But Which Math...?

The League of Ordinary Gentlemen is having a symposium series of posts about higher education in the 21st century.  James K, one of the regular front page authors, put up a post on the question of "Yes, But Which Arts?" with regard to what should go into a degree in the liberal arts.  One of the other regulars expressed the opinion that stopping at College Algebra was sufficient to "conceptually understand higher mathematical concepts, even if you have no idea how to work them."  James disagreed; I disagreed much more strongly.  Stopping at algebra leaves you woefully ignorant of almost all of the powerful ways that math can be used to think about the world.

I admit to being prejudiced on the subject: I'm an applied math guy and have spent essentially all of my adult life using math as a tool to think about the world.  Not the only tool, but an important one.  Several of the other commenters display their own prejudices (in the non-derogatory sense of that word).  Some advocate learning at least one, if not two, foreign languages.  Some emphasize philosophy.  Some English Lit.  I found the number of different reasons for including English Lit to be interesting, since they covered a gamut from analysis to composition.  I'm absolutely an advocate of a composition requirement, but would prefer for it to be done absent the literature component.  My high-school composition class was enormously valuable: writing every day, with requirements to write a variety of things, and feedback on what you had written.  If pressed, I'm on the side that says no one should be allowed to graduate with a four-year undergraduate degree without taking such a class.

James' question prompted me to think about what math material should a well-rounded college education include?  To put limits on things, I assumed six semester-hours, and six topics to be covered.  I also assumed a mastery of algebra as a prerequisite, on the theory that anyone starting a four-year degree program without that much high-school math is deficient.  In no particular order, here's my first cut at a list of six topics.

• Enough probability to understand why my favorite bar bet is a sucker bet: We'll go around this crowded bar and ask people their birthdays (month and day, ignore the year).  I'll bet that at least some two of them will have the same birthday.  Loser buys the next round.  If you take me up on it, I'll be buying less than a third of the time.  Why?

• Enough statistics to understand that all of the questions you should be asking involve "What's the distribution?"  In the real world, descriptive statistics almost always provide an incomplete picture; you want to know about the distribution.  If there are 30 people in the bar, and one of them is Bill Gates, asking questions about wealth and income really do require you to know about Bill.

• Enough differential calculus to understand basic optimization.  I'm not particularly concerned about whether or not you can solve the problems; can you set them up?  An individual license for Mathematica doesn't cost much more than a high-end graphing calculator.  In addition to doing everything that the calculator can, Mathematica is also far better at taking derivatives or integrals than you or I will ever be.

• Enough integral calculus to understand the "infinite summing up" aspect.  I've always felt that this was the important aspect of the integral as applied to real life.  You can answer an enormous range of "how much" questions this way.

• An introduction to something like discrete dynamic systems, including feedback loops.  I think that sort of thing is much more accessible than writing systems of differential equations.  For some of the packages implementing this type of discrete model, there are graphical tools that help put things together.

• Finally, a module on graph theory with an emphasis on algorithms.  A lot of graph theory problems are "non-math" math -- no numbers involved at all.  I think it's important that non-mathematicians understand that math doesn't have to be just about numbers.