments on mathematics education for breadth rather than for speed through the standard curriculum, by a Fields medalist.
- timwiseman 17y ago
  I would second that, and add a recommendation for "Chapter Zero" which is a book targetted at the high school/early college level which provides an introduction to those fundamentals that are often skipped in the rush towards calculus.
  - tokenadult 17y ago
    This Chapter Zero, http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Mathematics/dp/0201437244 http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math... right? I have not read this particular book, but in general most students who desire to study math should take a look at a "transitions course" textbook or two, to fill in the gaps left by studying only in the fast lane to calculus. http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/ http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/ http://www.amazon.com/Mathematical-logic-and-foundations/lm/3L9OBW0IXF414/ http://www.amazon.com/Mathematical-logic-and-foundations/lm/... http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV24TPSZ/ http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV...
mitko 17y ago
I agree agree that probability and statistics are much more important in life than just calculus. That is why in college I am studying Artificial Intelligence and Machine Learning - these are all math and probabilities. Yet, I think Arthur Benjamin is right but for the wrong reasons: 1. People should know statistics instead of Calculus. To know probability well you need to have mastered calculus. All the distributions, all the formulas build on things like sequences, and integrals. 2. If everybody knew statistics we wouldn't be in the economic mess. Plain speculation. The majority people who have important part in the management of banks know statistics. 3. from continuous mathematics to discrete mathematics... Probability and statistics involves tons of continuous mathematics and Analysis. Is a Gaussian distribution discrete? Is the Central Limit Theorem discrete? The reason probabilities and statistics is important is because it lets people better understand why stuff around them happens and lets them reason about uncertain processes. Life is behavior under uncertainty.
- brianto2010 17y ago
  I believe it is important to remember that Arthur Benjamin is referring to High School Mathematics. To know probability well you need to have mastered calculus. I won't argue that point itself. However, given that the audience for this mathematics education is High School Students, we won't have to go that far. Regression lines, standard deviations, and Simpson's paradox can be understood without Calculus. Personally, more time should be focused on the analysis and application (again, at a High School Level).
joe_the_user 17y ago
Hmm, Oh where to begin? The guy's rhetoric is pretty divorced from the sad state of actual math education today. The way that the standard math curriculum today heads up to a point in a pyramid is a problem in itself since a lot of the topics are made dull in themselves ("This is just something you need to learn for calculus" is a terrible answer for "why should I learn this?" but substituting "Statistics" wouldn't change things much). It doesn't really matter what the point of the pyramid is when most people never get there. Just as much, the latest economic meltdown was engineered and believed in by statistics experts. For every Mandlebrot debunking the events, there were ten Myron Scholes basking in the glory of validating Wall Street's delusions with some mathematical magic. Statistics doesn't protect from wishful thinking outside of controlled, experimental situations. And statistics in daily life? One might use some basic probability but the only other use it would have would sorting the pseudo-statistical rhetoric used by the media. A simple course in mathematical literacy with an emphasis on fallacies would be best for sorting this stuff. BUT again, no course can protect from wishful thinking, can protect people from the fallacies that let them ignore possible later dangers for immediate apparent gain. Further, a non-calculus-based statistics or mathematically literacy course isn't a basis for further scientific study the way calculus is, and believe-it-or-not some students still become physicists, chemists and engineers where calculus is indeed the foundation. I could narcisistically say that my favorite, evolutionary game theory, would make a much better "point" for the curriculum pyramid but really, what is needed is to make every math class interesting in and of itself. With TV-dazed kids and math-apathetic teachers, I don't know if any curriculum could change things BUT I would want to have the curriculum of every class interesting and mentally challenging - taught axiomatically, Algebra and Geometry ARE interesting in and of themselves and a student needs no background at all for them. Math should be rigorous, conceptually challenge and optional past the basics. It seems like we'd need a different world for this but small steps are being made.
- RiderOfGiraffes 17y ago
  Firstly, Art is actually a math professor, and dealing with the low standards of incoming students is a major concern. I don't think he is "divorced from the sad state of actual math education today." But even more, you say: > ... taught axiomatically, Algebra and Geometry > ARE interesting in and of themselves Perhaps to you, but I deal with students for whom this is absolutely not interesting. They're not interested in puzzles, they're not interested in challenges, they're not interested in anything except texting their friends and talking about films, TV, clothes, football, etc. They don't want to be challenged. I think we all agree that classes in general should be stimulating and interesting, andshould be better tailored to suit the needs of the individuals, but that's the problem. For a given child we don't know what they'll need, and we don't know what they'll like, and every answer will be different. The problem is "one class fits all" and that's not going to change simply by re-writing the curriculum.
perkoff 17y ago
Maybe it's built in "pyramid" like this because: 1. Knowledge is hierarchical 2. Calculus has a longer list of skills that needs to be mastered before learning the subject. 3. Historically, calculus came around first (am I right?). This might not be a coincidence.
- seregine 17y ago
  Understanding (the path to knowledge) might be hierarchical but knowledge in general isn't. Organizing knowledge in hierarchies is a conventional teaching technique, and that has some self-fulfilling qualities: if you teach people in hierarchies, they'll see the world that way.
marze 17y ago
I would go further and not only teaching probability and statistics, but also teach basic calculus with simple polynomials, programming, Monti Carlo simulations, sines and cosines, computer simulations, all starting around age 13. Advance algebra, geometry, and full trigonometry are not required for any of that. Why make students wait until senior year or college to experience the fun stuff?

37 points by timwiseman 17y ago

See also: http://news.ycombinator.com/item?id=680063 http://news.ycombinator.com/item?id=680063 http://news.ycombinator.com/item?id=701367 http://news.ycombinator.com/item?id=701367

arketyp 17y ago
Why? There are no discussions there.
- deleted 17y ago
  [deleted]
- RiderOfGiraffes 17y ago
  Hmm. Maybe what I really meant was: this has been submitted at least twice before, three and four months ago or so, and in neither case did it generate any discussion. I hope it does this time, because it's been shown to be interesting enough to be "discovered" by more than one participant. And I do. I've never understood the US schools emphasis on calculus, and building it up to be such a huge deal. I first did calculus aged 16 in Year 10, and it was just another part of the syllabus. One more step to understanding the way stuff works. Replacing it by another "BIG IDEA" seems not necessarily to be a Good Thing(tm). Perhaps there should be more of a plain, and less of a pyramid. I agree that more elementary stats done earlier would be a really Good Thing(tm), but replacing one pyramid with another does not seem to me to be so.
  - rgoddard 17y ago
    First there is the recognition that there is no pyramid. Since math is so highly interrelated, learning about one area almost always helps understanding in a different area. Granted some areas require a certain prerequisites, hence you will encounter peaks, but I agree aiming for only one peak does not make all that much sense since calculus is only useful in certain situations.

amichail 17y ago

Bad idea. Some consequences: * People would stop voting. * Crime would probably increase. * People would be less friendly/helpful to strangers in large cities. * People would be less willing to drive (causing chaos in cities without sufficient public transportation). * People would no longer care about environmental issues. * Entrepreneurship would decrease. * Stock trading would decrease.

chollida1 17y ago
Are you sure you replied to the correct story. This story is about teaching more statistics and less calculus to students as it is vastly more practical for them.
- amichail 17y ago
  These are consequences of a greater understanding of probability and statistics.
  - crux_ 17y ago
    No, not really. You seem to be assuming that a greater understanding of probability would lead to decision-making that takes only probabilities into account and not, say, one's morality. But that's a ridiculous thing to assume.
chipsy 17y ago
Following the same "logic" you have: * People would follow healthier lifestyles. * People would avoid gambling. * People would avoid bad marriages. * The government would be more efficient. An understanding of statistics wouldn't realistically change all of these things. In practice, people can always be swayed by a powerful rhetoric, emotions, and cultural habit. Statistics lend more weight to arguments, but only a tiny fraction of people will go back and verify a statistic for accuracy.
- amichail 17y ago
  Losing democracy is a big deal though. While it is true that many people vote due to social pressure, I suspect that this social pressure would decrease with a better understanding of probability and statistics. You could try mandatory voting, but people could just vote randomly to avoid wasting any time thinking about the matter.

WalkingDead 17y ago

Best advice of this century in it's class. I don't know why we are still wasting our time with calculus while I see a lot of people suffering in their daily life because of their lack of understanding in probability theories and statistics. Statistics is easier and even more fun to learn than calculus.

kscaldef 17y ago
Sadly, calculus is a pre-requisite for anything more than very basic probability and statistics.
- dagw 17y ago
  True, but you don't need a lot of calculus to do a large amount of interesting statistics and probability. And the little you do need is probably better being taught as needed in response to actual problems showing up. Most people learn better when they see a clear reason for what they are being taught. Start with discrete probability, no calculus needed. Then when you move to continuous case show how integrals show up naturally in place of sums, and hey presto a clear and obvious intro calculus which far more people will understand the use for.
- winthrowe 17y ago
  I think this is directed mostly at the people who will never take another math class after they leave high school. Speaking from a canadian perspective, I had Grade 11 Math, Grade 12 Math, and Calculus classes. about half of the work in grade 12 was directly preparatory for the calc class. Students persuing a trade/vocational school or no post secondary education would only have taken 11 and 12 math. if the half of the math class that was 'wasted' calc prep was replaced with 'probability in the real world' type material, you'd still be in the basic precalc stats area, but possibly giving something more useful to the kind of people that are not well represented at a site like this.
bbgm 17y ago
Because calculus is beautiful, and if you are even remotely interested in the physical sciences, absolutely necessary. I can't even imagine a high school curriculum without calculus. It's what made some of the decisions I took about grad school possible.

tokenadult 17y ago

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/ http://www.amazon.com/gp/product/0387982892/ See also http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf with comments on mathematics education for breadth rather than for speed through the standard curriculum, by a Fields medalist.

timwiseman 17y ago
I would second that, and add a recommendation for "Chapter Zero" which is a book targetted at the high school/early college level which provides an introduction to those fundamentals that are often skipped in the rush towards calculus.
- tokenadult 17y ago
  This Chapter Zero, http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Mathematics/dp/0201437244 http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math... right? I have not read this particular book, but in general most students who desire to study math should take a look at a "transitions course" textbook or two, to fill in the gaps left by studying only in the fast lane to calculus. http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/ http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/ http://www.amazon.com/Mathematical-logic-and-foundations/lm/3L9OBW0IXF414/ http://www.amazon.com/Mathematical-logic-and-foundations/lm/... http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV24TPSZ/ http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV...

mitko 17y ago

I agree agree that probability and statistics are much more important in life than just calculus. That is why in college I am studying Artificial Intelligence and Machine Learning - these are all math and probabilities. Yet, I think Arthur Benjamin is right but for the wrong reasons: 1. People should know statistics instead of Calculus. To know probability well you need to have mastered calculus. All the distributions, all the formulas build on things like sequences, and integrals. 2. If everybody knew statistics we wouldn't be in the economic mess. Plain speculation. The majority people who have important part in the management of banks know statistics. 3. from continuous mathematics to discrete mathematics... Probability and statistics involves tons of continuous mathematics and Analysis. Is a Gaussian distribution discrete? Is the Central Limit Theorem discrete? The reason probabilities and statistics is important is because it lets people better understand why stuff around them happens and lets them reason about uncertain processes. Life is behavior under uncertainty.

brianto2010 17y ago
I believe it is important to remember that Arthur Benjamin is referring to High School Mathematics. To know probability well you need to have mastered calculus. I won't argue that point itself. However, given that the audience for this mathematics education is High School Students, we won't have to go that far. Regression lines, standard deviations, and Simpson's paradox can be understood without Calculus. Personally, more time should be focused on the analysis and application (again, at a High School Level).

joe_the_user 17y ago

Hmm, Oh where to begin? The guy's rhetoric is pretty divorced from the sad state of actual math education today. The way that the standard math curriculum today heads up to a point in a pyramid is a problem in itself since a lot of the topics are made dull in themselves ("This is just something you need to learn for calculus" is a terrible answer for "why should I learn this?" but substituting "Statistics" wouldn't change things much). It doesn't really matter what the point of the pyramid is when most people never get there. Just as much, the latest economic meltdown was engineered and believed in by statistics experts. For every Mandlebrot debunking the events, there were ten Myron Scholes basking in the glory of validating Wall Street's delusions with some mathematical magic. Statistics doesn't protect from wishful thinking outside of controlled, experimental situations. And statistics in daily life? One might use some basic probability but the only other use it would have would sorting the pseudo-statistical rhetoric used by the media. A simple course in mathematical literacy with an emphasis on fallacies would be best for sorting this stuff. BUT again, no course can protect from wishful thinking, can protect people from the fallacies that let them ignore possible later dangers for immediate apparent gain. Further, a non-calculus-based statistics or mathematically literacy course isn't a basis for further scientific study the way calculus is, and believe-it-or-not some students still become physicists, chemists and engineers where calculus is indeed the foundation. I could narcisistically say that my favorite, evolutionary game theory, would make a much better "point" for the curriculum pyramid but really, what is needed is to make every math class interesting in and of itself. With TV-dazed kids and math-apathetic teachers, I don't know if any curriculum could change things BUT I would want to have the curriculum of every class interesting and mentally challenging - taught axiomatically, Algebra and Geometry ARE interesting in and of themselves and a student needs no background at all for them. Math should be rigorous, conceptually challenge and optional past the basics. It seems like we'd need a different world for this but small steps are being made.

RiderOfGiraffes 17y ago
Firstly, Art is actually a math professor, and dealing with the low standards of incoming students is a major concern. I don't think he is "divorced from the sad state of actual math education today." But even more, you say: > ... taught axiomatically, Algebra and Geometry > ARE interesting in and of themselves Perhaps to you, but I deal with students for whom this is absolutely not interesting. They're not interested in puzzles, they're not interested in challenges, they're not interested in anything except texting their friends and talking about films, TV, clothes, football, etc. They don't want to be challenged. I think we all agree that classes in general should be stimulating and interesting, andshould be better tailored to suit the needs of the individuals, but that's the problem. For a given child we don't know what they'll need, and we don't know what they'll like, and every answer will be different. The problem is "one class fits all" and that's not going to change simply by re-writing the curriculum.

perkoff 17y ago

Maybe it's built in "pyramid" like this because: 1. Knowledge is hierarchical 2. Calculus has a longer list of skills that needs to be mastered before learning the subject. 3. Historically, calculus came around first (am I right?). This might not be a coincidence.

seregine 17y ago
Understanding (the path to knowledge) might be hierarchical but knowledge in general isn't. Organizing knowledge in hierarchies is a conventional teaching technique, and that has some self-fulfilling qualities: if you teach people in hierarchies, they'll see the world that way.

marze 17y ago

I would go further and not only teaching probability and statistics, but also teach basic calculus with simple polynomials, programming, Monti Carlo simulations, sines and cosines, computer simulations, all starting around age 13. Advance algebra, geometry, and full trigonometry are not required for any of that. Why make students wait until senior year or college to experience the fun stuff?