Donald Knuth's letter about teaching O notation in calculus classes

http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/

by plinkplonk • 18 years ago

50 25 18 years ago

micromath.wordpress.com

dissenter• 18 years ago

Calculus is designed as a gateway for all of our quantitative students, our future engineers and physicists in addition to our computer scientists, and so it's already forced to wear many hats. Changing it isn't something we can do lightly, as it's integral in everything from standardized testing to college admissions (in the case of high school students); and nearly every quantitative track (in the case of college students).

What good does splicing Big-Oh into the process do, when computer science enrollment is in steady decline, and understanding Big-Oh is pointless without first understanding the costs of polynomial algorithms? It would be nice if our calculus students knew Lebesgue integration too, but I would offer that the calculus sequence is not the right time. If the kids are in college, they can take Intro. to Comp. Sci., and if the kids are in high school, they probably shouldn't worry about Big-Oh at all.

The biggest revision high school math needs is 1. A course which instills students with a basic statistical fluency, so they can read the newspaper and endure advertisements with a critical mind, and 2. A course for advanced students which teaches discrete math and basic non-geometrical proofs, for our increasingly discrete world.

LogicHoleFlaw3• 18 years ago

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.

amichail2• 18 years ago

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

brentr2• 18 years 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.

kalid• 18 years 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-ex...

yummyfajitas2• 18 years 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.

stcredzero• 18 years ago

I really wish that a lot of the non computer-science people I have to work with knew at least a smattering of what it means that something is O(n^2) versus O(n log(n)). In fact, I sometimes wish that programmers who I've had to work with understood!

johnm• 18 years ago

Indeed!

I was "interviewing" with the head developer at a media startup who thought he knew his stuff and after handing him his hat on a number of things, he tried one last "zinger" question on O(). Turns out that neither he nor the head of engineering knew the difference between O(n^2) and O(n log n) but they were so confident that they did. Unbelievable.

freax• 18 years ago

Don Knuth wants to see your "O" face.

Donald Knuth's letter about teaching O notation in calculus classes

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