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In-service/Methods course documentation
Identifying characteristics of math inquiry experiences (D5.1)
Additional essays on inquiry available in these materials

(This essay was adapted from: R. Borasi (1996). Reconceiving mathematics education: A focus on errors. Ablex. Chapter 2, pp. 15-26, for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional development" [Borasi & Fonzi, 1998])

Introduction

Teaching mathematics through inquiry represents a radical departure from traditional mathematics instruction. Most math lessons follow the predictable sequence of review of homework, teacher presentation of new material, in-class practice exercises, and assignment of similar exercises for homework (NCTM, 1989; U.S. Department of Education, 1996). The pervasiveness of this practice is not so surprising once we realize that it is the logical consequence of the following set of assumptions informing most traditional schooling (often referred to as a "transmission paradigm"):

• A view of mathematical knowledge as a body of established facts and techniques that are hierarchically organized, context­free, value­free, and thus able to be broken down and passed along by experts to novices.
• A view of learning as the successive accumulation of isolated bits of information and skills that are achieved mainly by listening/observing, memorizing and practicing.
• A view of teaching as the direct transmission of knowledge that can be achieved effectively as long as the teacher provides clear explanations and the students pay attention to them and follow them with memorization and practice.

It is obvious, therefore, that attempts at changing the way math classes are currently taught and at introducing alternative pedagogies -- such as the inquiry approach promoted by these materials -- are not likely to succeed unless we can provide a cogent critique of these assumptions and offer alternative views of knowledge, learning and teaching that are grounded in theory and research. The scope of this essay is to develop such an argument. We hope that, in the process, this will also contribute to a deeper understanding of the characterizing elements of teaching mathematics through inquiry and their rationale.

A critique of the transmission paradigm from multiple perspectives

Although each of the assumptions of a transmission paradigm may at first seem dictated by common sense, they have all been criticized on several grounds by scholars from a variety of fields. A first critique of the transmission model found in both the NCTM Standards and reports published by National Research Council (NCTM, 1989, 1991; NRC, 1989, 1990, 1991b) builds on economic reasons, as it argues that the kind of mathematical knowledge and skills that have traditionally been the goals of direct teaching (i.e., some basic factual knowledge and computational skills) are no longer what our society requires, given its rapid changes and the availability of more and more sophisticated technology. Rather, to fully function in today's world, students should become good mathematical problem solvers and critical thinkers, confident in their mathematical ability and able to apply what they know in novel situations and to learn new content on their own.

Defining the goals of school mathematics purely in terms of acquiring some specific techniques has also been criticized, although from a different perspective, by a group of mathematicians and mathematics educators supporting a more "humanistic" view of mathematics (e.g., Brown, 1982; Lerman, 1990a, 1990b; White, 1993). These scholars have argued that, in order to portray the true nature of mathematics as a discipline, learning mathematics in schools should not be reduced to technical content, but rather should also explicitly address elements such as the history and philosophy of mathematics, applications of mathematics that reveal the social, political, and ethical dimensions of this discipline, and even affective issues connected with the learning of mathematics. An awareness of these elements (ignored in traditional curricula informed by a transmission approach) is crucial if students are to challenge the common belief that mathematics is a deterministic, black­and­white and cut­and­dried domain where there is no place for reasoning, creativity, or personal judgment (Borasi, 1990; Schoenfeld, 1989). This dualistic view of mathematics has proven dysfunctional for many students, potentially causing math avoidance and anxiety, as well as expectations and behaviors that are likely to reduce a student's chances of success in the discipline. Explicitly addressing the elements identified by the supporters of a humanistic view of mathematics in the mathematics curriculum would thus help counteract these negative effects. It would also enable students to become aware of some important aspects of the mathematical culture and, in turn, make them feel more a part of the mathematical community of practice.

A second kind of critique of the transmission model is instead based on more philosophical grounds. Peirce's arguments against the expectation that "absolute knowledge" can ever be achieved presents a challenge to the first assumption of a transmission paradigm. Peirce rejected the idea that knowledge is stable and certain and proposed, instead, that knowledge is processual, ever open to doubt. However, he also suggested that the uncertainty permeating human knowledge has some positive implications, because it can cause the kind of "doubt" that promotes continuous inquiry in the effort to achieve more and more refined explanations of the world around us. Peirce's alternative view of knowledge as a "process of inquiry motivated by uncertainty" is well illustrated by the following metaphor of "walking on a bog":

[W]e never have firm rock beneath our feet; we are walking on a bog, and we can be certain only that the bog is sufficiently firm to carry us for the time being. Not only is this all the certainty that we can achieve, it is also all the certainty we can rationally wish for, since it is precisely the tenuousness of the ground that propels us forward.... Only doubt and uncertainty can provide a motive for seeking new knowledge. (Skagestad, 1981, p. 18)

A similar view of knowledge as inquiry also informed the work of Dewey on logic, as reflected by his definition of "reflective thought" as involving "(1) a state of doubt, hesitation, perplexity, mental difficulty, in which thinking originates, and (2) an act of searching, hunting, inquiring, to find material that will resolve the doubt, settle and dispose of the perplexity" (Dewey, 1933, p. 12).

This dynamic view of knowledge finds support in the works of Kuhn (1970), Lakatos (1976) and Kline (1980) on the history of science and mathematics. These philosophers of science have provided several historical examples that show how some scientific theories and mathematical concepts were challenged and changed over time (as discussed in more detail in the essay entitled "Rethinking the nature of mathematics," also included in this CD-ROM). By showing the fallibility of results held "true" by great scientists for long periods of time, these historical analyses also warn us that the body of knowledge we currently rely on in any subject matter (even mathematics) may not be as "secure" as we would like to believe. In fact, it is always possible that new events and discoveries may challenge even today's most taken for granted "truths."

A similar position about the nature of mathematical knowledge characterizes mathematics educators who identify themselves as "radical constructivists" (e.g., von Glasersfeld, 1990, 1991). These mathematics educators argue that even in mathematics (perceived by many as the "discipline of certainty" par excellence) knowledge is socially constructed and therefore neither predetermined nor absolute. It is important to note, however, that even when the notion of absolute knowledge is rejected, this does not mean that "anything goes" in mathematics. Rather, radical constructivists suggest that the whole notion of "mathematical truth" needs to be reconceived as the result of social negotiations within the mathematical community of the time.

[Radical] constructivism can be described as essentially a theory about the limits of human knowledge, a belief that all knowledge is necessarily a product of our own cognitive acts. We can have no direct or unmediated knowledge of any external or objective reality. We construct our understanding through our experiences, and the character of our experience is influenced profoundly by our cognitive lenses. ... Mathematicians act as if a mathematical idea possesses an external, independent existence; however the constructivist interprets this to mean that the mathematician and his/her community have chosen, for the time being, not to call the construct into question, but to use it as if it were real, while assessing its worthiness over time. (Confrey, 1990a, pp. 108-109)

The important role played by the mathematics community and by the set of shared values, beliefs and practices that characterizes it, has recently received increasing attention in mathematics education (e.g., Ernest, 1991; Resnick, 1988; Schoenfeld, 1992; von Glasersfeld, 1991). This theme is further explicated in the following description of mathematics proposed by Schoenfeld (in press):

Mathematics is an inherently social activity, in which a community of trained practitioners (mathematical scientists) engages in the science of patterns-systematic attempts, based on observation, study, and experimentation, to determine the nature or principles of regularities in systems defined axiomatically or theoretically ("pure mathematics") or models of systems abstracted from real world objects ("applied mathematics").... Truth in mathematics is that for which the majority of the community believes it has compelling arguments. In mathematics truth is socially negotiated, as it is in science.

Several psychological studies on how individuals learn and attain knowledge have provided a third major critique of the transmission paradigm, as they radically challenge some of the behavioristic assumptions about learning on which such a paradigm relies. The 1980s and l990s, especially, have seen a wealth of research on children's mathematical learning and problem solving based on the premise, derived largely by Piaget's (1970) model of cognitive development, that children construct concepts and cognitive structures through interactions with their environment. These studies (e.g., Baroody & Ginsburg, 1990; Ginsburg, 1983, 1989; Steffe, van Glasersfeld, Richards, & Cobb, 1983) suggest that in order to learn mathematics effectively, students need to make sense and construct a personal understanding of specific concepts, rules or algorithms-they cannot simply "absorb" these from teacher's explanations or even demonstrations. The role played by individuals' background knowledge, cognitive structures, interests, and purposes, in increasing such personal understanding has also been emphasized:

Many decades of research on human learning of complex subjects suggests that people do not always learn things bit by bit from the ground up.... They often jump into any situation with some knowledge, however rudimentary or inaccurate, and, even before they have mastered specific techniques, they begin fitting their knowledge into a larger picture. Students bring their own interpretation of tasks and concepts to the instructional process. (Silver, Kilpatrick, & Schlesinger, 1990, p. 6)

Research on learning informed by the work of Vygotsky (1962, 1978), although sharing the fundamental constructivist assumption that learners have to construct their own knowledge, has pointed out the importance of social interactions on learning and thus added another crucial dimension to the study of students' mathematical learning.

[S]ocial interaction is not a source of processes to be internalized. Instead it is the process by which individuals create interpretations of situations that fit with those of others for the purpose at hand. In doing so, they negotiate and institutionalize meanings, resolve conflicts, mutually take others' perspectives and, more generally, construct consensual domains for coordinated activity (Bauersfeld, 1988; Bishop, 1985; Blumer, 1969; Maturana, 1980; Perret-Clermont, 1980). These compatible meanings are continually modified by means of active interpretative processes as individuals attempt to make sense of situations while interacting with others. Social interactions therefore constitutes a crucial source of opportunities to learn mathematics in that the process of constructing mathematical knowledge involves cognitive conflict, reflection, and active cognitive reorganization. (Cobb, Wood & Yackel, 1990, p. 127).

Social constructivist theories of learning and development are receiving increasing attention from the educational research community and have provided a framework for valuing social norms and classroom dynamics.

This emphasis on the social nature of learning well complements the critiques of the transmission model by philosophers and historians of science discussed earlier in this section, and resonates with the results of studies of learning and thinking undertaken within an anthropological framework (e.g., Lave, 1988; Lave & Wenger, 1991; Rogoff & Lave, 1984). The latter studies have also made mathematics educators more aware of the crucial role played by both the context and the community of practice (Lave & Wenger, 1989) within which the learning act occurs. By providing the ultimate purpose for a learning or problem solving activity, the context can in fact inform the definition of a problem, the choice of one solution approach versus another, as well as the evaluation and use of the final solution/response. The community of practice within which the learner is situated also can shape the learners' goals and expectations and, even more importantly, implicitly provides a set of viewpoints, perspectives, and values that inform the learner's activity as well as his or her perceptions and interpretations. This awareness has led several mathematics educators (e.g., Bishop, 1988; Lave, Smith, & Butler, 1988; Resnick, 1988; Schoenfeld, 1992) to posit that mathematics education should be reconceived not so much in terms of instruction, but rather as a process of enculturation or socialization where acquiring a "mathematical viewpoint" becomes central. This new emphasis is well articulated by Resnick (1988):

[T]he reconceptualization of thinking and learning emerging from the body of recent work on the nature of cognition suggests that becoming good mathematical problem solver-becoming a good thinker in any domain-may be as much a matter of acquiring habits and dispositions of interpretation and sense­making as of acquiring any particular set of skills, strategies or knowledge. If this is so, we may do well to conceive of mathematics education less as an instructional process (in the traditional sense of teaching specific, well­defined skills or items of knowledge), than as a socialization process.... If we want students to treat mathematics as an ill­structured discipline-making sense of it, arguing about it, and creating it, rather than merely doing it according to prescribed rules-we will have to socialize them as much as to instruct them. This means that we cannot expect any brief or encapsulated program on problem solving to do the job. Instead, we must seek the kind of long-term engagement in mathematical thinking that the concept of socialization implies. (p. 58)

Once the positivistic assumptions about "absolute knowledge," the behavioristic assumptions about learning, and even many of the traditional goals for school mathematics are challenged, "direct teaching" loses much of its common-sense appeal. The connection between this approach to teaching and the other assumptions of a transmission paradigm has been made explicit by Neilsen:

Our current educational system has been largely shaped by ... mechanistic assumptions ... [that] equate knowledge with facts, skills and procedures. ... Having subscribed (however unconsciously) to a view that sees knowledge as objective, atomistic, and hence portable, educators have developed a system dominated by a pedagogy which places overwhelming emphasis on teaching and considerable faith in direct instruction (lectures, readings, and drill exercises) as the chief means of transmitting the facts and skills that students will need to understand in order to operate in the world. This transmission pedagogy assumes that knowledge can be passed along from one person to another. (Neilsen, 1989, pp.3-4)

In contrast to direct teaching, the following instructional implications of assuming a constructivist view of knowledge and learning have been suggested in the literature:

• teachers should listen to their students so as to get clues about how to best support their learning (Confrey, 1991);
• teachers should value the knowledge learners already possess, and find ways to set up dialogues to help enhance and reshape such knowledge (Barnes, 1985);
• teachers can no longer be seen as the ultimate source of knowledge and truth; this, in turn, shifts more responsibility on the students' themselves for monitoring and justifying their own work (Borasi, 1992);
• since inquiry doesn't "just happen," especially within school mathematics, teachers should design situations that can stimulate and support their students' own inquiries and construction of meaning (Borasi, 1992; Richards, 1991);
• teachers should value uncertainty, and even confusion (Brown, 1993) and errors (Borasi, 1994, 1996), as a positive element in math lessons;
• teachers (and schools) need to develop appropriate means to assess and reward students' learning, so as to convey to students a clear message about what kind of learning processes and outcomes are really valued (NCTM, 1995).

Taken together, the critiques summarized in this section have not only uncovered some serious weaknesses in the assumptions at the basis of the transmission paradigm, but also suggest an alternative set of assumptions about mathematics, teaching and learning which are at the basis of the inquiry approach promoted in these materials.

Summary of key assumptions of an inquiry approach

Based on the arguments developed in the previous section, we can now articulate the theoretical assumptions informing an inquiry approach to mathematics instruction as follows:

• A view of mathematics as a humanistic discipline; that is, the belief that mathematical knowledge is socially constructed rather than deterministic, and is shaped by cultural and personal values like other products of human activity.
• A view of knowledge more generally as constructed through a process of inquiry where uncertainty, conflict and doubt provide the motivation for the continuous search for a more and more refined understanding of the world.
• A view of learning as a generative process of meaning making, requiring both social interaction and personal construction, and informed by context and purposes.
• A view of teaching as stimulating and supporting the students' own inquiry and establishing a learning environment conducive to such inquiry.

Taken together, these assumptions offer an alternative theoretical framework for mathematics instruction, whose theoretical and empirical bases are at least as compelling as those behind a transmission paradigm. The characteristic features of teaching mathematics through inquiry discussed in Section C.1 of the main text are all consistent with these assumptions and could be considered the logical implications of applying such a framework to mathematics instruction.

References

Barnes, D. (1985). From communications to curriculum. NY: Penguin Books.

Baroody, A. J. & Ginsburg, H. P. (1990). Children's learning: A cognitive view. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 19-29). Reston: VA: National Council of Teachers of Mathematics.

Bishop, A. (1988). Mathematical enculturation. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Borasi, R. (1990). The invisible hand operating in mathematics instruction: Students' conceptions and expectations. In T.J. Cooney & R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (pp. 174-182). 1990 Yearbook of the National Council of Teachers of Mathematics, Reston, VA: National Council of Teachers of Mathematics.

Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann.

Borasi, R. (1994). Capitalizing on errors as "springboards for inquiry": A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166-208.

Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Norwood, NJ: Ablex.

Brown, S. I. (1982). On humanistic alternatives on the practice of teacher education. Journal of Research and Development in Education, 15(4), 1 - 12.

Brown, S. I. (1993). Towards a pedagogy of confusion. In A.White (Ed.), Essays in humanistic mathematics. Washington, D.C.: Mathematical Association of America.

Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers. In R.B. Davis, C.A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 125-146). Dordrecht, The Netherlands: Kluwer.

Confrey, J. (1990). What constructivism implies for teaching. In R.B. Davis, C.A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 107-122). Dordrecht, The Netherlands: Kluwer.

Confrey, J. (1991). Learning to listen: A student's understanding of powers of ten. In E.von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111-138). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Dewey, J. (1933). How we think. Boston: D. C. Heath.

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Ginsburg, H. P. (1989). Children's arithmetic (2nd ed.). Austin, TX: Pro-Ed.

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Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory and applications. NY: Preager Scientific.

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von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 19-29). Reston, VA: National Council of Teachers of Mathematics.

von Glasersfeld, E. (Ed.). (1991). Radical constructivism in mathematics education. Dordrecht, The Netherlands: Kluwer Academic Press.

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White, A. (1993). Essays in humanistic mathematics. Washington, D.C.: Mathematical Association of America.

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