Intuition, Rigor, and Visualization

Way back when, during my first quarter of graduate school in fall 1976, I took what was then the traditional calculus based microeconomics course where we had two textbooks. One was a theory book by somebody named Kogiku (a text I’ve never seen anyone else teach with elsewhere. Indeed, many of the students in the class got a copy of Henderson and Quandt, which was the standard at the time.) The other was an “intuition book” by Gary Becker – his lectures for teaching this course. At least since I got started learning economics, there has always been this tension between rigor, represented by highly mathematical formulations of the economic models, and intuition, which is done in a more discursive style and typically makes anecdotal reference to real world situations.

It may seem strange that an economist would become a teaching with technology guy, there are many more faculty from the humanities, particularly English and there particularly Writing Studies, who have taken this path. But it seems to me not so surprising really, at least in my case, because of seeing so many ordinary students react negatively to the way economics is taught and in trying to work through this tension on intuition versus rigor.

I started out as a complete rigor guy and over the years have softened into an intuition guy. My simple justification for this is that to go the rigor route students have to be able to see through the math into the underlying ideas. If the students can do this, it is a remarkably good way to understand things, because then the students will get grounded in the theoretical justifications for why economists hold such and such to be true. Since I essentially missed undergraduate economics, this is how I learned in graduate school. So I certainly think there is value to the approach.

But, truthfully, many students can’t see through the mathematics (and many more don’t even want to try). So teaching this way at the undergraduate level leads to a schizophrenic type of outcome. The engineering students and the trickle of math majors who take the course are fine with the approach. The business students and those from political science, urban planning, or elsewhere usually are frustrated with it. Teaching the intermediate micro course that so many of us teach either leads to an approach that alienates the students or to the alternative that alienates ourselves in that it is unlike how we learned the subject. Let me get back to that point.

It is fairly standard that topics which were core in graduate school twenty or thirty years ago become part of the undergraduate curriculum today. Nowadays, there are many instructors who will teach the intermediate course at approximately the level of my first quarter graduate course. There is extensive use of LaGrange Multipliers and Bordered Hessians (those are matrices of partial derivatives, not German soldiers). A somewhat more modern version of this theory that emphasizes the duality between allocation and valuation derives many of the fundamental results without calculus (and the Implicit Function Theorem that is the basis for the standard Slutsky decomposition). The duality approach is a little less notation heavy and, I believe, more powerful in developing intuition. It is useful for those who want to do empirics based on the theory, because it emphasizes the testable hypotheses. But it still doesn’t go far enough, in my opinion, to make it really teachable at the undergraduate level.

Understanding economics means seeing the implications of the theory and my argument is that sometimes the theoretical apparatus itself blocks that seeing. Perhaps in my first or second year of graduate school I read a paper by Roy Radner, a very well known theorist, which appeared in Econometrica, the top theory and econometrics journal. The paper was from 1965 or 1966 and was about general equilibrium (that means looking at all markets simultaneously) when different agents in the economy had different pieces of information. It talked about partitions and probability distributions over them and how in being consistent with this differential information, market prices are constrained to be functions only of the coarsest partition that is consistent with each individual’s information. This was the primary result of the paper as I recall. But if asked by a non-economist what was the implication for reality, I couldn’t have told them a thing.

A little while later I read George Akerlof’s famous paper, the Market for Lemons, which appeared in the Quarterly Journal of Economics, 1970. Most people, including many in the profession, probably don’t know that Radner and Akerlof were modeling essentially the same phenomenon. And Radner was clearly first with the result. But it was Akerlof who won the Nobel prize (to be fair both published extensively on other topics). And it was Akerlof who is associated with the idea called “adverse selection,” that bad risks in the market often drive out good risks. The Market for Lemons is mostly a very simple example that plainly illustrates the result in a partial equilibrium (supply and demand) framework. It is teachable at the undergraduate level. And it is an incredibly important insight for a lot of real world economics (like why the unemployed have to pay such high premiums for health care).

I’ve written before about my colleague in the Math department, Jerry Uhl, and on his paper about getting away from lecturing and using Mathematica to teach calculus via giving students examples to play with. Almost all of us would rather learn from example than from theory. That goes for Economics, Calculus, and I believe Pedagogy as well. The example is easier to penetrate. The learner can draw conclusions from it. So the could teacher comes up with examples, to help the students visualize. The difficulty in doing so is that those examples may very well not come from how the instructor learned the subject. The instructor must be creative and generate those examples in a fresh way under the constraints that the students must find them accessible and that they indeed are exemplary of the core idea. This is hard to do, mostly because it requires an appropriate mindset. Some instructors make this leap to hyperspace. Many others don’t.

I wish I could say it ends there and it does for me in teaching a general education course that is outside the major, where the intuitions learned in the course either last a lifetime or are lost. But what of the course that is in the major, where the student is expected to get a deeper understanding of the subject, where rigor is a requirement for obtaining the degree. Does this intuitive approach spoil the student for the harder, more rigorous alternative? How does one reconcile promoting the student curiosity and demanding that arguments be presented in a rigorous fashion?

I don’t know the answer to that, but here is what I’d like to see. From a good example, one should draw an intuition. (If not, what was the point of the example?) The student needs to develop the habit of mind to ask, “Is that really right? Do I really believe that?” And then the student needs to spend some time, perhaps quite a bit of time, answering that question. How does the student do that? By constructing an analysis and reasoning through it. The student needs to be skeptical. The argument could be flawed in some way. And the intuition itself might be erroneous. So the student needs to be patient and skeptical, yet at the same time excited by coming to the possible realization that the example illuminated.

Is this what we teach in the major? If we peel off the disciplinary trappings, does it look at all like I’ve described? If not, is there something else there that allows the students to grow and thrive as the discipline changes? Or do they become used goods as the knowledge they acquire in school depreciates.