Tuesday, April 30, 2013

Math Modeling for the Math Phobic Student - The Dilemmas*

*Subtitle - Drill or Meta Skill?

Since 2011 in my Econ classes  I have embraced a "dual track" approach to the material.  One track is narrative-discursive and I have the students blog, with an attempt to tie their own experiences to the topic under consideration.  This track has worked reasonably well and based on responses from the students they like and appreciate that part.  The other track is a math modeling-analytic approach.  This has worked much less well. Teaching math modeling in Econ courses has always been difficult for me dating back to 1980, with the exception of the Math Econ and Honors Intermediate Micro courses where the students had the requisite proficiency.  The rest of the time a handful of students would get it, but the rest would not.  In my last offering this past fall, the better students began to catch on a bit around the middle of the course.  The not so good students never caught on at all.  I have been noodling on what to do about it since I got the course evaluations.  Several of the students wrote that they would have liked to been drilled on the math more.  I found that odd and I've been conflicted about how to respond.  I will teach the same course again this coming fall.  I want to maintain the dual track approach but do a better job with the math track.  I'm still not quite sure what I will try that is different from the last time.  Below are the aspirations and issues.  Taken together they are mutually contradictory, at least in my experience.

Teach in the Language of the Discipline

When I first got started with learning technology I read multiple times that a significant part of undergraduate instruction in the major should be to enculturate the students into the discipline.  In my case this means student should learn to "think like an economist" and it means students should develop some fluency in the language of economics, which is math.  It is this imperative which keeps me with the dual track, even when the math track doesn't seem to work.

A related issue is at what level the math should be presented.  My sense of the answer to this is represented by this video on Jensen's Inequality, where the entire presentation is graphical and the random income has a two-point support, and this follow up video, where the same material is analyzed in an algebraic way.  Students need facility with both approaches, but there is no need to go beyond a two-state world to get at the issues.

I am leaning toward doing this with the rest of the math in the course and take a flipped classroom approach: micro-lectures online and some homework online too but then additional problems for students to work in class for practice.  I believe that one of the big issues is that either students won't practice on their own or they don't know how to do this with the math. 

Provide Students with Durable Human Capital

My class last fall had seventeen students.  Two asked me to write letters for them to graduate school.   I assume that most of the rest will not go to graduate school, at least not immediately after graduation.  (Some of the students were juniors and might ask me for a letter next year, but I can imagine only one or two in that category.)  So there is an issue of what durable value, if any, the math modeling gives to the students who will not study economics beyond the bachelors. 

For an specific model the answer may very well be, none whatsoever.  I do believe, however, that the ability to reason through about reality by focusing on an abstract representation and reasoning about that is a very useful skill for a manager.  I know that when I had my administrative positions on campus I would do that fairly often and I believe it helps to make sense of what is going on.

But, to be fair, then a simple abstract model would have to be constructed to match the situation.  It wasn't that you could simply pull a known model from some list of models and apply it to the situation, though perhaps with budgeting issues you can do something of that sort.

In conjunction with this skill there is a need to develop a healthy skepticism about the predictions of the model.  Abstract models may very well omit features of reality that are actually important to consider.  Expected Utility Theory and what risk aversion means in that setting (the why for students needing to know Jensen's Inequality) provides a very strong sense of rationality.  It is known that most people don't act in accord with the predictions of expected utility theory.  They are instead subject to various "mistakes" due to how a particular gamble is framed. 

A more realistic approach, known as Prospect Theory, takes into account the framing.  But it is more complex to present and it is non trivial what is to serve as the status quo in any particular instance.  So my approach had been not to do prospect theory but instead only do expected utility theory and then provide examples where is sure seems that individuals and groups are not acting rationality.  We use a non-economics textbook for those.  The goal is for students to learn the predictions of the model can be used as a baseline, and is useful for that, but not always as predictions of actual behavior. 

I suspect that getting students to come to such a realization is beyond the capacity of any single course, unless the student already has some fairly sophisticated math modeling capabilities.  This gets me to the next point.

Create Value Add for Students Based On Where They Are

It's no mystery that students are heterogeneous in their prior preparation and commitment to their studies.  With the blogging, I'm able to react to what they write and give feedback situated in their post, the sort of feedback that encourages them to think a bit harder on what they've already said.  It is harder with the math to customize feedback in this way. 

If I were tutoring a student individually and the student got stuck on a problem, I would begin by noting that most of being stuck is not understanding what the question is asking.  Students are usually pretty good at applying a procedure they've been taught.  It is not understanding which procedure is relevant to be applied that is usually the issue.  So I might then offer a suggestion aimed to partially get the student to consider what the question is asking and see if that does the trick.  The goal is to have the student herself reason through as much of the problem as is possible.  In other words, the hints shouldn't be near to a full solution of the problem. 

So I think I could offer the requisite customization if done one-on-one.  But I don't teach the class that way.  At least I haven't so far.  I could try to do some online office hours and see if that would help.  Maybe the flipped classroom approach will be sufficient, though I'm skeptical on this front.  Students who don't get it tend to be shy about it and thus rather than ameliorate the situation they let it fester. 

Have Students Develop an Intuitive Understanding of the Theory

I believe that students should approach the math in much the same way that Sherlock Holmes approaches a mystery.  There are clues.  Those must be uncovered.  Then they must be pieced together to create a picture. 

I'll illustrate what I mean below, but first let me offer a caveat.  In the type of homework problems one can give in a course like this, deductive logic is all that is needed.  Conan Doyle wrote Sherlock Holmes in that manner - it's all deductive logic.  For a manager, however, even after examining the various clues, there remains a fair amount of uncertainty.  Statistical inference is necessary as much or more than simple deduction.  And in practice, administrators frequently make inference based on anecdote only, because statistical information is not readily available.   My point is not to oversell how much a course like this can prepare students for future management work.

The question below is from my second midterm last fall.  I will do a bit of Sherlock Holmes on it to analyze what's going on.  Some students got it.  It's for the ones who didn't that I'm writing this post.

The Sherlock Holmes like student will compare the alternative contracts to the optimal contract given in the problem.  The results is
1.  For the alternative contract both wL and wH are the average of what they were in the optimal contract.
2.  wL is lower in the alternative contract than in the optimal contract while wH is the same in both contracts.
3.  wL is the same in both contracts while wH is lower in the alternative contract.
4.  wL is the same in both contracts while wH is higher in the alternative contract.

Armed with the information and some rudimentary understanding of the theory, the student can perform the appropriate matching.  Without this information, there is no way to get at the right matching.  The intuition I mention will lead the student to make these comparisons.

In my view, it is obvious that these comparisons need to be made, so all the students should have gotten the question right.  Of course, since I wrote the question it is obvious to me.  The performance evidence from administering the exam is that the students didn't think it was so obvious.  Since the course evaluations are anonymous I don't know this for a fact, but my surmisal is that students wanting drill are the same ones who didn't think to make these comparisons on this question.   

I believe that students should be able to make some headway in understanding a situation even when there is novelty to it.  So while I understand that the student may need some practice to master the theory itself, I don't want to have to restrict the testing of that understanding to scenarios that the students have already seen.  Seeing a possible path in a novel situation requires intuition. 

How do we get over the hump to where the math phobic student shows some willingness to embrace the novel situation?  I wish I had a good answer for that question. 

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