Why teach calculus to everyone?
Stephen Preston
life science or liberal arts or social science majors. Many of these
people are required to take a 4-semester sequence of calculus (there's a
faster version, and there's other ways, but this is the one most people
choose).
As one who teaches these courses, I get asked why people are required to
take calculus. And although I can give a convincing answer (teaches
logical problem-solving, etc.), I'm not really convinced myself.
Calculus reform, it seems, has tried to cater to such students as we
have (e.g. Hughes Hallett), but it ends up seeming rather phony, and
since there is so little that is actually calculated, one wonders why
it's necessary to study calculus at all. Plus, of course, many
professors hate teaching out of these books, and many people see them as
dumbed down.
My proposal would be, instead of teaching calculus, teach instead
"Calculation", which would be composed, say, of a variety of fields of
math, not just calculus. E.g. discrete probability and combinatorics,
difference equations, geometry, number theory, plus light calculus and
other such things. (Not really sure what the ideal syllabus would be.)
Unlike the reform texts, you'd still focus on algebra, analysis, not
using computers as a crutch, and doing lots of calculations by hand.
But you still wouldn't need to spend so much time teaching techniques of
calculus.
I think the value would be that students would have a higher interest,
since you could restrict yourself to real-world problems instead of
contrived problems or physics problems. You could also teach most of
the things in a historical context (e.g. this technique was invented to
solve this problem), which might appeal more to the liberal arts
students. Plus many of them would be better able to use what they
learned in their future courses, or even in real life.
What are your opinions? Thanks for advice or thoughts.
--Steve Preston
bo...@rsa.com
Stephen Preston <pre...@math.sunysb.edu> wrote:
> I'm a grad student at a state university where most of the students are
> life science or liberal arts or social science majors. Many of these
> people are required to take a 4-semester sequence of calculus (there's a
> faster version, and there's other ways, but this is the one most people
> choose).
> As one who teaches these courses, I get asked why people are required to
> take calculus.
Why are people required to study
moral reasoning
literature
history
social (so called) science
etc. etc.
If people only studied what they needed for their day to day existence,
then most of us would be pretty ignorant.
Liberal arts means studying a variety of subjects.
Secondly, life science majors NEED calculus. How do they expect to
understand (for example)
population growth models
membrane diffusion
enzyme kinetics
Ph and LeChatelier's (spelling?) principle
etc. etc.
without an ability to set up AND SOLVE differential equations?
Social science majors need to study Calculus in order to understand the
underpinnings of statistics (i.e probability distributions). They need
to learn transformation of variable techniques among other things. How
else are they to know that adding together two normal populations gives
another normal population?
I agree that students studying the 18th Century American Novel (probably don't
NEED Calculus. But it won't hurt them either. One
could turn this around: why would a physics major need to take Shakespeare?
It seems to me that people always ask why they need to take Calculus because
Calculus is HARD, and that they are looking for any excuse they can find to
avoid having to do hard work in school.
Also, in a more general sense, people need to learn that the world is not
LINEAR and to understand rates of change. They learn this (or they should)
by studying Calculus and applying it within their own particular discipline.
Finally, I end by saying that undergraduates don't know enough in general
to be able to say what they will or won't need to learn to be able to deal
with their later professions. That is why faculty (older, more experienced,
more educated) determine courses of study and not the students.
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Stephen Preston
Of course. This was my point, that you could study a variety of mathemaical
subjects as well, not just the relatively rather narrow field of calculus.
> Secondly, life science majors NEED calculus. How do they expect to
> understand (for example)
>
> population growth models
> membrane diffusion
> enzyme kinetics
> Ph and LeChatelier's (spelling?) principle
> etc. etc.
IIRC, most or all of these differential equations are of the form
dy/dx = ky,
or something similar. They're certainly not so difficult. One could just study
linear, constant-coefficient DE's instead of all the very general techniques.
> I agree that students studying the 18th Century American Novel (probably don't
> NEED Calculus. But it won't hurt them either. One
> could turn this around: why would a physics major need to take Shakespeare?
A physics major doesn't need to take Shakespeare; a physics major may need to take
SOME humanities course, but he/she can generally choose which one.
> Also, in a more general sense, people need to learn that the world is not
> LINEAR and to understand rates of change. They learn this (or they should)
> by studying Calculus and applying it within their own particular discipline.
Yes, but this can be done very simply. One does not need very advanced techniques
of calculus. And people should also understand probability, but most of them
don't. They should certainly understand linear algebra, but most of them never
get around to it. You end up teaching people a bunch of algorithms which they'll
forget soon after the course. A lot of people don't remember how to solve
differential equations, even easy ones (even engineers). I have friends in
engineering who describe their classes as "And this situation is modeled by this
differential equation. The solution is this formula.", etc. They're expected to
memorize the solutions, not the method of getting the solution.
> Finally, I end by saying that undergraduates don't know enough in general
> to be able to say what they will or won't need to learn to be able to deal
> with their later professions. That is why faculty (older, more experienced,
> more educated) determine courses of study and not the students.
I'm not an undergraduate, if that's what you're suggesting. I love calculus. I
liked taking it, I like teaching it, I like doing integrals in my spare time. But
I still think most of the rationalizations we give for why we need to teach
calculus, above all other math subjects, are contrived, and it would be nice to
re-evaluate our reasons.
--Steve
Benjamin P. Carter
world implies some understanding of the historical background of that
technology. Much of modern technology, unlike ancient technology, is
closely related to science. It follows that an educated person must know
something about the history of modern science. Central to this history is
the development of astronomy, physics, and mathematics from Copernicus to
Newton. You don't absolutely have to know calculus in order to study this
history, but it helps. For one thing, by studying calculus, a student
will finally master such prerequisites as high-school algebra, geometry,
trigonometry, and analytic geometry (in the 20th century meaning as
opposed to what DesCartes meant by analytic geometry). These intellectual
by-products, more than calculus itself, will help the humanities students
understand the related astronomical developments of the 16th and 17th
centuries.
--
Ben Carter
Virgil
Ken.P...@vuw.ac.nz (Ken Pledger) wrote:
> But I'd go much further. Why teach _mathematics_ to everyone?
>Before the whole news group rises up in righteous indignation, let me
>qualify that a little! I can see that the everyday lives of most citizens
>make frequent use of basic arithmetic, so it makes good sense for children
>in the first few years of schooling to learn about numbers and their use.
>But what happens after that, say in high schools? My experience is mainly
>in New Zealand, so people from other countries may like to compare these
>remarks with what they know of their own school systems.
>
One of the problems with the current school format of teaching mathamatics
is that the mathamatics is isolated from everything else.
Most areas of study today use a little mathematics, and many of them use
quite a bit, but somehow all that mathematics gets bypassed in learning
about the areas where the mathematics naturally occurs. The mathematics is
segregated into courses in which its uses are ignored in favor of its
mechamics.
This is great for those few who enjoy the mechanics, as I do, but is rough
on everyone else.
The universal plaints of those not interested in mathematics are "What
good is it?" and "I'll never use it".
I predict that the problem of what mathematics to require,and of what
students to reuire it, will remain a largely unsolved problem until
mathematics is largely integrated with other studies.
--
Virgil
vm...@frii.com
Ken Pledger
> I'm a grad student at a state university where most of the students are
> life science or liberal arts or social science majors. Many of these
> people are required to take a 4-semester sequence of calculus (there's a
> faster version, and there's other ways, but this is the one most people
> choose).
> As one who teaches these courses, I get asked why people are required to
> take calculus. And although I can give a convincing answer (teaches
> logical problem-solving, etc.), I'm not really convinced myself....
>
> My proposal would be, instead of teaching calculus, teach instead
> "Calculation", which would be composed, say, of a variety of fields of
> math, not just calculus....
>
> What are your opinions? Thanks for advice or thoughts.
>
> --Steve Preston
I whole-heartedly go along with your doubts. Any education system
can easily get into a rut, and some habit of thought such as "everyone
needs calculus" can become accepted without question.
But I'd go much further. Why teach _mathematics_ to everyone?
Before the whole news group rises up in righteous indignation, let me
qualify that a little! I can see that the everyday lives of most citizens
make frequent use of basic arithmetic, so it makes good sense for children
in the first few years of schooling to learn about numbers and their use.
But what happens after that, say in high schools? My experience is mainly
in New Zealand, so people from other countries may like to compare these
remarks with what they know of their own school systems.
Our schools have a severe shortage of adequate mathematics teachers.
This is not new, it does not depend on the economy, and it does not depend
on syllabus changes: there has been a shortage for at least 40 years, and
perhaps much longer. Each generation seems to produce just a certain
number of good mathematics teachers, not nearly enough to go round. Yet
most of our teenagers are compelled to take the subject whether they like
it or not. So the schools are cursed with many mathematics classes in
which a teacher of something else (e.g. geography) is unhappily planted in
front of reluctant pupils, so they can all learn to hate mathematics
together.
This dismal regime is imposed with the best of intentions, in the
belief that everyone needs mathematics, but in practice how can it achieve
anything but misery? It's no use demanding something we can't have. A
good mathematics teacher for every teenager is something we can't have.
Why not face it? If mathematics were an _optional_ subject in high
schools and beyond, many people would be spared useless suffering, and
there would be more chance of good teaching and good learning for those
interested and willing to benefit.
You may ask, "What about people who give up mathematics early, then
later find that they need it?" They would need catch-up courses just as
we have now. The big difference would be that instead of being immunized
against mathematics and approaching it with loathing, they could come to
it fresh. I suspect that many of them might cope a lot better.
O.K. - now tear me to shreds! :-)
Ken Pledger.
Nico Benschop
Stephen Preston <pre...@math.sunysb.edu> wondered:
[..on the problem of teaching Calculus in a relevant & exciting way..]
My proposal would be,
instead of teaching calculus, teach instead "Calculation", ..(#)
which would be composed, say, of a variety of fields of math,
not just calculus. E.g. discrete probability and combinatorics,
difference equations, geometry, number theory, plus light calculus and
other such things. (Not really sure what the ideal syllabus would be.)
Unlike the reform texts, you'd still focus on algebra, analysis, not
using computers as a crutch, and doing lots of calculations by hand.
But you still wouldn't need to spend so much time teaching techniques
of calculus.
I think the value would be that students would have a higher interest,
since you could restrict yourself to real-world problems instead of
contrived problems or physics problems. You could also teach most of
the things in a historical context (e.g. this technique was invented
tosolve this problem), which might appeal more to the liberal arts
students. Plus many of them would be better able to use what they
learned in their future courses, or even in real life.
What are your opinions? Thanks for advice or thoughts.-- Steve Preston
-------------------------------------------------------------------**]
Re(#): Good idea. Why not do a bit more of discrete modeling, say
the finite state model of any sequential behaviour with memory:
("internal state" concept, Mealy, 1956 -- also psychologically!).
This allows to treat, in a simple and intuitive way, all associative
algebra concepts, as well as tied-in with math historic development:
1. Such as logic (Boole's idempotent brach of arithmetic: read his
"The Laws of Thought" 1854, available as Dover publ. pocket - the
very best as teaching material, I tell you!)
= The birth of Set Theory, mid last century.
2. And arithmetic: commutative (as Boolean algebra!) but allowing the
concept of "iteration" (= counting: Peano's naturals N = {+1}* and
the old Greek's induction method).
Take Gauss(1801) residue arithmetic n= c.m +r (modulus m, carry c,
rest r < m) preserving the known arithmetic laws: assoc've, cmt've,
distribution (^) over (*) over (+), but NOT (^) over (+) --> Pascal
Triangle --> Fermat & Pascal (1640) --> the birth of Statistics:
Gauss' Normal distribution in the Pascal triangle: (a+b)^n
And consider Z(.) mod 10: the table of 10 for one digit!
Look at it as a Closure, with a minimal set of Generators {2,7,5}*
with drawn structure (VISUALIZE!!!!): 2* = {2,4,8,6} = {2,4,-2,-4}
7* = {7,9,3,1} = {-3,-1,3,1}
Integerated as one "closed system":
7 --> 9 --> 3 --> 1
/ \
/ \
2 --> 4 --> 8 --> 6 5
\ /
\ /
0
With idempotents 1,0,5,6 <--> 1.1=1, 0.0=0, 5.5=5, 6.6=6, ordered.
Bringing you back to Boole: a 2^2 Boolean Lattice of ordered idts.
Algebraic structure of the primary school, as it were.
What more do you want?? Arithmetic 'structure' discovered by
Fermat(ca. 1640):
Fermat's Small Thm (FST, : n^p = n mod p (prime p)
with p-1 cyclic structure, say p=5: Z_5(.) = 2* = {2,4,3,1}
2 --> 4 --> 3 --> 1
|
0
and it's connection to that infamous FLT: x^p + y^p <> z^p
(see http://www.iae.nl/users/benschop/ferm.htm
-- Could Fermat have found the cubic roots of Unity mod 7^2 ?-)
"On Fermat's marginal note: a suggestion"
http://www.iae.nl/users/benschop/marg-flt.dvi )
3. Function compostion f(g(x)) <> g(f(x)): associative only.
The basis of state machines: networks of smaller FSM's,
with the 5 basic FSM's as fundament (basic = minimal = no
subclosure: the 5 FSM with order 2 semigroup: <--> "isomorphism"
as crucial concept of comparing "same-structure"
like Z10 = Z2 x Z5 where Z5 occurs twice as subclosure in Z10
(see above): so 7* amd 2* in Z10 are isomorphic, both as Z5 in
separation.
Etc, etc -- All very trivial, *much* simpler and *much* more
instructive & exciting (include history!) than calculus as it
is normally taught.... don't you think? However, minor drawback:
there may not be a book on this integrated view of early & basic
discrete math ;-( But the material can be found & collected in
the first chapters of the many disciplines within mathematics..;-)
Ciao, Nico benschop (email\X) -- http://www.iae.nl/users/benschop
http://www.iae.nl/users/work.htm for a Research program frame:
The triple sandwich of 3 main & practical Algebras / Objects:
functions > numbers > sets
associative > commutative > idempotent (composition syntax)
sequencing > iteration > combination () > (+,*) > (&,|)