These are some excerpts from my talk today in Western Illinois University annual Teachers conference and also links to some aditional material.

In an ordinary mathematics class, the program is
fairly clear cut. We have problems to solve, or a method of calculation to
explain, or a theorem to prove. The main work to be done will be in writing,
usually on the blackboard. If the problems are solved, the theorems proved, or
the calculations completed, then teacher and class know that they have
completed daily task. Is this teaching how to think mathematically? Getting to
know new mathematical facts and there applications - it is creating new
knowledge in students heads - but is it creative thinking?

Everybody at least once has had an
experience when you have to talk in a language which is not native for you and
it is hard to find correct words to express what are you thinking. Mathematics
also has its own language (may be languages?) and we have to listen to another
person carefully to understand what he/she wants to say. Are we patient enough with
our students? How do we help students to express their ideas in a
"foreign" language? We can understand a person talking even with
mistakes if we understand the ideas the person is talking about. And no damage
is done to these ideas because of some grammar mistakes. Why do we want
mathematics to be an exception? Why is formal language so sacred?

Here
is a quote from George Orwell:

*It is instructive sight to see a waiter going into a hotel dining room. As he passes the door a sudden change comes over him. The set of his shoulders alters; all the dirt and hurry and irritation have dropped off in an instant. He glides over the carpet, with a solemn, priest-like air... he entered the dining room and sailed across it, dish in hand, graceful as a swan.*

Keith Devlin in

*American Scientist*compares learning mathematics to learning how to play piano:

Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society. In both music and mathematics, the symbols are merely static representations on a flat surface of dynamic mental processes. Just as the trained musician can look at a musical score and hear the music come alive in her or his head, so too the trained mathematician can look at a page of symbolic mathematics and have that mathematics come alive in the mind. So why is it that many people believe mathematics itself is symbolic manipulation? And if the answer is that it results from our classroom experiences, why is mathematics taught that way? I can answer that second question. We teach mathematics symbolically because, for many centuries, symbolic representation has been the most effective way to record mathematics and pass on mathematical knowledge to others.

A necessary (though certainly not sufficient) condition for significant teaching is the provision of emphases; if everything is important then nothing is important. - Abe Schenitzer

1964 book

*The Act of Creation*ArthurKoestler attempted to develop the general theory of human creativity. His concept of*bisociation*has been adopted, generalized and formalized by cognitive linguists Gilles Fauconnier and Mark Turner, who developed it into conceptual blending. Koestler defined*bisociation*as “the creative leap [or insight], which connects previously unconnected frames of reference and makes us experience reality on several planes at once.” How to realize it? Koestler offered a suggestion in the form of a triptych, which consists of three panels…indicating 3 domains of creativity which shade into each other without sharp boundaries: Humor, Discovery, and Art.
The first is
intended to make us laugh, the second make us understand, the third make us
marvel Or for short: Ha-ha-ha! – Aha!
– Ah!

But there is another word – Oh! – when
things go wrong. If math is to be a creative subject then we have to regard it
as a subject where it is ok to get things WRONG. If you have never made
mistakes, you are never discovering anything new.

Mathematics is not about numbers, equations, computations,
or algorithms: it is about understanding. - William Thurston

If you haven't seen yet then do join almost 10 million viewers of Sir Ken Robīnson's talkChanging Education Paradigms. Another one of his talks is The World We Explore.

Some more interesting talks

Fun to Imagine: http://www.youtube.com/watch?v=dQai9QikTBI

Are Mathematicians Creative? http://www.youtube.com/watch?v=1z1ct7Ru0GE

What Mathematicians Actually Do? http://www.youtube.com/watch?v=aaUvPYPARf8

I Want to Be a Mathematician: http://www.youtube.com/watch?v=ONvYPldXoZs

Andrey Cherkasov Math Jokes collection

Feynman and Computing: http://www.youtube.com/watch?v=9miKIWIYi4w

Mysteries of Mathematical Universe - talk from World Science Festival

Mathmagic with Arthur Benjamin

Sangaku