Wednesday, September 28, 2011

Competative Math: Time for the Putnam

Welcome back to everyone who has been here at Hopkins, and welcome to those who are new. Yes, it is that time of the year again.... to start getting in shape for a little competitive mathematics.

Early December again brings the

a locally held, national competition in undergraduate-level mathematics. Highly competitive and highly prestigious, the Putnam offers cash prizes as well as a very strong resume/CV credential to those who master the 6-hour two part exam. In fact, the Math Department recognizes the best from JHU in the exam each year with an award and cash prize. Recent JHU best-performers have included students who achieved recognition from the Putnam Committee. And our best school ranking in the last few years was 21st (out of upwards of 500 institutions that take part).

Registration for the 72nd national Putnam exam closes sometime around mid October, and the exam will be held on Saturday, December 3, from 10am-1pm and 3pm-6pm.

If you are interested in participating (and as a math major, I highly recommend that you consider these exams part of your training as a mathematician), please contact me in any way you can.

We are setting up training sessions for this exam on Wednesday evenings from 5-7pm. We will get a room once we have a head count and start a week from today.

Keep looking here for more announcements and news as the schedule develops.

Thursday, September 15, 2011

Math Play: Every Natural Number is Interesting.

I just read this in the first chapter of the text we use for our senior class Advanced Algebra. The book is Introduction to Advanced Algebra by W. Keith Nicholson. I like it, so I will pass it on.

If you study math, you have probably heard of the Well-Ordering Axiom, a property of the integers which is equivalent to the Principle of Induction. The Axiom states: Every non-empty set of non-negative integers has a smallest member.

Intuitively clear, no? Here's how it works in practice:

I claim that every positive integer is interesting. To show this, let's assume it is false (this is a proof technique known as proof by contradiction, in which one assumes the claim is false, then works by deduction until something absurd follows, a clear contradiction. If the logic is solid, only the assumption can be flawed. Thus your original statement must be true. )

Since the statement is assumed false, there must be a non-empty set of uninteresting positive numbers. But by the Well-Ordering Principle, there then must be a smallest uninteresting number. But an extreme element of any ordered set is automatically interesting (in that it is special)! Hence we arrive at the contradiction, making our assumption false and the original statement true.

Silly, yes?

Thursday, September 8, 2011

How to (what?) Our Math Education?!!?

Having concern for the state of mathematics education here in America is such a common thing among the population that there are probably almost as many ideas for a solution as there are people with concern. And while the issue is unsettled, it is great to have voices loud enough to keep the discussion lively and vogue. A recent (August 25) addition to the discussion is the New York Times Op-Ed piece
How to Fix Our Math Education
by Sol Garfunkel and David Mumford. I guess it is a hit piece on the No Child Left Behind initiative, but more it is an indictment on the standard idea of teaching math for math's sake at the elementary and secondary level. Their view seems to be that since mathematics was developed in tandem with science and applications, it should be taught that way. Bringing in the deep conceptual beauty of mathematical relationships in a class focusing on engineering or finance would better serve the students' educational needs rather than teaching the pure elements of, say, algebra in their own right.

I will let you read the article and form your own conclusion. My take? Little can be farther from the truth! The applications of mathematics are many, varied and beautiful. But the essence of mathematics is the study and development of pure rational thought. It is precisely the abstract nature of pure mathematics that should be taught to young students in our schools. And it should be taught as an art at every level, with applications only to serve as neat ways to display its innate beauty. The lack of a cohesive story about abstract mathematical relationships and patterns in our math class sequences is what fails our educational systems today. And not the fact that we do not apply math correctly. This is just my opinion.

I found the letters in rebuttal to this article of most interest to me: Read here for some of them:
Math = The Practical and the Beautiful
One real money quote that gives away my take on this whole business? From the Computer Scientist Jonathan David Farley's letter at the end:
You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things.
Pure candy, that quote is!!