Abstraction and Consolidation

Abstract

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can be used to create new constructions. Newly formed constructions are fragile entities. In the course of consolidating a construction this student creates connections between the new construction and already established mathematical knowledge and develops a language to describe and guide mathematical actions related to the construction. The resulting abstraction is a more resilient form of the construction and the student is able to justify his assertions. The paper also examines issues in designing tasks to consolidate a construction.

Construction Process of the Length of 3√2 by Paper Folding

“…Most of the researchers carried out researches on abstraction in context "Recognizing", "Building-with" and "Constructing" as epistemic actions, (Altun & Kayapınar, 2011;Kidron & Dreyfus, 2010), while several researchers emphasized the consolidation process (Dreyfus & Tsamir, 2004;Tabach, Hershkowitz & Schwarz, 2006;Monaghan & Ozmantar, 2006;Dreyfus, Hadas, Hershkowitz & Schwarz, 2006) and characteristics of consolidation (Dreyfus & Tsamir, 2004, Tsamir & Dreyfus, 2005Anabousy & Tabach, 2015). Dreyfus and Tsamir (2004) carried out a study which revealed the characteristics of consolidation that emerges from a sequence of interviews about the comparison of infinite sets with a talented student.…”

A Historical Analysis of Primary Mathematics Curricula in Terms of Teaching Principles

“…However, studies conducted especially over the last fifteen years have shown that consolidation of newly constructed mathematical knowledge plays an essential role in future learning (e.g., Tabach et al, 2006, Monaghan & Ozmantar, 2006. These studies suggest the importance of consolidation through activities focusing on the use of newly constructed mathematical structures in different contexts and with different dimensions.…”

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

“…This cognitive process repeated in different situations leads to the possibility of consolidating mathematical conceptions. Therefore, the iteration of these phases may help in the consolidation of such mathematical structures (MONAGHAN;OZMANTAR, 2006). From this perspective, the consolidation process of mathematical conceptions is another manifestation of reflective abstraction that leads to a new cognitive state.…”

“…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 any event, more research is required to provide information that will be useful in reaching a clearer theoretical understanding of task-design, with all the obvious implications for the improvement of teaching methods.…”