Re: abstract for roundtable
psmagorinsky who-is-at uoknor.edu
Wed, 21 Feb 96 14:11:06 -0600
>
>Dear Vygotsky Listers
>
>In lieu of an abstract I am sending the original proposal (2 pgs) for a
>roundtable discussion submitted by Vicki Zack. Please accept her apologies
>for getting it to the list at this late date but she only noticed the
>request when revewing her mail this morning.
>
>
>"Appropriating Mathematical Discourse: Fifth Graders Make Connections between
>Invention and Convention"
>
>Vicki Zack, Roundtable, Vygotsky Centennial, Chicago, February 23-25, 1996
>
>This paper investigates how the discursive practices of Grade 5 students
mediate
>their learning processes. The aim of the current research-in-progress is to
>understand how mathematical meaning is constructed and shared (Bishop, 1984).
>Specifically it explores the role of explanations during paired and small-group
>discussions of challenging mathematical problems to understand the changes in
>students' conceptual knowledge. The underlying assumption is that a "good
>explanation" is achieved when there is a fit between what a speaker wants to
>convey and what the listener needs to hear at any given moment. This shifts the
>focus away from the notion of a good explanation as rhetorically defined,
>that is as a discourse object having a number of specified features, and
>investigates it as an interactive exchange.
>
>This study explores the implications of Vygotskian ideas in the author's own
>inquiry-based mathematics classroom. Most salient are the ideas that (1)
meaning
>comes from a dialectic of thought and language (Vygotsky, 1934, 1986); (2)
there
>is semiotic mediation by means of the verbal--spoken and written-- and
gestural,
>and by means of multiple representations such as diagrams, charts and symbolic
>notation, which mediate both understanding and communicating; and (3) there
is a
>zone of proximal development where "skills . . . and understandings are
achieved
>in interaction with others before the children can do them on their own"
>(Newman, Griffin & Cole, 1989, p. 15). A specific goal of this study is to
>investigateintersubjectivity, and appropriation of new understandings,
>which is similar to
>the question investigated by another teacher-researcher, Roth (1995), as to
"how
>much students appropriate from collaborative achievements so that they can
claim
>this knowledge as personally meaningful" (p. xv).
>
>The school and classroom learning site is a community of practice which
Richards
>(1991) has called inquiry math; it is one in which children are expected to
>publicly express their thinking, and engage in mathematical practice
>characterized by conjecture, argument, and justification (Cobb, Wood, & Yackel,
>1993, p. 98). Twenty-five Grade 5 students participated in this research.
Almost
>all of the children have been part of this school culture since their entry
into
>the school 5 years ago. In this setting there are multiple opportunities for
>diverse peer exchanges as the children discuss in pairs, in small groups of
four
>or five participants, and in larger groups of approximately twelve. Data
sources
>include: (1) videotapes of the students' discussions, (2) the students' written
>comments detailing whose explanations were helpful to them while working on the
>problem, and (c) retrospective interviews with students one year later while
>viewing selected segments of the videotapes.
>
>Preliminary findings for discussion:
>
>The roundtable presentation and discussion will focus upon one non-routine
>problem. It was chosen as an example of the dialectic process of action and
>reflection: the linking of actions described verbally, for example
"overlapping"
>and "double-counting" with the corresponding mathematical operation, i. e.
>"divide in half", with the mathematical symbol - 2 (divide by two), and
finally,
>with the algebraic notation representing generalization. The patterns of
>discourse reveal that the initial talk tends to be descriptive, indexical and
>situated, and is subsequently elaborated by means of culture-specific
>terminology, namely the mathematical conventions. There were nine children of
>the twenty-five who not only derived the algebraic expressions, but could also
>explain why it was so. This was exciting and memorable because it is learning
>rarely attained in elementary school settings. Part of the discussion will
>include vignettes revealing how the children contributed to each other's
>learning; conceptual change is facilitated by interaction (Zack, in press).
>
>Pivotal for this discussion is Britton's idea regarding "shaping at the
point of
>utterance" (1982b, p. 141), as well as a twofold use of the term invention.
They
>would include LeFevre's (1987) description of "invention as a social act", as
>well as Lampert's (1990) assertion that in an inquiry-based classroom the
>teacher's role is to help the students explore their personal
>understandings, and to assist them in connecting their "inventions" or
>personal constructions with the conventions of the culture.
>
>Specific topics for exploration will be: (1) how understanding proceeds in the
>context of the children's explanations; (2) the topic of ownership of
ideas; and
>(3) the directionality of the learning which takes place in this community of
>practice and occurs not only between children and from adult to child, but
which
>moves also from child to adult.
>
>A good explanation resides in the interaction. There are numerous examples in
>these data of strong, clear explanations (in adult assessment terms) given
>to two or more listeners, which evoked diverse responses and did not always
>facilitate understanding. At the same time there were other instances where
>an 'incomplete' explanation proved a catalyst and promoted learning in the
>ZPD. The latter instance may indicate congruency with Britton's (1982a, p.
>135) idea that when we are listening we do not focus upon the words--we
>look through the words to focus upon the meaning that emerges.
>
>With respect to the ownership of ideas, Rogoff (1990) has stated that in
>collaborative exploration "individuals' ideas are tested and stretched and
>become part of a joint construction. The participants gain in understanding
>and may have
>difficulty determining 'whose' idea an insight was" (1990, p. 196). The
findings
>suggest, however, that some students can indeed trace the source of salient
>ideas, and can pinpoint the contribution that either they themselves, or a peer
>they recall by name, made to the joint construction of an idea; moreover, they
>can do so even after an interlude of a year.
>
>Lastly, Vygotsky never spoke of learning moving in a child-to-adult direction
>(Confrey, 1993). The children's phrasing and representations are at times novel
>and evocative, and have been used successfully by the teacher as mediating
>devices for other students. In addition, although atypical, it was a
>student-devised approach which precipitated the above-noted evolution to an
>algebraic expression. More generally, this has upon several occasions produced
>genuine mathematical learning on the part of the teacher (cf. Confrey, 1991, p.
>115). The acknowledgement that not only children but also their teachers may
>appropriate knowledge as a result of interaction with vigorous minds speaks to
>the roles of students and teachers in classroom interactions.
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>---------------------------------------------------------------------
>Barbara Graves Email: CXCH who-is-at musica.mcgill.ca
>Laboratory of Applied Cognitive Science Off: (514) 398-4256
>Department of Educational Psychology Fax: (514) 398-6968
>McGill University
>
>
>
Peter Smagorinsky
University of Oklahoma
College of Education
Department of Instructional Leadership and Academic Curriculum
820 Van Vleet Oval
Norman, OK 73019-0260
(405)325-3533
fax: (405)325-4061
smagor who-is-at aardvark.ucs.uoknor.edu
psmagorinsky who-is-at uoknor.edu