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Teaching Radical Math: Taking the Numb Out of Numbers

by Marilyn Frankenstein

‘Science for the People’ Vol. 15, No. 1, January/February 1983, p. 12 — 17

Marilyn Frankenstein teaches mathematics at the College of Public and Community Service (University of Massachusetts/Boston Downtown). She is on the board of directors of the Group School, an alternative high school in Cambridge, MA. She is also a member of the editorial collective of Radical Teacher.

In 1980, at a workshop I gave on “Political Math” at the National Council of Teachers of Mathematics conference, most of the audience objected strenuously to my claim that all math education is biased. I argued that even a trivial math application like totaling a grocery bill carries the non-neutral message that paying for food is natural, and that society should be organized in such a way that people must buy food from grocery stores. I argued further that traditional math courses which use no real-life data carry the non-neutral hidden message that learning math must be divorced from helping real people understand and control the real world. I wondered if those same teachers objecting to my discussion of “Political Math” also objected to the 1981 conference workshop which examined “the activities and applications of applied mathematics in the mapping processes at the Defense Mapping Agency” or the workshop on “Mathematics and the Military” which discussed “the importance of mathematics to the nation’s readiness. ” ¹

Although I am sure most of the teachers who attended my workshop would disagree, I believe that traditional mathematics education is often used against people on both personal and political levels. Many people (especially women) are made to feel inferior and incompetent because they “cannot do math.” Many people cannot make intelligent consumer decisions, such as choosing which loan to obtain, as well as they could if they were confident in their understanding of mathematics. Politically, people can be more easily oppressed when they cannot break through the numerical lies and obfuscations thrown at them on a daily basis. A mathematically illiterate populace can be led to believe, for example, that welfare programs are responsible for their declining standard of living. Lack of math knowledge will inhibit them from researching the numbers to find out that “welfare” to the rich dwarfs any meager subsidies given to the poor. (Here are a few examples: in 1975 the maximum payment to an Aid for Dependent Children (AFDC) family of four was $5000 and the average tax loophole for each of the richest 160,000 taxpayers was $45,000; ² in 1980 $510 million of our tax money paid for new airports so that private pilots would not land their planes at large commercial airports; ³ and, our taxes also pay for “defense company lobbying since lobbying costs can be included in a contract …. five [defense] companies charged the Defense Department $15.8 million of the $16.8 million they spent to maintain Washington offices between 1974 and 1976. ⁴) A mathematically sophisticated populace has an important tool with which to fight back. When the Coalition for Basic Human Needs, for example, prepares statistics to show that actual shelter costs in every major Massachusetts city exceed the AFDC welfare grant, ⁵ they have a powerful argument to use to prevent cut-backs and to heighten people’s awareness of the living conditions of welfare mothers.

Basic Mathematics for the people means more than the ability to calculate. It means the ability to reason quantitatively, the ability to use numbers to clarify issues and to support or refute opinions. Mathematics education for the people must also be mathematics education with the people. It cannot be taught using what Paulo Freire calls “banking” methods: “expert” teachers depositing knowledge in the presumably blank minds of their students who memorize the required rules in order to get future dividends. ⁶ It must be taught by students learning math together; by students creating as well as solving problems, so they can control how math is used, and control their own learning process.

This article describes a non-neutral basic math course that I developed to be a “mathematics for and with the people.” ⁷ The content teaches arithmetic while simultaneously raising political consciousness. The methods try to break down traditional authoritarian teacher-student relationships by giving students meaningful control over their own learning. The aim of the course is to educate people to understand the need for radical social change, while giving them both the math literacy tools necessary to challenge ruling class ideas, and the cooperative learning experiences necessary to create and live in a new society.

The Critical Curriculum

Although most of my twelve years of teaching have been in alternative schools, it is only over the last few years that my politics and my teaching have become truly integrated.

When I taught at Park East, an alternative high school in East Harlem, my students knew my political beliefs only because of the posters in our classroom and the announcements I made at town meetings. My math courses had nontraditional content, such as math magic and math-art, but no radical political content. I was actually quite angry with leftist colleagues who spent their class time talking revolution to kids who needed to improve reading, writing, and math skills. As a consequence of this anger, I refused to read education theory; I felt that intellectuals read and talked too much, and did too little.

When I began teaching at Stockton State Colege in New Jersey, originally an experimental “1960s-style” institution, my students enjoyed the magic and art. But they consistently asked “Why do we need to learn this?” I wanted to be able to answer with more than, “It teaches you to think abstractly.” And, because I did not want to build a course solely around survival in a capitalist society, I tried to find nonconsumer, daily-life uses of math. I then realized what seems so obvious now-that many newspaper articles contained numbers, and that in order to analyze them carefully people needed a critical understanding of basic math. I started to design a curriculum using The New York Times as a text.

When I began my current job at the College of Public and Community Service (CPCS) at the University of Massachusetts/Boston’s Downtown Campus, I had an ideal situation in which to expand the idea of using math to understand current events into using math to analyze critically the condition of society. The average age of our urban adult students is 36, and most are currently employed or preparing to work in public or community service. Most of the students, therefore, want to examine political issues and are interested in finding out how math can be used to understand more about those issues.

At CPCS I also had an ideal situation in which to begin reading radical education theory. Socialist colleagues who were both intellectuals and activists encouraged me to read Paulo Freire’s work. His ideas convinced me that theory and practice could be valuably intertwined, so that reflection by reading and talking with students and coworkers improved my daily work, and modified and clarified my reflections. Freire also made me aware of the political implications of my teaching methods. During most of my years of teaching, I had thought my methods were those of any dedicated teacher who respected his/her students. Because I did not emphasize the radical philosophy behind my methods, they didn’t have as strong a political impact as they could have. Now, I stress to students that it is not because I am “nice” that I treat them and their work with respect; it is not because I am “modest” that I do not view myself as superior to them or as an expert who understands things they never will; it is not because I am “idealistic” that I am confident they know more math than they think and will be able to understand math they felt they never could; it is not because I am “lazy” that I do not simply lecture, but encourage them to listen carefully and learn from each other. I act the way I do because in a socialist society the relationships between teachers and students will be restructured. Although we are presently living and struggling under capitalism, we can begin to explore and develop new social relations as part of creating socialism.

Math and Political Consciousness

The following sample problems illustrate how to integrate the teaching of basic mathematics with the raising of political consciousness, and how to foster critical thinking by expanding traditional problem-solving techniques to include definition of problems and gathering of required information. ⁸ All the problems have a purpose; performing the math operations clarifies the data or presents it in a more forceful way. Also, teaching basic math by using it to analyze complex issues increases students’ intellectual self -image. Since the applications come from a wide variety of areas, students will probably raise subject-matter questions the teacher cannot answer. They will realize that the teacher is not an “expert” with all the answers and they will gain self-confidence and experience in searching for information to answer their own questions, or in becoming what Freire calls “critical co-investigators in dialogue with the teacher.” ⁹Finally, the content of this radical math course challenges the fragmented view of society presented by a curriculum which breaks knowledge into separate, unrelated issues, to be discussed only by specialists. When math is taught as a necessary part of a careful analysis of the conditions of society, students develop a clear sense of how knowledge of specific subjects can be integrated to give a critical understanding of the world. ¹⁰

Example 1 (used after basic operations with whole numbers have been studied):

A. The Empty Pork Barrel: Unemployment and the Pentagon Budget by Marion Anderson uses numerical arguments and charts to document the fact that as the military budget goes up, the number of jobs lost in civilian goods and services (because of tax monies going to the military) exceeds the number of jobs generated by military contracts. Military spending at a rate of $78 billion a year is responsible for the annual loss of 907,000 jobs. Every additional $1 billion of Pentagon spending causes [an average] loss of 11,600 job opportunities … “How many jobs will Americans lose this year from our current level of military spending?

Solving this problem involves using many of the whole number operations and finding information. It can also lead to a discussion of the politcal bias of statistics. Exact military expenditures are difficult to determine because “by custom and accounting practice, national military budgets usually do not include expenditures for veterans’ benefits, interest on war debts, civil defense, and outlays for strategic industrial stockpiling. Military budgets may also exclude all or part of national intelligence expenditures … there are also substantial social costs which are extra-budgetary, including … [such things as] tax exemptions accorded military priorities … ” ¹¹

B. Write a brief statement of your opinion about military spending. List the kinds of numerical data that would support your opinion. Find at least one of the facts that you feel would support your opinion and describe how you would find the others.

The goal of this exercise is to make students aware of how people find and use numbers to support their arguments. For example, to argue against military expenditures, in addition to the number of additional jobs created if the military budget were spent on the civilian sector of our economy, one could find numerical information on the enormous amount of overkill both major military powers have; on the holocaust effects of nuclear war; on military-related expenditures, such as our support for the Phillipine and South African dictatorships; and, other uses of resources, such as “in an oil-short world, the newest military tanks will consume 1.9 gallons of gas per mile” and 20 times as much U.S. public research money goes for transportation into space as for mass transit on earth.” ¹²It is also important to stress that a detailed argument against military expenditures involves more than just numbers; it involves a discussion of the necessity of imperialism in advanced capitalist development and corresponding necessity of maintaining a huge military force to protect capital.

Thinking of the kinds of numerical data that could support an argument comes from experience with the types of questions that can be asked. For example, once the idea of comparing the results of military versus civilian spending on jobs is introduced, you think of asking that same question of other government spending. One instance is described in an article called “The Nuclear Numners Game” ¹³ : “A Senate Commerce subcommittee staff proposal sent to President Carter shows that a $1.65 billion investment in conservation, using public service workers, would create 100,470 new jobs and save over 2000 million gallons of oil per year … a $3.4 billion investment in a nuclear power plant … would create, at most, 11,000 jobs and save the equivalent of only 28 million gallons of oil.”

Finally, as a source of information, you can introduce the students to various local action groups working on peace conversion, groups trying to link social service cuts with military budget increases, and groups connecting solidarity with Third World Liberation struggles to fighting the U.S. military build-up.

TEACHING COLLECTIVE ACTION
By Paul Rowland

During the past two decades, Paulo Freire developed a radical pedagogy for adult literacy programs in Third World countries. His work is based on the assumption that education should empower the learner to analyze the world and then act on this analysis. As a radical teacher of high school science, I have attempted to adapt Freire’s method to the science classroom. Although my teaching methods are still evolving, I hope that the following description of activities will be useful to other teachers and learners who want to use our schools more effectively.

An important first step in radical education is to create a classroom atmosphere in which students are aware of the fallibility of the teacher and value their own opinions. To become what Freire calls the “teacherstudent,” the teacher must encourage students to critically examine both of these roles. The rule. making, authority figure must give way to a partnership; in other words, the teacher must give up control of the student. Many teachers argue that giving up control is impossible, illegal, immoral or just suicide. My experience is that the anticipation is worse than the act. Real institutional constraints need to be recognized by both teacher and student but they should be clear from the start and not imposed later on.

A next step is to choose a social issue in the discipline and use it as a first attempt at problem-posing. Problemposing involves facilitated discussion in which students develop an analysis of a socially relevant issue. The challenge for problem-posing teachers is to provide students with alternative ideas that allow the students to take actions (preferably collectively) which leave them feeling powerful and not defeated.

My first problem-posing discussion involved a tenth grade biology class in rural New York. After showing a filmstrip about life support systems, I posed the problems of who should have access to machines that save lives, who has access under our present system, who benefits from the use of extraordinary means for life support, and how these systems influence health care in general. My seemingly sullen biology class came alive and asked more questions than we answered. Over the years I developed similar sessions for my biology classes.

In 1976 I began teaching an environmental elective for high school seniors (still in rural northern N.Y.). This course went beyond dialogue and into action. We examined waste (electrical, food, paper) in the schools and presented recommendations to the school principal. We sampled a local stream above and below a village with no sewers (and suprisingly found no difference). Our visit to a nuclear power plant ended in a session with the public relations officer. Without any prodding or prior discussion, the students quizzed and discredited the company flack. Our busride home was dominated by a discussion of how outrageous it was for the consumers to have to pay to have people lie to them about the hazards of nuclear power.

The following year the class looked at land use and made recommendations to local planners on how they could control development in ways that would benefit the people of the community and preserve the uniquness of the area.

Paul Rowland is a doctoral student in Curriculum and Instructions at New Mexico State University. He is currently developing progressive approaches to educational uses of computers.

Example 2 (used to review percent): According to “Eating Better for Less” by Lucille Sandwith, ¹⁴50 out of the 32,000 U.S. food manufacturing firms make 75% of the net profits. Of these top 50 corporations, 31 bought 63% of the national media advertising, or roughly $5 billion in 1977. Of the top 25 advertisers from all industries, 18 were food companies.

A. What percent of the U.S. food manufacturing firms make 75% of the net profits?

This question requires careful reading since the many given percents might be confused with the percent asked for. And its solution serves a purpose: changing 50/32,000 to 0.20% highlights the fact that only a tiny percent of the firms make most of the profits. The information in the question can lead to a political discussion of agribusiness and corporate monopoly in general, as well as to a math-related discussion of the advertising industry. (For example, 70% of television food advertising promotes low-nutrient, high-calorie foods, whereas only 0.7% promotes fresh fruits and vegetables.)

Analyzing Error Patterns: All wrong answers involve some correct, logical reasoning. For example, there is logical thinking behind these subtractions:

48.37
– 5. 4

43.33

23.45
– 2. 8

21.37

128.423
– 82. 22

46.401

This person subtracted correctly in relationship to basic subtraction facts and “borrowing.” However, he or she did not understand the decimal place-values and therefore treated the decimal parts like whole numbers. The class not only analyzes this student’s reasoning, but also discusses how to convince him or her that the method was wrong and how to teach him or her correct methods. Analyzing error patterns provides nonrote reinforcement of computation skills, and shows students that you respect their intelligence and will not think they are stupid when they make errors. This, in turn, encourages students to respect their own and each other’s intelligence.

Keeping a Math Journal: Journals serve as vents for students’ feelings about math and act as concrete records of progress for students who too often belittle their own successes and focus on what they cannot do. The journals help students realize that they can now accomplish what one month ago they thought was impossible and helps them clarify which learning techniques worked best and where they use math in real life. I collect the journals frequently and comment on them, offering encouragement, alternative solutions or perspectives, and explanations of how students’ remarks on learning math often apply to learning in general. Students’ comments on the class are very helpful in my lesson planning. I find time to read and comment on journals because I do not collect homework assignments, but instead give students the answers to homework problems and encourage them to work together and evaluate their own learning.

Students Teaching: In order to teach a math problem to someone, you must be able to recognize all the correct methods of solving it as well as the logic behind incorrect methods. As various students practice teaching, they begin to involve other students, asking them to justify their answers. The class checks itself and rarely lets a mistake go by. The students get very involved, arguing constructively and thinking creatively about solutions to the problems. The student teachers effectively involve even the quiet students, who are more willing to participate when asked by a classmate. A feeling of solidarity develops in the class as students, learning from each other, come to respect one another. After many students have had a chance to teach problems at the board, the class attitude begins to reflect their greater understanding of the role of the teacher. Students realize how difficult it is to think on one’s feet, to write at the board, and to talk to people who are not paying attention. Having students teach helps break down the authoritarian image of the teacher and simultaneously builds true respect for the hard job good teachers do.

Students Working in Groups: In order for students to work in math study groups, the misconceptions they might have about math learning need to be dispelled, and they must realize how much they can learn from sharing their knowledge. They must have some understanding of why some people are quiet and others talk too much in groups, and through this understading they need to work together to make their group learning experience help everyone. ¹⁵Some examples of suggested group tasks are:

B. Based on the information given, create and solve a math problem whose solution involves using percents.

Students will fully understand percents when they understand which percent problems can be created from given information. For example, here students must realize that you cannot find out how much profit the top 50 firms make, but you can find out how much money is spent on national media advertising. Also, it is unclear whether the national media advertising figure refers to the total spent by food manufacturing firms or by all industries. More information must be found in order to clarify this.

C. Read the entire article. Discuss how at least three points in the article are supported by use of percents.

The sample problems also illustrate the basic idea behind politicizing the content of any course: find political applications for each concept in the curriculum, teach those concepts in the context of the applications, create nonrote assignments which gradually involve the students in asking and answering their own questions, and, wherever possible, include information about local groups fighting to change the situation. In biology, for example, students can study the facts of reproduction through examining issues ranging from sterilization abuse to statistics on infant and maternal mortality rates among various races and classes. In chemistry, the process of molecular interaction can be studied through the specific science behind how corporations are polluting our environment for profits. An introduction to general science can include data on the numbers of blacks, hispanics, and women employed in different fields and anecdotal reports about the conditions of working in science.

Alternative Math

As Freire says, “a project’s methods cannot be dichotomized from its content and objectives;”¹⁶ new teaching methods, as well as course content, are important in teaching mathematics for the people. The methods that follow are intended to counter the misconceptions about learning mathematics that are often part of traditional schooling. Students begin to realize that they are not “stupid” if they make a mistake, that people learn from analyzing their mistakes, asking questions, and evaluating exactly what they know and what they need to find out. Students start to understand that everyone learns at different rates, that learning is not linear, that going on to a new topic allows them to then review the old topic with deeper understanding, that stopping work on a problem to rest and later returning gives fresh insight, and that using math involves slow, careful thinking, not quick, immediate answers. Students discover that there are often many equally good ways to solve a math problem, that it is within their control to present numerical data effectively to prove their points, and that, depending on their assumptions and the real-life situation, there can be more than one correct answer to a specific math problem. Also, the following methods are intended to encourage students to share what they know with others and to work together to accomplish the task at hand.

Group Evaluation of Homework — Working in groups of three or four, determine which homework problem was easiest and which was hardest. Evaluating homework questions is a good lead into having students create their own math problems. Also, this task shows students that because people learn in different ways, they find different problems easy or hard.
Group Creation of Quizzes — Working in groups of three or four, create two review questions based on the previous lesson. Hopefully, the more practice students have in creating questions, the more they will become accustomed to asking questions, both in school and in their daily lives.

Currently, I am trying to create a better balance among problems which help students focus and document their criticisms of life under capitalism, and problems which show the victories that have been won against oppressors and in the fights that are now taking place. Only over the last few years, after reading Paulo Freire, have I overcome my own pessismistic feelings about our ability to change society radically. In addition, conversations in a class on “Politics and the Education of Oppressed Communities” have underlined for me the fact that one of the main obstacles to creating a socialist world is a popular feeling of powerlessness to change the existing social conditions. We radical educators must further the development of a “knowledge for the people” that will challenge basic assumptions, critically analyze society, and instill hope and the energy to act in our students.

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REFERENCES

Conference Program, 1981 Annual Meeting, National Council of Teachers of Mathematics.
Steve Babson and Nancy Brigham, What’s Happening to Our Jobs? Popular Economics Press, 1978, p. 37.
John Judis and David Moberg, “Some Other Ways to Cut the Budget.” In These Times, March 4-10, 1981, p. 37.
Environmental Action, July I August, 1980.
Coalition for Rasic Human Needs, April 1980 Report. Boston, MA.
Paulo Freire, Pedagogy of the Oppressed, Seabury Press, 1970, Chapter 2.
Another description of this course, ”A Different Third R: Radical Math” appeared in Radical Teacher #20 (P.O. Box 102, Kendall Square Post Office, Cambridge, MA 02142). I am currently writing a text for this course and am very interested in having people class-test it.
The examples are some of the hardest problems in the course. I chose them to illustrate the goals of the course that are gradually built-up to during the term; and, to show the different types of critical, creative thinking used throughout the term. Compacting all these aspects into two examples meets length requirements for the article, but distorts the more gradual learning of problem-solving and problem-posing.
Paulo Freire, Pedagogy of the Oppressed, p. 68.
In addition to Science for the People, some excellent sources for a political curriculum are:

Food and Nutrition Group of Boston Science for the People, Feed, Need, Greed: A High School Science Curriculum.
Science for the People, 1980.
Rita Arditti, Pat Brennan, and Steve Cavrak, eds., Science and Liberation, South End Press, 1980.
David Weir and Mark Schapiro, Circles of Poison: Pesticides and People in a Hungry World. Institute for Food and Development Policy, 2588 Mission St., San Francisco, California, 94110, 1981.
Food Monitor (a monthly magazine about the politics of hunger). 350 Broadway, Suite 109, New York, New York 10013.
Environmental Action (a monthly magazine about the politics of the environment). 1346 Connecticut Avenue NW, Washington, DC 20036.
Dollars and Sense (a monthly popular socialist magazine about the economy). 38 Union Square, Room 14, Somerville, Massachusetts 02143.
Ruth Leger Sivard, World Military and Social Expenditures. World Priorities, Box 1003, Leesburg, Virginia 22075, 1980.
Ibid.
Environmental Action, March 1979.
Food Monitor, Sept.-Oct., 1980. pp. 8-12.
David Reed, in his Education for Building a People’s Movement, South End Press, 1981, discusses the experience of a group that “challenged the notion of calling (some people) the ‘silent ones’ and affirmed that their form of participation in the organizing work was a result of the dynamics of the entire group, not just individual problems. The underlying problems were of style, vocabulary and kinds of analysis used primarily by more experienced leaders. One conclusion of the discussion was that if the problem of some people not speaking-up were to be resolved it would be through a collective effort including those who were more aggressive and vocal.” (p. 95).
Paulo Freire, “Cultural Action for Freedom.” Harvard Educational Review, 1970, p. 44.