Fall 2008 Course Syllabus
Course: 0824.001.
Course Title: Mathematical Patterns in Everyday Life.
Time: 10:40 - 12:30, Mondays and Wednesdays.
Place: Be 164.
Instructor: Paulos, John A.
Instructor Office: 542 Wachman Hall.
Instructor Email: paulos@temple.edu
Instructor Phone: x-5003.
Office Hours: 9:30 - 10:30, 1:00 - 2:00, Mondays and Wednesdays (plus CAs' hours).
Prerequisites: An ability to think critically.
Textbook: The course will be based on notes presented during class. See course description.
Course Goals: Roughly the goal is to to impart an understanding of some important mathematical notions and to provide an appreciation for their role in everyday life, particularly in the media. See course description below.
Topics Covered: Quotidian Quantities or Mathematics and Everyday Life by John Allen Paulos There will be three units in the proposed course: Basic Numeracy, Informal Logic and Philosophy of Science, and Probability and Statistics. The approach will be primarily via suggestive stories that illuminate the ideas in question rather than through general principles or excessive computation. Consider unit three on probability, for example. There will be some calculation in it, but many of the ideas and problems in the subject can be more easily communicated via standard vignettes. These include stories such as the gambler's fallacy and gambler's ruin, the Banach match box problem, the drunkard's random walks, the Monty Hall problem, the St. Petersburg paradox, the random chord problem, the hot hand, monkeys randomly typing on a typewriter, the Buffon needle problem, and many others. Even without much supporting context or argument, these stories are such that any fuller discussion or theory must accommodate and account for them. They provide part of the raw material that any reasonable mathematical theory must make sense of and thus, ideally at least, should be part of the intellectual gear of all curious students. A similar story-based approach will be taken in the units on basic numeracy and logic and the philosophy of science. Another sort of story will also play an important role in the course. The lectures and readings will examine the mathematical angles of stories in the news and endeavor to suggest novel perspectives, questions, and ideas on such stories. Mathematical naivete (innumeracy, if you will) can put readers at a disadvantage in thinking about many issues that, on the surface, don't involve mathematics at all. In many ways "number stories" complement, deepen, and sometimes undermine "people stories." The notions of probability and randomness can enhance articles on crime, health risks, or other societal obsessions. The logic of self-reference may help to clarify the hazards of celebrity and spin-doctoring. Business finance, basic combinatorics, and simple arithmetic point up consumer fallacies, electoral tricks, and sports myths. Quite accessible ideas from chaos and non-linear dynamics suggest how difficult and frequently worthless economic and environmental prediction is. And mathematically pertinent notions from philosophy and cognitive psychology provide perspective on a variety of public issues. To reiterate, the course will be focused on paradoxes and illustrative vignettes as well on mathematical applications to everyday life and social issues. There will be an attempt as well to impart some writing skills. One likely assignment, for example, will be to pick a topic of interest and do a column similar to one of my ABCNews.com Who's Counting pieces. (abcnews.go.com/Technology/WhosCounting/) Other assignments will involve writing up little amplifications of, or calculations suggested by, the ideas discussed in class. The texts for the course will be, as of this writing, paperbook versions of several of my books, the daily newspaper, and a more standard tome to be determined. Once again, the three units of the course are: I. Basic Numeracy, II. Logic, Philosophy of Science, and Mathematical Models, and III. Probability and Statistics. Some of the topics to be discussed are listed below, most of them in raw shorthand form. A few are developed a bit more fully. I. Basic Numeracy Essential numbers, coast to coast distance, population of the US, the world, approximate number of deaths annually from various common diseases contrasted with the number of deaths in more dramatic contexts. Basic arithmetic including fractions, percentages, decimals, ratios, percentage of what, "of" versus "more than" (200% more than vs. 300% of). One fourth of the world's population is Chinese, and one fifth of the rest is Indian. What percentage of the world's population is Indian? The percentage of employees who subscribe to a particular medical plan has risen from 4.2% to 5.6%. By what percent has this figure risen? By what percent must it be cut to reduce it to its former level? Not always so elementary. For example, you can't average percentages. I'll discuss the well-known sex discrimination case in California. Looking at the proportion of women admitted to the graduate school at the University of California, some women sued the university claiming that they were being discriminated against by the graduate school. When administrators looked for which departments were most guilty, however, they were a little astonished to find that there was actually a positive bias for women. To keep things simple, let's suppose there were only two departments in the graduate school, economics and psychology. Making up numbers, let's further assume that 70 of 100 men (70%) who applied to the economics department were admitted and that 15 of 20 women (75%) were. Assume also that 5 out of 20 men (25%) who applied to the psychology department were admitted and 35 of 100 women (35%) were. Note that in each department a higher percentage of women was admitted. If we amalgamate the numbers, however, we see what prompted the lawsuit: 75 of the 120 male applicants (62.5%) were admitted to the graduate school as a whole whereas only 50 of the 120 female applicants (41.7%) were. Such odd results can surface in a variety of contexts. For example, a certain medication X may have a higher success rate than another medication Y in several different studies and yet medication Y may have a higher overall success rate. Or a baseball player may have a lower batting average than another player against left-handed pitchers and also have a lower batting average than the other player against right-handed pitchers but have a higher overall batting average than the other player. Of course, these counter-intuitive results don't usually occur, but they do often enough for us to be wary of uncritically combining numbers and research studies into a so-called meta- analysis. Estimating and comparing: How much human blood in the world? It would fill Central Park in NY to a depth of approximately what? Anorexia deaths example - 150,000 originally claimed versus only 70 documented. Homelessness in NYC. Number of battered women.1.7 million college students received bachelor degrees last year. How many footbal stadiums.? Related to this are so-called Fermi problems. Physicist Enrico Fermi was known for challenging his classes with problems that, at first glance, seemed impossible. One such was estimating the number of piano tuners in Chicago given only the population of the city. Let's do it for New York: We know that New York has a population of about 8 million people. Now, assume that an average household has four members so that the number of households in New York must be about 2,000,000. If one in ten owns a piano, there will be 200,000 pianos in New York. If the average piano tuner serviced four pianos every day of the week for five days rested on weekends, and had a two week vacation during the summer, then in one year (52 weeks) he would service 1,000 pianos. Since 200,000/(4 x 5 x 50) = 200, so that there must be about 200 piano tuners in New York. Given its position as an entertainment capitol, let's up it by 50% and say there are 300 piano tuners in the city. This method does not guarantee precise results, of course, but it does establish a first estimate which might be off by no more than a factor of 2 or 3 - certainly well within a factor of, say, 10. We know, for example, that we should not expect 30 piano tuners, or 3,000 piano tuners. A factor of 10 error, an error of one order of magnitude, that is, is referred to as being 'to within cosmological accuracy.' Rounding rules. Significant digits. Museum, Olympics skiing and sledding. Dimensional analysis and basic conversions: scientific notation, million, billion, trillion. How fast does human hair grow in miles per hour? Cost of beef in drachma per kilogram, many disparate conversions. Scaling examples: string around equator, basic formulas for length, area, volume, models, people, trees, health care in Hawaii extended nationwide. Number of nickels in tower equal to Empire State Building or under some table. More examples on lengths, areas, and volumes and the tendentious way these simple notions can be used. Voting: Olympics (judges, precision), Florida election, Lani Guinier, Oscars, Enron. Example of Borda, Condorcet, with the five actors or movies. Application to presidential primaries. Few Examples. Annual spending on ice cream - servings per person x price per serving x population - 50 x $1 x 300 million = $1,5 x 10^10 dollars annually, or $15 billion. More serious: Ocean contamination calculations. 4.5 billion years versus 10,000 years of written human history on time line one mile long. Often arithmetic enough if reconceptualize problem: train and canary problem, coffee and milk, pigeon-hole principle, $30 hotel problem. Other numerical oddities. II. Logic, Philosophy of Science, and Mathematical Models Scientific method. Design. Observational study versus experiment (treatment and control group), single-blind, double-blind, placebos, in between: case-control study, an observational study with controls. Logical arguments. Inductive vs deductive. Standard fallacies, self-reference and logical paradoxes: Russell's barber, etc. Other paradoxes: grue-bleen, raven, crocodile, the unexpected test, etc. Stories and mathematics - mindsets, suspend disbelief and accept something false vs suspend belief and reject something true, suggestion vs proof, substitutibility and extensional logic, non-substitutibility and intensional logic, narrative vs number. Models of humor. Incongruity in an appropriate emotional climate., juxtaposition, reversals, substitutions, word play, self-reference, non-standard models. Desiderata in evaluating study: 1. Identify goal, population, and type of study. 2. Consider the source. 3. Look for bias. (selection bias, participation bias) Literary Digest, 900 polls. 4. Look for problems defining varaibles of interest. (love, wealth, illegal, ...) inappropriate comparisons 5. Watch for confounding variables. (capuccino, shoes, rural less likely to divorce because older) 6. Wording of polls. The Yankelovich polling organization, for example, asked the following question: Should laws be passed to eliminate all possibilities of special interests giving huge sums of money to candidates? To this 80% of the sample responded Yes, 17% No. They then posed the question: Should laws be passed to prohibit interest groups from contributing to campaigns, or do groups have a right to contribute to the candidate they support? In this case only 40% said Yes, while 55% answered No. Presumably the latter, less provocative question made the opposing argument more available to the respondents. 7. Results presented fairly. 8. Conclusions make sense? (alternative explanation, confirmation bias, practical significance vs. statistical significance) Extraordinary claims require extraordinary evidence. A discussion of various pseudosciences, their appeal despite their baselessness. From biorhythms to astrology to psychic predictions. Falsifiability, other notions from the philosophy of science. Theoretical versus observational, bridge principles, reductionism, induction and "justifications" of it. Mathematical models, trade-offs between accuracy and feasibility. Rates. How fast, rising, falling, pitfalls in the notion. S-curves: This curve characterizes, or at least seems to characterize, a variety of phenomena, including the demand for new toys. Its shape can most easily be explained by imagining a few bacteria in a petri dish. At first the number of bacteria will increase at a rapid exponential rate because of the rich nutrient broth and the ample space in which to expand. Gradually, however, as the bacteria crowd each other, their rate of increase slows and the number of bacteria stabilizes. What is interesting is that this curve (sometimes called the logistic curve) appears to describe the growth of entities as disparate as Mozart's symphony production, the rise of airline traffic, new mainframe computer installations, and the building of Gothic cathedrals. Other common curves: normal curve in statistics, linear and polynomial curves, parabolic trajectories, the curve for the exponential growth of money, exponential versus linear. Malthus. The very important anchoring effect, availability error, and confirmation bias. Examples from Tversky, Kahneman, and other cognitive psychologists. Relevance to the stock market. Defining and using randomness. Stock scam, Apple's iPod, etc. Data mining: data, data everywhere. Scouring databases. Kaplan service. Dennett dream game. III. Probability and Statistics Sample spaces, examples (coins, dice, heights, incomes) Probability rules (sum rule, product rule, at least one rule). Expected value. Conditional probability. Racial profiling and false positives. Bayes' theorem. Cancer test (98% accurate, 1 out of 200 with cancer, 10,000 tests administered.). Lie detector tests. Expected value, insurance, blood tests, Pascal's wager, etc. Probability of a particular event vs probability of some event of a general sort. JFMAMJJASOND, MVEMJSUNP. Significant? No. And, of course, the birthday problem. The various puzzles and stories mentioned at the top of this proposal: Monty Hall, gambler's ruin, etc. Lotteries, a tax on innumeracy. Rare events in general. How likely to inhale a molecule exhaled by Julius Caesar in his dying breath? A special case of people's attributing meaning to phenomena governed only by chance is provided by regression to the mean, the tendency for an extreme value of a random quantity whose values cluster around an average to be followed by a value closer to the average or mean. Very intelligent people; Very short people; Twenty darts This phenomenon leads to nonsense when people attribute the regression to the mean as due to something real, rather than to the natural behavior of any random quantity. If a beginning pilot makes a very good landing, it's likely that his next one will not be as impressive. Likewise if his first landing is very bumpy, then by chance alone, his next one will likely be better. Tversky and Kahneman studied one such situation, in which after good landings pilots were praised whereas after bumpy landings they were berated. The flight instructors then mistakenly attributed the pilots' deterioration to their praise of them, and likewise the pilots' improvement to their criticism; both, however, were simply regressions to the more likely mean performance. "Behavior is most likely to improve after punishment and to deteriorate after reward. Consequently the human condition is such that ... one is most often rewarded for punishing others, and most often punished for rewarding them." Sequel to a great movie; Great season by a baseball player in his prime; the novel after the best-seller; the album that follows the gold record, or the proverbial sophomore jinx. Should be carefully distinguished from the gambler's fallacy, to which it bears a superficial resemblance. Distributions: mean (expected value), median, mode; standard deviation (as average deviation). five number summary, percentiles. uniform distribution, skewed distribution, other distributions. Normal distribution. 50-50 chance of rain, rare events and hysteria. Simpson paradox update: medicine, marathons, crime stats. Meta-analyses. The standard deviation is very useful, however, when the quantity in question can assume many different values and these values, as they often do, have a normal bell-shaped distribution. In this case, the expected value is the high point of this distribution of the possible values and approximately 2/3rds of the values lie within one standard deviation of the expected value, and 95% of the values lie within two standard deviations of the expected value. Age-specific heights and weights, natural gas consumption in a city for any given winter day, water use between 2 AM and 3 AM in a given city, thicknesses of a machine part, I.Q.'s (whatever it is that they measure), the number of admissions to a large hospital on any given day, distances of darts from a bullseye, leaf sizes, nose sizes, the number of raisins in boxes of breakfast cereal, and, as mentioned, possible rates of return for a stock. If we were to graph any of these quantities, we would obtain the bell-shaped curve. Let's take length of leaves for a particular sort of tree. If the expected value of this quantity is 2 inches and the standard deviation is .4 inches, then the high point of the graph would be at 2 inches. Furthermore, about 2/3 of the leaves would be betwee 1.6 inches and 2.4 inches, and 95% of them would be between 1.2 inches and 2.8 inches. Or consider the rate of return of a stock. If the possible rates are normally distributed with an expected value of 8.2% and a volatility (standard deviation, that is) of 1.4%, then about 2/3 of the time, the rate of return will be between 6.8% and 9.6%, and 95% of the time the rate will be between 5.4% and 11%. In general, the larger the standard devi Whether the quantities are nose sizes or water use in a city, note that all of the above normally distributed quantities can be thought of as the average or sum of many factors (genetic, physical, social, or financial). Not surprising since the so-called Central Limit Theorem states that averages and sums of a sufficient number of random quantities are always normally distributed. Samples. Sampling variation. Confidence interval equals mean plus or minus 1/ n .5. What's more accurate: a random sample of 400 from a small town of 2500 or a random sample of 1000 from New York City? "So what is the value of an opinion poll? Provided the sample is representative of the population and does not favour voters with particular views, the question is dispassionately worded, and opinions are not changing too rapidly, an opinion poll can be very informative. But if these conditions are not satisfied, it might not be worth anything." Coincidences, correlation and causation, conspiracies, scams, con games, hoaxes Finally, among the news stories covered in the three units will be many of the following: air safety, relative risks; scoring streaks, "hot hands," and records in sports; health hazards of all sorts; statistics, expert witnesses, and the courtsl; the U.S. Census; redistricting, elections, especially 2000 and 2004 presidential election; use of DNA and, more generally, (conditional) probability in the courts; randomized clinical trials; "studies show," "many," and "may be linked"; epidemiology, AIDS; the stock market and the random walk hypothesis; economic statistics and reporting; environmental concerns, contamination reports; conspiracies, hoaxes, especially on the internet; demographic variations, social work issues; reliability of political polls; lotteries and other gambling issues; junk science of all sorts. The quizzes and the final will, as of this writing, contain three sorts of questions: basic spit-backs of the ideas and examples discussed, applications or extensions of these ideas to slightly new situations, and a choice of a more extended response to one of two open-ended questions.
Course Grading: Tentative: 3 quizzes, final.
Exam Dates: TBA.
Attendance Policy: Strongly encouraged.
Any student who has a need for accommodation based on the impact of a disability should contact me privately to discuss the specific situation as soon as possible. Contact Disability Resources and Services at (215) 204-1280, 100 Ritter Annex, to coordinate reasonable accommodations for students with documented disabilities.
Freedom to teach and freedom to learn are inseparable facets of academic freedom. The University has adopted a policy on Student and Faculty Academic Rights and Responsibilities (Policy # 03.70.02) which can be accessed here.
Students will be charged for a course unless a withdrawal form is processed by a registration office of the University by the Drop/Add deadline date given below. For this semester, the crucial dates are as follows:
- The first day of classes is Tuesday, September 2.
- The last day to drop/add (tuition refund available) is Monday, September 15.
- Thanksgiving is Thursday, November 27.
- The last day to withdraw (no refund) is Monday, November 3.
- The last day of classes is Wednesday, December 10.
During the first two weeks of the fall or spring semester or summer sessions, students may withdraw from a course with no record of the class appearing on the transcript. In weeks three through nine of the fall or spring semester, or during weeks three and four of summer sessions, the student may withdraw with the advisor's permission. The course will be recorded on the transcript with the instructor's notation of "W," indicating that the student withdrew. After week nine of the fall or spring semester, or week four of summer sessions, students may not withdraw from courses. No student may withdraw from more than five courses during the duration of his/her studies to earn a bachelor's degree. A student may not withdraw from the same course more than once. Students who miss the final exam and do not make alternative arrangements before the grades are turned in will be graded F.
The grade I (an "incomplete") is reserved for extreme circumstances. It is necessary to have completed almost all of the course with a passing average and to file an incomplete contract specifying what is left for you to do. To be eligible for an I grade you need a good reason and you should have missed not more than 25% of the first nine weeks of classes. If approved by the Mathematics Department chair and the CST Dean's office, the incomplete contract must include a default grade that will be used in case the I grade is not resolved within 12 months.