Summary
The main focus of this chapter is to “[cope] with multiple and frequently conflicting theoretical perspectives” in the field of math ed, and specifically “how we might make and justify our decision to adopt one theoretical perspective rather than another” (p. 3).
The Positivist Epistemology of Practice
This first section is dedicated to discussing what Donal Schön calls the positivist epistemology of practice, drawn from his 1983 book The Reflective Practitioner. The idea is that abstract theoretical principles are applied in practice through practical reasoning. A common practice would be seen in teaching preparation programs, wherein appropriate theory (psychological, sociological, etc.) is first discussed and absorbed by the student, then it is applied in methods courses before being deployed with the prospective teacher in practicum experiences. Cobb claims that this epistemology of practice has implications for mathematics teaching and learning specifically.
Fundamentally, though, the positivist epistemology relies on the idea that theory runs in the background and is brought to the forefront through practice; in our field, this means that the theory is brought to life directly and without intermediary through instruction and instructional techniques. However, Cobb contends that this approach does not account for inherent contradictions between theories. He uses this as a jumping-off point to reference Thomas Kuhn’s book The Structure of Scientific Revolutions, in which Kuhn proposes that theory and theoretical reasoning engage in a dialectic not just within research tradition/focus, but across traditions and focuses. In this way, theory necessarily takes a circuitious path through inconsistencies and uncertainties on its way to practice and application, but also sees individual communities of practice rather than abstract adherents.
Cobb also extracts the idea that questions raised by Kuhn and others imply that we must “see research, theorizing, and indeed philosophying as distinct forms of practice rather than activities whose products provide a viable foundation for the activities of practitioners” (p. 6). To that end, he proposes two criteria to compare/contrast theoretical perspectives:
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“…in terms of the manner in which they orient and constrain the types of questions that are asked about the learning and teaching of mathematics, and thus the nature of the phenomena that are investigated and the forms of knowledge produced” (p. 7)
- “…their potential usefulness given our concerns and interests as mathematics educators” (p. 7)
Mathematics Education as a Design Science
The second criterion that Cobb proposes addresses the question of this section, as he contends that “mathematics education can be productively viewed as a design science, the collective mission of which involves developing, testing, and revising conjectured designs for supporting envisioned learning processes” (p. 7). He argues that the mission of math education frequently requires designing processes or procedures at all levels of the educational experience. All of this occurs with the endgoal of improving students’ mathematical experiences and learning; in this way, design science is one potential approach for mathematics education as a scientific endeavor.
Despite identifying design science as one potential framing, Cobb argues that this framing identifies so-called ‘core aspects’ of the discipline: “specifying and clarifying the prospective endpoints of learning”; “formulating and testing conjectures about both the nature of learning processes that aim towards those prospective endpoints and the specific means of supporting their realization” (p. 8). In working toward those endpoints, he asserts that “the process of formulating instructional goals should involve delineating central mathematical ideas in particular mathematical domains and clarifying more encompassing activities such as mathematical argumentation and modeling” (p. 8). Additionally, these endpoints should envision students’ roles. However, no matter the scientists’ preferences, goals cannot be chosen based on whim.
Ultimately, the deliberative nature of the direction of research, for Cobb, aligns with Kuhn’s proposed reasonings for science, as well as the pragmatism of Richard Rorty. For him, the second of his criteria “reflects the view that the choice of theoretical perspective requires pragmatic justification” (p. 9).
Pragmatic Realism
Interestingly, Cobb begins this section with a few comments about radical constructivism. Citing von Glasersfeld & Cobb (1984), he notes that radical constructivists “speak of knowledge that has proven viable as fitting with, rather than matching, external reality in that it satisfies the constraints of that reeality as much as a key satisfies the constraints of a particular type of lock” (p. 10) because radical constructivism adopts the psychological notion that “learning is an active, constructive process” and the epistemological notion that “it is impossible to check whether our ideas and concepts correspond to external reality” (p. 10). This is in contrast to pragmatic realism, which asserts that it is possible to “bridge the gulf between knowledge and reality” (p. 10). Specifically, for Putnam (presumably the one who coins the “pragmatic realism” term), pragmatic realism is a response to the idea that perspectives (scientific or not) must be static, unchanging, and inflexible, as traditional epistemology ascribes one viewpoint to Reality (“Reality” as opposed to “reality” meaning objective truth versus observed truth). Rather, “questions concerning the existence of abstract mathematical and scientific entities should be addressed not by making claims about Reality… but by examining disciplinary practices” (p. 11) in the same way that scientists test different assumptions against observed reality. This excerpt strikes me:
Putnam takes care to differentiate conceptual relativism, the notion that the phenomena that are treated as real differ from one perspective to another, from the view that every perspective is as good as every other. “Conceptual relativity sounds like ‘relativism’, but it has none of the ‘there is no truth to be found… “true” is just a name for what a bunch of people can agree on’ implications of relativism” (1987, pp. 17-18). Putnam instaed characterizes truths as fallible, historically contingent, human productions that are subject to correction. His goal in adopting this view is to rehabilitate the notion of truth while simultaneous rejecting the view that Truth should be ascertained in terms of correspondence with Reality. In his view, it is imperative to preserve the notion of truth given that people make and risk acting on the basis of their truth claims in both mundane and scientific realities. (p. 11)
The utility, then, of pragmatic realism is that it gives a starting point from which he can address the “compare/contrast” question of different theories/perspectives.
The Notion of the Individual as Conceptually Relative
This section is somewhat limited, I think. It boils down to the commentary that, at the outset of mathematics education as its own field, cognitive psychology was the primary source of theoretical frameworks. However, as both fields matured, pushback (attributed to De Corte et al. 1996) made a distinction between ‘first wave’ theory which models “teachers’ and students’ individual knowledge and beliefs by positing internal cognitive structures and processes”, and ‘second wave’ theory which notes the “limited emphasis on affect, context, and culture” in the first wave (p. 12). This distinction is amplified when considering other background theories such as distributed cognition; consequently, the notion of an “individual” enacting mathematics becomes relative to the theoretical framework. This could range from the individual as an atomic entity to the individual as a collection of cooperating persons across cultures and contexts (which would be in line with a Vygotskian interpretation of learning).
Usefulness and Truth
Utility of different perspectives, Cobb says “carries with it the implication that they can be viewed as conceptual tools” (p. 13). He calls Prawat’s (1995) “three types of pragmatic justification” to help here:
- “Actions are judged to be true if they enable the achievement of goals”
- Some sort of inductive inferential justification, in that if a hypothesis results in a consistent outcome, then that hypothesis can be judged to be true
- ‘Qualitative confirmation test of truth’: this seems to be more deductive, in that there is a scientific process developed and followed to hone understanding of a situation
The importance of “truth”, “ideas are potentially useful to the extent that they give rise to conjectures about envisioned learning processes and the specific means of supporting them” (p. 14). But because conjectures require adjustment, Cobb arrives at this criterion (influenced by Dewey): “the truth of fallible, potentially revisable ideas is justified primarily in terms of the insight and undersanding they give into learning processes and the measn of supporting their realization” (p. 14). However, undergirding this criterion is the problem that it is we who have to decide what works and what doesn’t; theories don’t speak back and confirm or deny their truth- humans, fallible humans, are the ones who make the inference.
Comparing Theoretical Perspectives
Cobb’s overarching goal for the chapter was to take the theoretical perspectives of experimental psychology, cognitive psychology, distributed cognition, and sociocultural theory, and compare/contrast them after framing the necessity for the endeavor. He gives the following brief introductions/explanations for each:
- Experimental psychology “refers to the psychological research tradition in which the primary methods employed involve experimental and quasi-experimental designs, preferably with the random assignment of subjects” (p. 15)
- Cognitive psychology “is limited to theoretical orientations that involve what MacKay (1969) termed the actor’s viewpoint… to account not merely fo rteachers’ and students’ observed behaviors but for their inferred interpretations and understandings in terms of internal cognitive structures and processes” (p. 15)
- Sociocultural theory draws on the work of Soviet psychologists Vygotsky and Leont’ev and views “individual activity as situated with respect to social and cultural practices” (p. 15)
- Distributed cognition, described earlier, is lumped in with sociocultural theory in its initial explation.
Experimental Psychology
Cobb identifies the contributions that experimental psychology has made in mathematics education by pointing out the “development of assessment instruments, norm-referenced tests, and the findinds of studies that have assessed the relative effectiveness of alternative curricular and instructional approaches” (p. 15). His descriptions seem to assert that experimental psychology- at least in the sense that he uses it in this chapter- is ruled by proper experimental design and random assignment. Ultimately, this approach is used to determine ‘what works’ for students in various educational settings and contexts.
Here Cobb asserts that experimental psychology (presumably as opposed to or as contrasted with other theories) relies on a notion of the individual is a “statistical aggregate that is constructed by combining measures of psychological attributes of participating students” rather than the actual participants (p. 16). This approach permits or even encourages recognizing the participants as “error variance…characterized by the extent to which their performance deviates from group norms that measure the performance of the collective individual” (p. 16). This approach assumes that students are made of “discrete, isolatable attributes” and that “the environments in which students acquire these capacities are composed of independent features that the investigator can manipulate and control directly” (p. 16).
This methodology sees pushback about dehumanization and objectification of the students under study. Interestingly to me though, Cobb cites Danzinger (1990) in claiming that experimental psychology is by nature shallow because it “separates people from [their] social contexts…and eschews the study of the interpretations that they make in those contexts” before dismissing the critique as using theory to attack theory. For Cobb, the depth appears in not in the experimental design, but in the analysis that must be done to construct meaning from the data.
As far as usefulness, Cobb again cites Danzinger (1990) in observing that, as a consequence of the co-evolution of the United States educational system with the science of statistics, experimental psychology functions as “an administrative science in that it investigates the reality that administrators seek to manage” (p. 18).
My Thoughts
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Citations
Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F.K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (Vol. 1, pp. 1-38). Greenwich, CT: Information Age Publishing.