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Sapare aude! (Dare to know)
Immanuel Kant

We must know.
We will know.

David Hilbert Radio Address

The art of teaching is the art of assisting discovery. Mark Van Doren

Alfred North Whitehead, The Aims of Education:
“But what is the point of teaching a child to solve a quadratic equation?”


Teaching mathematics through a problem solving approach:

Math is an imaginative endeavor; it goes beyond the symbols, but every symbol has a meaning. The teaching of mathematics is the teaching of abstract ideas through reasoning, logical arguments, meaningful symbols. Iit is the only pure language we learn at school. Math is the signature of critical thinking.

A math teacher?  A critical thinker ! A problem solver. There is a book out there that tells the story of problem solving as nobody has done yet: Pólya: How to solve it. The preface to the first edition reads:
(…) a teacher of mathematics has a great opportunity. lf he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent  thinking.” Link to the full text here.

By most accounts, math education is in crisis. Here and there, everywhere. Also in the past, it was always the same picture: students hate the subject; teachers don't get how to teach these abstractions, symbols, connections. Today the situation is different from yesterday's story: now we need more mathematical knowledge, we are more technological, society demands this day a better instruction on this subject. Schools do not deliver however.

A very expressive document on this topic --today's math education in America- was written by Paul Lockhart: A Mathematician's Lament. According to MAA (Math Association of America) Mr. Lockhart is a mathematics teacher at Saint Ann's School in Brooklyn, New York. He earned a Ph.D. from Columbia in 1990, taught mathematics at University of California, Los Angeles and then decided to teach at high school level. Follow the link to read the article: A Mathematician's Lament.

Mr. Tom Davis, a mathematician, has also interesting suggestions to improve the mathematics curriculum in secondary schools: A Better Mathematics curriculum.

Here a brief but compelling proposal about Math Curriculum by Professor Arthur Benjamin: TED.COM Arthur Benjamin's formula for changing math education.

The teaching of Mathematics and Math Foundations: a series of lectures discussing the very basic concepts of Mathematics that affect the teaching of the subject in the secondary school, given by Mr. N J Wildberger, Associate Professor in mathematics at UNSW (University of New South Wales) in Sydney Australia.
Insights into mathematics, njwildberger's Channel: Math Foundations.

Last but not least, let's read Alfred North Whitehead, The Aims of Education: “But what is the point of teaching a child to solve a quadratic equation?”
Full text by Stanford university site:http://edf.stanford.edu/sites/default/files/whitehead.pdf


Culture is activity of thought, and receptive ness to beauty and humane feeling. Scraps of information have nothing to do with it. A merely well-informed man is the most useless bore on God's earth.
In training a child to activity of thought, above all things we must beware of what I will call "inert ideas" -- that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh  combinations. 
 In the history of education, the most striking phenomenon is that schools of learning, which at one epoch are alive with a ferment of genius, in a succeeding generation exhibit merely  pedantry and routine. The reason is, that they are overladen with inert ideas. Education with inert ideas is not only useless:  it is, above all things, harmful -- Corruptio optimi, pessima.
Education is the acquisition of the art of the utilization of knowledge. This is an art very difficult to impart. Whenever a textbook is written of real educational worth, you may be quite certain that some reviewer will say that it will be difficult to teach from it.
I appeal to you, as practical teachers. With good discipline, it is always possible to pump into the minds of a class a certain quantity of inert knowledge. You take a text-book and make them learn it. So far, so good. The child then knows how to solve a quadratic equation. But what is the point of teaching a child to solve a quadratic equation?
The difficulty is just this: the apprehension of general ideas, intellectual habits of mind, and pleasurable interest in mental achievement can be evoked by no form of words, however accurately adjusted. All practical teachers know that education is a patient process of the mastery of details, minute by minute, hour by hour, day by day. There is no royal road to learning through an airy path of brilliant generalizations. There is a proverb about the difficulty of seeing the wood because of the trees. That difficulty is exactly the point which I am enforcing. The problem of education is to make the pupil see the wood by means of the trees.

Alfred North Whitehead


PISA web site: http://www.pisa.oecd.org/pages/0,2987,en_32252351_32235731_1_1_1_1_1,00.html

Results presented and discussed by Bruce A. Fuchs, Ph.D.
Director of the NIH Office of Science Education

Part 1

Part 2


A Tool Kit of Dynamics Activities

Introducing  the  students to iteration, fractal geometry, linearization, and chaos with this accessible, engaging series: Chaos, Fractals, Iterations and Julia Sets –Middle/High school level  By  Mathematician Robert L. Devaney. Texts available at http://www.keypress.com/x8192.xml and amazon.com; also available at amazon.com the video lecture: Chaos, Fractals and Dynamics: Computer Experiments in Mathematics by Robert L. Devaney

Paradoxes and mathematics

Math instructors have used paradoxes’ unique intellectual appeal to introduce complex ideas and concepts in class.

Since ancient times men amuse themselves with paradoxes. A paradox, said flatly, consists of a statement that implies an insoluble contradiction.
In Mathematics paradoxes defy the very core of the science, its consistency. This is, the statement S and the statement non-S cannot be both true
in any logical system. Most paradoxes are still discussed today. They all have a consistent explanation within mathematics.  

Some of the most known and recurrent paradoxes are:

1.- Zeno’s paradoxes of motion: The Dichotomy, Achilles and the Tortoise and The Arrow.

Zeno’s paradoxes are related to the following concepts: Infinity, infinitesimal, series (as an infinity sum of terms) and the notions of continuous and discrete space. For a detailed description of the three paradoxes you may search the Stanford Encyclopedia of Philosophy and Philpapers.org, Philosophy of Mathematics. Most Calculus textbooks make reference to Zeno's paradoxes.

2.- Galileo’s paradox of infinity, offers an opportunity to introduce Cantor’s transfinite numbers.

3.- Russell Paradox: related to set theory, infinite sets, and self-reference sets.

4.- The Hilbert Hotel-Series related to infinite sets:

Two Geomerty Textbooks:

Nothing is harder than finding a great Geometry textbook. Of course, there is Euclid’s Elements, a nice recommendation to be on the teacher’s desk,
but it is not proper for a middle or high school student. By most accounts, some of the best options available are:

1. Kiselev's Geometry,  Book I. Planimetry by A. Kiselev. Adapted from Russian by Alexander Givental.

Amazon.com’s Product Description:

This is an English translation of a classical Russian grade school-level text in plane Euclidean geometry. The book dominated in Russian math education for several decades, was reprinted in dozens of millions of copies, influenced geometry education in Eastern Europe and China, and is still active as a textbook for 7-9 grades. The book is adapted to the modern US curricula by a professor of mathematics from UC Berkeley.

2. Lessons in Geometry by Jacques Hadamard

Amazon.com’s Product Description:

This is a book in the tradition of Euclidean synthetic geometry written by one of the twentieth century's great mathematicians. The original audience was pre-college teachers, but it is useful as well to gifted high school students and college students, in particular, to mathematics majors interested in geometry from a more advanced standpoint. The material also includes introductions to several advanced topics. Much of the value of the book lies in the problems, whose solutions open worlds to the engaged reader. And so this book is in the Socratic tradition, as well as the Euclidean, in that it demands of the reader both engagement and interaction.

Brain at Work !

Taxi Drivers' Brains Grow to Navigate London's Streets.
Memorizing 25,000 city streets balloons the hippocampus, but cabbies may pay a hidden fare in cognitive
skills,  By Ferris Jabr At Scientific American

National Geographic Video: London Taxi Drivers' Brains