College Math Teaching

February 9, 2016

An economist talks about graphs

Filed under: academia, economics, editorial, pedagogy, student learning — Tags: , — collegemathteaching @ 7:49 pm

Paul Krugman is a Nobel Laureate caliber economist (he won whatever they call the economics prize).
Here he discusses the utility of using a graph to understand an economic situation:

Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101. I say production possibilities yes, Edgeworth box no — which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box — and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days.

There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.

OK, this is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.

Beyond that, pictures are often the best way to convey global insights about the economy — global in the sense of thinking about all possibilities as opposed to small changes, not as in theworldisflat. […]

And it probably doesn’t hurt to remind ourselves that our students are, in general, NOT like us. What comes to us naturally probably does not come to them naturally.

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March 21, 2014

Projections, regressions and Anscombe’s quartet…

Data and its role in journalism is a hot topic among some of the bloggers that I regularly follow. See: Nate Silver on what he hopes to accomplish with his new website, and Paul Krugman’s caveats on this project. The debate is, as I see it, about the role of data and the role of having expertise in a subject when it comes to providing the public with an accurate picture of what is going on.

Then I saw this meme on a Facebook page:

These two things (the discussion and meme) lead me to make this post.

First the meme: I thought of this meme as a way to explain volume integration by “cross sections”. 🙂 But for this post, I’ll focus on this meme showing an example of a “projection map” in mathematics. I can even provide some equations: imagine the following set in R^3 described as follows: S= \{(x,y,z) | (y-2)^2 + (z-2)^2 \le 1, 1 \le x \le 2 \} Now the projection map to the y-z plane is given by p_{yz}(x,y,z) = (0,y,z) and the image set is S_{yz} = \{(0,y,z)| (y-2)^2 + (z-2)^2 \le 1 which is a disk (in the yellow).

The projection onto the x-z plane is given by p_{xz}(x,y,z) = (x,0,z) and the image is S_{xz} = \{(x,0,z)| 1 \le x \le 2, 1 \le z \le 3 \} which is a rectangle (in the blue).

The issue raised by this meme is that neither projection, in and of itself, determines the set S . In fact, both of these projections, taken together, do not determine the object. For example: the “hollow can” in the shape of our S would have the same projection; there are literally an uncountable. Example: imagine a rectangle in the shape of the blue projection joined to one end disk parallel to the yellow plane.

Of course, one can put some restrictions on candidates for S (the pre image of both projections taken together); say one might want S to be a manifold of either 2 or 3 dimensions, or some other criteria. But THAT would be adding more information to the mix and thereby, in a sense, providing yet another projection map.

Projections, by design, lose information.

In statistics, a statistic, by definition, is a type of projection. Consider, for example, linear regression. I discussed linear regressions and using “fake data” to teach linear regression here. But the linear regression process inputs data points and produces numbers including the mean and standard deviations of the x, y values as well as the correlation coefficient and the regression coefficients.

But one loses information in the process. A good demonstration of this comes from Anscombe’s quartet: one has 4 very different data set producing identical regression coefficients (and yes, correlation coefficients, confidence intervals, etc). Here are the plots of the data:

And here is the data:

Screen shot 2014-03-20 at 8.40.03 PM

The Wikipedia article I quoted is pretty good; they even provide a link to a paper that gives an algorithm to generate different data sets with the same regression values (and yes, the paper defines what is meant by “different”).

Moral: when one crunches data, one has to be aware of the loss of information that is involved.

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