
The Calculus Trap - tokenadult
http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php
======
tokenadult
"If ever you are by far the best, or the most interested, student in a
classroom, then you should find another classroom."

This advice generalizes to many subjects besides mathematics.

AFTER EDIT:

Here's what one math professor says about the delightful aspects of
mathematics to learn besides calculus as it is now taught. 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/>

~~~
mnemonicsloth
In principle I'm sympathetic to this idea, but attempts to make this work in
the past have been worse than useless:

<http://en.wikipedia.org/wiki/New_Math>

Stillwell is wrong, or at least not doing his homework, when he says "Everyone
concedes that [ideas of number and space] are fundamental, but they have been
scandalously neglected..." Back in the early sixties, when Bourbakism was hip
and new (as mathematical ideas go at least), everyone agreed that set theory
and mathematical logic were fundamental, whence the debactacular attempt to
begin teaching first-order logic to seventh graders.

The uncomfortable truth is that math is like a language in that it requires a
student to put together a large and messy collection of terms and concepts to
express any kind of thought at all. The fact that all of the thousands of
entries in this vocabulary can be derived _in principle_ from maybe a dozen
axioms is what gives math its power, but nobody learns it that way, proving
this or that fact as needed. Real skill comes from practice, which starts out
as imitation and then generalizes into problem-solving and proof.

All the emphasis on The Fundamentals of math is really just evidence that most
mathematicians are bad at teaching -- it's an attempt to impose mathematical
order and rigor on the fundamentally messy and organic process of human
learning.

~~~
RiderOfGiraffes
I know John Stillwell, and I think his opinions and beliefs are being mis-
quoted by being taken out of context. I think he is right that ideas of number
and space are fundamental, and I think he's right that they are being
neglected, which is a scandal. I agree that "New Math" was a complete debacle,
but I think it's not relevant to this question.

I agree that math is like a language, but I disagree that it requires the
student to put together a large and messy collection of terms and concepts.
That may be the way you view it, but it's not the way I view it, it's not the
way I teach it, and I get pretty good results in masterclasses, and more
generic classes, by helping the development of the web of inter-related ideas,
without large numbers of apparently unmotivated definitions.

In short, I think Stillwell is right, and I think you're missing his points
because you appear not to have read his original works.

------
lutorm
This was interesting, and speaks to my own experience.

In high school, I did the local quals for the international math, physics and
chemistry olympiads. I did quite well on the physics and chemistry ones and
went to the international versions as part of the Swedish teams. On the math
one, though, I plain sucked. In post-processing, I've realized that what
happened was exactly what this article talks about. The math questions were
questions where I just went "Huh? I have no idea how to even approach this,"
because my math education had largely only prepared me for solving specific
kinds of problems. I'd never been exposed to things like proving theorems or
other kinds of creative thinking.

~~~
req2
The first math competition I entered was an interesting experience. There were
so many completely foreign concepts like sigma notation that precluded me from
even trying to answer the 'tricky part' of the questions. Luckily, the proofs
came out on a later competition, after some good math competition coaching by
the quizmaster.

------
g_
I read this article in high school, and it permanently changed my math life.
Without it, I would continue to think I know derivatives because I learned
twenty formulas and did forty exercises (which in fact every idiot could do).

I'll allow myself to give some advice for those who are interested in problem-
solving, but have no experience. If you have some, then this should be well
known.

* Keep problem / exercise ratio as high as possible. This is impossible with many calculus books; find a book with hard problems. An "exercise" is something which checks your understanding of definitions and theorems; a "problem" is something which exercises your skill and forces to think. Do exercises if the theory is unclear. If a task starts with "using mathematical induction prove that..." then it is an exercise. A problem forces you to think _how_ to do it.

* Doing differentiation exercises will give you some speed, but after five-twenty minutes your brain will stop thinking and start to rot. Healthy mathematics - just like programming - hates doing the same thing again. Of course you have to learn some algorithms, but this is a tip of the iceberg.

* Always take 20 minutes (some say more) on a problem, unless you think it is ill-posed; giving up early is stupid. If you think the problem is impossible, try proving it. Think about some way of solving, reject it quickly if you made a thinko; if you sense "this might work" go deeper. Use paper.

* Don't read too much on philosophy of mathematics or biographies; this isn't deep from the mathematical side. Other articles ([http://www.artofproblemsolving.com/Resources/AoPS_R_Articles...](http://www.artofproblemsolving.com/Resources/AoPS_R_Articles.php)) are also worth reading. Check their forum (<http://www.artofproblemsolving.com/Forum/index.php> or www.mathlinks.ro).

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WilliamLP
Why make a big stink about calculus?

To a smart person, after learning about it, calculus is simple and self-
evident. The rate of change of something is the slope, and to find it at a
point you take two points arbitrarily close together. The rate of change of
the area under a curve IS the curve. That's calculus, and the details of the
rest follow.

Why not give a bright young person those insights and let them play?

They never teach the really interesting and cool parts about calculus. How
many people know that the rate of change of volume of a sphere _is_ the
surface area, for instance? It makes wonderful sense.

~~~
Retric
_the standard curriculum is not designed for the top students_ it’s not a
question of learning calculus that's easy the question is how to keep up with
your potential.

If you understand calculus and diff EQ you can do 95% of the useful math for
most walks of life. But getting to that point at 16 is not much more useful
than getting there at 20. What’s often harder and more interesting is to get
into topology and number theory so you can start to explore higher math.

PS: High level math completions in the US are also focused on a wider range of
math skills. With a little effort you can start to step out of the "normal
mold" and compete at that level.

~~~
WilliamLP
> What’s often harder and more interesting is to get into topology and number
> theory so you can start to explore higher math.

If you learn the basic ideas of calculus when you're 12, this is not going to
dissuade you from learning about topology and number theory too. Prime numbers
pretty much sell themselves.

------
defen
One of the best classes I took in college was Abstract Algebra, and I only
took it because my math-major roommate kept telling me that I needed to "learn
more math." My favorite part was coming to the realization that linear algebra
was not just a set of tools for solving particular problems, but rather an
instance of a much more general mathematical structure with a beauty and
simplicity of its own.

~~~
alain94040
First day of college: "prove that 0 times x is equal to 0". That got rid of
everyone who wasn't really into maths :-)

That exercise was literally the first question, on the first hour of back to
school, after 5 minutes of a welcome speech.

I can't even figure out the proof anymore, although it really can't be
complicated given the ridiculously small amount of tools you are given
(basically x + -x is 0, and (a+b) times c equals a times c + b times c. You
are given nothing else.

~~~
antiform
From the real number axioms:

Let x \in R. Then we have that

    
    
      x*0 + x*0 = 0*x + 0*x = (0+0)*x = 0*x = x*0
    

by commutativity of multiplication, distributivity, the existence of a neutral
element 0 for addition, and the commutativity of multiplication once again.
Thus, we have that

    
    
      x*0 + x*0 = x*0.
    

For clarity, define

    
    
      A = x*0.
    

Then we have that

    
    
      A + A = A
      A + A + (-A)  = A + (-A), by the existence of additive inverses.
      A + (A + (-A)) = (A + (-A)), by associativity of addition
      A + 0 = 0, by the definition of additive inverses
      A = 0, by the definiton of the neutral element 0 of addition
    

which gives us our desired result.

------
dlevine
So what he's saying here is, "surround yourself with people smarter than you."

This is good advice. Whenever you feel like you're not being challenged any
more, it's time to move on.

~~~
lutorm
I read a book called "My Job Went to India" (it's a tongue-in-cheek tutorial
of how to stay competitive in the computer industry) and one of the advice
was:

 _Always be the worst on your team._

The idea is, of course, not to suck but to put yourself on a team with people
that are more awesome than you, because that's how you grow. People usually do
a double-take when I quote this, though. :-)

~~~
gms
Easier said than done no? Outside of school, where do you find people willing
to work with someone worse than them?

~~~
zkz
The smart people always need other less smart people to do the things they
don't want to do. There is always space for a worst of the team.

~~~
gms
That is what I thought. But not how it is playing out when applying for jobs
that put me as the worst one (from my perspective anyway). Perhaps other
factors are at play.

------
InkweaverReview
Very fascinating article. I fell into the Calculus trap myself, and I admit
that it definitely would have been more interesting to have avoided it.

As it was I got an A in Calculus, breezed through the course as an early
college course student.

How much better it would have been to have enjoyed the alternate possibility
presented:

"Going from ‘top student in my algebra class’ to ‘average student in my city’s
math club’ is a huge step forward in your educational prospects. The student
in the math club is going to grow by great leaps, led and encouraged by other
students."

------
10ren
I lost interest in maths in high school when I realized that the problems we
were solving were specifically designed to be solvable with the techniques we
were taught. Their generality was specious. This was at an expensive private
school, in the so-called advanced maths stream...

I much preferred programming, because things made sense there. You could work
things out. Even the instruction set for the Z-80 processing I was coding for
was more consistent and general than the maths I was taught. The compound
instructions were obviously compound, and - and this is the best part - the
aspects that I didn't understand enriched me when I finally did understand
them (I realized they had machine code support for structs, before I'd heard
of structs - I 'invented' structs because my code needed the concept).

Today, after bachelor, masters and not-yet-submitted doctorate in Computer
Science, the maths I see in papers still seems technique-based rather than
about the truth. I use set-builder notation, and grammar notation (regular
expression and CFG syntax), but only because it is logically consistent, not
because it's "mathematics".

It seems to me that there isn't any way to learn the true mathematics - the
mathematics that is our heritage as human beings - except by the good fortune
of finding someone who does understand it, or long and personal study, using
historical sources, because these are the only ones that show the point of the
techniques - _why_ they are useful.

There must be good text books out there, but the ones I've seen either dive
into obscure notation, or use inadequate metaphors. Disclaimer: The only maths
I took at uni were taught by the engineering dept. Maybe maths taught by the
maths department might have helped me. Actually, my very best comp sci
lecturer was from the maths dept - he took care to make it logical.

------
bpyne
There's another more fundamental issue with education at the high school level
in the US: scheduling. Usually high school students go through 6 periods of 50
minutes each daily, often with one of them being a study period alternating
with physical education.

Putting the students truly curious about math, who usually make up a small
percent of the students in the college bound math classes, and aligning them
with a teacher capable of keeping them engaged is possible. The problem lies
in having them attend their other subjects at a level appropriate to them.
Keep in mind that each member of this subset of students may not be at the
same level as their peers in other subjects.

If these curious math students are not segregated out, which often happens,
then a teacher has 20-30 good math students with a few being curious enough to
be elite. Keeping the two distinct groups of that class engaged requires
distinct lesson plans for each group. The goals of each day's lesson plan need
to be met within no more than a 40 minute window - I'm being generous that
taking attendance and assigning home work take up no more than 10 minutes of
class time.

I admit it's a worthwhile goal to have the curious math students more engaged.
However, I don't see it being possible unless a more fundamental restructuring
of high school education occurs. Perhaps have it structured more like a
university schedule in which students take 3 subjects daily with each class
being 1-1.5 hours.

------
eru
Why so much focus on calculus? I found algebra much more entertaining,
abstract and challenging.

~~~
tokenadult
Are you talking about the school algebra that for most students precedes
differential and integral calculus, or are you talking about the abstract
algebra that in most curricula follows calculus as an upper-division
undergraduate course?

~~~
eru
Yes. The latter. However we had Analysis (Calculus) and Lineare Algebra at the
same time. Algebra came afterwards.

Calculus in school was as boring as everything else. School math is mostly a
basic version of engineer's math.

------
lacker
This is definitely true. Number theory is more interesting than calculus.

The difference is, if you are not good at math, number theory is impossible.
But anyone can learn calculus if they try hard enough.

~~~
shaunxcode
I don't buy that at all. That attitude is precisely what makes people think
they can't "get it". As if they lack some sort of natural aptitude or genetic
predisposition towards numbers.

~~~
lacker
I don't know about genetic stuff, I'm just trying to make a relative statement
between calculus and number theory.

Who knows what causes it, but there are a lot of smart 15 year olds who are
less good at math than they are at most things. If those people really work
their asses off at calculus, it works better than working their asses off at
number theory. That's because calculus contains more memorization and less
problem-solving.

------
mgenzel
I also didn't really enjoy Calculus in high school, only learning to love it
in hindsight, when I took Real Analysis in college. Then again, I was always
interested in theoretical mathematics (poetry, abstraction, & all that) rather
than practical (yucky numbers & approximations :); so maybe pragmatists enjoy
Calculus as taught?

------
christopherolah
I haven't taken a Calculus course yet... But I've thoroughly enjoyed learning
it on my own. I've proved the laws I work with (except chain rule...).

I think it's probably been the richest part of my math experience.

------
dmfdmf
I love math, unfortunately they don't teach it in the schools any more.

------
pageman
I'm convinced that young people should be learning descriptive statistics and
move on to inferential statistics online like in:
<http://davidmlane.com/hyperstat/index.html> it's easy now to learn from
online data sets and you can collect more information through social media
(vs. using physical paper questionnaires.) I learned these kinds of statistics
when I was 12 years old - I wish someone told me what I could do multivariate
analyses ...

