David Draper–Teaching Statement 2006-07
Professor in Applied Mathematics and Statistics

The kinds of courses I teach.

I teach statistics, a discipline that many people approach with fear and loathing. For me, it’s a rewarding subject, potentially useful to just about everybody. I define statistics as the study of uncertainty: how to measure it, and what to do about it. As we know too well from experience, it’s part of the human condition to be forced to make choices with insufficient or imperfect information. But, interestingly, it’s not part of our species’ hard-wired evolutionary heritage to reason correctly in the face of uncertainty. That’s my starting point.

Since arriving at UCSC in January 2001 I’ve taught five different classes: ENGR/AMS 5, a lower-division introduction to statistical ideas and methods (F 2001–02, Sp 2004–06); ENGR 88A/AMS 88B, a discovery seminar on statistical ideas for first- and second-year undergraduates (Sp 2004–05); ENGR 181, an upperdivision introduction to Bayesian statistics (Sp 2001); ENGR/AMS 206, a more advanced graduate-level version of ENGR 181 (W 2002–05); and AMS 7/7L, a lower-division course on statistical methods for the biological, environmental and health sciences with a computing lab component (F 2006, W 2007). AMS 7 used to be a course on biostatistics with no computing lab; I’ve redesigned it from the ground up to better meet the needs of Biology, Environmental Studies and Health Sciences majors. ENGR/AMS 5 already existed on campus as MATH 5 before the AMS Department began, but I completely revamped it, and the others are all new to the campus. Enrollments have risen steadily: for example, from 99 to 124 to 168 to 209 to 315 in ENGR/AMS 5, and from 64 to 113 in AMS 7/7L. My courses are attended regularly by faculty as well as graduate and undergraduate students.

At every level, the content of my teaching falls into three main categories: probability, which is the part of mathematics devoted to quantifying uncertainty; statistics, which is about reasoning backwards from data to make intelligent guesses about the causes of the underlying processes at issue; and decisionmaking, which is about actually making behavioral choices in the world based on data (and despite the uncertainties).

Approach to teaching, and teaching philosophy.

The first overall principle that guides my teaching is that my job is not simply to teach facts, methods and theories; it’s to help people learn, encourage them to think more clearly, and engage them in a process of teaching themselves how to learn. Most of the ideas and methods people use on a daily basis are things they’ve taught themselves since finishing their schooling. For this reason I structure my classes so that — when things go as planned — students come away with more than just notes on paper: they gain insights into the discovery process itself. For me, one secret to good teaching is to remember what it was like not to know something useful that I now take for granted. With this in mind, my goal is to help people at all levels of understanding to find a personal path from lack of knowledge to insight and clarity.

My second overall principle is that the kinds of ideas I try to share are best conveyed through a case-study approach. My version of this teaching method has four steps (see below) but its most important feature is that the cases are interesting. They reveal the utility of statistical techniques in uncovering important patterns in nature as well as in social systems and institutions. My case studies have ranged from HIV screening methods to nuclear groundwater contamination to racism in the Chicago fire department of 1968; my supporting document (homework assignment 2 from AMS 7 in winter 2007, also available at www.soe.ucsc.edu/classes/ams007/Winter07/, which provides an example of the web pages I create for my classes) gives some problems from experimental design and probability.

Each statistical study unit begins with

(1) A scientific or decision-making case study, with sufficient contextual details for the real-world issues to be clearly in focus (Example: In problem 4 in the supporting document, we consider the relationships among age, mortality and smoking habits in a sample of British women who were studied in 1972 and revisited in 1995). Then

(2) The statistical methods that are the point of this unit can be developed in the context of the case study (Example: We use conditional probability to discover that smoking seems at first glance, and counter-intuitively, to be protective: overall the smokers had lower mortality than the non-smokers), after which

(3) These methods can be applied to solve the real-world problem in step (1) that prompted the inquiry in the first place (Example: A closer look reveals that after age is taken into consideration, smoking was actually associated with higher (not lower) mortality; this is an example of a Simpson’s Paradox, and the answer accounting for age is likely to give a better idea of whether smoking really does cause an increase in mortality). After this the unit is concluded with

(4) An investigation of the general properties of the methods developed in step (2). (Example: Q: Why did the Simpson’s Paradox occur in this problem? A: Because the smokers were considerably younger on average, and young people have lower mortality; this made smoking (falsely) look protective. Q: What are the general conditions that can lead to a Simpson’s Paradox? A: Given an outcome variable Y (mortality), a treatment variable X (smoking), and a potential confounding factor Z (age), in an observational study (in which the subjects in the experiment choose their own treatment groups instead of having them assigned at random) it’s possible for the association between Y and X to go one way when Z is not accounted for but to reverse direction when Z is (correctly) taken into consideration).

I undertake steps (2) and (4) in an interactive way, by asking the students to suggest ideas for how progress might be made in solving the problem in (1), developing the method adaptively based on the suggestions they give me, and interactively exploring the general attributes of the method we’ve “created.” When someone suggests an idea that’s only partially successful, we go down the indicated path until we hit a brick wall, and then we figure out how to climb over the wall. The case studies I use come directly from my research and consulting work, and from other sources with which I’m sufficiently familiar to be able to construct a good case. I developed the above four-step approach myself. In my 28 years of teaching I’ve used it in classes ranging in size from 1 to 500, and at levels ranging from first-year undergraduate courses to the most advanced graduate seminars. It’s worked well.

Teaching Accomplishments.

(a) When students increase their aptitude for critical thinking (Example: when AMS 5 students noticeably improve at reading newspaper and web articles and spotting flaws in someone else’s reasoning about what can validly be concluded in a cause-and-effect way from a given body of evidence);

(b) When I’m reading homework assignments or tests and students offer ideas/lines of reasoning/arguments that are completely original in relation to the material I’ve presented (in other words, when I observe at a meta-level that the big picture described above has sunk in along with the specific ideas and methods);

(c) When my large AMS 5 and AMS 7 classes evolve from {75% or more of the students saying at the beginning of the quarter that the only reason they’re taking the class is that they’re required to do so} to {75% or more of the students saying at the end that the class was a genuinely exciting learning experience}; and

(d) When UCSC graduate students tell me that the ideas and methods I teach them have deepened their understanding of their own fields of specialization (and, in four cases since I got to UCSC, even caused them to switch their focus to statistics).

Numerical teaching evaluations are part of the story; the rest is sketched in with open-ended comments, hints that I may have reached something deeper. I strive to convey life lessons along with math lessons, about problem-solving, the value of perseverance, and the dignity of striving toward a worthy goal even if we don’t get all the way there.

When people are asked how they picked the line of work they chose as adults, they often point to an encounter with a charismatic teacher as pivotal in their choice. I’m trying to be that kind of teacher.

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