
Donald Knuth's letter about teaching O notation in calculus classes - plinkplonk
http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/
======
dissenter
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.

~~~
LogicHoleFlaw
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.

~~~
kalid
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...](http://betterexplained.com/articles/an-intuitive-guide-to-exponential-
functions-e/)

~~~
yters
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
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.

~~~
yters
The tricky thing is that infinity isn't a number as we normally think of
numbers. There are different kinds of infinity, so infinity / infinity does
not necessarily equal 1. It could be any number between 1 and infinity,
inclusive.

To get this intuition, think of the integers. There are obviously an infinite
number of them. Now think of the rationals. There are now an infinite number
of numbers between 1 and 2, so there are more rational numbers than there are
integers; in fact infinitely more.

~~~
rms
I believe set theory is the only area of mathematics where different orders of
infinite actually mean anything.

So, in set theory, cardinality is a measure of the elements of a set. The
cardinality of the infinite set of natural numbers is aleph-0. An infinite set
has cardinality aleph-0 if it can be put in one to one correspondence with the
naturals. <http://en.wikipedia.org/wiki/Bijection> This counterintuitively
includes the rational numbers also.
<http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument>

The real numbers can't be put into one-to-one correspondence with the rational
numbers. So the infinite of the reals truly is bigger than the infinite of the
rational numbers.

Also see this ppt:
[http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/di...](http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/discretemath/Lectures/cantor.ppt)

~~~
yters
You don't think mathematics is unified, i.e. something true in one area may
not be true in another? I don't think a lot of mathematicians believe this,
and it isn't clear to me that "infinity" refers to two different things in the
two different realms (the other possibility).

------
freax
Don Knuth wants to see your "O" face.

