Dear Melvin,

I am touched to know that you have found my web
site useful and inspiring. I have a few responses from graduate
students from California who have said how much it has helped them in
Mathematical Analysis. My modest aim is to share what I have
found useful, what I have found it difficult to understand and how I have
strived to understand, and from quite a different perspective to provide a
selected window, probably with a somewhat biased emphasis on the
mathematics that I have tried to present as clearly and as logically and not
always in a very polished elegant way (when most of the detail would have been
left out). I have also been in correspondence with a retired gentleman
from Belgian who (since 2000) started to learn calculus upon the retirement at the
age of 75! Good for him!

Yes, mathematics is about finding truth and
sometimes truth is not what it seems just like real life events.

Calculus is particularly difficult because it deals with the real numbers and the the set of real numbers or its many manifestations is not easy to understand. Lots of questions can be asked and many can be directed towards a logical foundation of mathematics. Many structures are built up from the real numbers, first complex numbers, then the quaternions, the diviison algebras, vectors spaces, abstract algebras, and other structures that are built upon theses structures and further structures on these structures. It is not surprising when we have time to ponder about the real numbers and its immediate structured domain, calculus that we find there are still many questions one can ask and try to understand, without ever finding them stale and that each time the question is refreshing even though it may have been known and one's own inner working in arriving at the understanding is as invigorating as life itself.

As now is the Christmas Season, I would like to
share some of my philosophy or what guides me in my teaching and learning and in
life itself. It is to share ones knowledge for the betterment of
human kind and NEVER for the harm or destruction of life in whatever
form. I believe in sharing knowledge for the good of the community
to spread the understanding as wide as possible, to solve problems,
resolve conflicts, eradicate misery, sorrow, hunger, diseases, wanton
hopelessness, right injustices and boldly to reach out to mend the very
fabric of peace and to bring love to this very very troubled world, where human
conflicts abound and meaningless death are very much a human weakness
and more so of world leaders.

And I would also like to share this. Very often acquiring knowledge seems to be an end in itself and now it has
become a competitive tools for advancement and climb up the academic
ladder. (Like the Babel Tower, heaps and heaps of research papers are
written every year, at a modest count at least 30,000 a year.) We
have forgotten the reasons or purposes for acquiring knowledge or have
replaced them by self-absorbed, self-serving ones propelled by
careerism and academic pecking order. It is not how much mathematics
we can create but how well we use the mathematics and it is not always the
mathematics that is on the boundary of what mathematicians consider as frontier;
it is about how well and how we can impart the use of mathematics towards a
better understanding of our world for us to share and to live in.
Mathematics understanding can have its simplest and humblest beginning in the
smallest of steps one takes. The very first steps one takes to
understanding how diseases spread, how to control the spread of cancer cells by
building a mathematical model, the mathematics will present itself in ones
search for solution. The very first steps one thinks of how to bring
employment to a work deprived neighbourhood will bring together the knowledge of
local conditions, existing infra structures, natural resources,
geography, historical, social and cultural background and knowledge of the
people of the neighbourhood and the beginning of a sophisticated
economic/industrial model built from those input and with a very different kind
of mathematics that finds logics, combinatorics, discrete mathematics,
statistics, operations research, business mathematics and hence calculus
and even geometry together in a myriad and probably unstructured
form. Mathematics should beckon you and ask you questions and along
the way the training and the care with which one answers those questions are
solid foundation for the use of mathematics effectively in solving problems
(not necessarily mathematical in nature) and also in mathematics
itself. Mathematics is reflective in nature and undergoes a lot of
self correction very much like human thoughts ( a la Morris Kline).
My advice to students is always this: Be true to yourself; if you do not
understand, seek understanding; ask your teacher, you will probably find
that many fellow students do not understand too ; work in a group if
you can, share your thoughts or understanding or what you find conflicting,
collectively you will probably be able to come to a better understanding
or to a better position to judge how far you have succeeded in
understanding; try to solve the problem or answer the questions yourself;
ask questions and answer your questions yourselves, very often your own
questions are the landmarks of your own understanding; if all else fails, ask
your teachers politely or someone whom you think can explain to you in
detail.

Warmest Regards,

Merry Christmas

God Bless

Tze Beng

Greetings from a fellow mathematics teacher in Billings, Mt. I teach AP Calculus at Billings Central Catholic High School in Billings, Montana

and I am always looking for sites for my students to check out. I found yours last year and was impressed with your philosophy statement, which

read to my class. I just finished your Definite Integral quiz and am proud to say that I scored 5/5!! Not bad for an old calc teacher. It

is amazing how much you continue to learn about mathematics, years after having taken the course work. I just figured out the formula for

finding F'(x) last year and gave it to my students. I used this formula to solve

the last integral on the quiz.

F'(x) = f(g(x))*g'(x) where g(x) is the upper limit.

I'll have to admit that most of your problems are far above my high school students capabilities at this time. I do have some great raw

talent but it will take some time to develop it. I will give them your web address so they can see what a tough calc course would look like.

Keep up the good work. Any ideas you might have for my students please feel free to share.

Sincerely,

Mel Wahl

BCCHS Mathematics Dept.