Adopted by the SAS Faculty, February 2, 1997
At the request of Dean Rescorla, an ad hoc faculty committee has over the last two years been investigating the quantitative skills of our students. This committee, chaired first by Paul Rozin and then by Ingrid Waldron, tested several samples of College students to evaluate their quantitative skills. On the basis of that survey and many discussions, they came to the conclusion that the College should do more to help students become more sophisticated in applying mathematics to other domains of knowledge. They pointed out that this generation of students must be able to interpret and evaluate quantitative data intelligently since, in contemporary society, citizenship, work and personal decision-making all require sophisticated thinking about quantitative evidence. Yet the committee found that students were not sufficiently skilled in interpreting quantitative evidence. For a fuller account of their findings, please see Appendix 1.
For this reason, they made two suggestions, which CUE endorses and recommends, for faculty approval.
1. Departments and faculty should be encouraged to expand the discussion and analysis of quantitative evidence in their courses. The committee felt that a major way to increase our students' understanding of quantitative evidence and mathematical analysis is to have repeated exposure at multiple levels of the curriculum. Therefore CUE
a. asks departments and faculty across the School, including departments in the humanities, to consider bringing the analysis of quantitative data into their courses and to increase the discussion and evaluation of quantitative evidence, where appropriate.
b. asks the Dean of the College to take steps, including the making available of course developments funds, to support faculty efforts to incorporate more quantitative material into their courses.
c. recommends that the Dean and a Quantitative Skills Committee, to be appointed by the Dean, should identify and develop sources of assistance for faculty and teaching assistants who wish to incorporate analysis of quantitative data in their courses. This effort might be done in cooperation with the Mathematics Across the Curriculum Initiative.
2. The College should take steps to ensure that every Penn student develops not only an appreciation for quantitative analysis but also the ability to apply quantitative analysis in the service of understanding other areas of knowledge. Students should learn to think critically about quantitative information and the inferences that can be drawn from it. This involves the ability not only to perform computations but also to consider alternative hypotheses and understand the relation between quantitative data and those hypotheses. It is therefore especially important that students have experience with the use of quantitative analysis in interpreting empirical data.
Consequently, the College should establish a requirement that students complete at least one course which uses mathematical or statistical analysis of quantitative data as an important approach for understanding another area of knowledge. Students in these courses should:
a. learn how to summarize quantitative data (e.g. in graphical or tabular forms) and learn how to interpret summaries of quantitative data
b. understand concepts of random variability and an elementary level of statistical analysis of data
c. develop sophistication in generating alternative hypotheses and learn how to analyze quantitative data to evaluate alternative hypotheses
d. actively analyze quantitative data (e.g., through work with data sets), using the skills outlined above, as well as other relevant skills
e. critically evaluate reports of inference from quantitative data by others.
Courses that fulfill the Quantitative Requirement would not necessarily include all of these characteristics, but would include a substantial component of quantitative analysis both in class time and in assignments and/or exams. Courses designated as satisfying this requirement would be selected from across the curriculum at introductory, as well as intermediate and advanced levels. Students will be encouraged to select a course at a level that allows them to enhance their skills beyond those that they bring to Penn. In many cases, such courses will be lodged in the major of the student, but many students will also fulfill this requirement by selecting designated courses in the General Requirement. In the typical case, this requirement would be satisfied by appropriate selection among courses that also fulfill other requirements.
Courses that fulfill this requirement would be identified by the previously mentioned Quantitative Skills Committee. This committee, appointed by the Dean, would include SAS faculty from the three divisions and two student members. This Quantitative Skills Committee would be charged with reviewing existing courses for inclusion in the list, as well as encouraging appropriate departments to consider ways of augmenting and modifying their offerings to provide greater emphasis on quantitative material. Courses would be reviewed on a case by case basis, with the possibility that the same course number would satisfy the requirement when taught by some instructors but not others.
The Quantitative Skills Committee should also work with the Dean, the Mathematics Department, the Department of Academic Support Services and other relevant parties to ensure that sufficient support is provided to students who are developing quantitative skills. Useful types of support would include walk-in math clinics, problem solving/discussion groups, a help line and newsgroup for math-related questions, and counseling for "math anxiety."
Examples of existing General Requirement courses that might fulfill this Quantitative Skills Requirement are listed below.
1. Physics 150 Lab: Principles of Physics I. Gladney, Soven. In this laboratory course students collect their own data, make repeated observations, summarize those observations graphically, evaluate the variability, and consider what inferences can be made to formal theories. Heavy use of computers.
2. Sociology 4: The Family. Watkins, Morgan. In this lecture course students consider information contained in large data sets, develop hypotheses about interrelations, and evaluate those hypotheses.
3. Statistics 111: Introductory Statistics. Hildebrand. In this methodologically oriented course students also work with data sets that they bring in, considering how to summarize those data and make statistical inference from them.
4. Psychology 1: Introduction to Experimental Psychology. Rozin. This lecture course includes in-class design of experiments, collection of data, and analysis of those data.
Many additional courses at both introductory and advanced levels would be expected to satisfy the Quantitative Skills Requirement. Based on the information provided by faculty questionnaire responses, it appears that the following courses would be suitable for this requirement. These courses would require additional evaluation based on more complete information (including a syllabus, copies of assignments, and responses to a revised questionnaire). We anticipate that further inquiry would reveal other courses which would also be suitable for this requirement.
Anthro 210 - Anthropology and Biomedical Science
Anthro 554 - Quantitative Analysis of Anthropological Data (to be
renumbered Anthro 454)
Biol GH121 - Molecular Biology of Life
Biol 423 - Plant Ecology
Biol 425 - Molecular Biology and Genetics Laboratory
Biol 446 - Statistics for Biologists
Biol 480 - Advanced Cell Biology
Chem 1 - Introductory Chemistry
Chem 53 with Chem 54 - General Chemistry Lab
Chem 53 with Chem 54 - General Chemistry Lab
Chem 101 and 102 - General Chemistry I and II
Econ 4 - Macroeconomic Theory (Computer section - Adams)
Econ 5 - Statistics for Economists
Econ 6 - Econometrics
Econ 37 - Population Economics
Econ 113 - Topics in Economic Measurement and Econometrics
Econ 115 - Topics in Econometrics: Microeconometric Techniques and
Applications
Gen Hon 208 - Diet and Health
Geol 100 - Introduction to Geology
Math 151 - Calculus for the Social and Biological Sciences, II
Math 170 - Ideas in Mathematics
Math 475 - Statistics of Law
Physics 1 and 2 - General Physics
Physics 150 and 151 - Principles of Physics
Physics 170 and 171 - Honor Physics
Physics 250 - Principles of Physics III - Modern Physics
Pol Sci 298. Introduction to Political Research
Pol Sci 498 - 301 - Selected Topics in Political Science (Golden)
Psych 1 - Introduction to Experimental Psychology (Rozin)
Psych 151 and GH 151 - Cognitive Psychology
Psych 153 - Thinking and Decisions
Psych 157 - Human information Processing
Psych 192 - Psychological Testing
Psych 311 - Research Experience in Perception
Psych 321 - Research Experience in Learning
Psych 343 - Research Experience in Neuroethology
Psych 351 - Research Experience in Cognitive Psychology
Psych 362 - Research Experience in Abnormal Psychology
Psych 453 - Research Experience in Decision Analysis
Sociol 4 - The Family (Watkins, Morgan)
Sociol 7 - Population and Society
Sociol 10 - Social Stratification
Sociol 41 - Topics in Sociology (Regional Analysis or Urban Analysis with
computers - Douglas)
Sociol 100 - Introduction to Sociology Research
Sociol 128 - Introduction to Demographic Methods
Sociol 302 - Community Research and Community Service: Senior Projects
Stat 111 and 112 - Introductory Statistics
Additional specific policies:
1. The Quantitative Skills Requirement would be in effect beginning with the class of 2002.
2. AP credits and "credit away" could not be used to satisfy this requirement. The proposed Quantitative Skills Committee would develop a policy for implementing this requirement for transfer students.
3. The Quantitative Skills Committee should report to CUE each spring for evaluation of the progress in improving the quantitative skills of College students.
Appendix I
Report on 1995 Survey of Numeracy and Critical Thinking Skills of College Students at the University of Pennsylvania
In order to evaluate the Numeracy and Critical Thinking Skills of College Students at Penn, a questionnaire was mailed to a random sample of 60 graduating seniors in the spring of 1995 and 120 entering freshmen in the fall of 1995. Responses were received from 35 seniors and 61 freshmen. The respondents seem to be reasonably representative of their classes. The seniors in the survey were very similar to the whole class of graduating seniors in GPA and percent who had a major in the natural sciences or a quantitative discipline such as economics. However, the seniors in the survey had somewhat higher SAT scores than average for their class. The SAT scores for the Freshman in the survey were similar to the SAT scores for the whole class of entering freshmen. (See Table at end.)
For the Numeracy portion of the questionnaire, seniors got 73% correct and freshmen got 67% correct, on average. Both seniors and freshmen did well on questions which required basic graph-reading or quantitative skills. However, in both senior and freshman samples, a third to a half of respondents missed various questions that required quantitative graph reading or an understanding of probability. For example, one question asked "If on a given trading day the same number of stocks gained and lost value (and none stayed the same), what is the probability that a person picking two stocks at random that day will get one winner and one loser (in either order)?" Only about half of seniors or freshmen got the correct answer (1/2).
Seniors did better than freshmen on several questions that involved an understanding of basic statistical concepts, such as statistical significance or standard deviation; about three-quarters of seniors vs. one-half of freshmen answered these questions correctly. Roughly a third of College students take at least one semester of statistics during their years at Penn, and this presumably contributed to the better performance of seniors on these questions. For both seniors and freshmen, over two-thirds of respondents missed questions that required the application of the principle that results from smaller samples are more variable.
For the Critical Thinking portion of the questionnaire, seniors got 74% correct and freshmen got 71% correct, on average. Both groups did well with many questions which required logical inference or the critical evaluation of evidence and the conclusions that can be drawn from particular types of evidence. A few questions were missed by half or more of the respondents, apparently because many respondents failed to think of possible alternative interpretations of the evidence presented and therefore drew erroneous conclusions from the evidence.
As would be expected, higher Math SAT scores predicted higher percent correct on the Numeracy portion of the questionnaire for both seniors and freshmen. In addition, among seniors, those who had taken a natural science or quantitative major had higher Numeracy scores; however, this association was much reduced and not significant in analyses that controlled for Math SAT scores.
We also asked seniors which course at Penn they believed had helped them most in critical thinking. In response, seniors mentioned a wide variety of courses: eleven in the natural sciences, six in mathematics or statistics, four research experience courses, and eleven in a variety of other disciplines.
Some of the major conclusions derived from this survey are supported by findings from an earlier survey. In January, 1995, 161 College freshmen who were taking Psych 1 completed a similar Numeracy and Critical Thinking questionnaire. Most of these freshmen showed good basic graph-reading skills and generally good skills in evaluating evidence, but they showed weaknesses in understanding probability and insufficient caution in drawing conclusions from evidence, probably because of failure to consider alternative interpretations of the evidence. Thus, the general pattern of strengths and weaknesses in Numeracy and Critical Thinking appears similar in the two surveys.
Acknowledgments: These surveys were planned and executed in large part by Paul Rozin, Clark McCauley, Peter Marvit, Caroline Szeto, Kent Peterman, Doris McLeod, and Kim Sheppard. We very much appreciate their work, and we thank the students who participated.
Table Mean Math SAT Mean Verbal SAT Seniors in survey 688 609 Entering freshmen in 1991 660 590 Freshmen in survey 689 614 Entering freshmen in 1995 680 604