I'll be on Cogito.org next week....

I will be hosting a discussion forum on the website www.cogito.org for the next couple of weeks. Cogito is a math and science website and online community for talented youth, and part of the Center for Talented Youth (CTY) family here at Hopkins. I'll be taking questions and offering advice on whatever I can (involving mathematics, I suspect).

It sounds like it will be a lot of fun. I'll post my thoughts here in the interim.

My response to the NYT Op-Ed on Math Ed

Well, it has been a while since I commented here on the New York Times Op-Ed on Math Education and its ills. My rebuttal here in this blog caught the interest of a talk show in California, though I did not reply in time to attend the discussion. I did write a rebuttal to the article and submitted it to the NYT. Alas, it was ignored. Oh, well.

Here it is, though. At least I accept my own submission. Enjoy:

Why Not Teach Math for Math’s Sake?

It is quite conventional wisdom, with lots of supporting evidence, to believe that the way we teach primary and secondary mathematics here in the US is generally failing our young students. This was detailed in the recent article in this forum “How to Fix our Math Education”, by Sol Garfunkel and David Mumford, and I see the general effects of pre-university education in students daily from my perspective as the director of an undergraduate program in mathematics at an American university. I believe the problems discussed in that article are real and demand action, and I applaud the authors for writing the piece. I disagree, however, with the conclusions of Professors Garfunkel and Mumford.

From my perspective, students come to university with a view of mathematics as a giant tool box they carry around with them, the tools being techniques useful to solve many kinds of diverse math-based problems. Pre-university education seems to be filled with disparate situations where a new concept is introduced (abstract or applied) to solve a certain kind of numerical problem, a technique is drawn up and the student receives a worksheet containing 40 or so variations of the same type of problem. Once completed, the class moves on to the next idea. While this assessment of pre-university education is simplistic, the outcome is that students never really learn how to think analytically, reason deductively, understand why these tools exist in the first place, or see just how each idea fits into the whole. Context via applications to real world phenomena (the kernel of the above authors’ proposed solution) may help in this regard, but there is a deeper problem with simply embedding math into applications to prove its usefulness.

The idea that mathematics lives only to serve its applications and functions only as the language of the sciences is absurd on many levels. More like music and poetry than physics or engineering, mathematics is an art, the art of pure reason. One could say that mathematics is the distillation of pure rational thought. When we teach math, we are not teaching how to solve problems. Instead, we are teaching how to think analytically; how to analyze any given complex situation, discover and understand its underlying logical structure, and figure out how to abuse that underlying logical structure to say something useful or conclusive about that situation. Numerical problem solving is solely one manifestation of this process. In the general sense math has very little to do with actual numbers at all. It is just that the use of a number system as one of our basic building blocks allows for a natural logically consistent system. Instead of giving students tools for solving problems, we should be teaching them the very nature of how and why these tools exist and were developed. Questions like why there is a quadratic formula, and why does the sine function exist at all are much more thought provoking and fundamental than how they work. We should be engaging students to actually design and redesign the tools themselves, a process of self-discovery which enables them to own the math they create. We should be giving them the confidence and experience to be able to see a problem as an opportunity for creativity and ingenuity, rather than an obstacle to overcome. And we should be teaching them that mathematical constructions have an innate aesthetic quality. They exist simply for what they are: beautiful constructions, often useful, whose existence lies entirely in the imagination, but whose manifestations in the real world are everywhere.

Math, like music and poetry, has a few constituent parts (notes and keys in music, words with contextual meanings and rhyming schemes in poetry) and a few logical rules which they follow. But with these few rules and parts, no one questions the infinite beauty and variance of musical creations or the fact that a few well-placed and possibly rhyming words can draw such emotion (think Shakespeare). And no one questions the value of teaching primary and secondary students music for music’s sake. Why not teach math for math’s sake? View it as a ground up endeavor where applications can serve as motivations for new mathematical ideas, but where the math lives outside of any application; where the beauty of self-exploration and discovery of fascinating concepts arises simply out of the aesthetic appeal of the constructions; and where the process of developing the skills of analysis and deduction in abstract logical systems becomes the goal of mathematics at the primary and secondary level. The application-based problem-solving skills could come along for the ride, and be reinforced in the other science-based classes. But the math would exist on its own.

Someone once said to me, “When are we going to stop getting students to solve problems and start getting them to POSE problems?” At the research level in math, we design and use the tools we need to pose and solve questions and problems as we need them. Teaching children the rudimentary process of doing this would go a long way to curing our math education woes.

What is that old saying “Teach a student a technique, and she will be able to solve some problems. Teach a student how to develop techniques, and she will be able to solve any problems.” I just made that up. But if we can teach our children to think analytically (read mathematically) before they reach university, imagine what we can do with them in university and beyond.

Thinking of graduate school?? Think NSF first!

Hey graduating seniors!

Thinking of graduate school in mathematics or some math-centric science? How about a Fellowship as a good credential? The National Science Foundation is one place to look. I just got this announcement form a colleague here:

To senior math majors:

NSF will again be awarding Graduate Research Fellowships in math and science. Last year, 2000 fellowships were awarded, including 80 in mathematics. The fellowships are for 3 years and provide an annual stipend of $30,000. The deadline for applications is November 15. (Reference letters are due on November 29.) Information and online applications are available at http://www.nsfgrfp.org/.

Check into it: What is it they always say? "You can't win one if you don't apply" (okay, maybe not they. Maybe only I say that.

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

William Lowell Putnam Mathematical Competition,

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.

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?

A career with the NSA...?

The National Security Agency has just announced their upcoming Intelligence Community Virtual Career Fair on Tuesday, March 8, 2011. A chance to talk directly to NSA Recruiters about career opportunities. And a place that hires LOTS of mathematicians for careers doing pure mathematics (imagine that!). The posting, in its entirety is below. Check'em out.

Join the National Security Agency at the Intelligence Community Virtual Career Fair Tuesday, March 8, 2011 from 2 p.m. - 8 p.m. (Eastern)

Let’s talk about your future at the National Security Agency. Please join NSA March 8th at the IC Virtual Career Fair to explore exciting opportunities for your future. Reserve your spot today: http://www.aftercollege.com/op/op.asp?id=5307

Who: We are looking for technical career professionals (computer network operations, software/hardware design developer, software engineer, global network exploitation and vulnerability analyst, computer scientist, computer/electrical engineer, computer systems analyst, computer systems architect)

Language analysts and instructors (Chinese Mandarin, Dari, Farsi, Pashto, Punjabi, Somali and Swahili)

What: The Intelligence Community Virtual Career Fair

Where: The nearest Internet connection.

When: Tuesday, March 8, 2011, from 2 p.m. to 8 p.m. (Eastern)

Why: To discover the incredible benefits of working with the National Security Agency. You’ll be able to chat with recruiters. Explore exciting career fields. Enjoy live presentations. Mingle with like-minded folks. And just have an all around positive career-oriented event … all from the comfort of your own computer.

Register today: http://www.aftercollege.com/op/op.asp?id=5307

U.S.citizenship is required. NSA is an Equal Opportunity Employer.

All applicants for employment are considered without regard to race, color, religion, sex, national origin, age, marital status, disability, sexual orientation, or status as a parent.

Playful Math: Doodling to Aleph_null

I believe most people who really get mathematics are the ones who see the frivolity in much mathematical construction not as a flaw, but as a strength ( I am teasing my profession here). And sometimes presenting mathematics in playful ways is precisely the best way to expose deep meaning.

With a hat-tip to Engineering Innovation (@JHU_EI), a high school summer program in the Whiting School of Engineering here at Hopkins, I am reposting a video from one of their recent tweets.

Doodling in Math Class: Infinite Elephants

Have fun!

Oh, and BTW, we mathematicians tend to associate letters from other alphabets to important constants and concepts in our work. Aleph (the first Hebrew letter) is commonly used for measures of infinity. Aleph_null, or Aleph with the subscript zero, is used to denote a kind of infinity called countable infinity, and denotes the size of a set of objects that can be placed in a one-to-one correspondence with the natural numbers 1,2,3,... Jus'sayin'....

Math (not quite) in the Media... College not producing?

A new study has come out which seems to conclude that a college education may not be worth the many hundreds of thousands of dollars it is costing these days. Here is an article from the Huffington Post detailing some of the points coming from the data. I have not read the study. The article, I have, and am less than impressed.

My take: While there may be quite an important trend uncovered by this survey, I believe (from my position) that some of the assumptions may not be fully vetted.

45% of Students Don't Learn Much in College

The article starts off well enough (emphasis mine):

A new study provides disturbing answers to questions about how much students actually learn in college – for many, not much – and has inflamed a debate about the value of an American higher education.

The research of more than 2,300 undergraduates found 45 percent of students show no significant improvement in the key measures of critical thinking, complex reasoning and writing by the end of their sophomore years.

However, this part bothers me:

One problem is that students just aren't asked to do much, according to findings in a new book, "Academically Adrift: Limited Learning on College Campuses." Half of students did not take a single course requiring 20 pages of writing during their prior semester, and one-third did not take a single course requiring even 40 pages of reading per week.

Where is the metric measuring critical thinking and complex reasoning OUTSIDE of a writing class? Mathematics and physical science-based courses are prime venues for the development of rational and analytic reasoning, no? How come no mention of non-writing-based courses in the ENTIRE article? Perhaps no data from these courses is in the study?

More from the article:

Three of the five classes [Julia Rheinecker, a freshman at the University of Missouri,] took... were in massive lecture halls with several hundred students. And Rheinecker said she was required to complete at least 20 pages of writing in only one of those classes.

I am very dubious about any direct link between college class size and either the difficulty of a course or its perceived lack of rigor. Large-lecture courses can be quite challenging and yet every bit as personal and interactive as small seminar-type classes. I doubt this study had any focus on class size at all. So this part is not relevant, IMHO.

And finally, some additional conclusions in the study:

_Students who studied alone, read and wrote more, attended more selective schools and majored in traditional arts and sciences majors posted greater learning gains.

_Social engagement generally does not help student performance. Students who spent more time studying with peers showed diminishing growth and students who spent more time in the Greek system had decreased rates of learning, while activities such as working off campus, participating in campus clubs and volunteering did not impact learning.

Knowing something about how math is taught and learned at the university level, I am hard-pressed to believe any data showing that studying alone is better than studying with peers. On this point, I can write volumes! And there are data and programs (Think PLTL, for example) to support the opposite conclusion. In fact, we have a local PLTL program here at Hopkins, and our data do not support this study's conclusion in this point.

Overall, I see this study as being a bit alarmist. Critical studies on the effectiveness of college education are absolutely necessary. I will read this study. But I am already biased due to this article. Oh well....

Undergraduate Research Opportunity!

I just got a solicitation for a great summer program in research for undergraduates. Click on the flier to see the details. It is the Research in Industrial Projects for Students (RIPS) program at the Institute for Pure and Applied Mathematics (IPAM) at UCLA. Top-notch, IPAM is, and I am sure the opportunity is quite competitive. But the program is a good one, and will make for a great credential (not to mention the experience!). Check it out and let me know if you plan to apply: The deadline is February 10. Plenty of time to get your act together, no?