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Mr. McNealy, the fiery co-founder and former chief executive of Sun Microsystems, shuns basic math textbooks as bloated monstrosities: their price keeps rising while the core information inside of them stays the same. - More

Dan Meyer’s blog.

Interactive web page that illustrates domain, range, and inverse functions.

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With Winplot one may first graph, as an example, y=2x-3. Then draw its graph reflected in y=x and the implicit x=2y-3 where x and y has swapped places. The latter two will always be identical.

Winplot is a free graph plotter for Windows made by Richard Parris, Phillips Exeter Academy.

I have just started to scratch the surface  its possibilities, but what I have found so far is excellent. The PageUp and PageDown keys do the zooming and the arrow keys move the center of the graph. A right-click changes the center to where you clicked.

I use it with a projector in my class. Gone are the days where infinity had to be spent for students or teacher to produce a graph of dubious accuracy. (Needless to say, sometimes a sketch is to be preferred if one is analysing the general properties of a function.)

Mathematics can be used to predict things if you spot a pattern in one form or another. Sometimes patterns are best described as algorithms.

For 29 years, Bueno de Mesquita has been developing and honing a computer model that predicts the outcome of any situation in which parties can be described as trying to persuade or coerce one another.  - Read more

He charges 50,000USD per prediction and has done thousands. If you know what 50,000 x 1,000 is you may like to take up game theory and programming.

Can a fringe branch of mathematics forecast the future? A special adviser to the CIA, Fortune 500 companies, and the U.S. Department of Defense certainly thinks so. – Read more

The other day, via blog Edumate Perú, I discovered the dy/dan blog. There I found a philosophy of teaching math similar to mine. As an example, I fell for the post What I Would Do With This: Glassware.

Reading the comments to the post I noticed #38:

I think that Alan Kay and Kim Rose’s car game is similar. I thought a rolling glass was a bit boring, but it is a surprisingly rich vein for investigation.

I highlighted the part in bold and landed at the book Powerful Ideas in the Classroom Using Squeak to Enhance Math and Science.

I read the preface, downloaded the free program called Squeak and have decided to include it in my elective Programming course this fall. Above is my code for leading the car to the goal following the blue trail.

I have used Logo in the past, and have stumbled upon Squeak before, but never tried it out. It looks quite appealing.

In my browsing I also found the site http://waveplace.com/ which gives away OLPC computers in the Caribbean and use the Squeak software.

“We never learn to think for ourselves! The teachers don’t like to admit it, but I also have a brain. They always try to fill my brain with all kind of garbage, but I am not a garbage dump. It drives me mad!” Reaction from a student. (Matthew Lipman, page 24.)

Albert Einstein put it like this: “It is really no less than a miracle that the modern ways of teaching have not completely killed the inquirer’s sacred curiosity, since this fragile plant needs, in addition to stimuli, above all freedom; without freedom it will without fail be lost.” (Carl Rogers, page 4).

The school has always been more concerned with what the students should think than how they should think. The first can lead to indoctrination or brainwashing, while the second can lead to freedom. A freedom which is easier to praise in a graduation speech than to make happen in the classroom. It is especially difficult if one does not try.

“Indoctrination. What is that?” asks Jørgen Bruun Pedersen in an exciting, little book. He writes: ‘The communication that takes place between the indoctrinator and the student intends apparently to resolve a problem or seek the truth. However, the fact is that the indoctrinator has decided beforehand which opinion the student shall reach, and all his actions have the sole purpose of bringing forward the desired state of mind in the student. The crime must be considered as especially severe if it is carried out by a person that the society has given the mandate to educate children and young people.”

While in Denmark, we should also listen to these words of wisdom from Søren Kierkegaard: “The secret of communication is to liberate the other. Therefore one should not communicate in a straight forward fashion, yes it is even ungodly to do so.”

Inspired by Søren Kierkegaard and Karl Popper’s principle of falsification, I like to suggest this strategy: “At all times, try to prove to yourself that the students have not understood.” Looking for confirmations of your belief is a futile exercise as a million supporting cases do not constitute a proof. Looking for observations that do not support your belief is more exciting as one of them is enough to prove you wrong.

Thomas J Cooney gives an example of the use of this strategy in “The Issue of Reform: What Have We Learned From Yesteryear?” Cooney had carefully taught his students that if two chords intersect in a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the second chord.

He then asked the students: Which chord through point P in circle O will give the largest product of the lengths of the two segments? Surprising as it may sound, the students did not have a clue to what the answer was. Cooney then let his students study special cases until they reached a higher plateau of understanding. A plateu they would not have reached if he had not tried to shoot down his belief that they had understood in the first place.

John Locke said it a long time ago, “The child’s mind is a fire to be ignited, not a pot to be filled.”

References:

• Matthew Lipman. Harry Stottlemeier’s Discovery. Montclair State College: The Institute for the Advancement of Philosophy for Children, 1977.
• A good site about Søren Kierkegaard is Anthony Storm’s: http://sorenkierkegaard.org
• Carl Rogers. Freedom to Learn.
• Jørgen Brun Pedersen. Indoktrinering. Hvad er det for noget? En debatbog. (In Danish) Copenhagen: Gyldendalske Boghandel, Nordisk Forlag AS, 1976.
• Thomas J Cooney. “The Issue of Reform: What Have We Learned From Yesteryear?” Mathematics Teacher 81, no. 5 (1988): 352-363.

As explained in About Mumnet was a site I started ten years ago. Today starts the move of the content to this blog. The move will be a slow one and the original pages will still be available at their old urls for the time being.

Needless to say new content will be added. As with my other web sites its main purpose is to serve as external memory. My brain is equipped with 1KB and is leaking.