Tag Archives: Books

Books: “A Mathematician’s Lament” by Paul Lockhart

A Mathematican's Lament book cover
Anybody who teaches math should read this book!

This book perfectly summarizes my disgust with the state of “education” today. The author’s main thesis is that math is an art that should be taught in a way to cultivate appreciation and understanding. It is not merely a set of formulas and definitions that students should commit to memory. Here is a quote from page 29:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

And another choice quote about the evils of High School Geometry from page 67:

…never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous.

Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way.

The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”

This book is available as a freely downloadable PDF.

Advertisements

Books: “Particle Physics A Very Short Introduction” by Frank Close

Particle Physics book cover
I found this book in the library the other day and found it to be a very good introduction to the basics of particle physics. I liked the fact that it explains the basics of the experimental side of particle physics as well as introducing the three families of leptons and quarks. Too often, popular science books explain the various subatomic particles with out giving you any idea of how we actually know of their existence. Because of this, I think disbelief tends to creep in. But Close does an excellent job of explaining the basics of how accelerators operate and also the basics of how detectors work. And he does all this in only 129 pages! Very short indeed, and highly recommended.

Algebraic Geometry: “Ideals, Varieties, and Algorithms” C++ Code


Ideals, Varieties, and Algorithms book cover
I did an REU (Research Experience for Undergraduates) this summer and we used the textbook “Ideals, Varieties, and Algorithms” (isbn 978-0-387-35650-1) by David Cox, John Little, and Donal O’Shea. As the title mentions, algorithms are an important tool in the study of varieties via their corresponding algebraic ideals. One of the main tools used in algebraic geometry is what is known as a Groebner basis, which is somewhat akin to a basis set in Linear Algebra. Anyway, I wrote some C++ code which implements a few of the algorithms in the book, but stops short of implementing Buchberger’s algorithm for computing a Groebner basis. The code should be useful as a starting point because it implements a Monomial class and a Polynomial class both with lots of useful methods thus eliminating the drudgery of implementing the book’s algorithms in C++.

The code can be found at: my web site as a gzipped tarball. Please let me know if you find it useful. I developed the code on an Ubuntu Linux machine, but it is generic enough that it should compile on any platform. You will need to read the README to understand the simple input, output format.