David Kirshner

Louisiana State University

Submitted to JRME’s Forum for Researchers August 1999

Resubmitted December 2000

Accepted Pending Revisions June 2001

Revised August 2001

In Press January 2002

Direct inquiries or comments to David Kirshner, Department of Curriculum & Instruction, Louisiana State University, Baton Rouge LA 70803-4728. (225) 578-2332. dkirsh@lsu.edu.

Running Head: A Crossdisciplinary Strategy

David Kirshner

Louisiana State University

The Learning Principle propounded in the Principles and Standards for School Mathematics (NCTM, 2000) rehearses the familiar distinction between facts/procedures and understanding as a central guiding principle of teaching reform. This rhetorical stance has polarized mathematics educators in the "math wars," (Becker & Jacob, 1998), while creating the discursive space for mathematics teaching reform to be reified into a unitary "reform vision" (Lindquist, Ferrini-Mundy, & Kilpatrick, 1997)–a vision that teachers can all too easily come to see themselves as implementing rather than authoring. Crossdisciplinarity is a strategy for highlighting the discrete notions of learning that psychology thus far has succeeded in coherently articulating. This strategy positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities, at the same time marshaling accessible, theory-based guidance toward realization of its diverse possibilities.

[PK to NCTM: Author acknowledgment]

I am grateful to graduate students at Korea National University of Education in Chungbuk Province and at Dangook University in Seoul during the fall of 1997 for enduring far less coherent versions of this approach during its formative stages.

The remarkable success of the NCTM’s Standards documents in mobilizing a consensus about the need for deep and systemic reform of mathematics education is counterbalanced by considerable confusion among teachers as to what is the "reform vision." As Lindquist, Ferrini-Mundy, and Kilpatrick (1997) intimate, theory is complexly bound up both with the success of the reform movement in inspiring educators to seek richer learning outcomes for their students and with creating a false sense of a unitary vision that curtails the very exploration teachers need to engage in to make reform effective. My purpose in this Forum article is to introduce a strategy for recasting psychological theory in reform-oriented teacher education so as to highlight the diversity of visions for student learning toward which teaching may legitimately aspire. This strategy positions teachers as authors of reform, while at the same time marshaling accessible, theory-based guidance toward realization of its diverse possibilities.

How is it that the kaleidoscope of psychological theories in mathematics education research—for example, constructivism, sociocultural theory, cognitive science—has come to subserve a unitary vision of reform? To understand this development, we need to look back to the struggle of progressive educators to make a case for meaning rather than just skills against the hegemony of behaviorism (Brownell, 1935). This clarion call has resonated through the ensuing decades–most famously in writings by Bruner (1960), Ausubel (1963), and Skemp (1976)–culminating in NCTM’s (2000) propounding the distinction between facts/procedures and understanding as the guiding Learning Principle for reform teaching. There are two reciprocal effects of posing the reform problem as an opposition of learning goals–skills alone versus understanding (and skills). First, the opposition of learning goals contributes to a polarization of positions. This theme is taken up in the final section of this article. Second, encapsulating desirable learning outcomes under a single rubric elides the "rather eclectic set of views" (Lindquist, Ferrini-Mundy, & Kilpatrick, 1997, p. 394) that we know actually underlies reform theorizing. I suggest it is this rhetorical stance regarding the desired learning outcome that creates the discursive space for mathematics teaching reform to be reified into a unitary reform vision–a vision teachers all to easily can come to see themselves as implementing rather than authoring.

It is worth a brief digression delve further into the question of how a community of researchers pursuing such varied psychological paradigms has allowed a unitary notion of learning to become the emblem of mathematics teaching reform. I want to implicate in this puzzle two factors operating in the current academic arena. First, in a preparadigmatic field like psychology, the trajectory of each of the competing approaches is outward, away from its limited (but powerful and generative) foundational insights toward comprehensive explanation of the field (Kuhn, 1970). As Kuhn noted, each paradigm tends to become highly attuned to the concerns and advances of the others, which can lead to a homoginization of interests. Perhaps it is this phenomenon that revealed itself to Silver (1988) as a "hidden agenda" of research into mathematical problem solving: "Although it is scientific progress that drives each researcher's agenda, the reform agenda is evident in the background. Given the diversity of disciplinary perspectives represented in the authorship of chapters in this volume, it is quite remarkable that a fairly common reform agenda appears to be represented" (p. 279).

Second, the presence of locally coherent theoretical approaches has led some psychologists—especially those sympathetic to education’s need for full-bodied explanation of learning—to work toward integrative theories like situated cognition theory and social constructivism, which bridge diverse perspectives on learning. Such endeavors can give the impression that an integrative theorization of learning already exists to guide a unitary reform teaching agenda. But ongoing controversies about social constructivism (e.g., Lerman, 1996, 2000; Steffe & Thompson, 2000) and situated cognition (e.g., Anderson, Reder, & Simon, 1996, 1997; Cobb & Bowers, 1999; Greeno, 1997; Kirshner & Whitson, 1998) show the significant problems that remain in establishing coherent theoretical syntheses. And, although they have produced important and compelling theoretical analyses of classroom moments in which social and psychological development mutually support each other, I believe such theoretical efforts have been singularly unsuccessful in meeting Greeno, Collins, and Resnick’s (1996) challenge "to develop ... new possibilities for practice, not just to provide inspiring examples, but also to provide analytical concepts and principles for people who wish to use the examples as models in transforming their own practices" (p. 41).

Regardless of their theoretical orientation, in a preparadigmatic field like psychology, all theorists are engaged in the academic work of elaborating an intellectual landscape dominated by a single, transcendent theory. In itself, this forward looking posture may predispose learning theorists to endorse a reform movement predicated on the assumption that learning is accounted for within a unitary framework. In addition, theorists have a vested interest in promoting their own vision of learning as the goal for education. Success in this endeavor is a de facto indicator of the success of their theorizing; it provides, as well, a vast laboratory of practice for the ongoing research effort.

By the same token, the crossdisciplinary strategy introduced here, celebrating diverse visions of learning, may be resisted by researchers as contrary to their scientific interests. Indeed, "crossdisciplinarity" is a pragmatic strategy for marshaling theory-based perspectives and insights for use by educators. It is not a scientific agenda, nor is it intended to supplant scientific endeavors in psychology or educational psychology. Rather, it is offered as an interface between scientific and educational discourses designed to enrich education with what psychology has achieved in its fragmented accomplishments and to insulate education from unitary visions of learning psychology now only hopes to achieve.

The symptom in educational practice to which crossdisciplinarity is offered as antidote is the homoginization of teaching reform recommendations into an amorphous and undifferentiated "reform agenda," in which pedagogical means have become dissociated from particular learning effects. As Knapp (1997) observed in a review of systemic reform efforts, "the more easily imported practices (e.g., the use of manipulatives in mathematics in the elementary grades) have become part of teachers' repertoires, while the full understanding of what these practice may mean has not" (p. 255). O’Connor (1998) links this malaise to the way in which constructivist and social constructivist discourses have been appropriated in practice:

I see hierarchical relations between the communities of theory and practice as implicated in these problems, as teachers too readily cede to theorists the complexities of understanding how teaching subserves learning, and researchers too eagerly adopt the field of practice as laboratory for still-evolving theories of learning. Crossdisciplinarity seeks to repair the fabric of reform by posing values choices concerning the intentions of instruction as the teacher’s first obligation, and by drawing on basic, rather than cutting-edge, learning theory, to provide accessible theory-based guidance toward realization of diverse possibilities for learning.

Crossdisciplinarity empowers teachers and makes theory accessible by tapping into diverse metaphors that ground our cultural common sense about learning. A crossdisciplinary framework begins with a selection of metaphors that motivate our scientific work in learning and infuse our educational discourse. These metaphors then are used to organize a constellation of theory-based pedagogical perspectives to support the achievement of the discrete notions of learning that have been selected. The use of metaphor is a deliberate starting point for integrating theory and practice, as metaphors "cross the borders between the spontaneous and the scientific, between the intuitive and the formal.... [T]hey enable osmosis between everyday and scientific discourses" (Sfard, 1998, p. 4).

What follows is a crossdisciplinary framework based on metaphors of learning as habituation (informing behaviorist and information processing theories), conceptual construction (informing Piagetian constructivist learning theories), and enculturation (informing sociocultural theories). My collage of metaphors is not dissimilar from the framework of behaviorist, cognitive, and situative approaches chosen by Greeno, Collins, and Resnick (1996) to organize their analysis of current perspectives on cognition and learning. However, they note that "other organizing principles could be chosen, and that many of our colleagues would characterize the field in different terms" (p. 15). Indeed, my proposal does depart pointedly from theirs in some respects. The most significant departure lies in the cognitive/constructivist rubric which for Greeno, Collins, and Resnick combines "general cognitive abilities, such as reasoning, planning, solving problems, and comprehending language" with "understanding of concepts and theories in different subject matter domains" (p. 16). The constructivist rubric I offer here includes only understanding of specific conceptual content, with general cognitive abilities seen as arising from cultural enmeshment. This interpretation is consistent with Cobb and Steffe’s (1983) distinction between microschemes, which are "‘content’ oriented" and macroschemes, which are "‘thought’ oriented" (p. 87), of which only the former are investigated in constructivist teaching experiments. Correspondingly, my enculturation rubric extends beyond a situative "focus on processes of interaction of individuals with other people and with physical and technological systems" (Greeno, Collins, & Resnick, 1996, p. 17) to include the general cognitive development that accrues from such cultural enmeshment.

It is beyond the scope of this brief article to say much about the theoretical interpretations employed in the framework (see Kirshner, 2000). Suffice it to note that learning is seen to progress very differently in these three conceptions. Habituated learning develops incrementally, for instance as strengthening or weakening of stimulus/response bonds in behaviorism or as adjustment of connection strengths in the most recent version of the ACT-R information processing theory (Anderson & Lebiere, 1998). Conceptual construction as theorized in the Piagetian tradition involves transformation of existing conceptual structures from perturbations that arise out of reflective abstraction (von Glasersfeld, 1995). Enculturation features discontinuity between prior patterns of participation and new cultural patterns appropriated (Leont'ev, 1981) through cultural enmeshment (Newman, Griffin, & Cole, 1989).

The remainder of this section provides illustrative examples and pedagogical guidance toward each of these possible objectives for student learning. To be clear, it should be stated that these illustrative teaching approaches are not proffered as exemplary practice. Rather, they have been selected for their unifocal aspiration toward the specific learning objective. Such unifocal pedagogical interpretation disagrees with the spirit of previous theory-based pedagogical guidance intended to reflect good pedagogy in general rather than just good pedagogy toward a specific learning outcome. But highlighting the distinctive and contradictory qualities of "good teaching" emphasizes the need for teachers to resolve difficult values issues, and then to devise their own syntheses in case they opt to pursue multiple learning objectives. More importantly, framing pedagogical guidance toward independent learning objectives enables us to be hard-edged and specific in contrast to the diffuse and undifferentiated theory-based guidance that, thus far, has served the cause of teaching reform so poorly. Table 1 provides an overview of the crossdisciplinary framework.

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Insert Table 1 about here

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Whether for rote recall of facts as formulated through behaviorism (Skinner, 1953), or for skillful performance of algorithms or word problems as formulated through information processing theories like ACT (Anderson, 1976, 1983, 1993), the basic premise of habituation is that repeated practice of routine problems leads to gradual adjustment to task constraints. Although traditional curricular approaches incorporate repetitive practice of routine problems, most also include lecture or explanation of principles. As a consequence, such curricula are organized topically, including homogeneous grouping of exercises for simultaneous consolidation of skills and concepts. In contrast, the curriculum of John Saxon (e.g., 1990, 1991) pursues a pure habituationist agenda. His incremental approach tends to reduce the teacher’s explanation of principles: "You learn to work problems by working them repetitively, over a long period of time" (John Saxon, quoted in Hill, 1993, p. 26). And his method of gentle repetition downplays the topical focus on new content in favor of mixed practice of old problem types.

For purposes of skill acquisition, gentle repetition is an ingenious innovation. The daunting challenge in establishing skillful performance in complex domains like algebra is not learning what to do, but when to do it–that is, stimulus discrimination (Greeno, 1978; VanderStoep & Seifert, 1993). Such training must include ongoing practice in classification/recognition, as provided for in Saxon’s mixed problem sets: "As the problems become familiar students can look at a new problem and recognize it by type. This recognition evokes conditioned responses that lead to solutions" (Saxon, 1992, inside front cover). Thus Saxon’s organization of gentle repetition may be more effective in promoting habituation than the traditional, homogeneous organization of exercises, which limits opportunities for stimulus discrimination to review tests (presumably because traditional practice aims for a blend of habituation and conceptual mastery).

As Steffe and Kieren (1994) noted, constructivist theories of students’ conceptual understanding of mathematical topics have been developed through constructivist teaching experiments (CTEs) involving one or two students (Cobb & Steffe, 1983), a tradition which also serves to inform constructivist teaching in classrooms (Steffe, 1991). In keeping with CTEs, the "constructivist teacher" engages students in activities or tasks designed to cause perturbations in their current structures of knowledge, leading to conceptual restructuring (von Glasersfeld, 1989). There is broad agreement that to accomplish this, the teacher must have a model of the students’ knowledge, including specific expectations for students’ conceptual restructuring (Cobb & Steffe, 1983; Confrey, 1993; Maher & Davis, 1990; O’Connor, 1998; Simon, 1995; Steffe & D'Ambrosio, 1995). Simon (1995) elaborates the teacher's model as a

It is important to recognize that students’ conceptual construction comes about from engagement in the task environment created by the teacher. In this respect, the "close personal and trusting relationship" (Steffe, 1991, p. 178) formed with the student serves to engage the student fully and deeply in the teacher’s agendas. So Tomm’s (1995) distress about interpretations of constructivism that "undermine the creativity of students [and] ... justify the generation and maintenance of hierarchical relationships in teaching situations" (p. 117) are seen here as a conflation of enculturationist and constructivist concerns. The constructivist teaching relationship is hierarchical and non-symmetrical, an interesting discovery of which was reported by a second-grade teacher who worked with Paul Cobb and his research team for a full academic year (Wood, Cobb, & Yackel, 1995). At the culmination of this year, she finally came to realize that in the interest of students’ conceptual construction it occasionally is necessary to be "very directive"–a move she feared might thwart students’ development of autonomy and independence as creative mathematical investigators. In her own words, she did learn to "‘walk ... the pedagogical tightrope’" (p. 421) between her concerns for students’ intellectual development and her concern for their social development; but it was difficult for her to transcend the presumption of reform that diverse learning objectives always can be seamlessly meshed in good teaching. A crossdisciplinary education would prepare teachers to expect such contradictions should they aspire toward diverse notions of learning for their students.

To emphasize the hierarchical nature of constructivist teaching, I offer Socrates’ oft-quoted interrogation of the slave boy in Plato’s Meno as an illustrative (though nonexemplary) constructivist intervention. Socrates’ questions are posed with clear assumptions about the boy’s current understanding and a clear anticipation of an ensuing learning trajectory–notwithstanding Socrates’ aloofness toward the slave boy, including that 45 of his 50 questions called for yes or no answers or required only routine calculation (Fernandez, 1994).

Enculturation is the process of acquiring cultural dispositions through enmeshment in a cultural community. I interpret dispositions broadly as inclinations to engage with people, problems, artifacts, or oneself in culturally particular ways. Thus, the NCTM’s (1991) objectives that students come to "explore, conjecture, reason logically; to solve non-routine problems; to communicate about and through mathematics ... [as well as] personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions" (p. 1) all reflect an enculturationist learning agenda. The sociocultural notion of appropriation (Leont'ev, 1981; Newman, Griffin, & Cole, 1989; Rogoff, 1990) provides insight into the processes of enculturation. Enculturationist teaching involves identifying a target culture and target dispositions within that culture, and working gradually to shape the classroom microculture so that it comes to more closely resemble the target culture with respect to the target dispositions. Students "learn" (in this sense) from their participation in the cultural milieu of the classroom rather than from other students or the teacher per se (Yackel & Cobb, 1996).

Generally, the reference culture for mathematical enculturation is mathematical culture (Lampert 1990, Schoenfeld, 1994). Thus, Polya’s motivating concern for the mathematician's "inductive attitude" (1954, p. 7), including intellectual courage, intellectual curiosity, and wise restraint, betrays an enculturationist agenda, as does his pedagogical method of unobtrusively slipping potent heuristic questions into the student’s own struggle with challenging problems so that these orientations toward problem solving might be appropriated into their own approach (Polya, 1957).

Christopher Healy’s (1993a, 1993b) Build-A-Book geometry course provides a fascinating instance of a pure enculturationist teaching approach. Healy begins his course by presenting a few geometric statements as starting points for discussion. But from then on, his sole concern is with fostering the cultural development of the classroom community as a community of mathematicians engaged in producing a geometry book. His role is restricted to facilitating the interactive environment of the classroom. It is the students who determine what topics count as geometric, what conjectures are worthwhile to pursue, and what arguments are sufficient to establish truth. If topics of traditional geometry courses happen to be omitted or dealt with errorfully by the class, Healy (1993b) does not intercede:

Of course students in Healy’s classroom do engage with important mathematical content and do develop skills and concepts with respect to that content. Learning is never restricted to the modality singled out in a unifocal teaching environment. I use the term inadvertent learning for such learning that might be anticipated to happen in an instructional setting but for which the teacher does not take direct responsibility in instructional planning. In contrast with Healy’s disciplined and powerful unifocal approach, reform-oriented instruction, I believe, too often relies on inadvertent learning, making gestures toward diverse learning goals, but without systematically supporting students’ accomplishment of each. This leads to the major pedagogical lesson that crossdisciplinarity offers teaching reform: There is no magic "reform method" that addresses the multiple forms of learning that teachers may aspire to for their students. Integrative teaching toward diverse learning objectives succeeds only to the extent that teachers attend, individually, to the requisites of each learning modality. This requires understanding and mastery of the demanding teaching regimens outlined above, as well as expertise in "‘walking the pedagogical tightrope’" (Wood, Cobb, & Yackel, 1995, p. 421) between the competing priorities.

Inclusion of all that is desirable in mathematics teaching under the banner of "understanding" is emblematic of a unitary conception of learning motivating the mathematics education reform movement (NCTM, 2000). Untangling the enculturationist and constructivist strands knotted together in this unitary goal has been a major challenge for our community. We are only just now coming to the point where specific mathematical dispositions are being targeted for instruction through development of the classroom microculture (e.g., Yackel & Cobb’s, 1996, sociomathematical norms). Our prior reluctance to embrace enculturationist goals is well demonstrated in the transmutation of George Polya’s enculturationist problem solving agenda into a habituationist agenda by "reduc[ing] the rule-of-thumb heuristics to procedural skills. ...In a sense, problem solving as art gets reduced to problem solving as skill" (Stanic & Kilpatrick, 1988, p. 17). Thus from a crossdisciplinary perspective, the reform effort is only just beginning to get its bearings, and substantial problems remain in mediating conceptual and dispositional goals for student learning.

Equally ill-fitting is the characterization of the North American tradition of lecture and practice as serving students’ acquisition of facts and skills alone. Lecture, as a pedagogical practice, is oriented to conceptual development (albeit, without providing the level of support for students’ conceptual restructuring envisioned for constructivist pedagogy). And the topical (rather than heterogenous) grouping of problems employed in standard textbooks seeks to promote skillful performance at the same time that it supports students’ conceptual attainment of topical content. Thus, traditional practice can be critiqued as an ineffective attempt to mediate between goals of conceptual attainment and skillful performance. But it does not aspire to inculcate facts and skills alone. Conservatives in the so called "math wars" (Becker & Jacob, 1998) are justified in their complaint that traditional practice is being set up as a "‘straw man’ ... [on the basis of] unsupported characterizations of traditional texts as consisting of ‘drill and kill’ and/or an incomprehensible chain of rigorous proofs," arguing instead that "[t]he traditional calculus books we used as students and have taught out of as professors contained a fairly even mix of computation, conceptualization, and theory" (Klein & Rosen, 1997, p. 1324).

In his analysis of the rhetorical structure of the Standards documents, Dreher (1995) distinguished between innovational movements that seek gradual improvement of existing practices and transformational movements that seek a radical departure. While noting elements of each in the mathematics education reform movement, Dreher found that "ultimately we must conclude that the NCTM is not an effective innovational movement, despite their attempts to be so. They have violated Smith and Windes’ [1975] injunction against finding a villain" (1995, p. 96). The villain he identified was instrumental teaching toward rote learning, especially as personified in John Saxon’s curricular approach.

In this respect, the crossdisciplinary strategy can be seen as providing an opportunity to change the tone and tenor of reform toward a true innovational movement whose rallying cry is educational efficacy rather than orthodoxy. Crossdisciplinarity seeks to marshal the best possible guidance for teaching supported by the discrete notions of learning that psychology, in its fragmented diversity, thus far has succeeded in coherently articulating. This positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities, highlighting the special difficulties faced in opting for multifocal learning objectives. In the long term, I think integrative theories like social constructivism and situated cognition hold promise for creating a new cultural common sense about the possibilities for learning. Certainly such theories already have succeeded in providing compelling and inspiring instances of integrative teaching. But to found teaching reform on such vignettes of teaching commits a category error of serious dimension. To use Aristotle’s terms, phronesis that teachers and theorists achieve in their local, personal understandings of teaching can never substitute for episteme, our collective theoretical accomplishment, in founding a metadiscourse for teaching reform (cf., Korthagen & Kessels, 1999).