Abstract: Teaching courses involving liberal arts/non- science mathematics students has been a long-time problem due to poor student preparation and motivation. Several suggestions are given with the aim of revitalizing such courses by building on the interests and strengths of the students. These include (1) relating mathematics to current critical issues; (2) use of projects and open- ended problems; (3) providing time for written and oral communication; and (4) collaborative learning. Several examples related to each approach are given.
A major problem area in mathematics education today is effectively teaching liberal arts/non-science majors. These students generally are poorly prepared in mathematics and have negative attitudes about the subject. Every teacher of liberal arts mathematics courses has probably heard students ask questions such as, "Why do I need to learn this mathematics?", "When will I ever use this material?", and "What connections are there between this mathematics and my life?"
Traditionally, liberal arts/non-science mathematics courses have had a set of unrelated topics, such as set theory, logic, probability and statistics, and number theory. Recently, modern topics, such as management theory, social choice, and elementary computer applications have replaced some of the older topics. Generally, these courses have been taught by the typical lecture-discussion method.
The lack of coherence between topics, the perceived irrelevance of the material, and the scarcity of opportunities for student involvement often result in apathetic classes. Classes are typically large, for budgetary reasons, and this contributes to the problem. Since the students need the course only to fulfill a graduation requirement and not as a prerequisite for any other course, they usually try to obtain a satisfactory grade with a minimum of effort. As a result teachers are often reluctant to teach such courses. Students generally end the course with their negative feelings reinforced, thankful that they will never again be expected to do any mathematics.
The following suggestions aim to stimulate liberal arts students' interest in mathematics by building on their concerns and strengths.
1. Relate mathematics to current critical issues While liberal arts/non-science students are often resistant to mathematics, they are generally concerned with environmental and other social issues. Hence, their interest can be stimulated by a course that relates basic mathematical concepts and exercises to such important current issues as population growth, resource scarcity, international relations, hunger, the arms race, health concerns, such as AIDS and nutrition, and a wide variety of environmental issues, including air and water pollution, acid precipitation, ozone layer depletion, destruction of tropical rain forests and other habitats, potential global warming, and soil erosion and depletion.
Such a course can involve exercises that use real- world data connected to issues that impinge on daily life. It can go an important step further than most mathematics courses in urging students to consider the significance of data. To add to class interest, the course can be related to current newspaper and magazine articles and TV and radio reports and programs. Use of news reports concerning global issues shows the students the relevance of what they are learning. The course can also be related to events such as Earth Day (an annual event in many countries), and U. N. conferences related to population, hunger, ecology, resources, refugees, and the arms race. The many news reports about these conferences provide an abundance of data that can be converted into mathematical exercises. Films, videotapes, and slide shows can also be used to broaden coverage of issues.
There are many sources for mathematical exercises related to current issues. In addition to the graphs, charts, and statistics in daily newspapers and weekly newsmagazines, groups concerned about global threats are excellent sources of material. For example, the Population Reference Bureau produces an annual World Population Data Sheet which contains statistics for the world`s nations and regions for birth and death rates, per capita GNPs (gross national products), population doubling times, infant mortality rates, and other demographic data, that can be used for problems on ratios, percents, averages, and correlations, and for the construction of various graphs and charts. The annual Statistical Abstract of the United States also provides a variety of graphs and charts from which many mathematical exercises can be constructed. Material on particular countries can be obtained from their ministries.
Recently, there has been an abundance of published material that provide data, charts, and graphs that can be very helpful for creating mathematical examples and models. These include the annual editions of The State of the World and Vital Signs, both produced by the World Watch Institute, the annual Earth Journal, produced by Buzzword Magazine, and the annual Information Please Environmental Almanac, produced by the World Resources Institute.
Using such reference materials, for the past 18 years I have been giving a course at the College of Staten Island, "Mathematics and the Environment", which aims to show students that mathematical concepts and exercises can provide information, insight, and approaches related to today's global concerns. It is an elective course, designed primarily for liberal arts students.
To organize the tremendous amount of available material, I wrote a text, Mathematics and Global Survival. It presents a wide variety of mathematical exercises and two population models. The mathematics topics and some environmental applications are indicated below:
MATHEMATICS TOPIC APPLICATIONS
1. Basic calculations An acre (43,793 square feet) of (ratios, percents, and Central American rain forest, if percent changes) converted to pasture, can produce about 200 pounds of edible meat during its eight-year lifetime. On the average, how many square feet of rain forest are destroyed to produce a quarter-pound hamburger?
2. Line graphs Using data from Statistical Abstract of the United States, 1998, draw a line graph showing energy consumption in the U. S. from 1902 to 1996.
3. Bar charts Using data from State of the World, 1992, draw a bar chart showing per capita paper use and the share recycled for various countries.
4. Circle diagrams Circle diagrams can be drawn for contributions to global warming by human activity, U. S. household water use, and solid waste generated in New York City.
5. Histograms An ecologist has developed an instrument to indicate average air pollution readings. Given the readings for 50 consecutive days, organize the data and draw a histogram. 6. Scatterplots Using the scatterplot in Mathematics and Global Survival showing the relationship between fat in the diet and prostate cancer deaths, estimate the number of lives that could be saved in a country with 50 million adults, if the fat consumption per person per day changed from 120 to 40.
7. Sequences A natural resource is being depleted at the rate of 4% per year. If there were 200 million tons of the resource in 1997, and there were no new discoveries, how much of the resource would there be by the year 2030?
8. Sampling Find some books in the library about vanishing species. Look for applications of the "tag and recover" approach. Write a brief report.
9. Averages Use data from a survey of student smokers at your school to find the mean, median, and mode for the number of cigarettes per smoker.
10. Variability Given ten pollution readings, find the range and the standard deviation.
11. Probability If all women in a certain society had children until they had one son or a total of 6 children, what would be the mean number of children in that society?
12. The normal curve If the average amount of daily garbage and confidence intervals. collected in a town is 300 tons, and the standard deviation is 20 tons, find the probability that the amount of garbage collected on a specified day will be more than 320 tons.
Of course, this is just a small sampling of environmentally-related exercises that could be asked for these mathematical topics. Using sources such as those mentioned above, an extremely large number of significant problems can be set up. Students should be encouraged, after solving each problem to ask questions related to the importance of the result with regard to pollution, destruction of ecosystems, resource scarcities, and other local, national, and global environmental issues. For example, if the energy consumption of developing countries approached that of the United States, what would be the impact with regard to the environment, resource scarcities, and other quality of life issues? If the tropical rain forests continue to be destroyed to create pasture land and for logging, what is the possible impact on the earth's climate and the potential for serious global warming?
A variety of teaching approaches can be used in relating mathematics to environmental issues. One can use a mathematical concept, such as the computation of percents and percent changes, and use a variety of environmental situations as examples. Or, one can take an environmentally-related topic and use several mathematics topics to help explore it. For example, chapter 3 ("Population Calculations") of Mathematics and Global Survival, reprints the 1992 World Population Data Sheet, and then uses a variety of mathematical concepts and examples to study world population. I have used a mixed approach of intermingling the mathematics and the environmental issues. But what is constantly emphasized is that the mathematics is not presented for its own sake, but as a means of gaining greater awareness of significant issues.
Teaching global issues through mathematics has several important advantages:
1) Students' motivation toward mathematics is greatly increased, as they see how mathematics can provide knowledge and understanding of current critical issues.
2) Coherence is provided by focusing all the mathematics on global issues.
3) Students are not lulled by vague generalities; they are able to integrate hard data. For example, by plotting a graph of carbon dioxide in the atmosphere versus time and solving related mathematical problems, students obtain a far better understanding of the nature of potential global warming than they would get from a general discussion.
4) Students become aware that their studies can provide the valuable background information necessary for them to play an active role in helping to solve environmental problems.
5) The relationships between variables such as population, pollution, hunger, energy, waste, and the arms race are clearly shown. Stress can be placed on the ecological principle that everything is connected to everything else, and that each action has many ramifications.
As the world's attention is increasingly on environmental issues, the mathematical community has also recently focused on mathematical connections to the environment (See Fusaro/Kreith in ZDM 1995/1). Thus, relating mathematics to environmental issues seems to be "an idea whose time has come".
Some mathematics teachers may hesitate to teach a course relating mathematics to current critical issues because it is so different from conventional mathematics courses, and they may not initially feel at ease with the environmental issues considered. However, the following should be helpful:
1) While there are a few new definitions, the mathematical topics are conventional and basic, since the students that take the course are generally not strong in mathematics. Hence, as in other mathematics courses, a great deal of class time is still devoted to standard topics, such as creating and interpreting graphs and charts, computing percents, ratios, averages, probabilities, and standard deviations, and working with mathematical models.
2) One can fairly quickly become educated on issues through the discussions and solutions of mathematical exercises, the use of some of the many recent valuable books and other resources on current issues, and by considering relevant background information in daily newspapers and weekly news magazines.
3) Teachers can tell their students that they are not familiar with every global issue (these issues are so broad that no one can really be an expert in all of them), but that the issues are so critical to everyone's future well being that people must stretch beyond former limits to try to make a difference. Students will appreciate such honesty, and the enthusiasm behind newly discovered facts and concepts will compensate for any initial lack of knowledge. Also, teachers should tell their students that in mathematical classes only a few aspects and only a part of all the complex relationships can be considered.
4) This approach effectively responds to the National Council of Teachers of Mathematics (NCTM) Standards goal of enabling students to "appreciate the role of mathematics in the development of our contemporary society and explore the relationships among mathematics and the disciplines it serves".
5) While more time would probably be spent on this course than on more conventional mathematics courses, especially the first time it was taught, the rewards should make the time well spent. One would be teaching relevant material, increasing his/her awareness and understanding of many of today's most important issues, and helping others to become better citizens. The facts and concepts that were gained would enable one to become actively involved, if so desired, in helping move our society toward changes necessary for stability and survival.
2. Use of projects and open-ended problems Connecting mathematics to environmental issues provides many opportunities for open-ended problems and projects. For example, students can use data from the World Population Data Sheet mentioned on p. 4 and the abundance of data in other sources discussed above, to create charts, graphs, and mathematical exercises, and then to draw conclusions about environmental threats and ways to address them. It is also valuable to have several students organize a scrapbook of articles and/or graphs on topics being considered.
Other projects could involve students gathering data from surveys that they planned and carried out. They could use their data, plus material from the class and other sources to make calculations, draw graphs, compute averages, etc. For example, they could use survey information on commuting habits of students at their school to assess the feasibility of car pooling or the need for better bus transportation. Students are highly motivated when they know that their opinion or information is part of data being analyzed. Many more suggestions for "mini-projects" are presented in Mathematics and Global Survival.
3. Providing time for written and oral communication Instead of giving a final examination, students could be required to do research and report to the class on some aspect of a global issue. These reports enable the class to get an introduction to a wide variety of important issues, not otherwise covered, and to gain greater knowledge and understanding of topics which were covered. They also give students the opportunity to apply the mathematics they have learned to the analysis of an issue. While often weak in mathematical skills, liberal arts/non-science students are generally stronger in the preparation of reports. Students should understand that their report must be centered around the use of graphs, charts, calculations based on appropriate data, information obtained from surveys, and other mathematical concepts.
Student reports can be on topics such as solar energy, noise pollution, the greenhouse effect, ozone depletion, destruction of tropical rain forests, pesticides, world hunger, population growth for a country, connections between diet and various diseases, such as heart problems, osteoporosis, and various types of cancer, or the results of surveys of student opinion on various issues. Class interest can be further enhanced by debates between students, in which students use mathematics to support their positions. There are many environmentally related issues that could lead to stimulating debates, including: - Is population growth a prime cause of poverty and hunger, or do poverty and hunger contribute to population growth? - Should there be a mandatory deposit on beverage containers? - Does wastefulness in the wealthier countries help cause world hunger? - What is the best energy path: nuclear energy? solar energy? conservation? other approaches? - Should developed countries give more aid to underdeveloped countries? If so, under what conditions and what kind of aid? - Is vegetarianism the best diet for individuals and for the world? Once again, it should be emphasized that student success will be judged not by which side of an argument they take, but by how effectively they make an argument based on data and mathematical concepts.
Another approach that has been increasingly suggested recently is to have students keep a journal. This could involve at least one paragraph related to each class session, discussing the student`s reaction to the material being covered - his or her successes, frustrations, questions, comments on the environmental issues and on how the course is going, etc. This would help personalize the learning experience for each student. A periodic review of the journals helps discover any sources of students' difficulties. This approach can be used on a limited basis by asking students to write their reactions to the course periodically during the term. A review of their responses would enable teachers to sense how students are responding to the course.
4. Collaborative learning To maximize student involvement, the class can be divided into groups of three or four students each. In these groups, students can discuss the problem under consideration, consider strategies for solution, resolve difficulties, and think about the process of problem solutions. Each member of the group can effectively learn course material through his or her active interactions with other members of the group. The groups can sometimes be given the responsibility of putting solutions to homework exercises on the blackboard, and perhaps also explaining their approaches.
Small groups of students can put into practice cooperative skills that can lead to problem solutions. Another benefit of this approach is that it simulates conditions in industry and business, where people work on problems cooperatively. The classroom is an ideal setting for cooperative learning activities.
My recent experience with student teams has convinced me that students like to work on problems with other students, rather than passively listening to lectures. They like having opportunities to discuss points of uncertainties with others.
5. Conclusion I have found that relating mathematics to current issues has been a very important motivating factor for students. They become very aware of environmental and other global issues and the need for fundamental changes to avoid future crises. Many students have stated that their personal beliefs and behavior changed through participation in the course, and that they were often speaking to friends and relatives about the need to address global problems. Several stopped smoking or sharply reduced the number of cigarettes that they smoked. Many increased their efforts to recycle materials. One became so enthused that he later ran (unsuccessfully) for the office of Borough President of Staten Island, in order to attempt to more effectively address the borough's severe environmental problems related to air and water pollution, flooding, and the world's largest garbage dump. Several other students chose careers related to the environment in order to investigate possible solutions to the environmental concerns that they became aware of in the course. Students were virtually unanimous in feeling that the use of real-world data and situations were important in a liberal arts mathematics course. The students who also took the more conventional course for liberal arts students stated that they preferred the environmentally-oriented course.
In conclusion, I strongly believe that teachers who have been frustrated in teaching liberal arts students using conventional approaches will find that the approaches suggested here will help to revitalize their courses.
6. Bibliographical references
- Brown, L. (Ed.): State of the World 1998. - New York: W. W. Norton, 1998
- Brown, L. (Ed.): Vital Signs, 1998 - The Trends that Are Shaping Our Future. - New York: W. W. Norton, 1998
- Buzzword Magazine (Eds.): 1993 Earth Journal. - Boulder, CO: Buzzword Books, 1993
- Gagliardi, R.: The Mathematics of the Energy Crisis. - Westmond, N. J: Intergalactic, 1977
- Gross, F.: The Power of Numbers: A Teacher's Guide to Mathematics in a Social Science Context. - Cambridge, MA: Educators for Social Responsibility, 1993
- Horowitz, L.; Ferleger, L.: Statistics for Social Change. - Boston: South End Press, 1980
- Johnson, M.: Writing in Mathematics Classes: A Valuable Tool for Learning. - In: Mathematics Teacher 76 (February, 1983), pp. 117-119
- LeGere, A.: Collaboration and Writing in the Mathematics Classroom. - In: Mathematics Teacher (March 1991), pp. 166-171
- National Research Council: Everybody Counts: A Report on the Future of Mathematics Education. - Washington: National Academy Press, 1989
- National Council of Teachers of Mathematics: Curriculum and Evaluation Standards for School Mathematics. - Reston, VA: National Council of Teachers of Mathematics, 1989
- The Population Reference Bureau, Inc.: 1998 World Population Data Sheet. - Washington, D.C., 1998
- Schoenfeld,A. (Ed.): A Source Book for College Mathematics Teaching. - The Mathematics Association of America, Washington, D. C. , 1990
- Schwartz, R. H.: Revitalizing Liberal Arts Mathematics. - In: Mathematics and Computer Education Journal (Fall 1992), pp. 272- 277
- Schwartz, R. H.: A Simple Mathematical Model for Population Growth. - In: UMAP Consortium Newsletter (Summer 1992), pp. 6-8
- Schwartz, R. H.: Population, Tree-Diagrams and Infinite Series . - In: UMAP Consortium Newsletter (Winter 1992), pp. 1-6
- Schwartz, R. H.: Mathematics and Global Survival. - Needham Heights, MA: Ginn Press, 1998 (4th edition)
- U. S. Department of Commerce.: Statistical Abstract of the United States. - Washington, D. C.: Bureau of the Census, 1998
- World Resources Institute: The 1992 Information Please Environmental Almanac. - Boston: Houghton Mifflin, 1992
Richard H. Schwartz Mathematics Department Room 1S-222 X 3621 Fax number X 3631 Richard H. Schwartz Professor Emeritus, Mathematics College of Staten Island 2800 Victory Boulevard Staten Island, NY 10314 USA (718) 982-3621 Email address: Schwartz@postbox.csi.cuny.edu Fax: (718) 982-3631
Author of Judaism and Vegetarianism, Judaism and Global Survival, and Mathematics and Global Survival. Patron of the International Jewish Vegetarian Society.
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