Study tip for the day: Quality vs. Quantity

People often ask me how many hours one must spend outside of class per hour inside of class to succeed in a university-level math course. I hesitate to answer. The reason is that the question is ill-posed. To really understand the logical structure of mathematical ideas, how they work and fit together, why they exist and work the way they do, one must really spend the time to dig into every nuance of the idea. To do this on a research level, a mathematician understands that she must isolate herself from outside distractions for sufficient periods of time to fully explore the structure of a new idea. The isolation-booth manner is as vital to the process as deep, dreamless sleep is to the health of a person. I call the process of getting to the level where one can focus exclusively on the task at hand without distraction as "going deep". It is kind of a meditation-type thing, and only really works when one practices it regularly. It can be dangerous, though, as the day-to-day tasks tend to get neglected. But it is quality time for understanding complicated mathematics, and easily outdoes simple quantity time when the latter is filled with noise and attention-taking "shiny objects" (distractions).

It's loads more productive to spend an hour in an isolation booth environment focusing solely on your mathematics work instead of 3, 4, or 6+ hours poring over books and notes while checking your phone, listening to music or chatting with friends (or potential friends). Distractions keep you from "going deep" and really digging into the conceptual and logical structure of the math you are doing. And if you do not allow yourself to go deep to really get a concept or idea, you wind up simply memorizing facts and patterns. While this may work for problems just like the ones you have seen, the minute a problem of a different form comes up, you will be lost.

Hence, there are no good guidelines in the form of "6 hours of study per hour in lecture". Studying is a personal thing, and the studying environment matters. If you refuse to allow yourself NOT to fully understand a new concept, then any and all time spent in the pursuit of full understanding is worth the effort. To do it right, allow yourself the ability to "go deep". Then you minimize the time spent to only the quality time.

Some people brag about their ability to multi-task. To me, the ability to mono-task is the lost art in society.

How to Learn by Lewis Carroll

I have recently been talking to a student about the whole idea of how one learns, especially at the university level. This student is thinking of starting a club of Hopkins students dedicated to discussing the theory and practice behind how one learns. Really a great idea. I will advise and keep you posted. For now, I give you Lewis Carroll's (Of Alice in Wonderland fame) ideas for learning. Enjoy! What is that old saying: Much truth is said in jest....

How to Learn by Lewis Carroll

1. Begin at the beginning, and do not allow yourself to gratify mere idle curiosity by dipping into the book, here and there. This would very likely lead to your throwing it aside, with the remark `This is much too hard for me!’, and thus losing the chance of adding a very large item to your stock of mental delights . . .

2. Don’t begin any fresh Chapter, or Section, until you are certain that you thoroughly understand the whole book up to that point and that you have worked, correctly, most if not all of the examples which have been set . . . Otherwise, you will find your state of puzzlement get worse and worse as you proceed till you give up the whole thing in utter disgust.

3. When you come to a passage you don’t understand, read it again: if you still don’t understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy.

4. If possible, find some genial friend, who will read the book along with you, and will talk over the difficulties with you. Talking is a wonderful smoother-over of difficulties. When I come upon anything—in Logic or in any other hard subject—that entirely puzzles me, I find it a capital plan to talk it over, aloud, even when I am all alone. One can explain things so clearly to one’s self! And then you know, one is so patient with one’s self: one never gets irritated at one’s own stupidity!

If, dear Reader, you will faithfully observe these Rules, and give my little book a really fair trial, I promise you, most confidently, that you will find Symbolic Logic to be one of the most, not the most, fascinating of mental recreations! …

Mental recreation is a thing that we all of us need for our mental health. Symbolic Logic will give you clearness of thought—the ability to see your way through a puzzle—the habit of arranging your ideas in an orderly and get-at-able form—and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!

Employment Op: Business Intel and Math

People often ask me what kinds of jobs math majors can get other than science applications or as an actuary. Really, the answer is: All Kinds. Usually, one needs a good hook in an outside area to get into the interview. But then the analytical skill set of a mathematician can shine. However, sometimes recruiters simply understand well that someone skilled in mathematical analysis possesses the ability to learn most skills very quickly. These recruiters are willing to take a math major who can learn on the job, quickly and efficiently.

I just got this request from a recruiter. Read it and go for it. Sounds like an interesting career:

Thorogood Associates is currently recruiting college seniors for full time business intelligence consulting positions. We are contacting you, as the [Director of Undergraduate Studies] of the Mathematics Department, because we think you may know students that would be good candidates for this position.

As a business information consultancy, Thorogood helps its clients use their data to make better business decisions. Our work has both a business and technology aspect. We don’t necessarily look for education or experience in both of these areas but rather an interest and an aptitude that will allow a candidate to be successful in this type of work.

We are seeking candidates that have excellent problem solving skills, leadership qualities, and initiative. Candidates must be willing to take responsibility for the achievement of results, have self-confidence, and be energetic and friendly.

If you know of any students that have the qualities that would allow them to be successful in this position, please let them know about this opportunity. They can apply for this position via J-Connect. Applications are due on January 30^th, 2012. We will be conducting on-campus interviews at JHU on February 7^th, 2012. Any questions can be sent to USRecruitment@thorogood.com.

Check them out at: www.thorogood.com

Math in the Media: Eating Mathematics?

Alright..., just for fun.

If you are not yet convinced that mathematics is not a subject to study as much as it is the underlying logical framework for all that exists both in reality and in imagination, I give you another example.

The New York Times' Kenneth Chang has written a piece on the mathematics of pasta:

Pasta Graduates From Alphabet Soup to Advanced Geometry.

Those seemingly random and crazy shapes, designed specifically for texture, even cooking, and the ability to meld well with sauces and such, can be quite beautiful and subtle. This article exposes those who look for the mathematical structure behind the designs and the playful aspects of the shapes.

Take a look. But beware. You may never view a plate of spaghetti in the same way again!

NPR on the JMM

Well, here is something you do not hear every day: A human interest story on a national radio news program focusing on the joys and wonders of a national meetings of 6000+ mathematicians!

Go figure!!!!

National Public Radio decided to attend the Joint Mathematics Meetings of the American Mathematical Society and the Mathematical Association of America, the national gathering place for the year of all stripes of mathematicians, to see just what was happening there. The result was a report by Ari Daniel Shapiro entitled

A Unique Expression Of Love For Math

detailing the huge diversity of expression and study, both in the art and the science of mathematics, that mathematicians bring to their profession. The transcript and the audio of the piece is at the link.

What a nice way to view the world of mathematics that we see every day, but which most people never get a glimpse of.

Thank you, Ari and NPR!!

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.