The Teacher's Forum:

Teaching the Art of Modeling to MBA Students

Stephen G. Powell
Tuck School of Business, Dartmouth College
Hanover, NH 03755


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Teaching in one form or another is of vital importance to most of us in the MS/OR community, whether we are in industry, academia, or consulting. For the community as a whole, teaching pays many of our bills, informs our current and future clients, and trains our successors. Despite this, teaching is poorly represented in our journals and professional conferences and is often treated as the stepchild of the profession. Furthermore, our teaching is under unprecedented critical scrutiny and is simultaneously undergoing a revolution in methodology.

The purpose of this column is to provide a forum for the presentation and dissemination of new and effective approaches to teaching MS/OR. I will contribute an occasional column based on my experiences in teaching MBA students, and I will also solicit contributions from the many innovative teachers around the world who are, for the most part, laboring in isolation. I invite you to submit articles on your own ideas and experiences, as well as your comments on articles that appear here.

Teaching the Art of Modeling to MBA Students


For the past six years I have taught a second-year elective course on the art of modeling to MBA students at the Tuck School. This course has met with some success, and since my experience challenges the accepted wisdom that MBA students are not receptive to quantitative methods, I would like to share some of my experiences here. I want to describe the philosophy behind the course and the manner in which it is delivered, to attempt to account for its acceptance, and to encourage you to experiment with similar pedagogical approaches.


Let me begin by making distinctions between problem solving, modeling, and MS/OR tools. I will use the term problem solving broadly to describe all the activities between the initial problem identification and the ultimate problem solution (or recommendation to the client). Modeling, or model building and analysis, is one of several phases within a problem-solving effort. Depending on the problem at hand, modeling can range from the simple to the complex. Simple modeling can include mental models and basic spreadsheet modeling, while complex modeling can include statistics, optimization, or any other appropriate mathematical method. The MS/OR tools, on the other hand, are a collection of specific analytic methods (optimization, simulation, and so on) that can be applied at the analyst's discretion within the modeling phases of an overall problem-solving effort. Thus the tools of management science are contained within modeling, which is itself contained within problem solving. The overall goal of a course in the art of modeling is to improve the problem-solving abilities of the students. A secondary goal is to improve their modeling capabilities. Improving their abilities in applying MS/OR tools is only a tertiary concern.


If I were a dean reviewing a proposal for a new course in "The Art of Modeling," the first question I would ask would be, Why teach a course in modeling to MBA students? Then I would ask, What can we expect MBA graduates to do on the job with modeling skills? How will we teach them these skills? Finally, what is the connection between a course in modeling and existing management science courses? (At Tuck management science is a required, first-year course.)


My answer to the first question consists of three points. First, modeling is one of the fundamental ways in which human beings understand the world. Most modeling is informal, using mental models, but formal modeling is a powerful way to understand the world and to communicate one's understanding to others. Second, the typical MBA curriculum teaches a great deal about models (from LP, to Black-Scholes, to the EOQ, to macroeconomic models) but very little about the broader and more fundamental activities of modeling and problem solving. Finally, if we want our graduates to be flexible and original thinkers, we must teach them as much about modeling and problem solving as we do about models.


My short and only partially glib answers to the remaining questions are as follows: How will they use modeling on the job? They will think better, which I accept intuitively as a basic goal of education. How will we teach it? Like any art, we will teach it in a studio. How does it connect to management science? Management science teaches some of the basic tools of quantitative analysis but it cannot, in the short time available, give the students experience in the craft aspects of modeling: problem formulation, sensitivity analysis, use of modeling heuristics, presentation of results to clients, and so on. Thus a second-year course in modeling offers the students a chance both to learn the craft of modeling and to put into practice many of the specific analytic tools and models learned in the first year.


This course has its origins in a course begun by Seth Bonder and Steve Pollock at the University of Michigan 20 or 30 years ago. They called their course the "Modeling Studio" [Pollock 1976]; their motivation was to give OR students experience in applying their tools to real problems. They did this by posing a series of problems to the students and giving the students free reign to work out solutions. Mike Magazine subsequently picked up this idea, offering a course over a number of years at the University of Waterloo in which the entire class spent a term working on a real problem for a single client. I encountered the course as a graduate student in the mid-70s in the Engineering-Economic Systems Department at Stanford, where it was (and is) taught by Dick Smallwood and Pete Morris. Their course is built around a weekly modeling assignment, which students tackle on their own. At the end of the week the teachers present their approach to the problem and discuss the students' attempts. In my day, this was known as both the best and the hardest course in the department, not because of the technical difficulty of the material but because of the open-ended nature of the intellectual challenge involved in working out a complete model and problem analysis every week. As far as I know, my course is the only attempt to adapt these ideas to the MBA curriculum, except for a course given by Anirudh Dhebar (also an EES graduate) at the Harvard Business School several years ago. (Courses on modeling taught in other settings are described by Cross and Moscardini [1985], Miser [1976], and Willemain [1994]).


My course rests on two critical assumptions. First, I believe everyone (literally) can use the basic tools of modeling effectively in ordinary life. For example, one cannot think rationally about saving for retirement without being able to build and exercise a simple model (probably using a spreadsheet). Prospective managers are even more in need of these capabilities than the layperson, since they will be in the business of allocating resources and choosing alternative courses of action in a complex and uncertain world. My second assumption is that since so much of modeling is an art (or craft), it should be learned in the way one learns an art: by doing. Thus my course has almost no lectures and a maximum of learning by doing. My model for teaching the course is a teacher of painting or drawing in a studio course: I select the subject for the day, I assist and critique the students' attempts at representing it, and I track their progress and help them with continuing technical and emotional hurdles. But I never paint for them, to do so would intimidate them, reduce their motivation, and slow or reverse their learning. (Willemain, however, makes innovative use of audio tapes of expert modelers in a modeling class [Willemain 1994]). The ultimate goal of the course is for students to develop skill and confidence in using modeling to analyze important problems and especially to stand on their own feet intellectually.


The course is organized around two loosely interconnected activities: a weekly modeling assignment and in-class modeling sessions. I will describe each in turn. Each week’s assignment is the same for all students. I randomly assign students to teams of two to work on the common modeling problem. These assignments place the students in the role of consultant to a decision maker. In the course of the week they must complete the entire cycle from problem formulation through to the recommendation to the client. Each team writes a short, nontechnical report to the client, and several teams (usually three) make brief (15 minute) presentations of their work to the class. I am available during the week for the students as a senior consultant: I will listen to their approach, make suggestions, and provide technical assistance where needed. I make every effort during these sessions not to mold the students' thinking on the problem to my own but to act as a catalyst in their thought processes.


A typical week-long assignment is Smallwood and Morris's Racquetball Racquet case. You have been asked by a friend to invest in his new technology to produce balls for playing racquetball. His technology is cheaper than the existing method and produces balls with somewhat more bounce but less durability than the current ball. A limited amount of realistic market research is available on consumer preferences and sensitivity to price differences between the balls. Your task is to develop a credible analysis of the market potential for this new ball and to decide whether you should invest.


This simple-sounding problem contains plenty of challenges for the students. First, they need to think through how a market with one product is changed when a second product is introduced. Some historical data on price and sales are available, but to use the data the students must disentangle what they know about an own-price elasticity from the effects of market growth and cross-price elasticity. And they must use their common sense, which they often seem unwilling to do in the face of seemingly objective data. Another major challenge is to think through the likely reaction of the competitor to the introduction of a new product at a lower price. Well-trained students often see this as an exercise in game theory, only to discover that they don't know enough about game theory to use it to find a definite answer. Once again, they are forced to use both what they have learned and their common sense to bring some order to a frequently occurring problem: pricing in the face of entrenched competition.


The second major activity in the course is a series of short modeling exercises designed to give the students further practice in modeling and to reinforce modeling principles. In these exercises, I present a problem to the students at the start of class, and we spend a 90-minute class working on it as a group. This allows me to demonstrate facilitation techniques, to help the students observe the use of modeling heuristics in the heat of modeling (Simplify, simplify, simplify; Can you draw a graph?; Is there a special case?; What happens at the extremes?) and to demonstrate for them the interpersonal behaviors I think are effective in modeling, such as showing enthusiasm for others’ ideas.


One very rich source of these exercises is the book How to Model It - Problem Solving for the Computer Age [Starfield, Smith, and Bleloch 1990], which not only contains nine or 10 interesting problems but works out in great detail how a practical modeler might approach each problem. I recommend this book to anyone interested in modeling, whether or not they plan to teach the subject. My favorite exercise for class use is "The case of the hot and thirsty executive" (Chapter 4). Your boss has gone to his (or her) summer home on a hot day, only to find that the beer he so much needs has not been put in the refrigerator and the refrigerator itself is malfunctioning. His question to you is, How soon will the beer be cold enough for him to drink?


Again, a seemingly simple problem becomes a vehicle for illustrating some basic modeling precepts. First, in order to model the evolution of the temperature of the beer the students will need to rediscover difference equations, using common sense to formulate a model in which the temperature at one time period is the temperature at the previous time less the heat lost over the interval in between. Next they will come up with a multitude of factors that influence the heat loss (type of container, its shape, and so on), and they will then be in a position to discover the notion of lumping parameters. Finally, they will discover that one elegant way to determine which of an entire family of curves applies in this case is to call the boss and ask what the temperature is after some time has elapsed, illustrating the effective use of data collection within the context of a model built from first principles.


* * *


What are the key ingredients that make this course succeed in my environment? In no special order, I can identify seven:


(1). A focus on creativity,

(2). Supportiveness,

(3). No right answers,

(4). An emphasis on heuristics,

(5). Practice in written and oral presentation,

(6). The use of teams, and

(7). The use of spreadsheets.


One of the ways I have made this course intriguing and appealing to MBA students is to stress the role of creativity in problem solving and modeling. Jim Evans [1992] has written in Interfaces on this subject; see also his 1991 book Creative Thinking in the Decision and Management Sciences. In one of the early class meetings I present the Osborne-Parnes Creative Problem-Solving Process [Evans 1991]. This becomes the common structure on which I suggest the students pattern their weekly modeling activities. I also engage the students in a number of creativity exercises in class in the early part of the course. Finally, I repeatedly use the terms divergent thinking and convergent thinking to give form to the students' struggles to balance the need to be creative with the need to test and refine their ideas into a workable approach.


Just as I emphasize creativity, I make a considerable effort to be supportive of the students and to play down my role as judge and evaluator. This approach came about naturally from working with the students and sensing their real fears associated with creative activity. It is easy to be supportive if one has any sense of humility about modeling, and I have found nothing so humbling as modeling. I tell the students they will go through a predictable series of emotional ups and downs while they struggle with the problems I set them, and then I spend a fair amount of time counseling them through their down periods. I know I am succeeding when a student comes to me and says he or she is confused and depressed about this week's modeling case - and is really enjoying the course.


The principle of "no right answers" is another key to making this course succeed with MBA students. Although I help the students often during their efforts to model a problem, I never present to them the right solution or even my solution. In fact, by having three groups present their analyses on the same day, I hope to convince the students that there are many effective approaches to the same problem. Again, I know I am succeeding when three groups present equally convincing and totally different approaches and solutions to the same problem. (This is a profoundly disturbing experience for some students, who have trained themselves to seek the one right answer, usually meaning the teacher's answer.) Only then will the students really believe that problem solving is fundamentally a personal process, in which each person can come up with his or her own believable and valuable result. In my experience, it is often the most mathematically well-trained students who find this principle most difficult to accept. Many other students, however, must be freed from their misperception that all quantitative courses are just like their high school algebra course: right and wrong answers and no debate. This is not to say, however, that I never criticize mistakes or challenge their approaches; rather, that I try to give them maximum freedom to make their own assumptions and take their own approaches, and then I try to channel my criticism toward improving their work.


My skeptical colleagues like to ask of this and other unusual courses, What is the intellectual content? This is a legitimate question and one that can be hard to answer for any course whose method is essentially experiential. I have come to believe that the answer lies in the role that problem-solving and modeling heuristics play in the course. By heuristics here I mean rules of thumb for effective modeling. This is such an important topic that I plan a later column on it. My favorite examples are drawing a graph to conceptualize a relationship; testing a model's behavior with special or extreme cases; simplifying when the model is too complex to understand; and prototyping, that is, building a simple (even toy) model first to create a base for further refinement and improvement. The in-class modeling exercises are my major vehicle with which to point out and reinforce the effective use of heuristics. I have found that these rules of thumb offer the modeler a practical tool kit for dealing with the inevitable points of confusion or stuckness during the modeling process.


Recruiters of MBA graduates consistently stress the need for good oral and written communication skills. Most students, however, make no connection between quantitative courses and communications, while all of us in the profession know how important these skills are to effective interaction with our clients. So it is important that this course requires constant written and oral communication of often technical ideas. This makes the course appealing to the nonquantitative students; it also helps to break down the artificial division of the world into two cultures, one verbal and one quantitative.


Teamwork is another skill the recruiters cite as a major predictor of success in business. Teamwork is so ingrained at my school that it is difficult to get the students to work on their own. I have found that the challenge of solving a modeling problem each week is sufficiently high when the students work in teams of two that I have not tried to force them to work solo. Teams of three or more, by the way, seem to be less effective than teams of two.


Finally, I come to the use of spreadsheets. I am one of those who believe MBA students can be taught management science very effectively using only spreadsheets (with the built-in optimizers and add-ins for simulation). I have converted my first-year management science course to spreadsheets, so that it is natural to encourage the students in the modeling course to use spreadsheets as well. There must be cases in which useful models cannot be built in a spreadsheet, but MBA students accept the spreadsheet as a legitimate business tool (where they reject less familiar software), and the great majority of the problems they will actually model themselves can be handled quite effectively using a spreadsheet.


* * *


I do not claim that these principles apply or that this course will succeed in any environment other than my own. I have a notion that every successful course represents a series of adaptations to the local environment, and it is a difficult for those within a given culture to see its peculiarities. Thus I offer no guarantees, just an exhortation to give a modeling course a try and to let all of us in the teaching profession know of your successes and failures. I will say that this course is both the most challenging and the most fun of all the courses I have taught, and that one hour modeling with students does more to keep me interested in management science than do dozens of hours teaching optimization.





I would like to acknowledge helpful comments from Hugh Miser, Tom Willemain, and Mike Magazine.





Cross, Mark A. and Moscardini, A. O. 1985, Learning the Art of Mathematical Modelling, Ellis Horwood Limited, Chichester (John Wiley, New York).


Evans, James R. 1991, Creative Thinking in the Decision and Management Sciences, South-Western, Cincinnati, Ohio.


Evans, James R. 1992, "Creativity in MS/OR: Improving problem solving through creative thinking," Interfaces, Vol. 22, No. 2 (March-April), pp. 87-91.


Miser, Hugh J. 1976, "Introducing operational research," Operational Research Quarterly, Vol. 27, No. 3, pp. 655-670.


Pollock, Stephen M. 1976, "Mathematical modeling: Applying the principles of the art studio," Engineering Education, November, pp. 167-171.


Willemain, Thomas R. 1994, "Insights on modeling from a dozen experts," Operations Research, Vol. 42, No. 2 (March-April), pp. 213-222.


Starfield, Anthony M.; Smith, Karl; and Bleloch, Andrew L. 1990, How to Model It: Problem Solving for the Computer Age, McGraw-Hill, New York.