**חקירות דיאלקטיות של תובנות מתמטיות: התהוות תצורות דיסציפלינאריות בפעילויות אינטראקטיביות תומכות למידת פרופורציות**

Responding to recent calls to develop cognitivist-cum-sociocultural theoretical models of mathematical learning, we present case analyses of students’ creative appropriation of forms emerging from interaction with symbolic artifacts. In this bootstrapping procedure, problem solvers: (1) hook – when a symbolic artifact is first introduced into the interaction space, they engage it because they recognize its contextual utility for enhancing the enactment, explanation, or evaluation of their current solution strategy; then (2) shift – in the course of implementing these new affordances, they notice in these artifacts additional embedded properties as affording a new or reconfigured strategy that better meets domain-general criteria of conciseness, precision, prediction, and – in the case of co-production with a peer – communication, coordination, and collaboration. These new strategies are then sanctioned by the instructor, who views the shift as advancing the child’s process of mathematization closer to disciplinary structures and procedures, in accord with the intervention’s pedagogical objectives. We support and elaborate the proposed constructs with a set of selected episodes from videographed empirical data gathered in a design-based research study that investigated the emergence of mathematical concepts from guided embodied-interaction activity (n=22; ages 9-11). We list and explain critical interaction dimensions enabling such learning.

Dor Abrahamson (PhD Learning Sciences, Northwestern University, 2004) is an Assistant Professor of Cognition and Development in the University of California, Berkeley’s Graduate School of Education. He studies mathematics learning from the complementary perspectives of cognitive science, socio-cultural theory, and semiotics. Central to his research is the creation of mathematics learning systems. This craft is motivated by an emerging thesis that the human mind is embodied and, therefore, mathematics learning depends on stimulating, simulating, blending, and anchoring imagistic fragments drawn from simple sensorimotor being—students build personal meaning for disciplinary notation through guided interactions and abductive inference. Abrahamson directs the Embodied Design Research Laboratory. EDRL projects are typical of the design-based research multi-disciplinary approach: inspired by all students’ capacity to deeply understand mathematics subject matter, such as rational numbers, probability, and statistics, and driven by specific conjectures as how to engender such understanding, EDRL members build task-based activities involving mixed-media materials they engineer, construct, and iteratively modify. By analyzing student multi-modal behaviors—speech, gesture—as they engage in these activities, EDRL develops theoretical models of learning linked to pragmatic design frameworks. |