A brief post of a few thoughts and a link my friend Tom sent me to an interview with Randall Monroe, the author of XKCD. The interview was a timely read so I’ll start with a great quote:

A good rule of thumb might be, “If I added a zero to this number, would the sentence containing it mean something different to me?” If the answer is “no,” maybe the number has no business being in the sentence in the first place. – Randall Monroe

When we’re teaching mathematics, especially if we’re trying to teach with applications or plausible scenarios we are confronted by all sorts of non-integer craziness. How do we help ourselves and our students put these numbers into context. Is a million a big number? Depends. Depends on what? Teaching moments abound from there. In a similar sense when you have students work on realistic models that require technology the discussion of accuracy and round off errors inevitably comes up. Recently during an session of group work students used software to find the eigenvalues of a 3×3 matrix (of decimals). One of them was something like 7.891393×10^{-17}. Is this an actual eigenvalue? Is something growing extremely slowly? These are good questions that come up when using realistic numbers. Another day recently we looked at the classic predator-prey equations which are nonlinear. They found the two equilibria and, using the jacobian, linearized the system at each point. At the non trivial equilibrium you should end up with two pure imaginary eigenvalues indicating closed cycles orbiting the equilibrium. Many of them approximated the fractions in the coordinates for the equilibrium and because of that ended up with complex eigenvalues that suggested a decaying spiral. It was a matter of 1/30 vs. 0.033. Not a lot but a great conversation starter.

The other quote that stands out to me, from the interview, is

But I’m also wary of people saying “everyone should know” some skill from their area of expertise, because people have their own stuff to deal with. It’s easy for me to imagine an abstract person and then say, “Wouldn’t it be better if that person knew how to program?” And maybe it would. But real people are complicated and busy, and don’t need me thinking of them as featureless objects and assigning them homework. Not everyone needs to know calculus, Python or how opinion polling works. Maybe more of them should, but it feels a little condescending to assume I know who those people are. I just do my best to make the stuff I’m talking about interesting; the rest is up to them. – Randall Monroe

You hear a lot of wishful thinking amount some educators. It’s understandable but on the other hand, what is the point of complaining what they don’t know instead of figuring out what you could show them. I think sometimes it boils down to a complaint “why aren’t they just like me” which is unfortunate because if you take the time to talk to your students, let them write about their beliefs or ideas, even if you don’t agree with them, I think you’ll find they’re a clever bunch of folks interested in different things with different beliefs. In trying to understand those differences there are tremendous opportunities for your teaching, to teach to those who are trying to learn from you as opposed to teaching the way you like to teach.

One last comment on numbers and students. If you use blackjack as an example in a math class (and there is a lot you can do with blackjack) and if you teach them what basic strategy is you might notice some resistance. For example: hit 16 if the dealer has a 10. Not everyone will want to do this. You can try all you want to convince them with arcane symbols but it’s their gut, not their brain, doing the resisting. I have found some anecdotal success with the classes carrying out trials (always hit or always stand) and tallying up the results. Even still, the odds of winning (which are poor) on such a hand are fairly close and you have to wonder if their decision to buck statistics isn’t a good one possibly. Statistically you’ll always hit because you’ll win a couple more 16 vs 10 hands for every hundred such hands you come across. To optimize your long term bankroll the solution is clear, but what if you’re trying to optimize something else? What if you like the risk, the unknown, the fear of losing and the thrill of winning for no good reason. It’s not unreasonable. Even the most calculating among us act irrationally towards our love interest (did you check her investment portfolio and genetic history before marrying and reproducing?). The many conversations I’ve had with students about these topics leave me a bit perplexed but with a feeling that I’m not seeing the whole person because of my own perspective (I’ve happily memorized the relevant strategy chart and deviations, woohoo!)