
The Calculus Trap (2005) - tokenadult
https://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap
======
jey
Personally, I hated calculus. It turned me off of math for a long time --
there was too much emphasis on memorizing heuristics for solving problems,
like integral tricks and trigonometric identities. Worse, calculus was used as
the canonical example of "college-level math", so it seemed that further math
courses would just be about memorizing more and more problem solving tricks.

(In my view, solving integrals is a search problem, and search problems are
for computers. I want to understand the concepts and the algorithms, not be a
glorified calculator.)

~~~
tonyarkles
I half agree. I'm not sure if your or most calculus curriculums involved much
derivation, or whether I ended up lucking out and getting a great instructor.
We ended up deriving a number of the "integral tricks" from "scratch" (e.g.
Fundamental Theorem of Calculus). Sometimes they went way over our heads, but
I generally found that the derivation process really helped me understand at
more of a gut level what was going on.

------
smilliken
I was a student that skipped high school and went straight into my local
college curriculum. It actually worked out great in my case, albeit largely
because I chanced across several high quality instructors that were able to
challenge me.

I am glad I followed the curriculum as well, though. In the end, you'll need
to take calculus (et al) at some point, and taking it as soon as possible
meant that it could help form my mind earlier, and allowed me to progress
beyond even sooner. Specifically, I followed the sequence: differential
calculus, integral calculus, multivariate calculus, differential equations,
and linear algebra (and then I transferred to Berkeley's math program). That's
a lot for a young mind to soak in, and the type of student these classes
select for make an excellent social environment as well (inevitably, the type
that likes learning).

If I were to make a recommendation between following the curriculum, and
pursuing extra-curriculars, I'd say: do both.

------
dalke
"you’re in ninth grade and you’ve already taken nearly all the math classes
your school offers"

I thought most high schools taught calculus. Both my and my wife's did. Why is
this 15 year old going to a local community college or university for that?

[http://www.maa.org/the-changing-face-of-calculus-first-
semes...](http://www.maa.org/the-changing-face-of-calculus-first-semester-
calculus-as-a-high-school-course) agrees, and points out that 3x more people
take AP Calculus now than when I was in high school. Which is about the same
time that the author went to school.

(FWIW, I do realize that my high school was unusual, in that it also offered
linear algebra, differential equations, and modern algebra. There were enough
of us who had taken calculus and were interested in additional classes. It was
a community college course taught at the high school by a high school
teacher.)

"I met nearly all of them through activities or employment that selected for
thinkers. In school, these activities were (and still are in most schools)
extracurricular programs, not curricular ones."

Nope, don't recognize that in my experience. The class with the most
"thinkers" was probably AP European History. I can't think of an
extracurricular which matched it. (AP American History, AP English, and AP
Calculus were pretty close.)

How much of this essay is the author's hypothesis, and how much of it is based
on actual research?

~~~
GFK_of_xmaspast
My high school ran out of math classes for me and 10% of my classmates after
11th grade.

~~~
dalke
Ran out at which level? The author apparently went to a high school without
calculus. I'm trying to get a feel for how common that is, especially since it
looks like several times more schools offer calculus now than when the author
went to school in the late 1980s.

~~~
GFK_of_xmaspast
No calculus.

~~~
dalke
Tx

------
ivan_ah
I totally agree that calculus is unlike "real math" (reasoning from first
principles, proofs, abstractions). Perhaps learning linear algebra would be
time better spent[1]. Sill, calculus is the first _applied math_ course and is
very useful for many areas of science so I'm not altogether in agreement that
learning calculus is bad for you. On the contrary---in combination with a
mechanics class it can be very good in terms of learning how to model the real
world.

That being said, I think UGRADs spend waaaaaay too much time learning
calculus: Calc I, Calc II, multivariable, vector calculus, etc. That's like 4
semesters of calculus! This signals to the student that calculus is somehow a
big deal, when in fact it is not: it's just _calc_ ulation techniques. I think
this should be cut-down to 2 semesters: Calc I+II together, then
multivariable+vector calc. Just learn it quickly and get it over with[2].

_________

[1] Gilbert Strang saying we should learn more LA and less Calc. [http://www-
math.mit.edu/~gs/papers/essay.pdf](http://www-
math.mit.edu/~gs/papers/essay.pdf)

[2] My short book on Calculus and Mechanics
[http://minireference.com/launch40](http://minireference.com/launch40)
(disclaimer: self-plug, but on topic)

~~~
dalke
A minor correction. Calculus has loads of proofs. The calculus taught to high
school and non-math undergrads tends to avoid more than a handful of proofs
because most students don't need to know, nor care, about the actual math
involved. Indeed, it would likely drive more students away.

I would say that calculus is the first "real math" that students study which
is so complicated that they can't really learn the first principles without
years of effort. That's why classes teach the most immediately useful parts
instead of covering the details. Someone who only studies those basics might
even come out thinking that calculus is just a bunch of techniques, without
grasping how gorgeous the underlying concepts and abstractions are.

Take geometry as something I think you regard as "real math." How come
geometry classes never prove that it's impossible to trisect the angle? Or
double the cube? It turns out that compass and straightedge can only produce
quadratic constructions, and Wantzel showed that these problems require
cubics. This shows just how limited geometry really is, which is partially why
students can feel that they have a handle on the entire topic. (Also, few if
any secondary schools cover non-Euclidean geometries, or Euclid's book X as it
explores incommensurable magnitudes.)

I would also say that trigonometry is the first applied math course that
students learn, not calculus.

Let's go to calculus. Take for example the chain rule, D(g∘f)(c) =
Dg(f(c))∘Df(c). This is one of those rules everyone learns in Calculus I. Can
you prove it? I once could. Here's the start of the proof from my text book:

"The hypothesis implies the c is an interior point of the domain of h = g∘f.
(Why?) Let e>0 and d(e, f) be as in Definition 39.2. It follows from Lemma
39.5 that there exists strictly positive numbers g, J such that if |x-c|<=g
then f(x) is an element of B and |f(x)-f(c)|<=K |x-c|. For simplicity, we
write L_f = D f(c) and L_g = D h(b). By Theorem 21.3 there is a constant M
such that |L_s(u)|<=M|u| for all u in R^q. If |x-c|<infimum(g, (1/K) ..."

And so on for another 10 lines of the book.

Do you really want to subject all calculus students to that level of detail? I
don't. Who other than a mathematician needs to learn about Lebesgue
integration and measure theory, which are even more advanced topics in
calculus?

BTW, my undergrad had multivariable and vector calculus as a single Calculus
III class, so there were only 3 semesters of calculus before getting into the
foundations of calculus. Also, calculus books do have some proofs. For
example, every chapter I looked at from
[http://ocw.mit.edu/resources/res-18-001-calculus-online-
text...](http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-
spring-2005/textbook/) contains a few proofs.

~~~
eli_gottlieb
>BTW, my undergrad had multivariable and vector calculus as a single Calculus
III class, so there were only 3 semesters of calculus before getting into the
foundations of calculus.

So did mine, which is biting me on the ass now that I'm studying for my
Machine Learning exam in graduate school. This course expected matrix
calculus.

------
pje
Lockhart's Lament [0] comes to mind.

[0]:
[http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)

------
stevep98
Reminds me of Arthur Benjamin's talk on focussing on statistics instead of
calculus.

[http://www.ted.com/talks/arthur_benjamin_s_formula_for_chang...](http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.html)

------
mikeash
I may not be the article's target audience, since I didn't really take
calculus until I went to college at 18. (I say "really" because I tried some
distance learning in calculus in high school, but I wasn't ready for that sort
of self-driven environment and failed miserably.)

However, my personal experience was the exact opposite. Calculus classes,
along with the accompanying physics-with-calculus classes, were what
transformed my concept of math from a game you play with symbols into a
powerful way to describe the way the world works. Once I realized that
equations like d=1/2at^2 just fell out of applying calculus to the idea of
change in position over time, everything suddenly made sense.

------
EdgarVerona
I feel like Calculus was very eye-opening for me, personally, and I felt like
it was a refreshing finisher to the traditional math courses. When you're in
the lower level classes, it's often hard to see where the practical
applications are going to start coming around... for me at least, Calculus was
where it all came together. I started to see just why I'd taken so many years
of math: they all built up to this very practical and fundamental set of
mathematics. That isn't to say that there's not a place for other math courses
- of course! But Calculus in particular is the "money shot" of math, in my
opinion. The applications are firm, and build the fundamentals of physics
among other subjects. If I hadn't taken Calculus, I don't think I'd have been
as positive about math as I ended up being. That's just my two cents, however.

------
ChristianMarks
Although I have a PhD. in mathematics, I found the field needlessly,
gratuitously competitive--nasty even--and I left. Math education--another
related subject which mathematicians often feel competent to pronounce upon--
seems to be a mine field [1]. There are only a few thousand mathematicians who
are paid to do mathematics--for the rest of us, the opportunity cost is almost
infinite. This is, for me, the real disincentive: the light at the end of the
tunnel is hardly more than a few photons.

[1] A double entendre. A _mine field_ is a set with the standard field
operations, which belongs only to the educator.

------
repiret
People studying hard sciences, engineering, statistics and mathematics should
study calculus, because they need it to understand and solve problems in their
domain.

High school students would be much better served in life studying statistics
instead. Far more important decisions in life and public policy are informed
by statistics than by calculus.

There was a time when I thought more breadth - number and set theory, proofs,
and so on - would be better in place of calculus. And while I think it would
be an improvement on teaching calculus, it would not be nearly as useful for
most students as a better understanding of statistics.

------
alexhutcheson
Steve Yegge had a good essay that touched on this.[1] He argues that the right
way to learn math is breadth-first, rather than depth-first - basically, learn
the basics of a wide variety of fields within mathematics, and then you will
know where to look when you need to understand a given problem.

[1] [http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html)

------
est
This article is lacking examples. What exact case makes the standard
curriculum bad? What are the good examples of the alternatives?

------
tokenadult
Thanks for the interesting comments. Taking the top-level comments in order of
posting, I read

 _This article is lacking examples. What exact case makes the standard
curriculum bad? What are the good examples of the alternatives?_

The article mentions, "more importantly, the gifted, interested student should
be exposed to mathematics outside the core curriculum, because the standard
curriculum is not designed for the top students."

The whole site that the article comes from serves as an example of mathematics
teaching that goes deeper and connects topics together better than the
standard curriculum in United States schools. Other authors have written on
the same topic. 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."

 _Personally, I hated calculus. It turned me off of math for a long time --
there was too much emphasis on memorizing heuristics for solving problems,
like integral tricks and trigonometric identities. Worse, calculus was used as
the canonical example of "college-level math", so it seemed that further math
courses would just be about memorizing more and more problem solving tricks._

This second comment to be posted expresses what many students miss out on if
their secondary school curriculum rushes to get to calculus as early as
possible without also being designed to help them understand mathematics as
well as possible. That's what the submitted article is about.

 _If I were to make a recommendation between following the curriculum, and
pursuing extra-curriculars, I 'd say: do both._

Yes, the both-and approach is helpful. That's what the article says when it
says "Developing a broader understanding of mathematics and problem solving
forms a foundation upon which knowledge of advanced mathematical and
scientific concepts can be built. Curricular classes do not prepare students
for the leap from the usual ‘one step and done’ problems to multi-step, multi-
discipline problems they will face later on. That transition is smoothed by
exposing students to complex problems in simpler areas of study, such as basic
number theory or geometry, rather than giving them their first taste of
complicated arguments when they’re learning a more advanced subject like group
theory or the calculus of complex variables."

 _Lockhart 's Lament [0] comes to mind._

[0]:
[http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)

Lockhart's Lament is indeed also a response to an era (different from the era
I grew up in) when many high school students are rushed into a calculus class
before reaching a profound understanding of fundamental mathematics.

 _" you’re in ninth grade and you’ve already taken nearly all the math classes
your school offers" I thought most high schools taught calculus. Both my and
my wife's did. Why is this 15 year old going to a local community college or
university for that?_

The author is indeed writing for a particular audience (which, as you
correctly point out, is growing in size) of young people who have blazed
through the United States mathematics courses that are now typical at ages
once thought unimaginable. My late dad took his calculus course in the late
1940s as a second-year college student. I had just seven high school
classmates in the mid-1970s who took calculus in high school at all. Most
students in my generation who took calculus at all took it as a first-year
university course. My oldest son began a formal course in calculus at eighth-
grade age, through an accelerated local program that was founded in the 1980s.
My second son is taking AP calculus BC as high school junior (eleventh
grader). People are rushing into calculus much more rapidly than ever before
in the United States, but often lack "profound understanding of fundamental
mathematics (PUFM)" before starting the calculus course. A link that furthered
my process of pondering how students might learn mathematics better was
Richard Askey's review of the book Knowing and Teaching Elementary Mathematics
by Liping Ma.

[http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf](http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf)

Another review of that excellent book by mathematician Roger Howe

[http://www.ams.org/notices/199908/rev-
howe.pdf](http://www.ams.org/notices/199908/rev-howe.pdf)

is also food for thought. In some countries, elementary mathematics is not
considered "easy" mathematics, but rather fundamental mathematics, which must
be understood in full context to build a foundation for later mathematical
study.

 _However, my personal experience was the exact opposite. Calculus classes,
along with the accompanying physics-with-calculus classes, were what
transformed my concept of math from a game you play with symbols into a
powerful way to describe the way the world works._

There are definitely a lot of students who enjoy a calculus course for that
experience. That seems to be a form of enjoyment that especially comes to
students who have had time to learn about other topics on the way to learning
calculus. Russian mathematical instruction tries harder than instruction in
the United States to bring in examples from physical science at all ages, so
that the mathematics that explains physics is taught to students who have a
decent background in physics.

~~~
dalke
> My second son is taking AP calculus BC as high school junior (eleventh
> grader).

Your example has a 15 year old going to community college or university in
order to go to a calculus course. You named it the "calculus trap", which
includes as a negative the social problem of having a 15 year old in a class
full of 19 year olds. This implies that it's unlikely that the high school
offers calculus.

My observation is that more and more high schools offer calculus in the high
school, so the 15 year old you described, who was taking college courses to
learn calculus, is now more likely to be a 15 year old at high school taking
courses with 16 year olds (like your second son).

In that case, the severity of the trap is lessened, no? If only because the
age gap is so much less.

I'm not saying that you are wrong about how math knowledge should be
developed. I point out only that the arguments from your hypothetical case
feel a bit out of date.

Than again, suppose the 15 year old does take calculus at high school, then
takes a tertiary education class at age 16. What class might that be? That's
about the time the standard college curriculum branches away from calculus, to
include algebra, differential equations, or discrete math.

In that case, it's not really a "calculus" trap, no? :)

Also, I read the reviews of KTEM. Cross-cultural observational comparisons are
often enticing, but it's hard to draw firm conclusions from them. Had you
read, say, a comparison with the Finnish model then perhaps you might have
drawn different conclusions?

My hypothesis, btw, is that the US is entirely too car dependent.
Extracurricular activities like a city math club would be much easier if teens
had ready access to transportation independent of their parents.

