At the banquet I attended recently, the high school senior I sat next to asked me two very interesting questions about his future education, one I had thought a lot about and one that I hadn't.
I'll tackle the first one now: what math should he take in college?
My own B.S. program required only Calculus. To be honest, I've never actually had to use calculus in my studies; the one time I attempted to integrate a function it turned out to be one without a computable derivative! But I would hesitate to ditch Calculus altogether, because it is a powerful concept and the concept of integrating shows up so often.
The easy recommendation was statistics. Of course, one needs a good statistics course -- the one course I took was awful. The main problem it had been dumbed down to meet the perceived weaknesses of the business majors for which it was a requirement, spreading over three semesters what should have been no more than 3/4 of a semester. It was only later I realized that I should have taken the Psych department's stats class, as it emphasized experimental design. Ideally, a modern stats course for biology would give a heavy exposure to Bayesian statistics and touch on topics that may see limited relevance in other areas (such as the Extreme Value Distribution -- important to search programs).
IMHO biologists would also be served well by a survey of various mathematical disciplines, a survey that would emphasize understanding of the key concepts over being able to execute all the calculations. I realize that is probably heresy in many math circles, but it would be very useful. For example, basic topology and graph theory -- everyone should understand the difference between a Directed Acyclic Graph and an Undirected Cyclic Graph without being able to prove theorems about them. You're going to run into eigenvalues constantly in the bioinformatics literature; demystifying them early is important. At home we have a great textbook (my mother collected them for tutoring) called Mathematics, A Human Endeavour, which has just the right flavor.
It should come as no surprise that I believe every science student should have an exposure to computer programming. Even if you don't ever write code for fun or profit, the underlying thinking is very useful. Understanding the fundamental data structures (tree, linked list, etc) and some common algorithms should simply be part of everyones intellectual foundation.
Given the opportunity (make me education czar!), I would, of course, like to see these things pushed far lower. My own math education, which I think is representative of typical in the U.S., was far too unaggressive and bored the heck out of too many people too quickly. Subjects like plane geometry need to be distilled down to their essence, with the remaining time filled with a taste of programming, which involves similar thinking to mathematical proof execution. It appalls me now that at an early age I learned how to compute mean and median, but nowhere in grade school was it pointed out when one might be superior to the other (especially since this distinction is so routinely ignored by the media).
One last comment: if my math education was representative, then too many math programs fail to stimulate real excitement in math. I do feel it is important to learn to compute certain things, and you really need to have the basic multiplication tables memorized to have any luck at algebra, but too much drilling flattens the excitement. I was disappointed, but not surprised, that my tablemate had never heard of the Seven Bridges of Koenigsburg. I doubt he had been exposed to snowflake curves (a simple to sketch fractal) either, a concept that blew my mind back in 3rd grade or so -- but it wasn't in the curriculum. A little less drilling, a little more mind expansion & math education would be on the right track.
The college I went to had a similar stats course. It was required for psych majors and I think poli sci also. Lots of graduate nurses and biologists took it as well due to the focus on research design. It was great to take a course where you could see the applicability of the math immediately.
I was ill-served by both my stats and physics courses, which were perfunctorily designed to fill out a pre-med curriculum. My professors honestly didn't realize how important those fields had become to biology; none of them were molecular biologists. And I never had time in grad school to play catch-up, so I'm doing it now, more or less. I completely agree that an introductory survey of techniques is needed - one can always delve deeper in the areas relevant to one's own research later on. But it's very hard to self-direct study when one doesn't have any clue what, for example, "Bayesian reasoning" is (would you believe I first discovered it on the internet after my PhD was over?!?)
I didn't hear about Bayes until an American Scientist article I happened to pick up in grad school. Alas, one of the great illusions in life is that your need for education, even after a quarter century of formal instruction, can ever be sated
Was this the question you had thought a lot about, or the one that you hadn't? And what was the other question?
The one I had thought about; yes, I will soon get around to the other one...
Are you related to John Elder Robison, author of the book, Look Me In My Eye?
I do not know of a relation, though his book on Asperger's sound's interesting. First, I really should get through Animals in Translation.
When I first heard of people putting their genealogies on the Web, I joked I would stumble across a long lost cousin using Google. Several years later I did precisely that, as I found my great-great grandfather in someone's online tree.
Perhaps you would be more interested in "Born On A Blue Day" by Daniel Tammet...? He also has a website: Optimnem.co.uk
I'd be interested to know what your thoughts are on the way he thinks...
Keith, both Daniel Tammet and I have savant abilities. In my case, I could visualize complex calculus functions as implemented in the real world.
One of my gifts was in the area of "seeing" time delay filter systems.
For example, if you delay a signal stream in time, and remix the delayed signal with the original, you get an output that, viewed in the frequency domain, has zeros and peaks every 1/x delay period across the spectrum.
That's called a "comb filter" because the graph of frequency response looks like a comb.
As you can imagine, the math when you combine 4-5 or 10 of these combs at one time, and mix them in various combinations . . . it becomes extremely complex.
I was able to see and "hear" the outputs of those systems in both the time and frequency domains with "slide rule" precision.
And with that "rough answer" I could tune the filter parameters to achieve a particular signal processing goal, which in my case was the design of target search sonar or sound effects in the music recording industry.
Examples of my filter concepts, implemented digitally, can be heard in the sound effects that seem to float into the distance in some of today's games. They are - in more sophisiticated form - on countless record albums, too.
The "savant" concept comes from the fact that I have no formal education at all beyond the ninth grade, and I have no real math training beyond basic multiplication and division. So I cannot write the equations even though I can work them in my mind.
I cannot really tell you how that happens.
Daniel Tammet has similar natural abilities, but to a much greater degree.
As to your ancestral connection . . . I have an extensive genealogy database online called "Cubbie's Ancestors"
I am also part of the Robison Genealogy project at Family Tree DNA.
Join up - maybe you are related, from centuries past.
And you can order my book on Amazon and all the other online retailers. It's in stores September 25th.
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