Untangling Teachers’ Diverse Aspirations for Student Learning:
A Crossdisciplinary Strategy
for Relating Psychological Theory to Pedagogical Practice1
David Kirshner
Louisiana State University
Key words:Conceptual knowledge; Constructivism; Learning; Learning theories; Reform
in mathematics education; Research issues; Teaching effectiveness;
Teaching practice
Submitted to JRME’s Forum for Researchers
Resubmitted
Revised
In Press
August 1999
December 2000
August 2001
January 2002
DO NOT CIRCULATE PRIOR TO PUBLICATION
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
Note to NTCM Editors: To appear in Forum for Researchers
1
Untangling Teachers’ Diverse Aspirations for Student Learning:
A Crossdisciplinary Strategy for Relating Psychological Theory to Pedagogical Practice1
David Kirshner
Louisiana State University
Abstract
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.
2
Untangling Teachers’ Diverse Aspirations for Student Learning:
A Crossdisciplinary Strategy for Relating Psychological Theory to Pedagogical Practice
A Crossdisciplinary Strategy
The current Standards documents contain relatively little explicit discussion of the
theoretical perspectives they reflect and few citations of the research literature. Perhaps
consequently, advocates and critics alike have sometimes misinterpreted the approaches
taken in the Standards ..., seeing a unitary focus where more dispassionate observers
might have seen a rather eclectic set of views. (Lindquist, Ferrini-Mundy, & Kilpatrick,
1997, p. 394)
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
A Crossdisciplinary Strategy
3
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
A Crossdisciplinary Strategy
4
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.
A Crossdisciplinary Strategy
5
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:
The constructivist “mantra” –“Students construct their own knowledge”–is often taken to
mean that pedagogy must sanctify the student’s inventions and explorations at the
expense of teacher instruction. ... Self-labeled social constructivist approaches,
analogously, often sanctify the student’s interactions and group “collaboration” at the
expense of any deep consideration of what is being learned (and how) or of the nature of
the social interactions or larger social arrangements or institutions. (p. 43)
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.
A Crossdisciplinary Framework
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
A Crossdisciplinary Strategy
6
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.
A Crossdisciplinary Strategy
7
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 specific1 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.
It is worth pointing out that though comparatively “hard-edged and specific,” the pedagogical
guidance offered here traces only the general contours of the teacher’s responsibility. Prescriptions
for teaching cannot issue forth from psychological theories of learning (Bernstein, 1996; Konold,
1995). My thanks to Stephen Lerman for raising this point.
1
A Crossdisciplinary Strategy
8
__________________________________________________
Insert Table 1 about here
__________________________________________________
Teaching for Learning as Habituation
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
A Crossdisciplinary Strategy
9
discrimination to review tests (presumably because traditional practice aims for a blend of
habituation and conceptual mastery).
Teaching for Learning as Construction
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
hypothetical learning trajectory ... made up of three components: the learning goal that
defines the direction, the learning activities, and the hypothetical learning process–a
prediction of how students' thinking and understanding will evolve in the context of the
learning activities.... [T]he assessment of student thinking (which goes on continually ... )
can bring about adaptations in the teacher's knowledge that, in turn, lead to a new or
modified hypothetical learning trajectory. (pp. 136-137)
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
A Crossdisciplinary Strategy
10
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).
A Crossdisciplinary Strategy
11
Teaching for Learning as Enculturation
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
A Crossdisciplinary Strategy
12
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:
After each presentation and the ensuing questions, there is a vote on whether the material
presented is true and worthy of entry into the book. This process produces some of the
most difficult moments for me, because students have presented and voted down things
that I feel are significant parts of geometry. Still, I believe it imperative that I not
interfere. (p. 87)
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.
A Crossdisciplinary Strategy
13
THE RHETORIC OF MATHEMATICS EDUCATION REFORM
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
A Crossdisciplinary Strategy
14
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.2
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. 3 Certainly such
theories already have succeeded in providing compelling and inspiring instances of integrative
Indeed, it is an interesting historical footnote that a request for advice concerning Saxon texts
that prompted the NCTM’s initial moves toward promulgating the Curriculum and Evaluation
Standards (McLeod, Stake, Schappelle, & Mellissinos, 1995).
2
In this respect, I do not share Sfard’s (1998) belief in the inevitable “‘incommensurability’” (p.
11) of the perspectives, or Lerman’s (1996) assessment that “a merger of these two views is
incoherent” (p. 138). Indeed, my own theoretical work in situated cognition theory aspires to
create a new cultural understanding about learning through a poststructural reorientation of
theory (e.g., Kirshner & Whitson, 1997).
3
A Crossdisciplinary Strategy
15
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).
A Crossdisciplinary Strategy
16
References
Anderson, J. R. (1976). Language, memory, and thought. Hillsdale, NJ: Lawrence Erlbaum
Associates.
Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University
Press.
Anderson, J. R. (Ed.) (1993). Rules of the mind. Hillsdale, NJ: Lawrence Erlbaum Associates.
Anderson, J. R., & Lebiere, C. (Eds.) (1998). The atomic components of thought. Mahwah, NJ:
Lawrence Erlbaum Associates.
Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education.
Educational Researcher, 25(4), 5-11.
Anderson, J. R., Reder, L. M., & Simon, H. A. (1997). Situative versus cognitive perspectives:
Form versus substance. Educational Researcher, 26(1), 18-21.
Ausubel, D. P. (1963). The psychology of meaningful verbal learning. New York: Grune &
Stratton.
Becker, J. P., & Jacob, B. (1998, June). ‘Math War’ developments in the United States
(California). ICMI Bulletin No 44, 16-25.
Bernstein, B. (1996). Pedagogy, symbolic control, and identity: Theory, research, critique.
London: Taylor & Francis.
Brownell, W. A. (1935). Psychological considerations in the learning and teaching of arithmetic.
In The teaching of arithmetic. Tenth yearbook of the National Council of Teachers of
Mathematics (pp. 21-51). New York: Teachers College, Columbia University.
Bruner, J. S. (1960). The process of education. New York: Vintage Books.
Cobb, P., & Bowers, J. (1999). Cognitive and situated learning perspectives in theory and
practice. Educational Researcher, 28(2), 4-15.
Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder.
Journal for Research in Mathematics Education, 14, 83-94.
A Crossdisciplinary Strategy
17
Confrey, J. (1993). Learning to see children’s mathematics: Crucial challenges in constructivist
reform. In K. Tobin (Ed.), The practice of constructivism in science education (pp.
299-321). Hillsdale, NJ: Lawrence Erlbaum Associates.
Dreher, M. (1995). Counting on you: The rhetoric of the National Council of Teachers of
Mathematics Standards. Unpublished doctoral dissertation, Louisiana State University.
Fernandez, E. (1994). A kinder, gentler Socrates: Conveying new images of mathematics
dialogue. For the Learning of Mathematics, 14(3), 43-47.
von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80,
121-140.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. New York:
Falmer Press.
Greeno, J. G. (1978). Understanding and procedural knowledge in mathematics instruction.
Educational Psychologist, 12, (3), 94-143.
Greeno, J. G. (1997). On claims that answer the wrong question. Educational Researcher, 26(1),
5-17.
Greeno, J. G., Collins, A. M., & Resnick, L. B. (1996). Cognition and learning. In D. C. Berliner
& R. C. Calfee (Eds.), Handbook of educational psychology (pp. 15-46). New York:
Macmillan.
Healy, C. C. (1993a). Build-A-Book Geometry: A story of student discovery. Berkeley, CA: Key
Curriculum Press. (The identical book also is published by the Key Curriculum Press
under the title Creating miracles: A story of student discovery.)
Healy, C. C. (1993b). Discovery courses are great in theory, but... . In J. L. Schwartz, M.
Yerushalmy, & B. Wilson (Eds.), The Geometric Supposer: What is it a case of? (pp.
85-104). Hillsdale, NJ: Lawrence Erlbaum Publishers.
Hill, D. (1993). Math's angry man: John Saxon's back-to-the-basics textbooks elicit both praise
and scorn--and not much in between. Teacher,5, 24-28.
A Crossdisciplinary Strategy
18
Kirshner, D. (2000). Exercises, probes, puzzles: A crossdisciplinary typology of school
mathematics problems. Journal of Curriculum Theorizing, 16(2), 9-36.
Kirshner, D., & Whitson, J. A. (Eds.). (1997). Situated cognition: Social, semiotic, and
psychological perspectives. Mahwah, NJ: Lawrence Erlbaum Associates.
Kirshner, D. & Whitson, J. A. (1998). Obstacles to understanding cognition as situated.
Educational Researcher, 27(7), 1-7.
Klein, D., & Rosen, J. (1997). Calculus reform for the $millions. Notices of the AMS, 44(10),
1324-1325).
Knapp, M. S. (1997). Between systemic reforms and the mathematics and science classroom:
The dynamics of innovation, implementation, and professional learning. Review of
Educational Research, 67(2), 227-266.
Konold, C. (1995). Social and cultural dimensions of knowledge and classroom teaching. In L. P.
Steffe & G. Gale (Eds.), Constructivism in education (pp. 175-184). Hillsdale, NJ:
Lawrence Erlbaum Associates.
Korthagen, F. A. J., & Kessels, J. P. A. M. (1999). Linking theory and practice: Changing the
pedagogy of teacher education. Educational Researcher, 28(4), 4-17.
Kuhn, T. S. (1970). The structure of scientific revolutions (Enlarged Edition). London: The
University of Chicago Press.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer:
Mathematical knowing and teaching. American Educational Research Journal, 27(1),
29-63.
Leont'ev, A. N. (1981). Problems in the development of mind. Moscow: Progress Publishers.
Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical
constructivist paradigm. Journal for Research in Mathematics Education, 27(2), 133-150.
Lerman, S. (2000). A case of interpretations of Social: A Response to Steffe and Thompson.
Journal for Research in Mathematics Education, 31(2), 210-227.
A Crossdisciplinary Strategy
19
Lindquist, M., Ferrini-Mundy, J., & Kilpatrick, J. (1997). Guest editorial. Journal for Research
in Mathematics Education, 28(4), 394-395.
Maher, C. A., & Davis, R. B. (1990). Building representations of children's meaning's. In R. B.
Davis, C. A. Maher, & N. Noddings (Eds.) Constructivist views on the teaching and
learning of mathematics, Journal for Research in Mathematics Education, Monograph
Number 4 (pp. 79-90).
McLeod, D. B., Stake, R. E., Schappelle, B., & Mellissinos, M. (1995). International influences
on the NCTM Standards: A case study of educational change. In D. T. Owens, M. K.
Reed, & G. M. Millsaps (Eds.), Proceedings of the seventeenth annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics
Education (pp. 240-246). Columbus, OH: ERIC Clearinghouse for Science, Mathematics,
and Environmental Education.
National Council of Teachers of Mathematics. (1991). Professional standards for the teaching of
school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
Newman, D., Griffin, P., & Cole, M. (1989). The construction zone. Cambridge, UK: Cambridge
University Press.
O’Connor, M. C. (1998). Can we trace the “efficacy of social constructivism”? In P. D. Pearson
& A. Iran-Nejad (Eds.), Review of Research in Education, 23 (pp. 25-71). Washington,
DC: American Educational Research Association.
Polya, G. (1954). Induction and analogy in mathematics. Princeton, NJ: Princeton University
Press.
Polya, G. (1957). How to solve it: A new aspect of mathematical method. Princeton, NJ:
Princeton University Press.
Rogoff, B. (1990). Apprenticeship in thinking. New York: Oxford University Press.
A Crossdisciplinary Strategy
20
Saxon, J. H. (1990). Algebra 1: An incremental development. Norman, OK: Saxon Publishers.
Saxon, J. H. (1991). Algebra 2: An incremental development. Norman, OK: Saxon Publishers.
Saxon, J. H. (1992). Saxon philosophy. In Saxon Publishers brochure. Norman, OK: Saxon
Publishers.
Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld
(Ed.) Mathematical thinking and problem solving, 53-67. Hillsdale, NJ: Lawrence
Erlbaum Associates.
Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one.
Educational Researcher, 27(2), 4-13.
Silver, E. A. (1988). Teaching and assessing mathematical problem solving: Toward a research
agenda. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of
mathematical problem solving (pp. 273-282). Hillsdale, NJ: Lawrence Erlbaum
Associates; Reston, VA: NCTM.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
Journal for Research in Mathematics Education, 26(2), 114-145.
Skemp, R. R. (1976). Relational mathematics and instrumental mathematics. Mathematics
Teaching, 77.
Skinner, B. F. (1953). Science and human behavior. New York: Macmillan. (Chapters 1-3)
Smith, R. R., & Windes, R. R. (1975). The innovational movement: A rhetorical theory.
Quarterly Journal of Speech, 61, 140-153.
Stanic, G. M. A., & Kilpatrick, J. (1988). Historical perspectives on problem solving in the
mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), The teaching and
assessing of mathematical problem solving (pp. 1-22). Hillsdale, NJ: Lawrence Erlbaum
Associates; Reston, VA: National Council of Teachers of Mathematics.
A Crossdisciplinary Strategy
21
Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E.
von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 177-194).
Dordrecht, Holland: Kluwer Academic Publishers.
Steffe, L. P., & D'Ambrosio, B. S. (1995). Toward a working model of constructivist teaching: A
reaction to Simon. Journal for Research in Mathematics Education, 26(2), 146-159.
Steffe, L. P., & Kieren, T. (1994). Radical constructivism and mathematics education. Journal
for Research in Mathematics Education, 25(6), 711-733.
Steffe, L. P., & Thompson, P. W. (2000). Interaction or Intersubjectivity? A Reply to Lerman.
Journal for Research in Mathematics Education, 31(2), 191-209.
Tomm, K. (1995). Response to chapters by Spiro et al. and Steier. In L. P. Steffe & G. Gale
(Eds.), Constructivism in education (pp. 109-121). Hillsdale, NJ: Lawrence Erlbaum
Associates.
VanderStoep, S. W. & Seifert, C. M. (1993). Learning "how" versus learning "when": Improving
transfer of problem-solving principles. The Journal of the Learning Sciences, 3(1),
93-111.
Wood, T., Cobb, P., & Yackel, E. (1995). Reflections on learning and teaching mathematics in
elementary school. In L. P. Steffe & G. Gale (Eds.), Constructivism in education (pp.
401-422). Hillsdale, NJ: Lawrence Erlbaum Associates.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in
mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.
A Crossdisciplinary Strategy
22
Table 1
A Crossdisciplinary Framework for Application of Learning Theory to Pedagogical Practice
___________________________________________________________________________
Metaphor
_______________________________________________________
Feature
Habituation
Construction
Enculturation
___________________________________________________________________________
What is gained
Skills
Concepts
Learning theory
Behaviorism;
information
Psychological
Constructivism
Dispositions
Sociocultural Theory
(Appropriation
)
processing
Progression
Incremental
Unifocal
Instance
Saxon
curriculum
Pedagogical
Focus
Pedagogical
Objective
Repetitive Practice
Transformative
Plato’s
Meno
Discontinuous
Healy’s Build-a-Book
Geometry
Hypothetical learning
Classroom
trajectory
microculture
Proficiency with
Conceptual
Culturally appropriate
routine exercises
restructuring
participation_______________________________________________________
____________________