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2014 April 11

Arthur Benjamin: Teach statistics before calculus!

I rarely have the patience to sit through a video of a TED talk—like advertisements, I rarely find them worth the time they consume. I can read a transcript of the talk in 1/4 the time, and not be distracted by the facial tics and awkward gestures of the speaker. I was pointed to one TED talk (with about 1.3 million views since Feb 2009) recently that has a message I agree with: Arthur Benjamin: Teach statistics before calculus!

The message is a simple one, though it takes him 3 minutes to make:calculus is the wrong summit for k–12 math to be aiming at.

Calculus is a great subject for scientists, engineers, and economists—one of the most fundamental branches of mathematics—but most people never use it. It would be far more valuable to have universal literacy in probability and statistics, and leave calculus to the 20% of the population who might actually use it someday.  I agree with Arthur Benjamin completely—and this is spoken as someone who was a math major and who learned calculus about 30 years before learning statistics.

Of course, to do probability and statistics well at an advanced level, one does need integral calculus, even measure theory, but the basics of probability and statistics can be taught with counting and summing in discrete spaces, and that is the level at which statistics should be taught in high schools.  (Arthur Benjamin alludes to this continuous vs. discrete math distinction in his talk, but he misleadingly implies that probability and statistics is a branch of discrete math, rather than that it can be learned in either discrete or continuous contexts.)

If I could overhaul math education at the high school level, I would make it go something like

1. algebra
2. logic, proofs, and combinatorics (as in applied discrete math)
3. statistics
4. geometry, trigonometry, and complex numbers
5. calculus

The STEM students would get all 5 subjects, at least by the freshman year of college, and the non-STEM students would top with statistics or trigonometry, depending on their level of interest in math.  I could even see an argument for putting statistics before logic and proof, though I think it is easier to reason about uncertainty after you have a firm foundation in reasoning without uncertainty.

I made a comment along these lines in response to the blog post by Jason Dyer that pointed me to the TED talk. In response, Robert Hansen suggested a different, more conventional order:

1. algebra
2. combinatorics and statistics
3. logic, proofs and geometry
4. advanced algebra, trigonometry
5. calculus

It is common to put combinatorics and statistics together, but that results in confusion on students’ part, because too many of the probability examples are then uniform distribution counting problems. It is useful to have some combinatorics before statistics (so that counting problems are possible examples), but mixing the two makes it less likely that non-uniform probability (which is what the real world mainly has) will be properly developed. We don’t need more people thinking that if there are only two possibilities that they must be equally likely!

I’ve also always felt that putting proofs together with geometry does damage to both. Analytic geometry is much more useful nowadays than Euclidean-style proofs, so I’d rather put geometry with trigonometry and complex numbers, and leave proof techniques and logic to an algebraic domain.

2013 April 29

Scientists need math

Filed under: Uncategorized — gasstationwithoutpumps @ 14:28
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At the beginning of April (but not on April Fool’s Day), the Wall Street Journal published an essay by E.O. Wilson (a famous biologist): Great Scientists Don’t Need Math. The gist of the article is that Dr. Wilson never learned much math and did well in biology, so others can do so also:

Wilson’s Principle No. 1: It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations.

Wilson’s Principle No. 2: For every scientist, there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.

The first principle is probably true, but is more a sociological statement than one inherent to the disciplines: applied mathematicians and statisticians welcome collaborations with all sorts of scientists and are happy to learn about and work on real problems that come up elsewhere, while biologists (particularly old-school ones like Dr. Wilson) tend not to be interested in anything outside their own labs and those of their close collaborators and competitors.

The second principle is possibly also true, though much less so than in the past.  Biology used to be a major refuge for innumerate scientists, but modern biology requires a really strong foundation in statistics, far more than most biology students are trained in. The number of positions for innumerate scientists is rapidly shrinking, while the supply of innumerate biology PhDs is growing rapidly.  In the highly competitive job market for biology research, those who follow E. O. Wilson’s advice have a markedly smaller chance of getting the jobs they desire. Of course, Dr. Wilson seems to be unaware of the decades-long oversupply of biology researchers:

During my decades of teaching biology at Harvard, I watched sadly as bright undergraduates turned away from the possibility of a scientific career, fearing that, without strong math skills, they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.

An undergrad degree in biology (even from Harvard) has not gotten many students much more than low-level technician jobs for most of that time (admission to grad school is the better option, as biology PhDs have been able to get temporary postdoc positions at least).  Perhaps Dr. Wilson considers a dead-end job at little more than minimum wage a suitable scientific career—many others do not.

Dr. Wilson does make one unsubstantiated claim that I agree with:

The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, when tested, fail. Possibly no more than 10% have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.

Biology is a data-driven science, not a model-driven science (a distinction that physicists trying to jump into the field often miss).  Most of “mathematical biology” has been an attempt to apply physics-like models in places where they don’t really fit.  But there has been a big change in the past 10–15 years, as high-throughput experiments have become common in biology.  Now mathematics (mainly statistics) is needed to make any sense out of the experimental results, and biologists with inadequate training in statistics end up making ludicrously wrong conclusions from their experiments, often claiming high significance for random noise.  To understand the data requires more than Wilson’s “intuition”—it requires a solid understanding of the statistics of big data and multiple hypotheses, as humans are very good at perceiving patterns in random noise.

I was pointed to Dr. Wilson’s WSJ essay by Iddo Friedberg’s post Terrible advice from a great scientist, which has a somewhat different critique of the essay. He accuses Wilson of “not recognizing the generalization from an outlier cannot serve as a viable model, or even an argument to support his position.”  Iddo makes several other points, some of them the same as mine—go read his post! Of course, like me, Dr. Friedberg is a bioinformatician and so sees the central role of statistics in 21st century biology.  Perhaps the two of us are wrong, and innumerate biologists will again have glorious scientific careers, but I think the odds are against it.