Critical mathematics education addresses three intersecting issues in educational theory: the question of a disciplinary orientation to curriculum development and design, the notion of “critical” educational approaches, and the peculiarities of mathematics itself as a nexus of educational and/or “critical educational” issues. This entry looks at the definition of mathematics from a critical perspective, discusses how critical mathematics is implemented, and examines issues of power and the impact of the new perspective.

## Defining Terms

The modifier mathematics at once assumes a need for a special set of questions and issues peculiar to a subject area while also raising the specter of challenges to a disciplinary orientation to educational programs. Is there something special about mathematics that warrants a separate entry in this volume, distinct from, say, language arts, social studies, science, vocational education, and so on?

One might define mathematics as a subset of literacy, involving reading, writing, listening, speaking, and otherwise representing ideas that focus on patterns, numbers, shapes, space, probabilities, and informatics. If so, do we need unique approaches to teaching, learning, and assessment, or are the methods of instruction and theorizing pretty much the same as for any other subject area of a school curriculum? Such questions are of interest because of the specific history of (Western) mathematics, which has so often been used as a model of rational thought and knowledge in general. Because this history is so intertwined with the history of AmeroEuropean thought, mathematics is in its own way a symbol of the intellectual history of colonialism and imperialism, as demonstrated in the fields of ethno mathematics and in ideological analyses of mathematics as constructed in ways that mask its relationships with power and social inequalities.

The modifier critical is added by theorists who are concerned with the constraining and enabling functions of mathematics education within a democratic society: As with any form of education, mathematical experiences may empower and disempower at the same time. Many mathematical innovations have been created in order to serve those who control, for example, within the development of military weapons and strategies, governmental accounting systems, and business practices. Issues of access internationally and in local contexts set up systems of social reproduction of inequality, where students of mathematics at various levels of education experience different forms of mathematics curriculum, with further implications for equity and diversity.

On the other hand, movements to introduce critical mathematics education and mathematics education for social justice enable students to use mathematics as a tool for social critique and personal empowerment. And knowledge of mathematics, as with any specific knowledge area, can be used as a cultural resource for reading the world and in the process transforming one’s place within that world. Malcolm X’s use of the concept of a variable for his name both represented the legacy of slavery and defined a political, cultural, and historic moment in the history of the United States. Students who invent their own algorithms for distributing governmental funds are confronted with the limitations of “mathematizing” social decisions.

## Implementation

Critical mathematics education demands a critical perspective on both mathematics and the teaching and learning of mathematics. As Ole Skovsmose describes a critical mathematics classroom, the students (and teachers) are attributed a “critical competence.” A century ago, mathematics educators moved from teaching critical thinking skills to using the skills that students bring with them. It was accepted that students, as human beings, are critical thinkers and would display these skills if the classroom allowed such behavior. It seemed that critical thinking was not occurring simply because the teaching methods were preventing it from happening; through years of school, students were unwittingly “trained” not to think critically in order to succeed in school mathematics. Later, ways were found to lessen this “dumbing down of thinking through school experiences.”

Today, there is a richer understanding of human beings as exhibiting a critical competence, and because of this realization, it is recognized that decisive and prescribing roles must be abandoned in favor of all participants having control of the educational process. In this process, instead of merely forming a classroom community for discussion, Skovsmose suggests that the students and teachers together must establish a “critical distance.” What he means with this term is that seemingly objective and value-free principles for the structure of the curriculum are put into a new perspective, in which such principles are revealed as value loaded, necessitating critical consideration of contents and other subject matter aspects as part of the educational process itself.

Christine Keitel, Ernst Klotzmann, and Ole Skovsmose together offer new ideas for lessons and units that emerge when teachers describe mathematics as a technology with the potential to work for democratic goals, and when they make a distinction between different types of knowledge based on the object of the knowledge. The first level of mathematical work, they write, presumes a true-false ideology and corresponds to much of what is found in current school curricula. The second level directs students and teachers to ask about right method: Are there other algorithms? Which are valued to meet needs? The third level emphasizes the appropriateness and reliability of the mathematics for its context. This level raises the particularly technological aspect of mathematics by investigating specifically the relationship between means and ends.

The fourth level requires participants to interrogate the appropriateness of formalizing the problem for solution; a mathematical/technological approach is not always wise, and participants would consider this issue as a form of reflective mathematics. On the fifth level, a critical mathematics education studies the implications of pursuing special formal means; it asks how particular algorithms affect people’s perceptions of (a part of) reality, and how people conceive of mathematical tools when they use them universally. Thus the role of mathematics in society becomes a component of reflective mathematical knowledge.

Finally, the sixth level examines reflective thinking itself as an evaluative process, comparing levels 1 and 2 as essential mathematical tools, levels 3 and 4 as the relationship between means and ends, and level 5 as the global impact of using formal techniques. On this final level, reflective evaluation as a process is noted as a tool itself and as such becomes an object of reflection. When teachers and students plan their classroom experiences by making sure that all of these levels are represented in the group’s activities, it is more likely that students and teachers can be attributed the critical competence that is now envisioned as a more general goal of mathematics education.

## Issues Of Power

In formulating a democratic, critical mathematics education, it is also essential that teachers grapple with the serious multicultural indictments of mathematics as a tool of postcolonial and imperial authority. What was once accepted as pure, wholesome truth is now understood as culturally specific and tied to particular interests. The mathematics-military connections and applications of mathematics in business decisions are examples in this sense of fantasies of power and control rather than consistently literal forms of control and power.

Critical mathematics educators ask why students, in general, do not readily see mathematics as helping them to interpret events in their lives or gain control over human experience. They search for ways to help students appreciate the marvelous qualities of mathematics without adopting its historic roots in militarism and other fantasies of control over human experience.

One important direction for critical mathematics education is in the examination of the authority to phrase the questions for discussion. Who sets the agenda in a critical thinking classroom? Authors such as Stephen Brown and Marion Walter lay out a variety of powerful ways to rethink mathematics investigations through what they call “problem posing,” and in doing so they provide a number of ideas for enabling students both to “talk back” to mathematics and to use their problem solving and problem posing experiences to learn about themselves as problem solvers and posers, instead of as an indicator of their abilities to match a model of performance.

In the process, they help to frame yet another dilemma for future research in mathematics education, and education in general: Is it always more democratic if students pose the problem? The kinds of questions that are possible, and the ways that people expect to phrase them, are to be examined by a critical thinker. The questions themselves might be said to reveal more about people’s fantasies and desires than about the mathematics involved. Critical mathematics education has much to gain from an analysis of mathematics problems as examples of literary genre.

And finally, it becomes crucial to examine the discourses of mathematics and mathematics education in and out of school and popular culture. Critical mathematics education asks how and why the split between popular culture and school mathematics is evident in mathematical, educational, and mathematics education discourses, and why such a strange dichotomy must be resolved between mathematics as a “commodity” and as a “cultural resource.” Mathematics is a commodity in the consumer culture because it has been turned into “stuff” that people collect (knowledge) in order to spend later (on the job market, to get into college, etc.). But it is also a cultural resource in that it is a world of metaphors and ways of making meaning through which people can interpret their world and describe it in new ways, as in the example of Malcolm X. Critical mathematics educators recognize the role of mathematics as a commodity in society; but they search for ways to effectively emphasize the meaning-making aspects of mathematics as part of the variety of cultures. In doing so, they make it possible for mathematics to be a resource for political action.

## Potential Impact

The history of critical mathematics education is a story of expanding contexts. Early reformers recognized that training in skills could not lead to the behaviors they associated with someone who is a critical thinker who acts as a citizen in a democratic society. Mathematics education adopted the model of enculturation into a community of critical thinkers. By participating in a democratic community of inquiry, it is imagined, students are allowed to demonstrate the critical thinking skills they possess as human beings, and to refine and examine these skills in meaningful situations.

Current efforts recognize the limitations of mathematical enculturation as inadequately addressing the politics of this enculturation. Mathematics for social justice, as advocated by such authors as Peter Appelbaum, Eric Gutstein, and Sal Restivo, uses the term “critical competence” to subsume earlier notions of critical thinking skills and propensities. A politically concerned examination of the specific processes of participation and the role of mathematics in supporting a democratic society enhances the likelihood of critical thinking in mathematics, and the achievement of critical competence.

**Bibliography: **

- Appelbaum, P. (1995). Popular culture, educational discourse and mathematics. Albany: State University of New York Press.
- Brown, S. I. (2001). Reconstructing school mathematics: Problems with problems and the real world. New York: Peter Lang.
- Dias, A. L. B. (1999). Becoming critical mathematics educators through action research. Educational Action Research, 7(1), 15–34.
- Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s epistemology. Journal of Education, 165(4), 315–339.
- Gellert, U., & Jablonka, E. (Eds.). (2007). Mathematisation and demathematisation: Social, philosophical and educational ramifications. Rotterdam: Sense.
- Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York: Routledge.
- Keitel, C., Kotzmann, E., & Skovsmose, O. (1993). Beyond the tunnel vision: Analysing the relationship of mathematics, technology and society. In C. Keitel & K. Ruthven (Eds.), Learning from computers. Mathematics education and technology (NATO-ASI-Series, F 121; pp. 242–279). Berlin: Springer.
- Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer/Springer.

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