Monday, October 26, 2009

The Nobel prize in economics

There is no Nobel prize in mathematics and the rumor that Nobel didn't create a mathematics prize because his wife/mistress/fiance had an affair with mathematician Gosta Mittag-Leffler is in all probability false. There doesn't seem to be a clear explanation of why the field of mathematics is overlooked as prize category.

Mathematics is closely linked to many branches of science. Although not necessarily highlighted, the use of mathematics is recognized, especially with the physics and economics awards. I listen to NPR's Planet Money podcast which covers all things economics and they recently posted an interview with Elinor Ostrom, one of the winners of the Nobel prize in economics, where she explained some of her research.

Her research is about 'the tragedy of the commons' (highlighted in a 1968 paper by Garrett Hardin) which says that if there is a common resource (e.g. fishing grounds that the public makes their living from by harvesting fish), then if this resource is limited then it will ultimately be overused until it is destroyed even if it is not in the public's interest for this to happen. The only way that this can be prevented is for the government to step in and through enforcement, stop the overuse of this resource. In the example of the fishing grounds, the government must come in and put limits on the amounts of fish that people can catch otherwise the grounds will be over-fished.

The discussion in this podcast with Elinor Ostrom highlights that this is a mathematical model which can be used to predict the behavior of a group's action. As she says in this podcast, this model "can be used mathematically to predict outcomes when the problem is pure private goods and you have a highly competitive market." She then goes on to say that it is a challenge to prevent this tragedy from occurring, but since humans are highly complex, this model does not always apply and that solutions for solving the tragedy of the commons can come from within the group.

Some of the ideas about her research became crystal clear in this interview when she was asked about the communal fridge in the office. The 'tragedy of the common fridge' is that it must eventually degrade into a repository for moldy sandwiches. The research of Elinor Ostrom says that, no, in fact human beings are complex enough that they can develop a solution from within the group.

I'm a fan of the podcast because I would like to be more economics literate. Economics reasoning is complex and this podcast looks at a lot of different aspects of it. If you are interested in hearing another good story about how mathematical models can be used to predict behavior, the podcast a day earlier talked about monkey behavior seems to follow a mathematical model for supply and demand, even though monkeys probably don't understand the meaning of 'one half of.'

Sunday, October 18, 2009

Too many being left behind

Perhaps comparisons to the rapture are inappropriate, but recent news about math scores in the U.S. do not look good for future students. Once again a report about the progress of students in math gets every pundit wagging their finger at someone.

It is worth reading the blog entry posted by the New York Times because I found it was way better than any of the news articles that just summarized the test score results.

Here is my summary of the opinions that they highlighted on this blog post

* the education policy professor says: "The culture of standardized testing ... has served to de-skill and demoralize our best teachers." But he offers no real suggestions about what to do.

The other 4 seem to fall into two main groups. The first says improve the skills of teachers:

* the education studies director says: "Teaching methods, curriculum, lack of adequate subject matter knowledge among math teachers and lack of real consequences in school accountability systems, rather than tests and standards, could be the real culprits for low scores." He highlights research that says that eighth graders taught by math majors did better than eighth graders taught by teachers without a math major.

* the mathematician says: "Many elementary teachers have strong backgrounds in reading and writing, but will readily admit their discomfort with math." This is yet another call for ensuring that math teachers have a strong math background. This pundit also falls into the second group and has comments about the math curriculum.

The second group says improve the curriculum:

* the parent says: "If we want to improve mathematics education, we should banish nonsensical curricula like Trail Blazers, Everyday Math and Investigations and make sure that our teachers are properly educated and proficient in math content." To paraphrase, this 'fuzzy math' is a joke and it is not preparing students to succeed in math (what is the color of infinity?).

* the policy analyst says: "What is needed is not another test, but sound mathematics instruction that stresses content over process." He says that content is very important and we are not doing students any favors by not teaching concepts and just teaching problem solving.

My opinion is that both of these are correct. Just saying 'no child left behind' three times, clicking your heels together, and waving a bunch of money around isn't going to solve any problems. A real solution is improving teacher training and getting a content rich curriculum in place.

Saturday, October 10, 2009

Teach the rule or teach how to derive the rule

So after class on Thursday a 'fundamentals' student asked me a couple of questions that I now think are worth exploring a bit on the blog because they can lead to some interesting philosophical questions about math, although at the time I was too hungry to answer clearly.

"Why is a negative of a negative a positive?" and "why is a negative number times a negative number a positive number?" My first thought when she asked me these was 'because' and my second thought was 'chicken sandwich now or a donut to tide me over until I get home?' It was particularly ironic that she was asking me these questions because on the first day of the 'Fundamentals' class I ask at least one similar question to gauge how well students are able to respond as part of their math background (and I probably got a few answers like 'chicken sandwich').

These are probably types of questions that teachers of basic math get asked all the time. Mathematicians encounter statements like this all the time too and we need to justify them in terms of the axioms.

To answer the question "Why is a negative of a negative a positive?" we need to know what 'negative' means: for any element x, -x is the element for which x + (-x) = 0.

This also means that -(-x) + (-x) = 0, so x = 0 + x = (-(-x) + (-x)) + x = -(-x) + ((-x) + x) = -(-x) + 0 = -(-x).

The second question, "why is a negative number times a negative number a positive number?" can be justified using the fact that a negative of a negative is a positive: 0 = a*0 = a*(b + (-b)) = a*b + a*(-b)
so a*(-b) = -(a*b) because a*(-b) is the thing you add to a*b to get 0.
Now (-a)*(-b) = -((-a)*b) = -(b*(-a)) = -(-(a*b)) = a*b

When I was asked this question, the 'fundamentals' student stated it as "Why is 3-(-5) = 3+5?" And to me the good answer to tell a elementary/high schooler would be "well, because that is the rule" unless I really wanted to drive them away from math. To me students just have to follow the rules of basic arithmetic and they have to learn them forwards and backwards. Her explanation was something like "We know 3-(-5) = 3+(-1)*(-5) = 3+1*5 = 3+5." And this explanation was no more convincing to me than my "because that is the rule" answer.

The more I think about it, the more I realize I am not sure how I would respond when asked by an elementary or high school student.

So here is where I get all philosophical (or perhaps I am getting not philosophical, but practical). Should you teach your students how to derive the basic rules from the definitions so that they are easier to remember, or do you just tell them the answer is "because that is the rule" ? As a teacher, do you need to know enough basic algebra to derive these rules yourself or can you accept them "because that is the rule" ?