Friday, July 25, 2014

Teaching a Modern ODE Course

Attending MAA MathFest 2014 in Portland, Oregon? Stop by this informational session on teaching a modern ODE course.

Hilton and Executive Tower
22nd Floor
Mount St. Helens Suite
Friday, August 8

The differential equations course has changed radically over the last quarter century.  Easy access to powerful computation has enabled visualization to play a much larger role. The increasing mathematization of the life sciences has greatly expanded the kinds of models available for investigation. The advent of dynamical systems has made new kinds of questions imaginable and accessible. A modern ODE course has to take all this progress into account, though it is perhaps not clear exactly how to do so.  Anne NoonburgUniversity of Hartford and author of OrdinaryDifferential Equations from Calculus to Dynamical Systems―and Steve Kennedy―Carleton College and MAA Books Sr. Acquisitions Editorwill lead a discussion focused on how best to react to these changes in your ODE course. We will ask such questions as:
  • What is the appropriate role of modeling in the ODE course?
  • How do we balance the needs of physics, biology and engineering majors in the course?
  • What are good sources of deep and interesting models?
  • How much emphasis on numerical methods is appropriate?
  • Similarly how much dynamical systems theory and visualization?
  • What is the appropriate role of technology and what are good choices?
  • Is a Linear Algebra prerequisite necessary, or can the needed material fit in the ODE course?

Friday, July 11, 2014

William Satzer Reviews 101 Careers in Mathematics

William J. Satzer reviewed 101 Careers in Mathematics, 3rd Edition edited by Andy Sterrett as part of MAA Reviews. 

This is a wonderful book, potentially of great value to students and those who advise them. It has some frustrating gaps too, but in a way they also emphasize how useful it is and could be. In brief, this book presents a collection of profiles of people who have (or had) a career that involves some aspect of mathematics. Nearly all the people here have at least one degree in mathematics; the few exceptions have degrees in field like physics, operations research, or a statistics-related area. Short essays at the end of the book discuss the processes of interviewing and finding a job, and what it’s like to work in industry (or, more broadly, outside the academic community).

There are 25 new entries in this new edition that bring the total number of profiles to 146. The “101 Careers” of the title is best regarded as meaning “lots of careers”; even the first edition had more than 101 profiles. Counting careers is also a little funny: they don’t match up one-to-one with people. As many of the profiles demonstrate, many people have more than one career. Indeed it is increasingly uncommon for people to have a single career throughout their lives.

Read the full review here.

Monday, July 7, 2014

New: Mathematicians on Creativity

Mathematicians on Creativity

Peter Borwein, Peter Liljedahl, and Helen Zhai, Editors

This book aims to shine a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-caliber working mathematicians. In their own words, they discuss the art and practice of their work. This approach highlights creative components of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work. Mathematicians on Creativity is meant for a general audience and is probably best read by browsing.

Friday, June 27, 2014

Al Cuoco on Wild About Math

Sol Lederman of Wild About Math interviewed Al Cuoco on his latest book Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem (co-author Joseph Rotman).

Listen to the interview here.

Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem
by Al Cuoco and Joseph Rotman

This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily "end up on the blackboard."

Friday, June 6, 2014

MAA Books Beat: Playing to Learn Game Theory

Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUS. Playing to Learn Game Theory appears in the June/July 2014 issue.

Playing to Learn Game Theory
by Steve Kennedy

Suppose the graph depicted in figure 1 represents a network of communities. Two doctors are planning to set up practice in one of the nine communities. A practice will attract all the patients in every town closest to it (closest means in terms of number of graph edges joining the vertices). In the case of a distance tie, each doctor will attract half the patients in the town. Where should the doctors locate to ensure their best result? (They are not conspiring; each doctor is acting independently and in ignorance of the other’s choice.)
Of course, neither doctor, presumably desiring to maximize her practice, would choose one of the degree-two vertices. The other six vertices look equally valuable. Suppose Doctor A chooses vertex 2. Were Doctor B to choose any of vertices 3, 5, or 8, each doctor would get 4.5 towns’ worth of business. If Doctor B chooses vertex 6, she “wins” by gaining five towns to A’s four. (If she chooses vertex 7, the payoffs are reversed and A wins five to four.)

If we conceive of the competition as a game between the two doctors, then this game has no Nash equilibrium. Each doctor can guarantee herself only four-ninths of the available customers.

The example comes from the new MAA ebook, Game Theory through Examples by Erich Prisner. The Doctor Locator Game is one of scores of clever, rich, interesting games described and analyzed in this lively text. There are also scores of apps, linked to the etext, so that the reader can play the games, the best way to understand them. In the doctor game, the reader can, with two mouse clicks, choose vertices to represent the doctors’ choices and immediately see a color-coded graph giving the division and the count.

The example games introduce and exposit all the important concepts of game theory. The Doctor Locator Game, on a different graph, introduces the idea of a Nash equilibrium and explains how to find one. The obvious next question of their existence is answered by the above example.