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.

March 25, 2014

An example for “business calculus”

Filed under: applied mathematics, calculus, economics — Tags: , , — collegemathteaching @ 10:49 pm

Consider this article by Paul Krugman which contains this graph and this text:


On one side we have a hypothetical but I think realistic Phillips curve, in which the rate of inflation depends on output and the relationship gets steep at high levels of utilization. On the other we have an aggregate demand curve that depends positively on expected inflation, because this reduces real interest rates at the zero lower bound. I’ve drawn the picture so that if the central bank announces a 2 percent inflation target, the actual rate of inflation will fall short of 2 percent, even if everyone believes the bank’s promise – which they won’t do for very long.

So you see my problem. Suppose that the economy really needs a 4 percent inflation target, but the central bank says, “That seems kind of radical, so let’s be more cautious and only do 2 percent.” This sounds prudent – but may actually guarantee failure.

The purpose: you can see the Philips curve (which relates unemployment to inflation: the higher the inflation, the lower the unemployment) and a linear-like (ok an affine) demand curve. You can see the concepts of derivative and concavity as being central to the analysis; that might be useful for these types of students to see.

February 11, 2013

Gee, Math is Hard! But ignore it at your peril…

Via Slate Magazine: (Edward Frenkel)

Imagine a world in which it is possible for an elite group of hackers to install a “backdoor” not on a personal computer but on the entire U.S. economy. Imagine that they can use it to cryptically raise taxes and slash social benefits at will. Such a scenario may sound far-fetched, but replace “backdoor” with the Consumer Price Index (CPI), and you get a pretty accurate picture of how this arcane economics statistic has been used.
Tax brackets, Social Security, Medicare, and various indexed payments, together affecting tens of millions of Americans, are pegged to the CPI as a measure of inflation. The fiscal cliff deal that the White House and Congress reached a month ago was almost derailed by a proposal to change the formula for the CPI, which Matthew Yglesias characterized as “a sneaky plan to cut Social Security and raise taxes by changing how inflation is calculated.” That plan was scrapped at the last minute. But what most people don’t realize is that something similar had already happened in the past. A new book, The Physics of Wall Street by James Weatherall, tells that story: In 1996, five economists, known as the Boskin Commission, were tasked with saving the government $1 trillion. They observed that if the CPI were lowered by 1.1 percent, then a $1 trillion could indeed be saved over the coming decade. So what did they do? They proposed a way to alter the formula that would lower the CPI by exactly that amount!
This raises a question: Is economics being used as science or as after-the-fact justification, much like economic statistics were manipulated in the Soviet Union? More importantly, is anyone paying attention? Are we willing to give government agents a free hand to keep changing this all-important formula whenever it suits their political needs, simply because they think we won’t get the math?

Well, most probably won’t get the math and even more won’t be able to if some have their way:

Ironically, in a recent op-ed in the New York Times, social scientist Andrew Hacker suggested eliminating algebra from the school curriculum as an “onerous stumbling block,” and instead teaching students “how the Consumer Price Index is computed.” What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI, as well as other evidence-based statistics, is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including … advanced algebra.
Whether we like it or not, calculating CPI necessarily involves some abstract, arcane body of math. If there were only one item being consumed, then we could easily measure inflation by dividing the unit price of this item today by the unit price a year ago. But if there are two or more items, then knowing their prices is not sufficient.

The article continues on; it is well worth reading.

So why does Andrew Hacker suggest that we eliminate an algebra requirement from the school curriculum?

This debate matters. Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources.

The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

Shirley Bagwell, a longtime Tennessee teacher, warns that “to expect all students to master algebra will cause more students to drop out.” For those who stay in school, there are often “exit exams,” almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.

“There are students taking these courses three, four, five times,” says Barbara Bonham of Appalachian State University. While some ultimately pass, she adds, “many drop out.”

Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor’s degrees. The main impediment to graduation: freshman math. […]

In other words: math is too hard! 🙂

Well, “gee, I won’t need it!” Well, actually, math literacy is a prerequisite to understanding many seemingly unrelated things. For example, I am reading The Better Angels of our Nature by Steven Pinker. Though the book’s purpose is to demonstrate that human violence is trending downward and has been trending downward for some time, much of the argument is statistical; being mathematically illiterate would make this book inaccessible.

We some basic mathematics when in discussions on our economy. For example: how does one determine if, say, government spending is up or not? It isn’t as simple as counting dollars spent; after all, our population is growing and we’d expect a country with a larger population to spend more than a country with a smaller one. Then there is gross domestic product; spending is usually correlated with that; hence “government spending graphs” are usually presented in terms of “percent of GDP”. But then what if absolute spending hits a flat stretch and GDP falls, as it does during a recession? That’s right: a smaller denominator makes for a bigger number! You see this concept presented here.

But if you are mathematically illiterate, all of this is invisible to you.

Ever see the “jobs graph” that the current Presidential Administration touts?


What does it mean? It actually demonstrates a calculus concept.

What about risk measurements? You need statistics to determine those; else you run the risk of pushing for an expensive “feel good” policy which, well, really doesn’t help.

Politics? If you can’t read a poll or understand what the polls are saying, you are basically sunk (as were many of our pundits in 2012). Of course, if you can’t understand a collection of polls, you can be a journalist or a pundit, but there is limited opportunity for that.

Science? Example: is evolution too improbable to have occurred? Uh, no. But you need some mathematical literacy to see why.

January 17, 2012

Applications of calculus in the New York Times: Comparative Statics (economics)

Paul Krugman has an article that talks about the economics concept of comparative statics which involves a bit of calculus. The rough idea is this: suppose we have something that is a function of two economics variables f(x,y) and we are on some level curve: f(x,y) = C_1 at some point (x_0, y_0, f(x_0, y_0) = C) . Now if we, say, hold y constant and vary x by \Delta x what happens to the level curve C_1 ? The answer is, of course, C = C_1 + (\Delta x) \frac{\partial f}{\partial x} (x_0,y_0) + \epsilon where \epsilon is a small error that vanishes as \Delta x goes to zero; this is just multi-variable calculus and the idea of differentials, tangent planes and partial derivatives. The upshot is that the change in C , denoted by \Delta C is approximately (\Delta x) \frac{\partial f}{\partial x} (x_0,y_0)  .

It isn’t every day that someone in the mainstream media brings up calculus.

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