Steering (Dis)Course between Metaphors and Rigor: Using Focal Analysis to Investigate an Emergence of Mathematical Objects

This study deals with students' construction of mathematical objects. The basic claim is that the need for communication-any attempt to evoke certain actions by others-is the primary driving force behind all human cognitive processes. Effectiveness of verbal communication is seen as a function of the quality of its focus. Material objects may serve as a basis for creation of such a focus, but in some discourses, focus-engendering objects must be created. Such discursive construction is observed in analysis of one classroom episode. Special attention is given to metaphor, which is the point of departure for the construction process, and to the subsequent dialectical process of closing the gap between the metaphor-induced expectations and the need for a well-defined construction procedure to ensure effective communication.

[1]  Ed Dubinsky,et al.  Reflective Abstraction in Advanced Mathematical Thinking , 2002 .

[2]  P. Cobb,et al.  Discourse, mathematical thinking, and classroom practice. , 1993 .

[3]  H. Maturana,et al.  The Tree of Knowledge: The Biological Roots of Human Understanding , 2007 .

[4]  Götz Krummheuer.  The ethnography of argumentation. , 1995 .

[5]  Heinz Steinbring,et al.  Epistemological Constraints of Mathematical Knowledge in Social Learning Settings , 1998 .

[6]  L. Vygotsky.  Mind in Society: The Development of Higher Psychological Processes: Harvard University Press , 1978 .

[7]  Guershon Harel,et al.  The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts , 2002 .

[8]  Patrick W Thompson.  Experience, Problem Solving, and Learning Mathematics: Considerations in Developing Mathematics Curricula , 1985 .

[9]  Carolyn Kieran.  The learning and teaching of school algebra. , 1992 .

[10]  M. Bakhtin,et al.  Speech Genres and Other Late Essays , 1988 .

[11]  P. Cobb,et al.  Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. , 1996 .

[12]  Paul Cobb,et al.  Individual and Collective Mathematical Development: The Case of Statistical Data Analysis. , 1999 .

[13]  P. Cobb,et al.  Reflective Discourse and Collective Reflection , 1997 .

[14]  Anna Sierpinska,et al.  Language and Communication in the Mathematics Classroom. , 1998 .

[15]  Magdalene Lampert,et al.  When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching , 1990 .

[16]  N. Minick,et al.  Contexts for Learning: Sociocultural Dynamics in Children's Development , 1993 .

[17]  Brian Rotman,et al.  Towards a Semiotics of Mathematics , 1988 .

[18]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

[19]  A. Sfard.  On Two Metaphors for Learning and the Dangers of Choosing Just One , 1998 .

[20]  A. Sfard.  Reification as the birth of metaphor , 1994 .

[21]  Valerie Walkerdine,et al.  The Mastery Of Reason: Cognitive Development And The Production Of Rationality , 1990 .

[22]  L. Vygotsky.  Thinking and Speech , 1987 .

[23]  J. Bruner.  Actual minds, possible worlds , 1985 .

[24]  Magdalene Lampert,et al.  Talking Mathematics in School: Studies of Teaching and Learning , 1998 .

[25]  H.J.M. van Oers.  The appropriation of mathematical symbols. A psychosemiotic approach to mathematics learning. , 1999 .

[26]  Michael J. Reddy.  Metaphor and Thought: The conduit metaphor: A case of frame conflict in our language about language , 1993 .

[27]  David Tall,et al.  Ambiguity and flexibility: A proceptual view of simple arithmetic , 1983 .

[28]  Heinz Steinberg,et al.  Routine and Meaning in the Mathematics Classroom. , 1989 .

[29]  P. Hanks,et al.  Collins dictionary of the English language , 1979 .

[30]  L. Vygotsky,et al.  Thought and Language , 1963 .

[31]  Jorg Voigt,et al.  Negotiation of mathematical meaning and learning mathematics , 1994 .