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?

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!!

JHU Mathematics #1 in the world in research citational impact!!!

The Times Higher Education, a London-based publication that focuses on higher education issues, puts out a series of university rankings each year. JHU does okay, IMHO, with a ranking of 13th in the world, based on a multitude of criteria, and 14th based solely on teaching and research reputation.

But to measure the influence that a research-oriented faculty has on the general research community, it is proper to measure the citational impact of a department's publications; how often papers are cited by other papers. The times measured this citational impact in mathematics. And who came out on top in a survey covering this last decade? Funny you should ask....

Congratulations to our active research faculty here in Hopkins Mathematics! A job very well done.

Advice for an independently-learning pre-college student.

Back again, I am....

Here was a question I received recently:

Hello, ... I recently finished my sophomore [year in high school].... I have, for the past year, learned mathematics independently, taking trig, pre-calculus, calculus BC, and multi-variable calculus on my own. For the first two courses, I used an online provider. The third, I took an AP test to demonstrate that I have sufficiently learned the material so that I might receive credit for it when I go to college.

For the last course, multi-variable calculus, I would like to find a way to gain either recognition (such that I would not have to take the class in college) or credit for the course before I enter college. This is not the only class that I will have taken independently for which I cannot take an AP exam to be granted recognition. Also, I plan on starting other math courses independently (Linear algebra, differential calculus, etc.), so there will be multiple classes which I will have learned, but nothing to show for them.

Is there a way, through Johns Hopkins, I could acquire either credit for having learned college-level courses independently? If not through Johns Hopkins, do you know of a way to do this using different means?

A good question, as there are many pre-college out there struggling to quench their thirst for mathematical knowledge amidst a dry, arid environment void of opportunity. My reply:

While I like your initiative, and value your capabilities, I am wondering why you are trying to burn through all of this material at such a high speed. The AP exams, while a nice system for providing advanced training in mathematics to pre-university students, do not really measure proficiency in calculus. Rather, they measure your ability to apply proper techniques to appropriate problem types. While this is helpful, it is not really what mathematics is all about.

In your case, looking for opportunities outside high school for advanced training (as you are doing through self-study) is a good idea. But simply relying on an online course or a book and a standard exam may wind up giving you a false indication of your true knowledge base in these subjects. And if you foundation is not strong in basic subjects, you may find yourself faltering later on at the higher levels.

Some questions: (1) Do you have a mentor at your high school, or nearby, a math instructor, or mathematician to help guide you through your self studies? Someone who can see your "path" from above while you walk it is very important to your training. (2) Is there a goal in your life, which provides the reason for going from trigonometry to vector-calculus and beyond in a single year? These are beautiful subjects full of amazing insight and deep conceptual meaning. Burning through them at top speed is really selling the individual topics short. This is like driving through a safari park at 80 miles an hour. You have done the park, but have you really spent time learning about the animals. (3) Have you looked at simply taking courses on these topics at your local university, one at a time, and with live instruction? Even at the community college level, there are very good instructors whose lectures in class and conversations outside of class can be extremely helpful in seeing more then the techniques.

Yes, we here at Hopkins have many ways of evaluating the proper level for students to start at their first semester here. And we are committed to ensuring that students are not taking courses they are clearly too advanced to take. Acknowledging a students proficiency in a mathematics course may not always involves credits for the course (maybe just a waiver), but most of our evaluation involves some sort of comprehensive documentation of prior training, and not just an exam. Exams are not usually very good indicators of real understanding.

I hope this helps. Good luck in your training.

Spirit, initiative and resourcefulness are primary qualities of the budding scholar. Having and/or finding a mentor or guide is absolutely fundamental (even Harry Potter wouldn't have made it on his own!) And taking your time to digest what you are learning always leads to "better nutrition", no?

Math in the Media - Jump Math

A neat article appears in the Opinionator column of the New York Times by David Bornstein;

A Better Way to Teach Math

The article details a new attitude and focus in the education of mathematics at the primary school level. The organization, Jump Math, is based in England and is the project of John Mighton, a playwright and author, and seems to already be showing results.

Really, it sounds like Jump Math (as I write this, the link above to the organization is down) is not a new set of concepts to teach. Rather, it is simply an idea that the best way to teach mathematics (at any level) is to instill the idea that high level math is not just for those who have "the ability" to get it, but for everyone. Many of us who teach math really do understand that anyone can understand high level math. The problem is that many students have already concluded that they are not able to get math, so they do not have the confidence to really try to understand what is going on. Couple that with a sense that many teachers of mathematics do not really get the art and beauty of mathematics. So they teach a technique-based, problem-centric type of math that loses the deeper meaning. Without proper motivation, much mathematics loses its context, and hence much of its meaning.

From the article:

Imagine if someone at a dinner party casually announced, “I’m illiterate.” It would never happen, of course; the shame would be too great. But it’s not unusual to hear a successful adult say, “I can’t do math.” That’s because we think of math ability as something we’re born with, as if there’s a “math gene” that you either inherit or you don’t.

I have heard this ALOT, and my response is always something like "probably because you were taught by people who didn't get it. Anyone can do math...."

The article is nicely written, and quite pleasing to hear for someone like me. I will be probing this new set of ideas called Jump Math over the near future and report my finding here. To me, at least on the surface, something like this is exactly what I think pre-university teaching of mathematics needs.

The article promises more at the end of the week. We will await the continuance. For now, a good ending quote:

"Even deeper, for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice."

One has to love quotes like that....

Math in the Media - Algebra a leading indicator of success in life??

Whoddathunkkit? Well, except for most of us that do math for a living, you mean?

This article in the Washington Post:

Requiring Algebra II in high school gains momentum nationwide

by Peter Whoriskey, seems to be really an article on the debate of the merits of teaching high level mathematics as part of the core curriculum in high school.

Peter discusses a study that shows a correlation between successfully taking mathematics through Algebra II, where properties of functions like exponentials and logarithms are analyzed (along, I guess with complex numbers) in high school and continued success in college and through a career. Whether learning algebra is the reason people are more likely to succeed, or those more likely to succeed usually wind up taking the challenge of Algebra II, is not apparent.

But the study is interesting and should keep up the discussion.

My personal take. Forcing middle schoolers and high schoolers to master problem solving strategies using highly abstract models in mathematics is a way to wire their brains for the complexities of real life events that will present themselves in any and every career path choice, no?

One can teach strategy and problem solving in any specific discipline using the techniques of that discipline, and you get people well versed in that discipline. But mathematics is a 100% in-the-head discipline. Mastering the abstract complexities of mathematical structure and analysis means learning not just how to problem-solve, but it is like learning the actual art of problem solving. It becomes adaptable to any future discipline one winds up in.

Good sound mathematical training is like producing problem-solving stem cells. Later in life, when you need those stem cells to morph into good problem solving skills in some job, you will have them ready for use.

I also believe that Algebra II is attainable for every high school student. Some of the quotes in this article come from students who do not get the subject. It looks like they were/are not well-taught the subject. Perhaps THAT is the real problem? Non-uniformly good teaching.

STEM Over Spring Break....

Here is an interesting activity; a way to give back to those yearning for the kind of "fun" of mathematics that you feel and felt back then....

Christine Newman, the Assistant Dean for Educational Outreach and Dr. Meg Bentley, Program Manager at the Center for Educational Outreach in the Whiting School of Engineering at JHU, are organizing a day of fun math-centric informal activities for Baltimore City School kids during their upcoming Spring break next week. It's called STEM over Spring Break (the STEM part means, I believe, Science, Technology , Engineering and Mathematics) and is meant to be lighthearted and playful. if you are interested in taking part by working with city students on fun math-ish activities, or if you just have some bright ideas for playful math or interested math activities you would like to share, contact Dr. Bentley directly at meg(dot)bentley(at)jhu(dot)edu. Or click on the link to the Center above for more information.

It is always good to stop once in a while along your own path and give a hand to those struggling along the same one, no?