Wednesday, June 30, 2010

Mathematics as weightlifting

Every math teacher has heard it: "Why do I need to learn this."  Like nails on a chalkboard!  I find the question especially difficult to answer as a pure mathematician.  I enjoy mathematics particularly because it is not something that can be used -- the abstractness is exciting.  But students don't want to hear this.  Increasingly, students go to college for the purpose of getting a higher paying job, and have little patience for anything they will not need to know in the workforce.

To be clear, there are two very different questions of this type.  One is an honest question: "What can this mathematics be used for?"  This question deserves an honest and careful answer.  The other: "Why do I need to know this if I won't ever use it in my particular job?"  It is this latter question I want to answer here.  Of course, I realize that often students pose the question as rhetorical, as an exhibition of their frustration.  In that case, the following can be used to motivate them to get back to work.

A few years back I was watching a baseball game, and the commentators got to talking about a particular player and how dedicated he was.  They said that he spent even more time lifting weights than his teammates did.  The whole idea of baseball players weightlifting was perplexing to me: why not spend that time at batting practice, or running sprints?  Baseballs are not that heavy, and it's not like once reaching second base the runner must bench-press his bodyweight.  But of course upon further reflection, it became clear that while the players do not need to use the particular weightlifting skills they work so many hours on, doing so makes them stronger over all.  It makes them better athletes.  Similarly, doing mathematics makes (future) scientists better thinkers.

Baseball players are not the only people who lift weights.  Most everyone can benefit from weightlifting as it is a great way to stay fit and healthy.  Similarly, mathematics is a great way to stay mentally fit and healthy.  Like weightlifting, it can be difficult when you first start out, especially if you are not doing it correctly.  But once you get the hang of it, not only can you lift more and more weight, but it will become enjoyable.  In fact, there are people who enjoy weightlifting so much, they do it professionally. 

Tuesday, June 29, 2010

Probability is hard

There is a nice article on the site about the Tuesday birthday problem:

I have two children, one of whom is a boy born on a Tuesday.  What is the probability that I have two sons?

The article does a very good job of discussing the solution, as well as why it is difficult (and why "Tuesday" has anything to do with the solution).  Commenters have pointed out that there is an additional problem: what the meaning of "one of whom" is.  It could mean "at least one of whom," which is the strictly correct mathematical interpretation, or "exactly on of whom," which is what most people would read if they are not being careful. 

And we wonder why students hate word problems!  Problems like this are often offered as examples of how counter-intuitive probability (especially conditional probability) can be.  And probability can be counter-intuitive -- just see the Monty Hall Problem.  However, problems like these two children birthday paradoxes are often confusing mostly because of the wording.  We should be very careful to remove all ambiguity from the question and let the interesting mathematics stand on its own.