A Course Called "Chance"

J. Laurie Snell and John Finn

Two years ago, inspired by Chance magazine, Tom Moore at Grinnell College, Bill Peterson at Middlebury College, and John Finn and Laurie Snell at Dartmouth College decided to develop a new course, called "Chance." We have since enlarged our group to include Negambal Shah at Spelman College, Chris Thron at King College, Joan Garfield at the University of Minnesota, and Peter Doyle at University of California at San Diego. We have taught several pilot versions of the Chance course during the 1991-1992 academic year and have several more slated for 1992-1993.

The aim of the Chance course is to study important current news items whose understanding requires a knowledge of chance concepts. Such news items are regularly found in newspapers like The New York Times and in science journals and magazines such as Chance, Science, Nature, and the New England Journal of Medicine.

We do not intend for Chance to replace an introductory statistics or probability course; its aim is rather to encourage students to think more rationally about chance events and to make them more informed readers of the daily press. Of course, we do hope it will also encourage the students to continue their study of statistics and probability, either informally or in future courses. In our pilot versions, we have assumed only high school mathematics as a prerequisite.

Here are our answers to the kinds of questions we think readers of Chance might ask about our experiences.

Where Have You Taught Chance, and to What Kinds of Students?

In the fall term of 1991, Tom Moore taught the course at Grinnell, Bill Peterson at Middlebury, and Peter Doyle and Laurie Snell at Princeton; in the winter term of 1992, John Finn and Laurie Snell introduced at Dartmouth. The courses taught by Moore and Peterson were limited to first-year students and were part of their schools' programs to improve writing across the curriculum. The Princeton course had students from all four years; about half of them were social science majors, and a quarter each were humanities and science majors. The Dartmouth course was for adult graduate students in Dartmouth's Master of Arts in Liberal Studies program. Chance has also been offered this summer in Middlebury's alumni program, and some of our materials were used in a statistics course at Spelman College. The course will be offered to undergraduates in the coming year at King College and Dartmouth.

Grinnell College students Ross Johnson, Neil Wohlford, Rick Osborn, Jacob Willig (seated), Bertrand Behm, and Nate Brennaman are assisted by Professor Tom Moore in the red bead experiment. (Photo by Sarah Delong)

What Kinds of Topics Have You Investigated?

We try to choose topics that are currently in the news and are likely to remain so. Our topics so far will be familiar to Chance readers. They include DNA fingerprinting, the statistics of AIDS, the undercount in the 1990 census, the reliability of opinion polls, the Deming story, clinical trials, the ability of SATs to predict performance in college, streaks in sports, and Marilyn vos Savant's Monty Hall problem. The treatment of any particular topic tends to concentrate on questions that are in the news at the time the course is being taught. For example, as we prepare for our Chance course in the fall, we find several interesting questions being discussed in the news. Some of these are

By the time you read this we will probably have a new definition of AIDS, you will know if Ray Fair was correct in predicting a Bush victory, and Barbara Bryant will very likely have made her decision. We will then be on to new problems relating to these issues to be investigated in the next version of our course.

What Special Materials and Resources Have You Used?

Springer-Verlag donated five complete series of Chance magazine to each of the schools in our pilot program, making it possible for us to assign regular readings in this magazine. Each of the courses had the students write final projects and the Chance magazines provided interesting ideas and resources for these projects.

Our experiences to date have shown that a large proportion of the current news that might enhance a Chance course can be found in a small number of newspapers and journals. Specifically, good choices would be The New York Times, Science, Nature, Chance, New England Journal of Medicine, Journal of the American Medical Association, and The Lancet. Most colleges have these publications; they are often in the public library as well.

The basic texts for our courses were Statistics, by Freedman, Pisani, Purves, and Adhirkari, used by Doyle and Snell, and Statistics: Concepts and Controversies, by David Moore, used by Peterson and Moore. These books do an excellent job of treating serious probabilistic and statistical concepts with a minimum of mathematical jargon.

As suggested in an article in the Winter 1991 issue of Chance on resampling, computer simulations can be used to give the students a good feeling for the amount of variation to be expected in chance experiments. We used simple programming languages to illustrate basic concepts by simulations and numerical computations. We also used standard statistical packages for exploratory analysis of data.

The video seriesAgainst All Odds has minidocumentary segments shot on location, showing statistics as it has been applied: the physicians' study conducted to determine the effectiveness of aspirin in preventing heart disease; a Colorado Springs study that resulted in equal pay for men and women in city government; using control charts to improve potato chips; and the clinical trials study to determine the effectiveness of AZT in the treatment of AIDS. Seeing these experiments being carried out adds an additional sense of reality to our studies.

Some of us found it useful to invite experts from different fields to give some background to the class. For example, the Dartmouth course occurred at the time of the presidential primaries, so we invited Jack Ludwig, chief methodologist of the Gallup organization, to help us understand some of the practical and theoretical problems of opinion polls.

What Have You Learned So Far From Teaching Chance?

Of course, we have learned a lot about AIDS, DNA, educational testing, and many other topics. We also learned that to really understand an issue such as the effectiveness of cholesterol in prevention of heart attacks, you have to look at a diverse group of studies, not just a single study. The fact that people who lowered their cholesterol were more prone to nonillness-related deaths like suicide and homicide did not stand out in individual studies, where it might be mentioned only briefly as a "troubling observation." When you follow a series of experiments, you get a better idea of what the truth may be.

We learned that it is difficult not to lecture, and we had to adjust to the experience of responding to students after they have discussed an idea in small groups. We found that our traditional neat and tidy presentation of concepts is not what occurs naturally to students. They tend, for example, to approach a conditional probability problem directly rather than setting up an elaborate sample space for the unconditional probabilities. One member of our Chance group, Joan Garfield, has as a major area of her research interest the ways that students think about probability and statistics, and we expect her work to help us in our continuing struggle to lead students to think intelligently about chance events.

What Difficulties Have Arisen in Teaching Chance?

The most obvious difficulty is that the instructor has to learn enough about the topic to intelligently guide discussion. For instance, you cannot talk about DNA fingerprinting without knowing what DNA is. But "DNA didn't even exist" when some of us took our biology courses! Similarly, the very definition of AIDS is complicated and has changed several times through the years. Each subject seems to have its own specialized jargon, and we have had to learn to explain the key ideas using a minimum of new technical words, an approach which presents a real challenge. Of course, science writers for newspapers like The New York Times and journals like Science and Nature have to deal with the same problem, which is partly why their accounts are an important resource for our course. However, in the last analysis, we were inevitably led further back into the technical literature, and we had to figure out what was being said in these articles.

In many cases, we wanted to get the actual raw data for a study but found this surprisingly difficult. When we were successful— such as with the data for a major study comparing life expectancies of left- and right-handers, and with the study of SAT scores—it made a significant difference in our understanding and feel for the topic. We are hoping to help make this raw data more easily available.

Which Topics Have Succeeded and Which Have Failed? Why?

We found that Parade Magazine columnist Marilyn vos Savant's Monty Hall controversy created as much interest in our classes as it did throughout the rest of the country and got students quickly into the questions of appropriate probability models and the assumptions involved in modeling real-world problems. Beyond that, we each had our own successes and failures.

At Princeton, the examination of the question: "Do SAT scores underpredict college performance for women?" was particularly successful. A recent Princeton graduate had included this question in her senior thesis and analyzed Princeton data. We were able also to provide the students with data from a study done at Dartmouth. They explored the data using their statistical package and came to their own conclusions. In addition, statistician Charles Lewis from the Educational Testing Service (ETS) joined the class to give the perspective of large-scale studies done at ETS. This was a popular topic also for the graduate class at Dartmouth. Many of the students were school teachers who came with their own insights into this problem.

The biggest failure at Princeton was our attempt to explain the central limit theorem. Peter Doyle gave a nice intuitive explanation that might have satisfied the class but did not satisfy Laurie Snell. Snell then gave his first formal lecture, which culminated in the graduate assistant going to the board to try to explain what Snell had said.

At Grinnell, Tom Moore found that the Deming story worked well for his class. Students read Mary Walton's Deming Management Method, and Moore introduced the topic by showing the unit on quality control from the Against All Odds video series. Moore's class visited a local sportswear manufacturer, and the students wrote a short paper on their impressions of how well the company adhered to Deming's principles. The class participated in the famous red bead experiment, using a paddle from the game "Boggle" and marbles used to line the bottoms of clay flower pots obtained from the local discount store. Moore comments that "this set-up worked beautifully and provided us with a very lively class." He had a less successful experience with Deming's funnel experiment, however, reporting that "they saw through it; too many of them realized intuitively that the smallest variability would occur from the no-adjustment strategy."

At Middlebury, Bill Peterson was particularly happy with his class's discussion of the undercount problem in the 1990 census. He used material from the Quantitative Literacy Series booklet Exploring Information from Surveys and Samples, where the capture-recapture model is illustrated in terms of estimating fish populations. The class talked about what would happen if fish died, migrated, or lost their tags, and then tried to map these into corresponding difficulties with the postenumeration survey for the census.

Peterson felt that another successful part of his course was the discussion of the Kahneman-Tversky ideas. The students liked the authors' point that we tend to give more credence to more detailed predictions, even though the increased detail can lower the probability that the prediction will be correct. Peterson comments that "implications for law, medicine, etc., generated a good debate." Of course, we all enjoyed the series of papers in Chance magazine on streaks in basketball, inspired by the original Kahneman-Tversky article suggesting that streaks are difficult to verify.

In the Dartmouth liberal studies Chance course, DNA fingerprinting was a popular topic. A molecular biologist gave the students background on the technique of DNA fingerprinting; a law professor explained the legal principles that determine the conditions under which DNA fingerprinting is accepted in the courts and described specific cases where it has been used.

How Much Have You Covered in Chance? How Does It Compare to a Standard Introductory Probability or Statistics Course?

One enjoyable thing about our Chance courses is that we were not under any obligation to cover a specific number of topics. We tried to develop basic concepts of statistics and probability in the context of the applications. For example, when we are trying to explain what Dan Rather means by "this poll has a margin of error of plus or minus three percentage points," we obviously have to develop simple ideas about Bernoulli trials. Talking about the Monty Hall and related paradoxes led naturally to the concept of conditional probability. In studying clinical trials, we have to talk about the meaning of the infamous p values and the idea of confidence intervals.

A recent joint committee of the American Statistical Association and the Mathematical Association of America has recommended less lecturing and more hands-on analysis of data in the first statistics course. A Chance course provides a wonderful opportunity to try this out. Both Moore at Grinnell and Doyle and Snell at Princeton did away with lectures and relied on class participation. At Princeton, the students were asked at the beginning of each class to form small groups and to consider 2 or 3 questions for about 20 minutes. Each group assigned a reporter to present their conclusions, and most of the remaining class time was spent responding to the groups' findings. Students kept a journal in which they recorded their impressions of the readings and included comments and questions on anything that came up in class.

We feel that whereas a Chance course lacks the methodical presentation of a more standard introduction, it may more than compensate for this with its greater motivation.

How Can Others Benefit From the Chance Pilot Group's Experiences?

Under a grant from the National Science Foundation we are developing a Chance database at Dartmouth. In this database, we plan to maintain a "profile" of each topic, which will grow continually. A profile may include raw data, a bibliography, a summary report, an account of experiences teaching the topic, student projects, and so on. Our database will also include detailed descriptions of Chance courses already given, and copies of important news and magazine articles, once we have secured suitable copyright permissions.

We plan to make videotapes of some of our guest experts who give us background information and make these available to others teaching a Chance course.

The standard databases provided for libraries (e.g., MEDLINE and LEXIS/NEXIS) make it easy to find current articles on topics of interest to a Chance course. These databases have the complete texts of major newspapers available one day after publication and have abstracts of articles in the journal literature. Using these and our own review of the current journals, we have started a monitored weekly electronic newsletter giving brief summaries and references to articles of interest to a Chance course. You can get this Chance news by sending a request to dart . chance@dartmouth.edu.

We have in mind several ways that other teachers may make use of these materials. For one, we encourage others to design and carry out their own versions of Chance. These may be elementary courses, as ours have been so far, but there is also the possibility of more advanced Chance courses, requiring a working familiarity with probability, statistics, or both. For the less daring, there is the option of using some of our materials in a more standard course. We want to hear about others' experiences and plan to incorporate them into our data base.

Have Your Chance Courses Left the Students Motivated to Further Study in Probability and Statistics?

We certainly hope so. They have seen that the problems of careful statistical analysis are not easy and may require techniques beyond their grasp. They have also seen that they may well have a personal stake in gaining a better understanding of scientific studies than that offered in the daily press. The fact that those of us in the pilot group have definite and enthusiastic plans for future Chance courses must mean that we think we are getting through.

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