To make the ideas in these books one's own there are certain things one must do. Some of this is common to making any ideas one's own. One thing, in particular, is unique to doing math. First and foremost you must play with the elements of the model - get a sense of what they are about and familiarity with them, learn little tricks about them that might prove useful later, perhaps find multiple paths to the same conclusion. Developing this sort redundancy is a very good thing. To know something you have to know it in many different ways. Then you have to do proofs. Proofs entail a kind of creativity in coming up with an argument that works. It's not the same creativity as when coming up with fundamentally new results. A student reading one of these books knows that there were others who have come before them and who discovered the results and proved them. But the student isn't sure whether he is replicating the argument previously found or finding a new argument that nonetheless leads to the same conclusion, since the arguments made by the authors are not completely spelled out. There are end of chapter problems that are enlightening to solve. Some of those are proofs. Neither book comes with an answer key.
The third thing is to apply the results in a specific context, really many specific contexts. This is called transfer. To learning theory folks, transfer is the be all and end all demonstration that learning has occurred, to the learner and to any outsider who might attempt to measure the learning. Perhaps play is necessary at the beginning as a means of acquisition. But transfer is just as necessary to show that the play did not go for naught. This gives a two-prong view of learning. Debreu in writing Theory of Value makes the prongs explicit. The first chapter is on mathematics where the fundamental tools and results (convexity, upper semi-continuity of a correspondence, asymptotic cones, fixed point theorems) are obtained. The rest of the book is applications, using the tools already developed in the context of the General Equilibrium model. But Debreu believes in doing the proofs. So his really is a three-prong approach.
Ironically, early Sunday afternoon the family went out for brunch to celebrate my wife's birthday. My younger son is a freshman in Engineering and he said he liked his calculus teacher who makes fun of the engineering students because they want to learn the math without doing the proofs. The older son, a junior also in engineering, verified this proposition. I suppose it balances out because I want to be able to drive a car and I certainly don't care how it works under the hood. But I am a bit befuddled that the older one, in particular, seemed so accepting of this view. How can he be my son and not care about the proofs? Are these attitudes only due to acculturation?
Let me return to the economic theory. In both of these books convexity plays a crucial role, particularly in the form of the supporting hyperplane theorem, Farkas' Lemma in Gale and Minkowski's Theorem in Debreu. In two dimensions this says that through any boundary point of a convex set you can find a tangent line through the point such that the entire set lies on or on one side of the tangent line. The economic interpretation is as follows. If that boundary point is an optimal allocation, then since the tangent line is described by the point and the normal to the line, and that normal vector can be interpreted as a price system at which that allocation maximizes profit in the set or minimizes expenditure in the set, it follows that valuation is the dual of allocation. There is immense beauty in this result and much intuition ensues by applying it extensively.
The world of Adam Smith in the Wealth of Nations is convex. It is a primarily an agrarian world of small farmers and manufactures, shop keepers and small merchants, none big enough to influence the course of events in and of themselves. Only in aggregate does their power manifest. For the mathematics, it is a delightful world in which to reside.
Convexity is not the world of QWERTY. Nor is it the world of the Academic Calendar, where the summer term is of different length than the fall and spring terms. And it isn't the world of social networking and Facebook. Indeed the first two examples in this paragraph are exemplars of Darwinian competition. A certain solution to a particular problem takes hold. Other solutions might very well have been available at that time. Serendipity selects the one we now know about. Then a bunch of related economic activity emerges around those solutions and those solutions become locked in, essentially impossible to alter. The third example is meant to show that what we have now is a kind of Darwinian competition on steroids. The players understand lock in very well and after they have found their toehold many of their actions are made with the intent to accelerate and intensify the lock in.
Given that, I have wondered how these math/economic models have influenced our views of politics. It is well understood that the Adam Smith economic world provides the background setting for Jeffersonian Democracy, the coincidence that Wealth of Nations was published the same year as the Declaration of Independence buttressing this view. Prominent in the Jeffersonian conception is that Government must be severely limited, because it might be the agent of tyranny. It's a point well taken. But in a world where big corporations facilitate customer lock in, might there be tyranny by those corporations as well? This possibility seems exempt from consideration, at least by those libertarians who are so worried about government tyranny. Here the exemplar of the tyrannical corporate titan is probably not Mark Zuckerberg but rather Bill Gates. Yet he remains more of a folk hero than anything else. Might that exemption be because people haven't learned the economics of Darwin but only the economics of Adam Smith?
Much of what we do teach today at the undergraduate level is derived not directly from Smith but rather from Alfred Marshall, the father of marginal analysis. Marshall developed a mathematical approach that is (barely) tractable by the novice in the subject. Because it is intuitive for those who practice the art, marginal analysis dominates the way most people think about economics. Yet marginal analysis is steeped in convexity and hence it limits the possible worlds we might consider. To my knowledge, there is no tractable math version of Darwinian competition suitable for teaching at the undergraduate level. And producing one is no easy task. Marshall's model is static and that greatly aids in making the math tractable. A Darwinian approach is essentially dynamic and stochastic. This makes things very difficult to explain to the non-math person. As Daniel Kahneman convincingly argues in Thinking Fast and Slow, we humans are not hard wired to think about randomness. We impute causality everywhere. A math model that has randomness at its core will be non-intuitive.
What is a conscientious economics instructor to do? One possible approach is to do math modeling for the straightforward stuff and take a discursive but story telling only approach to anything that is more complex. Indeed, I tried that the last time I taught intermediate microeconomics, spending much time on marginal analysis but also covering Paul David's paper on QWERTY. Yet I've found there is a kind of tyranny of the math modeling. Questions about the math models are much easier to place on exams since these questions are "objective" and much more readily graded. Further there is a tradition of economics courses emphasizing the models, so that if one goes over to the narrative approach too far, then even if that makes sense in the context of the course it is apt to be confusing to the students in the context of the tapestry of other courses with some economics that student will take. For intermediate microeconomics and principles of economics too, which serve as prerequisites for subsequent economics courses, there is further an expectation by those downstream instructors that the students will have been drilled in marginal analysis. The upshot is that students treat the Paul David paper as a curiosity only and don't make the connection that the underlying economics in it explains much current economic activity.
Let me turn to a related issue. I taught an upper level undergraduate special topics course last spring on the Economics of Organizations and I'm teaching the same course now. In the spring version my enrollments were very low so I opted to teach the class as a seminar. I decided for the math that I would present some of it, but in order for the students to get their fingers dirty I would have them present on some of the fundamental papers in the area, doing the presentations in teams of three students. I would coach the teams outside of class on the presentations, discuss the ideas in the paper, help them get oriented. This approach proved an unmitigated disaster. Part of it was some team members missing those outside of class meetings. But the bigger part is that the students had no conception about how to present the mathematics. They showed no play with the content and no transfer of the math to related ideas in the paper. With respect to the equations, one team did the equivalent of text to voice software They simply read aloud what was printed in the article. In my view they provided zero value add to the other students in the class.
It has always been a struggle with the student's math ability in microeconomics courses. They've had the prerequisites but that matters not. The bulk of the non-engineering students have never learned how to take the ideas in a math model and make them their own. Many have trouble reading the graphs that are common in microeconomics classes. Because the activity is unpleasant for them, most don't persist with it enough to get familiar with the content.
In the late 1990s I came up with a mechanism that addressed this issue. Students did the written problems, from the end of the chapter in the textbook, in teams. They'd submit each problem online individually and usually in the first go round they allocated the problems by some division of labor among the team members. If they didn't get maximal credit on that initial submission, my mechanism allowed that some other member of the team could resubmit the problem for additional credit. And that cycle could repeat, until the deadline for the homework. The idea behind the mechanism was to encourage discussion among the team members about how to work the problems. Further, and more importantly, I had online TAs, students who had taken the class previously and done well in the class, available to answer questions from the students about how to improve their answers. The TAs had my written up solutions to the problems and they were instructed not to give the students the answers but rather to help them think it through. While the mechanism was far from perfect and we had some bumps along the way, it actually was quite effective. Many students made use of the online office hours and came to rely on their interactions with the TAs to learn the economics. Were I teaching a high enrollment class (that class had 180 students) I'd adopt something similar, even now.
But my class is one tenth that size and it is only me, no TA, not even a grader. So I needed to do something different. The philosophy behind my approach this semester is articulated in an early post to my students. We will do models, and the math with them, but we will critique the models for the tension between the modeling assumptions and reality or for the tension between the assumptions of one model and another. For working through the models themselves my solution has been to produce interactive Excel exercises, dialogic learning objects if you will but of a special type. The next part of this essay will discuss those in some length.
Before getting to that, let me explain my admittedly obscure title. When I was an assistant professor, I was very fortunate that one of my fellow assistant professors in the Econ department, also a theorist, was the daughter of the Belgian Ambassador to the U.S. We became good friends. She taught me many things outside of economics - what very well prepared food was like, how to consider political/social issues from a more global and less provincial perspective, and lots of clever little sayings. One of those was, With if's you can put Paris in a bottle, an admonition to beware of the conditional in one's thinking because anything might follow thereafter. In my use of Excel I make massive use of the conditional - the built in IF function is all over the place and conditional formatting is used as well when giving automated feedback to a student response to a question. I don't know that I've really put Paris in a bottle with these workbooks, but some of it is quite nifty in my view. So the title is a bit of over promotion of the work and at the same time offers a long overdue note of thanks to my friend.
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At this point I've made 3 of these homeworks. The first is a tutorial for the student to learn how they work - what the student must do, what type of feedback the student will get after inputting an answer, how they get credit for the assignment. It doesn't have the interactive graphs that are a feature of the real homeworks, but in other respects the structure is the same. I encourage you to try it. To do so you must download the file You need a current version of Excel but your knowledge of Excel can be minimal. Put in some string for in the NetID cell. Choose an alias. It doesn't matter which here. Know that the students are assigned aliases and they must choose the assigned alias to get credit. But you are doing it without caring about the credit so it doesn't matter. It should only take 5 minutes or so to complete. You have to get all the answers Correct to be able to submit it. There is no partial credit. There is a participation credit when a student gets it all right and then submits the key via the Web form.
The second is a review of efficiency concepts that students should learn in intermediate microeconomics. (It turned out that many students hadn't previously seen the Edgeworth Box, the topic of the second worksheet.) It has the interactive graphs so you might try this one too, at least answer a few questions correctly and see how the graph updates when you do. This is achieved by having the series that generate the graph to be conditional on whether the student has answered the appropriate question correctly. For example, this is the entry in the cell that gives the series name for the series called Demand Curve.
=IF(H32="Correct","Demand Curve"," ")
Let's deconstruct this one entry. We'll leave it to the reader's imagination to understand how the rest of that series and other series are constructed. In this case the student puts an answer into cell G32. The feedback the student gets is in cell H32. The content of cell H32 has one of three values. It is blank if the student has yet to put an answer in G32. It is Incorrect if the student has put in a wrong answer in cell G32. And it is Correct if the student has put in a correct answer in cell G32. The quotes you see denote a text string between the left quote and the right quote. Absent the quotes Excel interprets entries numerically. The IF statement above can then be read as: if the student answers the question correctly then title the series Demand Curve, otherwise title the series with a blank string.
The student doesn't see this IF statement. It comes from cell AC34. But that cell is locked, the font in the cell is white so can't be distinguished from the background, and the worksheet is password protected so the student can't access the cell's content. I hope this gives a little sense of how the worksheet is constructed. There are quite a few other graph plotting tricks that are used, so things get labelled in a readable way to relate what the student is answering to what is being plotted in the graph. I won't cover those tricks here. But I will say that others can be taught them so that if this approach is liked, it can be replicated after some instruction on how to do it.
The real key to whether this approach provides effective instruction is not the cutesy use of technology, however. It is the quality of the authoring. Do the students see the value add to them from doing the homework? One critical idea here is that the homework is not assessing what students should have already learned elsewhere. The learning should be happening while the student answers the questions. In that sense, the aim of the homework is to be like play in getting the student familiar with the issues and to encourage her to spend enough time on the content to get some familiarity with it. The further aim is to convince the student that the models provide insight that wouldn't be apparent from a text only approach. And, of course, I hope student gets some insight into the underlying economics.
It is hard to author this way, no doubt. But one does develop facility with the approach over time. So the bigger issue with the writing is coming up with what one wants to do conceptually and then how to illustrate the concepts. The actual placing entries into the cells then goes reasonably quickly after that.
I am optimistic about the approach and the early feedback I've gotten from the students indicates they like this too. We'll see if I can keep it up and sustain their interest in the course this way.
The third workbook is about coordination problems and coordination mechanisms. It's a live homework now. You are welcome to try it, but please don't submit your key. It is the first homework with content relevant for our class that they likely haven't seen in other classes.
I believe this approach can work in other courses that have some math in them but that aren't only about the math. Perhaps they could work in math only classes but, but the math instructors I know tend to favor their own tools (not Excel). Way back when I was interested in this sort of thing so the assessment would happen on the local computer rather than on the LMS server and thereby not clog the server near a homework deadline for a big class. That issue is somebody else's problem now. My interest now is the blend of presentation and assessment delivered in homework in a highly interactive way, one that students find compelling. The authoring environments provided to most instructors don't allow for his possibility. But these Paris in a Bottle Excel Exercises are meant to show it is quite possible to deliver very rich content in this way. We should be going after that in what we give our students, especially the implicitly math phobic ones, who won't make good progress with more traditional methods.