
Dismantling the calculus pyramid - timwiseman
http://wildaboutmath.com/2009/10/27/dismantling-the-calculus-pyramid/
======
tokenadult
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/>

See also

<http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf>

with comments on mathematics education for breadth rather than for speed
through the standard curriculum, by a Fields medalist.

~~~
timwiseman
I would second that, and add a recommendation for "Chapter Zero" which is a
book targetted at the high school/early college level which provides an
introduction to those fundamentals that are often skipped in the rush towards
calculus.

~~~
tokenadult
This Chapter Zero,

[http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-
Math...](http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-
Mathematics/dp/0201437244)

right? I have not read this particular book, but in general most students who
desire to study math should take a look at a "transitions course" textbook or
two, to fill in the gaps left by studying only in the fast lane to calculus.

<http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/>

[http://www.amazon.com/Mathematical-logic-and-
foundations/lm/...](http://www.amazon.com/Mathematical-logic-and-
foundations/lm/3L9OBW0IXF414/)

[http://www.amazon.com/How-solve-prove-books-
useful/lm/31B4QV...](http://www.amazon.com/How-solve-prove-books-
useful/lm/31B4QV24TPSZ/)

------
mitko
I agree agree that probability and statistics are much more important in life
than just calculus. That is why in college I am studying Artificial
Intelligence and Machine Learning - these are all math and probabilities.

Yet, I think Arthur Benjamin is right but for the wrong reasons:

1\. _People should know statistics instead of Calculus._

To know probability well you need to have mastered calculus. All the
distributions, all the formulas build on things like sequences, and integrals.

2\. _If everybody knew statistics we wouldn't be in the economic mess._

Plain speculation. The majority people who have important part in the
management of banks know statistics.

3\. _from continuous mathematics to discrete mathematics..._

Probability and statistics involves tons of continuous mathematics and
Analysis. Is a Gaussian distribution discrete? Is the Central Limit Theorem
discrete?

The reason probabilities and statistics is important is because it lets people
better understand why stuff around them happens and lets them reason about
uncertain processes. Life is behavior under uncertainty.

~~~
brianto2010
I believe it is important to remember that Arthur Benjamin is referring to
_High School Mathematics_.

 _To know probability well you need to have mastered calculus._

I won't argue that point itself. However, given that the audience for this
mathematics education is _High School Students_ , we won't have to go that
far. Regression lines, standard deviations, and Simpson's paradox can be
understood without Calculus. Personally, more time should be focused on the
analysis and application (again, at a _High School Level_ ).

------
marze
I would go further and not only teaching probability and statistics, but also
teach basic calculus with simple polynomials, programming, Monti Carlo
simulations, sines and cosines, computer simulations, all starting around age
13.

Advance algebra, geometry, and full trigonometry are not required for any of
that.

Why make students wait until senior year or college to experience the fun
stuff?

------
amichail
Bad idea. Some consequences:

* People would stop voting.

* Crime would probably increase.

* People would be less friendly/helpful to strangers in large cities.

* People would be less willing to drive (causing chaos in cities without sufficient public transportation).

* People would no longer care about environmental issues.

* Entrepreneurship would decrease.

* Stock trading would decrease.

~~~
Zaak
The rational reason to vote is not because your vote actually has an effect on
the election's outcome. Rather, it is because if nearly no one voted, then
politicians would have no incentive to take care of the needs of the people.
The reason why representative democracy works (when it does work) is because
the people have the power to remove politicians from office (or prevent their
reelection).

Of course, it is unwise to limit your involvement in government to merely
voting, but that is a separate topic.

~~~
amichail
This is a question about how much influence an individual has on others in
terms of networking effects.

Moreover, it may be sufficient to have a positive attitude toward voting
without actually ever voting.

~~~
Zaak
> Moreover, it may be sufficient to have a positive attitude toward voting
> without actually ever voting.

Nice. Sort of like an election typhoid mary.

> This is a question about how much influence an individual has on others in
> terms of networking effects.

I agree that the influence an individual has on his social contacts is
relevant here. However, I still insist that the individual's decision to vote
is itself important.

------
joe_the_user
Hmm,

Oh where to begin?

The guy's rhetoric is pretty divorced from the sad state of actual math
education today. The way that the standard math curriculum today heads up to a
point in a pyramid is a problem in itself since a lot of the topics are made
dull in themselves ("This is just something you need to learn for calculus" is
a terrible answer for "why should I learn this?" but substituting "Statistics"
wouldn't change things much). It doesn't really matter what the _point_ of the
pyramid is when most people never get there.

Just as much, the latest economic meltdown was engineered and _believed in_ by
_statistics experts_. For every Mandlebrot debunking the events, there were
ten Myron Scholes basking in the glory of validating Wall Street's delusions
with some mathematical magic. _Statistics doesn't protect from wishful
thinking outside of controlled, experimental situations_.

And statistics in daily life? One might use some basic probability but the
only other use it would have would sorting the pseudo-statistical rhetoric
used by the media. A simple course in mathematical literacy with an emphasis
on fallacies would be best for sorting this stuff. BUT again, no course can
protect from _wishful thinking_ , can protect people from the fallacies that
let them ignore possible later dangers for immediate apparent gain. Further, a
non-calculus-based statistics or mathematically literacy course isn't a basis
for further scientific study the way calculus is, and believe-it-or-not some
students still become physicists, chemists and engineers where calculus is
indeed the foundation.

I could narcisistically say that my favorite, evolutionary game theory, would
make a much better "point" for the curriculum pyramid but really, what is
needed is to make every math class interesting in and of itself. With TV-dazed
kids and math-apathetic teachers, I don't know if any curriculum could change
things BUT I would want to have the curriculum of every class interesting and
mentally challenging - taught axiomatically, Algebra and Geometry ARE
interesting in and of themselves and a student needs no background at all for
them. Math should be rigorous, conceptually challenge and _optional_ past the
basics. It seems like we'd need a different world for this but small steps are
being made.

~~~
RiderOfGiraffes
Firstly, Art is actually a math professor, and dealing with the low standards
of incoming students is a major concern. I don't think he is "divorced from
the sad state of actual math education today."

But even more, you say:

    
    
        > ... taught axiomatically, Algebra and Geometry
        > ARE interesting in and of themselves
    

Perhaps to you, but I deal with students for whom this is absolutely not
interesting. They're not interested in puzzles, they're not interested in
challenges, they're not interested in anything except texting their friends
and talking about films, TV, clothes, football, etc. They don't want to be
challenged.

I think we all agree that classes in general should be stimulating and
interesting, andshould be better tailored to suit the needs of the
individuals, but that's the problem. For a given child we don't know what
they'll need, and we don't know what they'll like, and every answer will be
different.

The problem is "one class fits all" and that's not going to change simply by
re-writing the curriculum.

~~~
joe_the_user
_Perhaps to you, but I deal with students for whom this is absolutely not
interesting. They're not interested in puzzles, they're not interested in
challenges, they're not interested in anything except texting their friends
and talking about films, TV, clothes, football, etc. They don't want to be
challenged._

Uh, yeah I've dealt with those students. Notice the last part I add -
_optional_. The thing that kills math interest utterly is those "required"
classes which teach nothing to the uninterested.

Modern schooling drags the uninterested through a process of making motions
towards understanding - we all know its a waste of time. It really would be
better to give up until the students are interested. A motivated student can
learn more in a day than the bored learn in the semesters of basic math. That
might put you out of a job but those jobs just shouldn't exist. Sorry.

~~~
RiderOfGiraffes
Won't put me out of a job - you've jumped to an incorrect conclusion. I'm not
a teacher. I run two companies, and I go out to schools to give talks on why
math is fun, interesting, useful, and occasionally exciting. Most of the
students I deal with are motivated and interested, but even then, some don't
like puzzles, and don't like starting from the ground up axiomatically.

This Saturday I'm talking about the Banach-Tarski theorem, and I'm starting
from the result, then wroking backwards, deciding what we need to know as we
peel it back. I've found that working backwards from a surprising result can
create motivation to understand, but sometimes it causes the students to
dismiss the whole thing as useless, pointless and irrelevant.

Sorry, I'm rambling. Reply if you're interested, ignore me if not.

~~~
joe_the_user
Sounds like fun, I'm happy to hear about it...

I shouldn't be dogmatic about asking for an axiomatic development.

At the same time, it seems like the social attitude towards mathematics has
reached the point where it would be useful for schools to ask students to put
aside some of their initial attitude towards math.

The best teachers I've had often demanded more than I was initially capable of
accomplishing. It's true that such teachers risked losing some of their
audience. But if we don't have such teachers we risk even more.

~~~
RiderOfGiraffes
As an aside, your profile says you're interested in contract work, but gives
no way to contact you. Is that deliberate?

------
WalkingDead
Best advice of this century in it's class.

I don't know why we are still wasting our time with calculus while I see a lot
of people suffering in their daily life because of their lack of understanding
in probability theories and statistics.

Statistics is easier and even more fun to learn than calculus.

~~~
kscaldef
Sadly, calculus is a pre-requisite for anything more than very basic
probability and statistics.

~~~
winthrowe
I think this is directed mostly at the people who will never take another math
class after they leave high school. Speaking from a canadian perspective, I
had Grade 11 Math, Grade 12 Math, and Calculus classes. about half of the work
in grade 12 was directly preparatory for the calc class. Students persuing a
trade/vocational school or no post secondary education would only have taken
11 and 12 math. if the half of the math class that was 'wasted' calc prep was
replaced with 'probability in the real world' type material, you'd still be in
the basic precalc stats area, but possibly giving something more useful to the
kind of people that are not well represented at a site like this.

------
RiderOfGiraffes
See also:

<http://news.ycombinator.com/item?id=680063>

<http://news.ycombinator.com/item?id=701367>

~~~
arketyp
Why? There are no discussions there.

~~~
RiderOfGiraffes
Hmm. Maybe what I really meant was: this has been submitted at least twice
before, three and four months ago or so, and in neither case did it generate
any discussion.

I hope it does this time, because it's been shown to be interesting enough to
be "discovered" by more than one participant.

And I do. I've never understood the US schools emphasis on calculus, and
building it up to be such a huge deal. I first did calculus aged 16 in Year
10, and it was just another part of the syllabus. One more step to
understanding the way stuff works.

Replacing it by another "BIG IDEA" seems not necessarily to be a Good
Thing(tm). Perhaps there should be more of a plain, and less of a pyramid.

I agree that more elementary stats done earlier would be a really Good
Thing(tm), but replacing one pyramid with another does not seem to me to be
so.

~~~
rgoddard
First there is the recognition that there is no pyramid. Since math is so
highly interrelated, learning about one area almost always helps understanding
in a different area. Granted some areas require a certain prerequisites, hence
you will encounter peaks, but I agree aiming for only one peak does not make
all that much sense since calculus is only useful in certain situations.

------
perkoff
Maybe it's built in "pyramid" like this because: 1\. Knowledge is hierarchical
2\. Calculus has a longer list of skills that needs to be mastered before
learning the subject. 3\. Historically, calculus came around first (am I
right?). This might not be a coincidence.

~~~
seregine
Understanding (the path to knowledge) might be hierarchical but knowledge in
general isn't. Organizing knowledge in hierarchies is a conventional teaching
technique, and that has some self-fulfilling qualities: if you teach people in
hierarchies, they'll see the world that way.

