Calculus is lucrative business. Calculus is made to gold in many industries. Today!
Geography (google earth), computer vision (autonomous driving of cars),
Photography (panoramas), artificial intelligence (optical character recognition),
robots (mars bots), computer games (world of warcraft),
Movies (avatar), network visualization (social networks).
We live in a time where calculus is made to gold.
If linear algebra is added to calculus (page rank example), then applications
get even bigger. There is a gold rush on calculus. It currently is
responsible for billions of profit.
Why calculus is a prototype
Calculus helped to understand of what we are and to plan where we go.
No other field of mathematics is so rich in history and
culture than calculus. From fundamental geometry like Pythagoras as part of vector calculus,
measuring volumes with ideas of Archimedes, to deal with
velocities and forces which was essential in the development of astronomy and to answer questions
about our place in the universe. The motion of planets, the development of galaxies and phenomena like
black holes all need calculus and the quest to understand our role in the universe has led to calculus.
Calculus could be essential for our survival since we need to develop and understand
climate or population growth models, spread of diseases or mechanisms to resolve conflicts or
deal with economic and financial crisis.
Curriculum developers: Teach it yourself first and look whether it is teachable.
There are other subjects which are important. Some of them are more difficult to teach.
Probability theory is an example where the subject can only be taught well if some
mathematical maturity is reached in which calculus plays an important part. It needs also
mathematical maturity from the teacher.
A core problem in curriculum development is that pioneers
developing the material are often not teaching it themselves or do not see the
difficulties a typical teacher has.
Curriculum developers are by nature optimistic and want things to work. For calculus, the
task is relatively easy. No other field of mathematics is so well developed pedagogically.
Math 19b, linear algebra and
probability theory is a wonderful course. Because there is not yet much culture yet
in teaching linear algebra and probability together, it has been more difficult to teach.
What should replace calculus?
Calculus is inherent in every other subject, even discrete structures.
Discrete mathematics comes in mind. But calculus is already inherent in
discrete mathematics. Combinatorics, set theory or graph theory are usually core elements in a discrete
math course. Unfortunately these are rather different beasts. Combinatorics belongs
to probability theory, set theory to the foundations of mathematics and graph theory
to topology. Newer models of calculus see discrete structures as special cases of a more
general calculus.
We need to continue to look out for modifications and alternatives and also for new ways to teach
the subject.
For example, I believe that
quantum calculus will become more and more
main stream in the next 100 years. I have experimented with some of these ideas in my
single variable
calculus course of spring 2011.
Is it needed in life?
Calculus develops the ability to think and solve problems.
Is the ability to move fast necessary? In fact, we do no more
need to hunt our own food any more and the ability to run fast is not essential anymore.
Still, all sports have walking components as part of the
exercise plan, for swimmers, rowers, boxers, tennis players or car racers.
Running in particular is a simple sport, you only need minimal equipment
and can be done anywhere. In principle, it can be replaced by biking, swimming, golf, or
walking. But not everything works universally. I myself would love to climb mountains and fly down
with a paraglider (as I did in graduate school) but it needs money and the right geography.
I can do a workout in 30 minutes which includes walking to the gym and showering
doing something for 10 minutes can already be effective.
Like walking or running stay part of the athletic mix, calculus will always be part of mathematics education.
Calculus has proven to help in any other field, like graph theory, game theory or statistical or
data visualization.
Here is a project
where calculus and topology ideas enter discrete mathematics.
Here is a readers note (March 9, 2016):
"Your short article on why we teach calculus is marred with flaws. Why
should we continue to teach something just because it has a long tradition?
And not having a thing to replace it is your most valid excuse? Also, since
when is calculus a part of everyday culture? It sounds like you just want a
reason to defend your profession."
Here is my response:
Its a valid point. As a teacher one is of course
biased and in general, everybody overestimates
and naturally hypes his or her own work or profession.
There is a general principle however: if one
wants to replace something, one has to constructively
build an alternative which works and demonstrate
that it can work on a large scale. Just calling for
a replacement is cheap. In the case of calculus, it
is not only the results which have an excellent
track record (major industries mine calculus today),
but also calculus as a "tool to sharpen thinking and
problem solving skills" and prepare for other fields.
As replacement of calculus, both discrete math or statistics
comes in mind. I myself know that both need solid calculus
skills to be used effectively. Whoever calls for using stats as
a replacement of calculus does not know stats. Whoever calls
for discrete math as a replacement does not understand
discrete math. Both fields really only shine if one knows
calculus. I love discrete math,
(work on it) and statistics
(example [PDF])
and even discrete or alternate versions of calculus like
here.
Unfortunately, in many of todays implementations of discrete math or
statistics, the replacement is used as an alternative, the
implementation is an excuse to stop practicing harder problems or
acquire more sophisticated problem solving skills
or to learn harder subjects. Many discrete math courses are a race to the
bottom, the reason being the lack of a clear goal to reach. Calculus has
the fortune to have a clear goal: the fundamental theorem of calculus (both
in single and multivariable calculus), as well as established
levels of sophistication like integration skills, knowledge about series and
the ability to solve differential equations. Yes, these skills can be hard to
reach, but it is worth it. If acquired, the usual discrete math or stats
curricula are more rewarding.
Maybe we have to look at history also to see what worked
and where things were successful. The biggest advancements in
discrete math, physics, statistics,philosophy or computer science were done by people
who knew calculus well: Euler invented graph theory and was a master in
calculus, Leibniz invented determinants and a computing device and a master in
calculus, Newton figured out the laws of gravity, and was a
master in calculus, Kepler figured out the laws with which the
planets move and was a master in calculus, von Neumann invented
modern computers, game theory and was a master in calculus,
Archimedes invented countless of machines and was a master in calculus. Kolmogorov
wrote the first textbook in probability and was the first put the subject
on a solid foundation and was a master in calculus. Riemann dove into
the deep mysteries of the prime numbers and was a master of calculus.