Commognition

• In Event: Dealing With Diverse Discourses (3D): Can We Build on Each Other’s Research Contributions?

Abstract

Theory
The eponymous commognition, a combination of communication and cognition, signals that the processes of (a) thinking and (b) communicating with others are different manifestations of one phenomenon.

Key words
Mathematics: a discourse regarded as mathematical by its expert participants.
Discourse: a special type of communication, made distinct by its keywords and visual mediators, routines and endorsed narratives (utterances considered as one’s that can usefully mediate practical activities).
Learning mathematics: becoming an agentive participant of mathematical discourse.
Discursive (mathematical) object: a signifier with all its realizations (possibly other signifiers), the realizations of these realizations, etc.
Objectification: Discursive construction of new objects.
Routine: a sequence of discursive moves the participant reproduces in situations she judges as involving the same task as a situation in which this sequence was performed it in the past. Routine may thus also be defined as a task, as seen by the implementer, together with a procedure she chooses for the implementation of that task.
Ritual or exploration: Routines that the performer views as aiming, respectively, at implementing a certain procedure or at obtaining a particular outcome

Basic Tenets
• Development of mathematical discourse involves objectification, that is, discursive construction of new mathematical objects. In most cases, this means introduction of a new noun that allows to compress the discourse by (a) reifying - speaking of a process as if it was an object; and (b) alienating - employing the new noun in impersonal utterances (as an active subject).
• Historically, mathematical discourse has been co-constitutive with practical activities, which it helped to expand.
• Mathematics develops by annexing its own meta-discourses and/or by subsuming several discourses under a single one. For instance, algebra is a formalized meta-discourse of arithmetic.

Typical Methods or Methodology
The non-dualist conceptual framework (Author, 2008) comes together with rigorous research methods, enabling a study of high-fidelity transcripts of classroom interactions with the help of high-resolution special-purpose discourse analysis.

Findings
• Years before they are taught formal algebra, children are often able to solve basic algebraic problems (expressing patterns; finding unknown) by presenting the necessary computations verbally. With time, their informal algebraic discourse goes through three stages: (1) processual: the processes and their products are treated as separate; (2) granulated: some intermediary processes are objectified (as in “the sum of a and b), and (3) objectified: the description of the process serves also as signifying its product. The transitions often happen in leaps.
• Routines of a discourse featuring new mathematical objects make their first appearance in the life of a child as rituals. With use and proper support, the rituals gradually turn into explorations.
• While introduced to the discourse on functions, the children initially have recourse to routines from discourses on objects they already know: curves, algebraic expressions and tables. With proper support, the learners gradually become aware of isomorphisms between these three discourses and starts clustering their corresponding routines into equivalence classes.

Scholarly Significance
Research within this non-dualist framework is scientifically rigorous and widely applicable. It has been making novel contributions to educational theory and practice.

Author

• Anna Sfard, University of Haifa