Formation of Mathematical Objects as Decision Making

Different moments in the participatory stage of the secondary students' abstraction of mathematical conceptions

“…This recommendation is compatible with that made by Tzur (2006) following a whole-class teaching experiment. Consequently, in order to give learners the opportunity to change their attention-focus and begin to see a set of activityeffect registers as a unified object (the identification of the regularity and/or the general aspect) (DÖRFLER, 2002) it will be necessary to create opportunities for the development of language-items for the new construction. This characteristic of the task has also been considered relevant in designing tasks to Different Moments in the Participatory Stage of the Secondary Students' Abstraction of... ROIG, A. I.; LLINARES, S.; PENALVA, M. C. consolidate a new construction (GARCÍA; LLINARES; SÁNCHEZ-MATAMOROS, 2010;MONAGHAN;OZMANTAR, 2006).…”

“…In this sense it is essential to have accurate fine descriptions of the processes by which mathematical knowledge is developed. This situation generates issues about what it means to know something about mathematical objects, and how the learner develops or constructs that knowledge (DÖRFLER, 2002;DUBINSKY, 1991;HERSHKOWITZ;SCHWARTZ;DREYFUS, 2001). Cognitive theories based on Piagetian stances assume that mathematical conceptions reflect regularities from human actions and mental operations.…”

“…These two phases in the construction process have been treated differently by different researchers (TALL; THOMAS; DAVIS; GRAY; SIMPSON, 1999). Dörfler (2002), for instance, postulates that the construction process must be supplemented by a deliberate decision to view, treat, use and investigate a structure or a collection of items as a unified object, in which the formation of mathematical objects is conceived as a point of view taken by the learner. Simon and his colleagues postulate the existence of a mechanism that they call Reflection on Activity-Effect Relationship to explain this construction process.…”

“…That is to say, the abstracted conception (the regularity) is provisional and unstable; since, in other contexts (problem situations) learners may not be able to recognise on their own the involvement of the conception and come to depend on external prompts. In this process, one key notion is the learners' awareness of the new structure (DÖRFLER, 2002;GARCÍA;SÁNCHEZ-MATAMOROS, 2010). The extent to which a learner is aware of a constructed conception is considered an indicator of the construction process.…”

Characterizing Thematized Derivative Schema by the Underlying Emergent Structures

“…The processes of construction and use of mathematical concepts in calculus have been the focus of numerous analyses (Confrey & Costa, 1996;Zandieth, 1997;Dörfler, 2002;Berry & Nyman, 2003;Meel, 2003;Mason & Jonston-Wilder, 2004;Edwards, Dubinsky & McDonald, 2005;Harel & Sowder, 2005;Habre & Abboud, 2006;Hähkiöniemi, 2006). The cognitive construction is a process that uses symbolic, schematic, and linguistic meanings to express changes in perspective and points of view (Dörfler, 2002;Harel, Selden & Selden, 2006).…”