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by Christopher Wolfe
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Approved by the faculty of the School of Interdisciplinary Studies in the spring of 1990.
Introduction. The goal of improving students' quantitative reasoning stems, in part, from a comprehensive assessment of the Western College Program, indicating that Western students are often uncomfortable using quantitative and numerical concepts in their reasoning and problem solving. The phrase "quantitative reasoning" is used here to refer to a wide range of mental abilities. Quantitative reasoning skills include facility with measurement and estimation, a sense of scale, an understanding of basic probability theory and statistics, and a subjective sense of ease in reasoning with numbers. For these purposes, we are less concerned with calculus and higher mathematics.
The ideas outlined in this proposal are designed to provide direction for our efforts to integrate quantitative reasoning more fully into the Western curriculum. At the heart of this proposal is the belief that no single class or computer facility can bear the responsibility for quantitative reasoning alone. Rather, our efforts should spring from a common commitment to making quantitative reasoning a meaningful part of our courses. It is my hope that this commitment extends to the Natural Sciences, Social Sciences, and Creativity and Culture. I am currently compiling a set of materials for facilitating quantitative reasoning. These materials will include computer courseware, on-line laboratories, statistics packages and graphics packages for the Center for Computer Assisted Learning. I am also collecting books, articles, and notes describing exercises and assignments that promote quantitative reasoning.
To aid our thinking about quantitative reasoning in the Western curriculum, it is useful to consider four interrelated aspects of quantitative reasoning: Learning from Data, Quantitative Expression, Evidence & Assertions, and Quantitative Intuition. Learning from Data refers to the skills associated with collecting and analyzing data, particularly in the natural and social sciences. Quantitative Expression is the ability to use and comprehend quantitative language in a variety of contexts. Facility with Evidence and Assertions allows one to comprehend which conclusions may be reasonably drawn from a body of evidence. Finally, developing Quantitative Intuition refers to developing heuristics that lead to a "feel" for numbers and other quantitative concepts. Although it is useful to consider these four aspects of quantitative reasoning separately, it is important to realize that they are inherently interconnected. For example, reading an empirical research article in a scientific journal requires the ability to learn from data, comprehend quantitative expressions, evaluate evidence and assertions, and applying quantitative intuitions.
For the sake of clarity, this proposal is divided into several sections describing these four aspects of quantitative reasoning, some suggestions for coordinating quantitative activities, and sample exercises. Examples of quantitative concepts in each of these four areas of quantitative reasoning are included in the appendix.
Perhaps the most exciting aspect of science is the ability to ask questions of nature, and get answers from data. Yet many students lack the basic conceptual and methodological skills for learning from data. Acquiring the tools for collecting and analyzing data in the natural and social sciences can be an empowering experience for students. It gives students the ability to test their own ideas about the world, and make new discoveries. Some of the skills associated with data collection include an understanding of the notion of hypothesis testing, and specific methods of inquiry such as experimentation and systematic observation. In both the natural and social sciences, descriptive and inferential statistics are powerful tools for analyzing data. Thus, it is useful for students to learn to use (and calculate) descriptive statistics such as means, medians and correlation coefficients, as well as basic inferential statistics such as t-tests and Chi Square.
Western students would benefit from greater exposure to statistical reasoning in several ways. First, because basic statistics are widely used in a number of domains, acquiring these skills will enable students to advance academically in a number of directions. Second, there is empirical evidence from the psychological literature indicating that even a small amount of statistical training can have a measurable impact on everyday reasoning. Nisbett, Krantz, Jepson, and Kunda (1983) report that "training in statistics has a marked impact on reasoning. Training increases both the likelihood that people will take a statistical approach to a given problem and the quality of the statistical solution." Fong, Krantz, and Nisbett (1986) found that a brief lesson in statistics produced significant transfer of training' to a ñwide variety of problems of an everyday nature." Finally, the ability to reason with statistical data is a vital component of effective policy making. Students who aspire to have an impact on political and administrative decision making would greatly benefit from a working knowledge of basic statistics.
Embracing the notion that good teaching facilitates 'learning by doing,' propose that students be given many experiences in developing hypotheses and collecting and analyzing data in the natural and social sciences. For example, such experiences may include experiments, observational studies, and surveys. Learning about statistics is more meaningful (and fun) when students are analyzing real data that they themselves collected. However, it can be frustrating when students are given tasks without adequate preparation. To provide students with adequate background, it may be useful to have students read journal articles describing empirical studies, with special attention given to methods of data collection and interpretation. To give students 'hands on' experience in using sophisticated methodologies, it may be useful to have students replicate classic studies early in their education, (particularly as posed in a problem solving framework).
Despite the potential for empowerment, for many students, a traditional course in statistics is a painful and meaningless experience. This is because traditional statistics courses are focused almost exclusively on computation, and rarely on what statistics can do for the student. The approach advanced here differs from typical statistics and methods courses in many important respects. First, there is a commitment to teaching in context. It does little good to give students tools without showing them how these tools are useful for solving their problems. By teaching in context, we 'create' the demand for statistical and methodological problem solving techniques. Second, there is a focus on student generated hypotheses. Students will be encouraged to develop testable propositions about the world. Learning to ask empirical questions in a testable form is an exciting process. Getting answers to those questions requires some degree of statistical sophistication. And both activities encourage students to make the tools of learning from data their own. A third way in which this approach differs from the traditional lies in the coordination of different courses. The format of our Western courses affords us the opportunity to structure the quantitative aspects of our courses in ways that complement one another. For example, a Natural Science population study determining the mean size (and standard deviation) of a sample of fossils may be complemented by a Social Science exercise on per capita and median income. Much could be gained by a deliberate effort to coordinate the quantitative aspects of our Natural Science, Social Science, Creativity and Culture, and Applied Field Laboratory courses.
In many domains, it is useful to express concepts in quantitative language. Thus, familiarity and comfort with quantitative expression is essential for understanding (and participating in) work in these domains. An important dimension of quantitative expression is a working knowledge of various units of measurement. For example, students should understand (and not be afraid of) units such as milliseconds, light years, calories, and constant dollars. Students would benefit from working with a variety of measures, and learning about the rationale for their use.
Another aspect of quantitative expression is the ability to express quantitative concepts visually. A collection of numbers is, generally, difficult to understand. However, to the trained eye, tables, graphs and charts often make intuitive sense. Thus, it is useful for students to learn to read, and produce, a wide variety of graphs and charts. For example, starting with raw data, students should learn to produce histograms, pie charts, and scatter plots -and know which are appropriate for which kinds of data. Maps are another powerful means of representing data in visual terms. Maps are extremely useful for expressing geographical, social, and ecological concepts in context. Yet students are generally unschooled at reading maps, and particularly at making them. Map related exercises, such as tracing the migratory range of monarch butterflies, sources of acid rain, or the location of natural resources build visual and quantitative skills, and help students see the world in new ways.
A third dimension of quantitative expression is an understanding of various scales and distributions. Because integers can be used on nominal, ordinal, interval, and ratio scales it can truthfully be said that 'one does not always equal one.' People frequently make inappropriate inferences due to misunderstandings about different scales. For example, when talking about the weather, 60 F is not twice as warm as 30 F. Similarly, people often think in terms of idiosyncratic examples rather than distributions. Our understanding of the world is strengthened when we realize that some variables, (such as peoples' height) are distributed normally, while other variables (such as peoples' income) are part of skewed distributions. Students should work with a variety of scales and distributions, with special attention paid to their similarities and differences.
Raw data, whether qualitative or quantitative, tells us nothing about the world by itself. It must be interpreted by human beings. Of course, there are no universal rules for arriving at Truth, but there are some general guidelines for evaluating assertions based on claims of evidence. One set of guidelines stems from conditional and syllogistic reasoning. These alert us to common fallacies in reasoning and decision making. For example, a common fallacy is the "conversion error" whereby people erroneously assume that statements such as "all dogs are animals" imply that "all animals are dogs" or that "the probability that an African-American is poor" is equal to "the probability that a poor person is African-American." Another set of guidelines stems from probability theory. A common fallacy in probabilistic reasoning is "the base rate fallacy." The base rate fallacy refers to people's tendency to ignore base rate information (describing the overall or unconditional probability of events) in making conditional probability judgments. For example, in determining the probability that a blue taxi cab rather than a green one was involved in a hit and run accident, people frequently fail to consider the base rate frequency of blue and green cabs, (Kahneman & Tversky, 1972; Wolfe, 1989). As readers and writers, we should pay special attention to the construction of arguments, and fallacious reasoning.
Diagnostic reasoning requires the careful consideration of evidence and assertions. Diagnostic reasoning refers to the ability to arrive at specific differential conclusions on the basis of evidence collected for that purpose. Diagnostic problem solving is an important part of becoming proficient in domains as diverse as auto mechanics and medicine. A conceptual tool of wide applicability in diagnosis is the 2 x 2 table. The two by two table is used to assess the accuracy of a 'test' in terms of the rate of hits, misses, false alarms, and correct rejections. For example, consider the following data pertaining to the accuracy of "lie detector" tests (polygraphs). Horvath (1977) studied the official polygraph records of a number of criminal suspects. Through subsequent confessions, the investigator was able to determine the actual truthfulness of these records. Cases were selected such that one half of the suspects (56) later confessed the crimes themselves (establishing guilt), and the other half (56) suspects were later absolved of guilt by the confessions of others (establishing innocence). The subjects of the study were "field trained polygraph examiners all of whom specialized in conducting polygraph examinations for a police agency," (Horvath, 1977, p. 130). This design allowed the researcher to compare the results of the polygraph test with the verified true state of the world. These results are expressed in the following 2 x 2 table.
Polygraph Test Result
Yes No (Lying) (Honest) True State Yes Hit Miss of the (Lying) 75% 25% World No False Alarm Correct Rejection (Honest) 49% 51%
It can be seen that although polygraphs do a fair job of correctly identifying liars, an alarming number of innocent people (49%) are unjustly accused of lying. The media frequently reports on tests for AIDS or drugs. Yet they seldom report that such tests are inherently susceptible to two types of errors: false alarms i.e. telling people they have AIDS when they really don't, and miss', i.e. telling people they don't have AIDS when they really do. Thinking in terms of the 2 x 2 table has important ramifications for issues as different as drug testing and decisions made by juries.
Attention to the sources of information is vital to an accurate assessment of evidence and assertions. People frequently parrot statistics such as '1% of the population owns 35% of the wealth' with no regard for where these figures come from. I believe that The National Inquirer is a less reliable source of information than Science. Thus I am more inclined to be persuaded by claims supported by Science, and weary of unsupported claims. The tendency to reason on the basis of erroneous or inadequate information may be reduced by a conscious effort to address sources of information, (and by conveying a healthy skepticism about the sources of some assertions).
Quantitative Intuition. It is widely believed that some folks have a good "feel" for numbers. Many people (including some skilled mathematicians) seem to think this quantitative intuition is a natural gift. It is my contention that such intuitions are learned, and that they can, and should be taught. Quantitative intuition refers to a subjective sense of ease and comfort with quantitative concepts, and a good sense of when numbers seem right. Part of quantitative intuition is an appropriate sense of scale. An important dimension of developing a sense of scale is grounding numbers in everyday experience. For example, people frequently talk about millions, and billions, and trillions as if they were practically synonyms for 'big number.' An interesting exercise to gain a better feel for these numbers is to calculate how old someone is after their first million seconds of life (in days), and then to calculate how old they are after a billion seconds (in years, try this!). Another means of acquiring a sense of scale is to put novel scales into familiar contexts. For example, an exercise developed by Scott Ritger and Hays Cummins for helping students come to terms with geological time is to ask them to develop their own analogues expressing the relationship between the age of the earth, the age of human beings, and their own age.
A second facet of quantitative intuition is the ability to make order of magnitude estimates. These 'ballpark' estimates are extremely useful checking the accuracy of assertions, checking our own calculations, and applying the power of quantitative reasoning to everyday problems. Yet students are rarely taught how to make such estimates. Order of magnitude estimation starts with a rough sense of the quantities involved in a problem, and a sense of scale. For example, if asked about the distance between Chicago and Seattle, a person may reason that it's about 3,000 miles from coast to coast, and that Chicago is about 1/3 of the way across, so the distance is roughly 2,000 miles, (the actual distance is 2052 miles). Frequently, students have the prerequisite knowledge to make such estimates, but fail to apply this knowledge. One approach to developing this skill is to specifically ask students to make rough estimates in a variety of contexts. Through practice and feedback, estimation skills can be improved.
Stochastic intuition is an important aspect of quantitative reasoning. The phrase stochastic intuition is used to refer to a sense of the probability or frequency of events. People are often impressed by 'amazing coincidences' that, upon reflection, really aren't so amazing after all. For example, the probability that two students in a class of 23 should share the same birthday is about 50% (Paulos, 1988). Yet such discoveries are often accompanied by a sense of awe. As with order of magnitude estimation, stochastic intuition can be improved by practice. In addition, learning basic probability theory helps develop an understanding of statistics, and can enrich students' everyday thinking.
A fourth dimension of quantitative intuition is the appropriate use of heuristics or short cuts in judgment and decision making. People often exhibit systematic biases in their judgments due to the use of faulty heuristics. For example, people often behave as if they believe in an erroneous 'law of small numbers,' the belief that a small number of observations adequately captures the variability of a larger population, (Tversky and Kahneman, 1974). This leads to prejudiced generalizations, and is in direct conflict with the statistical 'law of large numbers.' Through a deliberate effort to challenge these tendencies, we can have a positive and pervasive effect on students quantitative reasoning.
Since many dimensions of quantitative reasoning are interrelated, it is important to coordinate way the quantitative concepts are treated in the three core areas. Fortunately, the curriculum structure of the Western College program affords us the opportunity to coordinate the quantitative aspects of courses each semester. For example, examining notions of causation and correlation in both the Natural Sciences and Creativity and Culture may reinforce these concepts, and give students a better intuitive feel. I propose that each semester, the faculty teaching Natural Science, Social Science, Creativity and Culture, and Applied Field Laboratory meet to discuss the quantitative aspects of their courses. It is hoped that this approach will provide students with a rich and interconnected appreciation for the power of quantitative ways of knowing.
The following are some tentative notions about the kinds of activities that may be appropriate for each of the core areas. Some example quantitative concepts may be found in the appendix.
Fong, G. T., Krantz, D. H., and Nisbett, R. E. (1986). The effects of statistical training on thinking about everyday problems. Cognitive Psychology, 18, 253 - 292.
Horvath, F. S. (1977). The effect of selected variables on interpretation of polygraph records. Journal of Applied Psychology, 62, 127 -136.
Kahneman, D. and Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3, 430 - 454.
Nisbett, R.E., Krants, D.H., Jepson, C., and Kunda, Z. (1983). The uses of statistical heuristics in everyday inductive reasoning. Psychological Review, 4, 339-363.
Paulos, John Allen (1988). Innumeracv. New York: Hill and Wang.
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124 - 1131.
Wolfe, C. R. (1989). Information seeking in the context of Bayesian conditional probability problems. Unpublished Doctoral Dissertation, Pittsburgh Pennsylvania: University of Pittsburgh.