Allowing Unexpected: Creativity in Your Classroom

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.

  What does this quote tell us about the teaching mathematics? What is the purpose of separating front from back, kitchen from dining hall? It is not only to keep customers from interfering with the cooking. It is also to keep them from knowing too much about cooking.

The front and the back of mathematics are not physical locations like dining room and the kitchen. The front is mathematics in its finished form - lectures, textbooks, journals. The back is mathematics among working mathematicians. Which mathematics we are teaching to our students? Of course, the front one. Why? Because we feel safe there. We are teaching facts which are a priori acknowledged and if that happened more than 100 years ago then it is even safer. Looking over mathematics curriculum you are teaching have you pondered about the question which is the newest mathematics we are teaching? Are we telling our students that mathematics is changing all the time even that it is one of the oldest human activities?

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.

As Tomass Edison once said – I made a lot of mistakes. Later I patented most of them.

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 talk
 Changing 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

I found Ueno Masao!

I was looking for something else but stumbled on intriguing picture.

It prompted me to look up who is the artist and I found another picture:

He  called this "Rotating Ellipses" but of course I noticed hyperboloid!
This is how I found Fabuluos works by Ueno Masao like this Spring:

Japanese bamboo artist Ueno Masao (b. 1949) studied architecture in Shibaura Institute of Technology (class of 1972). From 1977 to 1983 studied traditional bamboo art and worked as apprentice with master craftsmen Konma Hazuaki and Kondou Shousaku.

How bamboo is processed for his works



Ueno Masao Swirl

Making Swirl

An Interview with Ueno Masao

Braiding by Ueno Masao

Earlier Works

His works are indeed The Feast for the Eyes:

Secrets of Mental Math

 The Kieval Lecture Series (Cornell Mathematics Department) is funded through a bequest of the late Dr. Harry S. Kieval ’36, a longtime professor of mathematics at Humboldt State University in Arcata, California, who died in 1994.
Innaugural speaker in 1998 was John Milnor. The full list of speakers is here.

Today in 2013 Kieval lecture series Arthur Benjamin (Harvey Mudd) was speaking about Secrets of Mental Math.

Arthur Benjamin grew up in Cleveland, Ohio, and earned his B.S. at Carnegie Mellon University in 1983 and his Ph.D. in mathematical sciences at Johns Hopkins University in 1989. Since then he has been a professor of mathematics at Harvey Mudd College, in Claremont, California, where he has served as department chair. He has written three books and is co-editor of Math Horizonsmagazine, published by the Mathematical Association of America (MAA). In 2000, the MAA awarded him the Haimo Prize for Distinguished Teaching.
Arthur Benjamin is also a professional magician, and frequently performs at the Magic Castle in Hollywood. He has demonstrated and explained his calculating talents to audiences all over the world and has appeared on numerous television and radio programs, including The Today Show, CNN, and National Public Radio. He has been featured in Scientific American, Omni, Discover, People, Esquire, The New York Times, Los Angeles Times, and Reader's Digest. In 2005, Reader's Digest called him "America's Best Math Whiz."

While he clearly has an extraordinary gift in the realm of mathematics, Benjamin also considers a lot of his numerical talent (and prodigious talent in general) to be gained through hours of practice. He describes the some practical drills and techniques for left to right calculation. He also explores visual, auditory and mental imagery that can assist in holding and manipulating large numbers in your head. Certain tricks work better for some than others, depending on individual traits and learning styles.
Secrets of Mental Math are revealed to readers, hours of practice now should be done.

 TED talk: Arthur Benjamin does Mathmagic

 TED talk:    Teach Statistics before Calculus

Arthur Benjamin on Colbert Report

How to memorize Large Numbers

Bache Auditorium today was full with excited audience ages 3 to ... well, let us say, very mature (it means including professors-emeritus).

 

 

First calculators had to be checked, so four volunteers came forward. You may recognize that one of them is Steven Strogatz.

 

As expected, these four guys could not use their calculators as fast as Arthur Benjamin used his mental calculation skills.

 



One of audiences most liked tricks was creating "personalized" magic square - based on your birthdate. Since this was mostly "mathematical" audience one of the questions was - how did you do this? Arthur said that explaining magic tricks takes away that wow! moment but agreed to show how this trick works. Of course, the following question was - how did you come up with the idea of this trick? The answer was - very much like it happens with mathemathematics: you read about something, then start thinking how this can be used in other ways, then some new idea may appear. He said that he was reading some publication on magic tricks, then he decided to try to make these magic squares using person's birthdate and figured out how he could perform it. Important part of magic tricks is the performance. Magicians have to be also actors and psychologists.

Arthur Benjamin also explained how he has memorized digits of pi - this time paper sheet could hold only 60 of them but he can go up to 100.
How could this mental math can be useful? Arthur told us that he got interested in numbers after he learned as a child to multiply three digit numbers by two digit numbers. He figured out that there are many different ways how one can do it and amazingly - result is the same! This fact made him to be excited about mathematics and up to this day he always has the urge to try different ways how to prove or solve something. (Earlier today he gave a talk in Math Department how to prove trigonometric formulas with combinatorial tools. He call it "combinatorial proof" or shortly in Harvey Mudd it is called "cool proof".)
As the "grand finale" he sang us a song dedicated to "pi day" which will be in two days. Melody was borrowed but rhyms were original and cool.
It was fun day with Arthur Benjamin!

Math in Movies

I tried to take a picture of Toni deRose when he was giving a talk but it did not work. So here is a snapshot of him already afterwards when we all enjoyed some refreshments.On March 6th MoMath organized Math Encounters celebrated two year anniversary and for the very first time were hosted in MoMath own place. It was windy and cold outside, so while waiting for prompt 3:45pm opening of the door for the talk Math in Movies by Toni deRose we were let inside MoMath store. This was the first time I was lucky my visit to NYC coincided with Math Encounters, so I was as promptly there as was suggested by e-mail from the organizers. As it happens when things are set somewhere for the first time - prompt opening time was a little bit late and the talk started late. It had an advantage of catching-up with some people I hadn't seen for some time.
While I was thinking of writing about this event Tim Carmody already did - read it here.

Pixar's policy is to use only in-house stories, they do not accept stories from outside the studio. I found interesting Pixar's 22 rules of story telling. Number 9 on the list - When you’re stuck, make a list of what wouldn’t happen next – is a great one and can apply to writers in all genres.

The road from the story to the film is four year's long and takes about 150 thousand sketches. Technology through the years has improved and actually these days anybody can make an animated movie using free software Blender. As Toni deRose said - we are just waiting when some kid will make a great movie, so we can hire him/her before we are put out of business.

Mostly the math behind the animations is coordinate geometry - positions of key points of movie characters are described by coordinates and then software just translates (addition, subtraction), scales (multiplication), rotates (trigonometry) these coordinates. More interesting math is used in designing the characters - animators use subdivision surfaces, which actually has developed themselves into interesting research field.

More about how Pixar studio is using math:
An Interview with Toni deRose in 2009 and this year in January before his invited address for Joint Math Meetings 2013.

Some of previous Math Encounters can be seen on YouTube:

Keith Devlin on Golden Ratio

Colin Wright on mathematics of juggling

Jeff Weeks on Shape of Space

Eric Demaine on Geometry of Origami

Carlo Sequin on topological sculptures

Some other  Math in Movies:

25:5=14

7x13=28

Math - Bunker Style

Good Will Hunting - math problem

The best list of Math in Movies is by Oliver Knill (Dept.of Mathematics, Harvard)

There was not much time (always not enough time in NYC!) to find some art exhibit connected with math but the one on my list was the exhibit in  Metropolitan Museum Cambodian Rattan: The Sculptures by Sopheap Pich. My favorite was Morning Glory in which can be seen pseudosphere continuing in ruffles. Since it is exhibit, visitors are not allowed to take pictures, so I could not take a picture to show it. More images of Sopheap Pich sculptures are on his webpage.