Gas station without pumps

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
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.

1. Looks solid to me.

I got to talk with Arthur Benjamin in person a few months ago and I asked him about the argument that Calculus is required to understand Statistics (for example, deriving the integral under the normal curve). He response was “to fully understand Calculus in the same way, you need Analysis”. You could take the argument further and say to understand Analysis you need Set Theory, and to understand Set Theory you need Category Theory, and then everything devolves into absurdity.

Comment by Jason Dyer — 2014 April 11 @ 13:15

• I like the idea in general, but my initial thought is that statistics gets really complicated and difficult really quick. So I wonder how much statistics can be taught at various levels in HS.

I’m a big fan of the idea of infusing logical and critical thinking as a different path within the scope of Math. I suppose it’s not numeracy but would arguably be more interesting and relevant to the general population.

Comment by bcphysics — 2014 April 14 @ 10:12

• I’m not talking about really complicated statistics—just as high school physics is not about the details of QCD, high school statistics can stick to the basics. What is currently covered in statistics for biologists or AP statistics is at about the right level for everyone. People who are going to use statistics professionally (bioinformaticians, actuaries, data miners, sociologists, … ) should go further, of course, but that can wait until college.

Comment by gasstationwithoutpumps — 2014 April 14 @ 10:36