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In my experience, Calculus in the university is solely used as weeder material, designed to bludgeon students with the notion that mathematics is dull, tedious,

In my experience, Calculus in the university is solely used as weeder material, designed to bludgeon students with the notion that mathematics is dull, tedious, and a necessary evil. Nothing did as much to destroy the joy I derived from math as did my college Calculus courses. The current pedagogy of Calculus is atrocious, and any method which helps impart the beauty of it without battering the souls of the students is worth pursuing. Feynman's great strength was his disregard for orthodoxy in the search for greater truth. Big-O notation is for more than just profiling the costs of computational algorithms. It is for reasoning about inexact quantities and seeing the greater patterns in the relationships between numbers and functions. Is using Big-O as a method of teaching Calculus better than the current process? Honestly, almost anything would be an improvement. Math and Calculus are both beautiful and practical. But the current way we teach them completely loses those notions in favor of rote memorization and techniques without context.

amichail 19y ago
Some places teach Calculus well. For example, check out the advanced Calculus courses at Waterloo.
- LogicHoleFlaw 19y ago
  How well do the introductory courses fare?
  - amichail 19y ago
    I don't know what it's like now, but when I was there, the advanced courses were very intimidating. This is the real deal.
  - mrtron 19y ago
    The introductory courses are quite tough. They teach the material well, but when it comes to assignments and exams they ask questions that tend to force you to apply the things you learned with some sort of twist. I always thought I knew the material inside out and got crushed on exams compared to how I thought I did. edit: Actually, 137/138 (which you will take first and second term) have recently been nerfed since they eliminated OAC and stopped teaching much calculus in high school.
brentr 19y ago
I will agree with you on the point that mathematics is beautiful. I had always appreciated mathematics for its logic. A proof is either right (it proves) or it is wrong (it does not prove), however, it was not until I was introduced to double and triple integration that I realized the raw power and beauty of calculus. I start working towards my computer science and math degrees this summer, and my registered courses include discrete mathematics and probability. I can be confident in the fact that I will learn some easily learned ideas that are able to do very powerful things.
- LogicHoleFlaw 19y ago
  Best of luck in your studies! Discrete mathematics was one of my favorite courses in school.
  - brentr 19y ago
    It's a requirement for the CS portion of my degree. I already hold a finance degree from the University of Florida (2004), but the credit crunch sidelined me. Fortunately, I had the option of pursuing another two majors while still finishing in less than two years.
kalid 19y ago
I couldn't agree more. People forget that the calculus of Newton and Leibniz was based on infinitesimals, not the "rigorous" epsilon-delta proofs and the concept of a limit. Calculus was found by intuition and experimentation (Archimedes' method of exhaustion to find pi), and not following the implications of "What happens if I arbitrarily define a limit?". Current calculus education is like teaching kids about color theory, photons, how the eye works without just letting them fingerpaint. Those details will come, but the vast majority of people just remember calculus as a painful memorization exercise. As an aside, e is the same way. Most people have it taught as an abstract limit concept: lim n->inf (1 + 1/n)^n without realizing it's actually about growth (and that's how it was discovered): http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ http://betterexplained.com/articles/an-intuitive-guide-to-ex...
- yters 19y ago
  Excellent article. I noticed recursively growing series all seemed to have the same ratio, but I didn't realize this was related to e. However, I did see e show up all over the place, which I didn't understand. Anyways, it's obvious I'll need to read more of your articles. I'd much prefer to understand the intuition behind mathematics than just learn the formulae so I can "get things done." Regarding calc, I had a strange experience where I kept insisting calculus only logically works if it is based on infinitesimals, which I thought was obvious, but the people I was talking with insisted that it was based on limits. They couldn't understand that an infinitely small value has to be more than zero if an infinity of them is to add up to anything more than zero. In general, my math education seems to have been particularly bad at dealing with infinity.
  - Raphael 19y ago
    If d is an infinitesimal value, I'm not sure what d * infinity would be. If d is defined as 1 / infinity, then it would equal 1, but that doesn't seem right.
    - rms 19y ago
      This textbook teaches calculus using the infinitesimal approach. To do rigorous infinitesimal calculus, you have to define the hyperreal number system. http://theassayer.org/cgi-bin/asbook.cgi?book=770 http://theassayer.org/cgi-bin/asbook.cgi?book=770
yummyfajitas 19y ago
Quite right. I've been teaching calculus for several years, and for the most part it is a soul-sucking experience. Big chunks of it are utterly pointless. The main goal of the class is attrition. Seriously, integrating 1/(1+x^2)? Surfaces of revolution? Trig/hyperbolic trig substitution? Calc is full of stupid tricks that buy you closed form solutions. Adding Big-O notation to calc might help a little, especially when you get to series (with Big-O notation, you can explain divergent series). It's certainly a useful topic, and I can think of a dozen topics from calc I'd love to replace with it. Eliminating calculus and designing a basic math class from scratch would probably help a lot more.
- Pioneer1 19y ago
  I agree with this. The same goes for physics as well. Both physics and calculus look like humongous legal code created by a bureaucracy. They were intended to solve 18th century problems. The idea of creating a new version of these from scratch is a good idea. Why not eliminate calculus altogether and teach computer science instead, including programming?
  - yummyfajitas 19y ago
    Don't throw out the baby with the bathwater. Calculus is a fundamental topic, and even in computer science. Trig substitution is the bathwater. Curvature, tangent and normal are the baby, and they can be handy when you need them.

amichail 19y ago

Some places teach Calculus well. For example, check out the advanced Calculus courses at Waterloo.

LogicHoleFlaw 19y ago
How well do the introductory courses fare?
- amichail 19y ago
  I don't know what it's like now, but when I was there, the advanced courses were very intimidating. This is the real deal.
- mrtron 19y ago
  The introductory courses are quite tough. They teach the material well, but when it comes to assignments and exams they ask questions that tend to force you to apply the things you learned with some sort of twist. I always thought I knew the material inside out and got crushed on exams compared to how I thought I did. edit: Actually, 137/138 (which you will take first and second term) have recently been nerfed since they eliminated OAC and stopped teaching much calculus in high school.

brentr 19y ago

I will agree with you on the point that mathematics is beautiful. I had always appreciated mathematics for its logic. A proof is either right (it proves) or it is wrong (it does not prove), however, it was not until I was introduced to double and triple integration that I realized the raw power and beauty of calculus. I start working towards my computer science and math degrees this summer, and my registered courses include discrete mathematics and probability. I can be confident in the fact that I will learn some easily learned ideas that are able to do very powerful things.

LogicHoleFlaw 19y ago
Best of luck in your studies! Discrete mathematics was one of my favorite courses in school.
- brentr 19y ago
  It's a requirement for the CS portion of my degree. I already hold a finance degree from the University of Florida (2004), but the credit crunch sidelined me. Fortunately, I had the option of pursuing another two majors while still finishing in less than two years.

kalid 19y ago

I couldn't agree more. People forget that the calculus of Newton and Leibniz was based on infinitesimals, not the "rigorous" epsilon-delta proofs and the concept of a limit. Calculus was found by intuition and experimentation (Archimedes' method of exhaustion to find pi), and not following the implications of "What happens if I arbitrarily define a limit?". Current calculus education is like teaching kids about color theory, photons, how the eye works without just letting them fingerpaint. Those details will come, but the vast majority of people just remember calculus as a painful memorization exercise. As an aside, e is the same way. Most people have it taught as an abstract limit concept: lim n->inf (1 + 1/n)^n without realizing it's actually about growth (and that's how it was discovered): http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ http://betterexplained.com/articles/an-intuitive-guide-to-ex...

yters 19y ago
Excellent article. I noticed recursively growing series all seemed to have the same ratio, but I didn't realize this was related to e. However, I did see e show up all over the place, which I didn't understand. Anyways, it's obvious I'll need to read more of your articles. I'd much prefer to understand the intuition behind mathematics than just learn the formulae so I can "get things done." Regarding calc, I had a strange experience where I kept insisting calculus only logically works if it is based on infinitesimals, which I thought was obvious, but the people I was talking with insisted that it was based on limits. They couldn't understand that an infinitely small value has to be more than zero if an infinity of them is to add up to anything more than zero. In general, my math education seems to have been particularly bad at dealing with infinity.
- Raphael 19y ago
  If d is an infinitesimal value, I'm not sure what d * infinity would be. If d is defined as 1 / infinity, then it would equal 1, but that doesn't seem right.
  - rms 19y ago
    This textbook teaches calculus using the infinitesimal approach. To do rigorous infinitesimal calculus, you have to define the hyperreal number system. http://theassayer.org/cgi-bin/asbook.cgi?book=770 http://theassayer.org/cgi-bin/asbook.cgi?book=770

yummyfajitas 19y ago

Quite right. I've been teaching calculus for several years, and for the most part it is a soul-sucking experience. Big chunks of it are utterly pointless. The main goal of the class is attrition. Seriously, integrating 1/(1+x^2)? Surfaces of revolution? Trig/hyperbolic trig substitution? Calc is full of stupid tricks that buy you closed form solutions. Adding Big-O notation to calc might help a little, especially when you get to series (with Big-O notation, you can explain divergent series). It's certainly a useful topic, and I can think of a dozen topics from calc I'd love to replace with it. Eliminating calculus and designing a basic math class from scratch would probably help a lot more.

Pioneer1 19y ago
I agree with this. The same goes for physics as well. Both physics and calculus look like humongous legal code created by a bureaucracy. They were intended to solve 18th century problems. The idea of creating a new version of these from scratch is a good idea. Why not eliminate calculus altogether and teach computer science instead, including programming?
- yummyfajitas 19y ago
  Don't throw out the baby with the bathwater. Calculus is a fundamental topic, and even in computer science. Trig substitution is the bathwater. Curvature, tangent and normal are the baby, and they can be handy when you need them.