Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind

September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Friday September 16, 2005
Learning Log Number 11

5:15 am Edmonton, Alberta

Yesterday we drove from Calgary to Edmonton. While in the West Edmonton Mall I bought a few books. Three are worth a brief mention in this journal:

  • Infinite Ascent (2005) by David Berlinski
  • On Intelligence (2004) by Jeff Hawkins
  • The Complete Book of Drawing Techniques (2003) by Peter Stanyer

The Berlinski book has the subtitle "A Short History of Mathematics" and lives up to this. It is a small book, and like his book on a history of the calculus, is delightful to read. There are 10 chapers:

  1. Number
  2. Proof
  3. Analytic Geometry
  4. The Calculus
  5. Complex Numbers
  6. Groups
  7. Non-Euclidean Geometry
  8. Sets
  9. Incompleteness
  10. The Present


This is a superb summary of the important ideas in mathematics. Number is the core concept. The Greeks introduced the idea of proof and Descartes the idea of analytic geometry which brought together algebra and geometry. This led to ideas about the calculus and this led to the idea of imaginary numbers and complex functions. Group theory is a way of bringing together the essential ideas of algebra and non-Euclidean geometry introduces us to Riemann and prime numbers. Set theory is a way of making functions and numbers more precise and incompleteness does the same for logic. Superb!

I have read the first two chapters of Berlinski.


One of the reasons I like the books by Berlinski is that he conveys an aesthetic sense of mathematics. I think this is critical to understanding mathematics. Mathematics is more than reason and cold logic, at an important and deeper level it is about harmony and interconnectedness and beauty. It is about fun.

The early Greeks played with numbers and looked for patterns among them. The only modern books that I have that seem to convey this are The Number Devil by Hans Enzensberger and The Book of Numbers by Conway & Guy. The Greeks also formalized logic at about the same time. Euclid's Elements provided the epitomy of such an approach. They were no longer just drawing figures in the sand.

Here are a few quotes from "Infinite Ascent " by David Berlinski.
"... the Pythagoreans were taken with the inexhaustible variety of the natural numbers, their personalities." [p. 5]

Perfect. Numbers are more than just for counting. They can become your friends.

"... there are square numbers as well as triangular numbers, and amicable relationships between numbers, ..." [p. 5] Lovely! Amicable relationships are the very heart of mathematics, transcending even numbers. Saying this brings to mind the concept of a transcendental number (which is not yet a clear concept in my mind). Unfortunately I am not connected to the Web at the moment and cannot pursue this.
"... the Pythagoreans groping their way toward the remarkable doctrine that the harmony between numbers offers a guide to the harmony of things." [p. 7] This is a very deep idea. Why is it that mathematics turns out to be useful in describing nature?
"The doctrine that number is the essence of all things ... remains the central insight of Western science, the indispensible key of coordination. ... The fact that this key opens so many locks has often been celebrated, but it has never been explained. ... In mathematics, it is always the key that counts." [p. 8 - 9] This is very close to my driving force: what are the essential ideas in mathematics, and how are they related? These essential ideas are not difficult, but they require effort to identify and to pin down. Number is one such idea. Proof is another.
"... every cathedral has its mice." [p. 13] I have seen this metaphor before, but can't recall the source. I wonder if Google would help?
"The Pythagoreans had been intoxicated by the natural numbers, ..." [p. 14]  
"... the Elements is a textbook. It proceeds from the simple to the complex. It is beautifully organized. It is very cleara, succinct as a knife blade. And like every good textbook, it is incomprehensible." [p. 14]

What is a textbook? What kind of a category is this? And why, if the object is to facilitate learning, are so many indeed incomprehensible? And why do we continue to produce them if they are so ineffective?

I am making good use of about six such books, yet I do not stay with just one book. I think this is the key. Any single book represents a particular viewpoint and way of representing and describing the important ideas. But the learner is trying to make sense of this and form their own ideas and understanding. This involves a synthesis of views. The Web is going to have a profound impact on the way we view and package education in the future.

The incomprehensibility is due to the amount of information that conforms to one author's perspective. The learner does not wish to be a clone of the author and at some level begins to resist the book in favor of their own ideas.

"Euclidean geometry calls for a collaborative effort between the initiated and the unenlightened, the teacher droning, the student drowsing, until mastery of the material builds slowly in the warm space beween droning and drowsing." [p. 15] This is fairly similar to what I have just written. It is not just Euclidean geometry, but ALL Learning.
"... the Elements is not only a book about geometry; it is as well, a book about how a book about geometry should be written, ... a meta-text." [p. 16] Is this web site another meta-text?
"A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules." [p. 17]  
"The axioms must meet two constraints: They must be rich enough so that everything important about the world of geometry may be derived from them; and they must be sufficiently self-evident so that they may be accepted without argument. ... There are only five axioms needed to make possible the creation of the Euclidean world." [p. 18] Similarly there are only a few axioms needed to create the world of positive integers, and only a few more to create the world of all numbers. Stunning!
"Mathematical curiosity died in the Roman Empire and it stayed dead in the Christian West for more than one thousand years." [p. 27] Are we entering an age of Roman Empire 2.0?

There may be little or no technique in Berlinski, but he conveys some important mathematical truths nonetheless.


Here are a couple of quotes from "On Intelligence" by Jeff Hawkins.
"The United States alone has thousands of neuroscientists. Yet we have no productive theories about what intelligence is or how the brain works. ... And although legions of computer programmers have tried to make computers intelligent, they have failed." [p. 2]

I agree. So how can this be? We also have millions of teachers and educators, yet we have no theory of education although I do believe that the label of constructivism has the potential to evolve to such a theory.

"Many of the individual ideas you are about to read have existed in some form or another before, but not together in a coherent fashion." [p. 4] This is the problem with mathematics as well. How does one tie together the many individual ideas into a coherent whole? Berlinski comes close in his short book. But we need to provide this framework, this picture of a coherent whole, to our students, so they can see how it all fits together.
"... the core idea of the theory, what I call the memory-prediction framework." [p. 5]

Excellent. Right at the outset the core idea is identified.

Every book should have a sentence like this. This is analogous to Richard Feynman's idea that one should be able to summarize physics in a single sentence. In his case it was "all matter consists of very small things that jiggle a lot".

For mathematics it might be Kronecker's quote: "God created the numbers, all else is the work of man."

Even a book on intelligence has much to say about mathematics. Of course.

Here are a couple of quotes from "The Complete Book of Drawing Techniques" by Peter Stanyer.
"Drawing, just like writing or speech, is a form of communication, and in the same way as these other forms of communication drawing can be multi-faceted, and very diverse as a means of expression of our observations, thoughts and feelings." [p. 6]

And so is mathematics. And so is this web site.

One of the weaknesses of mathematics textbooks is that they fail to convey "an expression of our observations, thoughts and feelings."

"This book has been brought together in a unique way, as it brings about for the beginner and the student of drawing not only the techniques, but also the analytical and emotive approaches and attitudes to dawing." [p. 8] Again. In mathematics education we need to incorporate the emotive approaches and attitudes as well as the analytic. Often this will include confusion, frustration and despair but also elation, joy and peace.

This book on drawing also has much to say about mathematics. Of course. And I may finally Learn to draw as well.

An exhilerating beginning to the day.

7:05 am

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