What does mathematics education have to offer civics education?
On one hand, if we believe that mathematics is harmless and innocent because it has little to do with the world we live in, then we might simply respond that mathematics education does not have a role in civic education outside of offering some form of mental stimulation or exercise for the mind. However, even the pure mathematician agrees that mathematics is part of our human experience (Hardy, 1967; Davis and Hersch, 1981). As Keith Devlin notes, "the study of mathematics is ultimately the study of humanity itself" (Devlin, 1998, 9).
On the other hand, the applied mathematician would claim that his or her work is all about the world in which we live (Davis and Hersch, 1981) and that the role of mathematics in society has grown more and more significant in recent decades (D'Ambrosio, 1999). At the same time, one could argue that the mathematization of society "has become more and more hidden from view, forming an invisible universe that supports much of our lives" (Devlin, 1998, 12). If either the purist or the applied mathematician's observations about mathematics reflect the nature of mathematics, then it is important to consider the role that mathematics education could play in citizenship education because either an education in mathematics will help us understand ourselves or it will help us understand the world in which we participate. Of course, it could help us understand both.
The mathematization of society
The relationship between mathematics and society is at once obvious and subtle. In Lynn Steen's, Why Numbers Count, there is extensive discussion about the prevalence of quantitative and statistical thinking in our world today (Steen, 1999). For example, in an essay by Theodore Porter we read:
|
Mathematics is central to our modern scientific understanding of the natural and social worlds. But our reliance on it is not simply a consequence of its perceived objective validity. Quantification is also a critical element in how we conduct our affairs, exchange goods and services, define and enforce regulations, and communicate knowledge. In all these senses, the world has become much more thoroughly quantitative. (Porter, 1997, 9) |
Steen notes that the flow of information in the form of numbers has been instrumental in the formation of modern nations; indeed, he claims that "statistics" developed as the science of the state (Steen, 1997, xvii). If we think about it, we begin to realize that there is much in our society which has been quantified- the gross national product, the DOW index, unemployment rates, the weather forecast, the smog index, the quality of a hockey player's game performance, a student's understanding of literature, and intelligence itself, with the Intelligence Quotient. We, in this society, are in the practice of assigning numbers to almost anything we encounter in our day-to-day living. Given the extensive quantification in our society, we might suggest that the mathematical knowledge, skills and processes we teach in school mathematics are essential for active participation in the world in which we live.
The National Council of Teachers of Mathematics clearly articulates their vision for mathematics education. Numeracy, or quantitative literacy, is one goal but there is more than just that. In the NCTM's Standards 2000 the following goals for mathematics education are stated:
|
Mathematics for life. Knowing mathematics can be personally satisfying and empowering. (NCTM, 2000, 4) |
As the NCTM suggests, teaching for quantitative literacy is an essential aspect of the mathematics education citizens must receive, simply so they to participate in the most common of daily activities. As well, it is understood that those in society who hold technical and scientific jobs also need a good mathematics education. However, many observers point out that mathematical forms and logic are embedded in more than just scientific and technological structures. They are also embedded in social, political and economic structures (D'Ambrosio, 1999; Davis, 1995; Skovsmose, 2000). "Mathematical modeling" is an example of the formatting power mathematics has on society today. That is, describing reality with mathematics and then manipulating the mathematics in order to understand and/or predict reality is a common process found in society. Such a process plays a more significant role in society than simply describing reality. When reality has been "modeled and re-modeled, then this process also influences reality itself" (Skovsmose, 2000, 4). In other words, mathematics takes on formatting power.
|
Mathematics applied, for instance, in a business does not consist of 'pictures' of reality which exists prior to and independent of the modeling process. Mathematical models of advertising, marketing, investments, etc. become part of the economic reality themselves. They serve as a basis for decision making and for economic transactions. In this way, mathematics has become part of the economic reality. This not only applies to business but to economic policy-making in general and not only to economy, but to categories like time, space, communication, transport, war. (ibid., 4) |
It is important that school mathematics is more than simply teaching students to think quantitatively or statistically. In school mathematics we must educate our youth, our citizens, so that they begin to understand and critique the formatting power of mathematics in society (ibid.).
School mathematics: exploring the implications of our practices
The school, as an agent of society, does not merely transmit the knowledge of one generation to the next; it participates in the transformation of that knowledge. In focusing on this idea and not that one, it is assigning a value to both; in teaching in this way and not that way, it is privileging particular ways of acting over others. (Davis, 1995, 8)
In what ways does mathematics education contribute to one's citizenship education? We need to consider both the content we explicitly teach and the hidden curriculum that is manifested in the ways we teach.
Instructional stances and strategies in mathematics that potentially conflict with citizenship education
1) Mathematics as a set of facts, skills and processes
If we teach mathematics as a set of facts and skills where there is an optimal way of finding the correct solution to pre-given questions or problems, then at best students leave with little more than algorithmic and computational skills fit to participate as consumers and workers in our society. With such skills young adults will be able to calculate the price of a garment reduced by 30% or compute a 15% tip on a restaurant bill; but will they be able to challenge management's claim that a 29% salary increase for a low paid auxiliary health care worker is despicable?
2) Mathematics as facts and fact
When we teach students the number "facts" or that context can be stripped from word problems reducing them to mathematical equations what else might students be learning? Frankenstein warns that when mathematics is treated this way it sends a hidden message to students that using mathematics is not useful in understanding the world; rather mathematics is just pushing around numbers, writing them in different ways depending on what the teacher wants (Frankenstein, 1997). Even at its "best" such mathematics teaching might create experts who can develop models to describe the world they live in and manipulate those models to control the world but to what extent do such practices educate for an awareness of the mathematization of our society. Do the people who create economic models, for example, reflect on the formatting power of those models within society?
3) Mathematics as either right or wrong
Instruction in mathematics where the emphasis is placed on completing pages of exercises with the primary goal of getting the right answer is common place. For many students correcting homework is a task which involves publicly displaying the efforts of their thinking only to have it judged as right or wrong on the basis of whether or not it matches with the answer at the back of the book or the answer called out by the teacher. Such practices have the potential to reinforce the notion that mathematics is not to be questioned or that when it is used one should have confidence in its results because mathematics can be unproblematicaly determined as right or wrong. Given that more and more statistical claims are being made in advertising and the popular press, such impressions of mathematics and mathematical processes are clearly not in the best interest of citizenship education.
Having offered a few examples of where mathematics education may subvert citizenship education, there is a need to make some suggestions as to how mathematics education could support citizenship education.
Instructional stances and strategies in mathematics that have the potential to promote active and critical participation in society
In this section, I would like to offer a couple of suggestions for mathematics teaching that I believe would serve the dual purpose of teaching mathematics, per se, and for educating youth for active participation in society. I would like to suggest that mathematics education and citizenship education need not be distinct tasks of the teacher; rather, appropriate mathematics teaching also prepares the student for citizenship.
1) Variable-entry prompts and investigations: posing problems
One of the things we might do in mathematics classes is turn our attention away from finding the right answer to pre-given questions and focus instead on the questions and problems that arise in student interaction with mathematics. One strategy that can be used to do this is posing variable-entry prompts (Simmt, 2000). These are prompts which allow students with various backgrounds in mathematics to enter into mathematical activity in a variety of ways. Such prompts encourage students to use their powers of patterning, generalizing, specializing and reasoning (Mason, Burton, and Stacey, 1982). When using variable-entry prompts in the mathematics classroom students must specify what is relevant in the moment and work in ways that are appropriate for the emergent context.
An activity known as rectangular numbers offers a simple illustration of the features of variable-entry prompts and their potential for educating for active participation. The prompt goes like this- Given square tiles, for which number of tiles between 1 and 36 can you create rectangles. For example, one can create a 2 x 3 rectangle with 6 tiles but cannot create anything but a 1 x 5 rectangle with five tiles.
As most students quickly notice, you can create rectangles for all numbers if the 1 x n case is acceptable. At this point students must begin to make some distinctions and need to begin to negotiate constraints for the task. For example, we can immediately state a theorem (and prove it under the conditions we are working within) that for any number of tiles, n there is a 1 x n rectangle. Notice how the teacher's question has been answered. "You can create rectangles for any number of tiles between 1 and 36." But this is just the beginning. Now it is up to students to find out more about rectangular numbers. Students might ask: Which numbers form squares? Which numbers have many different rectangles? In what ways is a m x n rectangle the same as a m x n rectangle? In what ways are they different? Although this is a very simple prompt and one that can be accessed by students with varying "skills, abilities and background knowledge" it positions the students as problem-posers, negotiators and evaluators. Students must pose, negotiate and judge the appropriateness and adequacy of their own and classmates' questions and solutions.
When students are occasioned by such prompts to act mathematically they specify and negotiate the problems they seek and the resolutions they come to. Because most problems that arise in our day to day living are not pre-specified but arise in our actions and interactions, active and critical participation in society requires citizens to specify and negotiate problems that are important and to evaluate resolutions. By using variable-entry prompts in mathematics classes such behavior is encouraged and developed.
2) The Demand for Explanation
One of the most effective ways of building community in the mathematics classroom is to insist that students are responsible for contributing to the mathematics lesson. Specifically, students must be given the responsibility for explaining the mathematics they construct in terms that others are able to understand and, in turn, listening for the explanations of others. One of the most important things we can do in our classes as mathematics teachers is to discourage the belief that mathematics is all about right answers. To discourage this belief we must focus on explanations and multiple and diverse solutions.
Mathematics is most certainly about "truth" but the kind of truth that is deduced within a system of constraints, thus, in one sense, mathematical truth is highly contingent-but it is not about the right answer. In fact, there are many examples that could be taken from mathematics to demonstrate how at once local and global truths about a particular phenomenon could indeed be different. Hence one of the important features of mathematics that schooling must stress is justification and explanation-that is, ways of proving one's assertions and articulating one's understanding within a community. In this way, mathematics education helps educate for citizenship.
3) Mathematical Conversations
Gordon Calvert suggests there are three modes of verbal interaction that might be found in the classroom: monologue, argument, and conversation (Gordon Calvert, in press) Monologue is for one's self and not directed to an other. Usually, this form of verbal utterance does not foster the development of a community of mathematicians. The second form of dialogue is argumentation. Although this is quite a common form of dialogue in mathematics (both professional and school mathematics), argumentation without respect for the other can fragment the community rather than build and sustain the community. The third form of dialogue is that of conversation. Gordon Calvert suggests that through mathematical conversation students, in relationship with each other, offer explanations, examples, conjectures, pose problems and make space for the contribution of the other (Gordon Calvert, in press). Through such interaction there is potential for the community to address and solve problems that arise for them in their activity.
Educating for citizenship in today's world
Within a society whose structures are largely mathematical it is important that citizens be educated in the methods of mathematics: first in terms of general numeracy but also in terms of understanding mathematics as a discipline which has formatting power in society. Teaching students to identify and pose problems, to explain themselves in terms others can understand and to question the invisible structures of mathematics is key to developing informed, active and critical citizens. Mathematics has a role in citizenship education because it has the potential to help us understand our society and our role in shaping it.
References
D'Ambrosio, Uribe. 1999. "Literacy, matheracy, and technocracy." Mathematical Thinking and Learning 1 (2), 131 - 154.
Davis, Brent. 1995. "Why teach mathematics: Mathematics education and enactivist theory." For the Learning of Mathematics 15(2), 2 - 9.
Davis, Philip J. and Hersh, Reuben. 1981. The Mathematical Experience London, Penguin Books.
Devlin, Keith. 1998. The language of mathematics: making the invisible visible New York: W. H. Freeman and Company.
Frankenstein, Marilyn. 1997. "In addition to the mathematics: Including equity issues in the curriculum." In Janet Trentacosta and Margaret Kenney (eds.), Multicultural and Gender Equity in the Mathematics Classroom, 10 - 22. Reston, VA: National Council of Teachers of Mathematics.
Gordon Calvert, Lynn. Mathematical Conversations. New York: Peter Lang, in press.
Hardy, G. S. 1967. A Mathematician's Apology. London: Cambridge University Press.
Mason, John, Burton, L. and Stacey, K. 1982. Thinking Mathematically. London: Addison-Wesley.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA.: National Council of Teachers of Mathematics.
Porter, Theodore. 1997. "The triumph of numbers: civic implications of quantitative literacy," in Lynn Steen, (ed.), Why Numbers Count: Quantitative Literacy for Tomorrow's America New York, 5-10. The College Entrance Examination Board.
Simmt, Elaine. 2000. Mathematics Knowing in Action Edmonton, Alberta: unpublished doctoral dissertation, University of Alberta.
Skovsmose, Ole. 2000. "Aporism and critical mathematics education." For the Learning of Mathematics 20(1), 2 - 8.
Steen, Lynn (ed.). 1999. Why Numbers Count: Quantitative Literacy for Tomorrow's America. New York: The College Entrance Examination Board.