Date: Wed, 22 Nov 1995
From: dykstrad@varney.idbsu.edu
Subject: Chapter 4, finally

"Discourse, Mathematical Thinking and Classroom Practice" by Paul Cobb, Terry Wood, and Erna Yackel (C, W, & Y)

Chapter 4 in Contexts for Learning, Forman, Minnick, and Stone (eds.) reviewed by Dewey Dykstra, Jr.

Introduction:
This review of C, W, & Y's chapter focuses on particular significant issues raised in the chapter as opposed to being a survey of everything in the chapter. In what follows, after a very brief description of their project, a case will be made that C, W, & Y offer a perspective of Vygotskian notions which is based on a radical constructivist foundation. Their perspective leads the authors to a different position with respect to learning and teaching than that which seems typical of the literature of the "Vygotskian community." This state of affairs supports the notion that the differences between radical constructivism and Vygotskian theory do not center on issues of individual vs. social.

(Note: It should be noted that the authors never use the expression "radical constructivism" in the chapter. At least one of those with whom they cite compatibility is von Glasersfeld, who not only uses the expression, but whose works are usually cited as being the source of specific defining descriptions of "radical constructivism." In so much as is possible, the reader is cautioned against accepting or using substitute descriptions of this philosophical position. Other definitions or notions of constructivism are both incompatible with the author's statements and their use can render both the chapter and this review unintelligible. It is under these conditions that the writer of this review employs the expression "radical constructivism" as descriptive of the position of the authors.)

Their project:
C, W & Y set out to explore new instructional settings for mathematics in second grade classes. Their intent was to induce and study changes in mathematical understandings of the students. These new settings are consistent with what can be described as "inquiry" mathematics as opposed to more commonplace "school" mathematics. Their "inquiry" approach operates without "the assumption that all the students should make certain predetermined mathematical constructions when they completed and discussed their solutions to particular instructional activities." (p. 93) Among other things they sought to avoid the "social guessing games" that occur "when teachers attempt to steer or funnel students to a procedure or answer they have in mind all along," during which "the construction of mathematical knowledge becomes an incidental by-product of the interaction." (p. 93) They maintain that in the context of the mathematics classroom "mathematics does not consist of timeless, ahistorical facts, rules, or structures but is continually negotiated and institutionalized by a community of knowers." (p. 112)

Reflexive relationships:
C, W, & Y discovered that their initial assumption that the broader system in which the classroom was imbedded, the school and the school system, would be flexible enough to accommodate these alternative instructional settings essentially without influencing the project was unfounded. Instead, at every level they experienced examples of dialectic relationships between the cognitive activities of the participants and the social settings in which the participants were immersed. The students' notions of mathematics changed as a result of the social interaction patterns in class, but these social interaction patterns changed as the students' notions of what it is to do mathematics changed. The teachers' notions about the processes of being a teacher in a mathematics classroom changed as they interacted with the students learning mathematics, but nature of their interactions with the students changed as a result of their views of teaching mathematics changing. Features of the project were influenced by the institutionally accepted assessment criteria for the students' learning in mathematics and but the accepted view of the teaching of mathematics in the school system seems to have changed in order to accommodate the project and its expansion. In each case it seems that neither did the social setting merely determine the cognitive changes nor did the cognitive changes merely determine the social setting. Each influenced the other. Affect was observed to have been a factor in both the cognitive changes and the changes in social practices at every level. These observations are consistent with Vygotskian theory. C, W, & Y state that they find that ideas from the work of Vygotsky and Leont'ev inform their own interpretation of their observations in the project.

Differences between C, W, & Y and Vygotskian literature:
C, W, & Y also point out certain differences which they see between interpretations of Vygotski and Leont'ev in the literature and their own view. For this reason, this chapter seems to differ in fundamental ways from typical "Vygotskian" literature. Because C, W, & Y start from a radical constructivist background, they do not view the meaning of any social practice or of any piece of mathematics as existing independent of the students and the teacher having constructed meanings for these social practices or pieces of mathematics. Furthermore, they do not accept any social conventions of school mathematics practice as pre-given. The authors express themselves on this issue most clearly when they say:

On page 95:
"...we find ourselves in agreement with a basic tenet of Leont'ev's activity theory...that an individual's psychological development is profoundly influenced by his or her participation in particular forms of social practice. We do, however, reject any analysis that takes a particular social practice such as schooling or the mathematics education of young children as pregiven."

on page 96:
"...we question [the] metaphor of either students or teachers being embedded or included in a social practice. Such metaphors tend to reify social practices, whereas we believe that they do not exist apart from and are interactively constituted by the actions of actively interpreting individuals. ...we are attempting to avoid any tendency that subordinates the individual to the social and loses sight of the reflexive relation between the two."

on page 100:
"Although they acknowledged that these practices evolve historically, there is nonetheless a tendency to reify them when conducting psychological analyses. For example, Vygotsky's (1934/1987) analysis of the dependence of scientific concepts on school instruction takes as a given a classroom scientific practice akin to what we have called the school mathematics tradition. ...we wish to avoid the reification of these practices and, instead, prefer the emphasize that a practice such as inquiry mathematics is interactively constituted in the classroom and does not exist apart from the activities of the individuals who participate in its constitution (Maturana, 1980)."

on page 104:
"It is, of course, possible to analyze this interaction in terms of the teacher scaffolding the children's activity. However, such an analysis would tend to lose sight of the influence of the classroom mathematics tradition and downplay both the children's interpretive activity and their active contributions to the interaction. It is for this reason that we prefer to emphasize that the teacher and children together constitute a novel form of joint mathematical activity while simultaneously acknowledging the institutionalized power imbalance between the teacher and children."

on page 105:
"Following Vygotsky (1933/1976), one can in fact argue that the children created a zone of proximal development for themselves... Such a view emphasizes that children's social realities are relative to their levels of development and stands in stark contrast to characterization of development as the transfer of transmission of culturally developed modes of thinking from those who know to those who do not."

on page 105:
"...what counts as science and as rationality cannot be specified in terms of a historical criterion. Instead, scientific knowledge and the encompassing research tradition are continually reconstructed together, with each informing the other."

on page 110:
"...qualitative differences in the emerging mathematical entities reflected and were reflected in qualitative differences in classroom mathematics. Once again, we stress that the students were not simply embedded in differing mathematical activity systems. Rather, they participated in the constitution [of] differing systems ... and thus in the constitution of the social situations of their development. ... By characterizing the situation in this way, we once again question the tendency to reify social activity systems and instead emphasize the reflexive relation between the individual and the social."

on page 111:
"Such observations again emphasize the children's active contributions to the interactions in which they participated, a point that tends to be overlooked if one focuses on the teacher and regards her as scaffolding children to higher levels of performance."

The assertiveness, specificity, clarity and number of such quotations in the chapter attest to the significance of this aspect of their position as a difference from what they see in the literature.

On the nature of social and mathematical/cognitive knowledge: Further evidence of a radical constructivist "take" by the authors is evident in their treatment of the nature of knowledge. This is seen in their discussion of "shared" knowledge of social norms where they say on page 106:

"...the most that can be said when interactions precede smoothly is that the teacher's and the children's beliefs fit in that each acted in accordance with the other's expectations... Situations in which social norms were renegotiated occurred when there was a lack of fit--when either the teacher's or a child's expectations were not fulfilled."

In typical radical constructivist fashion, they also treat "shared" mathematical knowledge in the same fashion as "shared" social knowledge, seen in the following from page 113:

"From the cognitive perspective, their conceptions of place value numeration were not shared but, instead, fit sufficiently for them to talk about mathematics without becoming aware of discrepancies in their mathematical interpretations. In short, they individually constructed conceptualizations that were adequate for the purpose of coordinating their mathematical activity with that of others as they completed tasks and engaged in collaborative mathematical activity."

These two examples of the author's views on the nature of social and mathematical knowledge seem to make the expression "taken-as-shared" particularly appropriate, superior to the term "shared," in reference to mathematical knowledge and social practices and their meaning. In both cases the notion of "fit" of the knowledge is used in the same fashion as von Glasersfeld has in describing the radical constructivist position on the nature of knowledge. In the first quotation from p. 106 about social knowledge the notion of change in social practices and their meaning being driven by failure of fit or by "breaches" in social norms is described. This is analogous to radical constructivist descriptions of changes in knowledge and would be a logical extension of the quote from page 113.

Vygotskian "scientific knowledge":
The authors only use the expression "scientific knowledge" once (see the quote above, the second one from page 105). They specifically refer to "scientific knowledge" as being "continually reconstructed." While one might still interpret C, W, & Y's isolated statement to refer to "organized, academic, disciplinary knowledge" or "school knowledge," it is hard, if not impossible, to see how they could mean this in the context of the chapter as a whole. It is the case that radical constructivism is founded on the notion that all meaning or understanding merely fits experience, that it is constructed by the holder of that meaning or understanding is a by-product of this fundamental tenet. (One could make a case here that the term "constructivism" is a potentially mis-leading label for this philosophical point-of-view.) In the context of C, W, & Y's examples, it is hard now to see how Vygotskian positions on the reflexive relationship between the learner's development and the learner's social environment and on what is called scientific knowledge are logically compatible.

In summary:
C, W, & Y found it essential to view their initial goal of understand children's mathematical learning in the context of reflexive relationships between individual and social, between cognitive, social and affective knowledge, between teachers and students, between the classroom and its broader social setting, a notion consistent with Vygotski and Leont'ev. At the same time their basic radical constructivist foundation leads them to be independent of the notion that particular social norms or mathematical knowledge exists independently of the action of the teacher and students. As a result the authors are able to view these important reflexive relationships as truly reflexive. At the same time, this has led the authors to note important differences between their understanding of their observations and certain representations typically found in Vygotskian literature.

Comments and suggestions concerning this review are welcome.

Dewey I. Dykstra, Jr.
Professor of Physics
Department of Physics/SN318
Boise State University
1910 University Drive
Boise, ID 83725-1570

Date: Thu, 23 Nov 1995
From: Genevieve Patthey-Chavez (ggpcinla@ucla.edu)
Subject: Re: Chapter 4, finally

Re: Some of the ideas expressed below...Bourdieu's notion of Habitus and Giddens' notion of the duality of structure have been around since at least the 70s. Then there's Lave's Cognition in practice and a lot of recent work on constitutive indexicality developing precisely to begin to map out relationships between knowledge, action, structure, activity. Finally, the "container perspective of context" has been challenged rather forcefully by both Engestrom and Ruth Smith. Well, here's a small quotation from the latter:

(W)hile structuration theory might at first sight appear able to deliver on this challenge, Giddens' treatment of system as a relational duality (individual/social) as opposed to a relational triad (subject/community/object) that embraces the material world prevents its realization.

The conceptual move that needs to be made is to recognize that context is both a medium and outcome that materially and semiotically mediates the activity of the subject, the social group or community and the material world of objects (which are themselves constituted by sociocultural, historical, meaterial and semiotic forces). We produce and reproduce contexts, we then use those productions as mediational means to exchange, distribute, and consume, material and semiotic phenomena and to produce new contexts and new contextual media.

Duranti & Goodwin appropriately suggest that, instead of viewing context as a set of variables that statically surround strips of talk, context and talk are now argued to stand in a mutually reflexive relationship to each other, with talk, and the interpretive work it generates, shaping context as much as context shapes talk.

From: R. C. Smith, 1990, In pursuit of synthesis: Activity theory as a primary framework for organizational communication, pp. 111-112.

Just to add my own two cents' worth: I find it really had to pursue synthesis in a discourse community whose basic argumentative style builds on an adversarial premise--find a hole in someone else's argument and stake your claim to fame, and so on. And yes, I know my own reaction above does not break the chain. I remind myself, When in Rome, do as the Romans, but then it occurs to me, where IS xmca? Where am I? An exciting thought, actually...

Genevieve

>In summary:
>C, W, & Y found it essential to view their initial goal of understand
>children's mathematical learning in the context of reflexive relationships
>between individual and social, between cognitive, social and affective
>knowledge, between teachers and students, between the classroom and its
>broader social setting, a notion consistent with Vygotski and Leont'ev. At
>the same time their basic radical constructivist foundation leads them to
>be independent of the notion that particular social norms or mathematical
>knowledge exists independently of the action of the teacher and students.
>As a result the authors are able to view these important reflexive
>relationships as truly reflexive. At the same time, this has led the
>authors to note important differences between their understanding of their
>observations and certain representations typically found in Vygotskian
>literature.

>Comments and suggestions concerning this review are welcome.

>Dewey I. Dykstra, Jr.

Date: Mon, 27 Nov 1995
From: dykstrad@varney.idbsu.edu
Subject: Re: Chapter 4, finally

>Just to add my own two cents' worth: I find it really had to pursue
>synthesis in a discourse community whose basic argumentative style builds
>on an adversarial premise--find a hole in someone else's argument and stake
>your claim to fame, and so on. And yes, I know my own reaction above does
>not break the chain. I remind myself, When in Rome, do as the Romans, but
>then it occurs to me, where IS xmca? Where am I? An exciting thought,
>actually...
>Genevieve

To all (and Genevieve):

It seems to me that all of what was brought up before this last paragraph is essentially consistent with the views expressed in the chapter.

The last paragraph of Genevieve's leads me to think I'm missing something or reading something in here or maybe Genevieve is reading in something I did not intend, nor, I think the authors of the chapter intended. I cannot see how there can be a synthesis involving things which are the same; what would there be to synthesize?

Now maybe I'm wandering too far in the darkness of my not understanding what the point is here, but maybe these comments will help someone to help me to "get" what I am missing...

It seems to me that the authors were saying that looking at Vygotski and Leont'ev helped them understand their experiences with their project better and that they had something to offer in return, which comes out of their original point-of-view.

I think that the authors are trying to "pursue synthesis in a discourse community." I did not see the chapter nor did I intend the review to be in a "basic argumentative style [which] builds on an adversarial premise--find a hole in someone else's argument and stake your claim to fame, and so on." There's a difference between this and attempting to clarify in what ways one agrees with the perceived point-of-view of others and how one disagrees and why and in what ways this lack of agreement is manifest.

Unfortunately, any situation in which there is some difference in pov could be handled in an argumentative, adversarial, aggresive style and it seems that regardless of how a difference of pov is addressed, someone could choose to "take" or view this handling as argumentative and adversarial. Neither I nor, I think, the authors intend to function this way. I do not like the zero-sum or even negative-sum game of one-upsmanship either, but then I do not think that the handling of differences has to be this way and that when there really is synthesis the result is a positive-sum. I hope that it is clear that I agree with Angel Lin's comments (on adversarial discourse vs. communication Thu, 23 Nov 1995 12:51:39) along these lines.

If we cannot clarify for ourselves and each other the differences in our pov's without getting distracted by applying names to what each other is doing, then I find it hard to see how we can ever achieve synthesis, because when we are busy labeling, we are usually not "hearing" what the other is saying. If we can never hear or express our differences, then what's to synthesize?

Now I'm back where I started. Have I totally missed the boat here? I always thought that xmca (xlchc) was a place that we didn't have to be Romans.

Thanks in advance for any help anyone has to offer.
Dewey