Friday, November 15, 2013

Common Core and the Law of Large Numbers

Common Core is a big story that needs to be addressed in depth (Valerie Strauss's column is a good place to start), but a lot of the devils are in the details and one of thee peskiest of those devils (call him the Mephistopheles of education reform) is what happens to high-sounding ideas on the way to actual classrooms.

Let's take the proposed standard that all students should understand the law of large numbers. This a wonderful goal, but before we add it to the curriculum, we need to think about the Luskin effect. Donald Luskin is the CIO for the consulting firm Trend Macrolytics. He's also a widely read columnist and commentator on financial matters. He's someone who ought to understand sampling and who thinks he understands it, but he really, really doesn't.

You will occasionally find an algebra teacher who obviously doesn't understand something like factoring trinomials, but that's rare. Finding a high school algebra teacher (or for that matter, a university math professor) who doesn't understand probability theory is not that uncommon and a sufficiently clueless explanation can be worse than letting the topic wait until college.

When I Googled common core "law of large numbers" this was the first non-video that came up:.
By (date), when given (5) problems involving interpreting results from a simulation using The Law of Large Numbers (i.e. (# of times an event happens) / (total # of trials) approaches the theoretical probability for the event as the # of trials grows large), (name) will correctly solve (4 out of 5) problems.

Example: A student rolls a fair, 6-sided die 10 times and gets the following results: 4, 2, 4, 3, 5, 6, 6, 2, 4, 6. How many times do you expect that the student will roll a 1 after 600 rolls?

Answer: P(rolling a 1) = 1/6, (1/6)*600 = 100 times
This is a terrible example though there's some ambiguity about exactly why it's so bad. If they mean 'expected' as in 'expected value' then the answer is technically correct but has nothing to do with the law of large numbers. If they mean 'expected' in the common usage sense, the answer is just wrong.

I checked few of the other links from my Google search. Lots had simulation results (which was a good first step) but I don't think I saw any that truly got the concept, at least not well enough to explain it. Better than this but not that much better.

Concepts like the law of large numbers are not deadwood -- they are important and useful and if you can actually find a way for students to master them you should do it -- but they share a common problem with jetsam like synthetic division. There is always an impetus to add them to a curriculum but little counterbalancing pressure not to waste students' time.

The announcement of a new curriculum is invariably followed by a round of hearty round of self congratulations and talk of "keeping standards high" as if adding a slide to a PowerPoint automatically made students better informed. It doesn't work that way. Adding a topic to the list simply means that students will be exposed to it, not that they will understand or master or retain it.

If we start talking about setting aside significant time to cover probability and statistics accurately and in reasonable depth and put the ideas in proper context, you have my enthusiastic support, but until then maybe we should focus on the understanding, mastery, retention of the stuff that's already in the curriculum.