Definition
Abstraction has been the focus of extensive interest in several domains, including Mathematics Education. Many researchers have taken a predominantly theoretical stance and have described abstraction as some type of de-contextualization. Abstraction in context (AiC; Hershkowitz et al. 2001) proposes a quite different approach to abstraction. The following is the operational definition of AiC:
Abstraction in Context is an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure.
The term activity above emphasizes that abstraction in context (AiC) is considered to be a process taking place in a specific context; it may capitalize on tools and other artifacts and it occurs in a particular social setting. The phrase previously constructed mathematics refers to the outcomes of previous processes of abstraction, which may be used during the current abstraction activity. The phrase reorganizing into a new structureimplies the...
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References
Davydov VV (1990) Types of generalisation in instruction: logical and psychological problems in the structuring of school curricula. Soviet studies in mathematics education, vol 2 (ed: Kilpatrick J, trans: Teller J). National Council of Teachers of Mathematics, Reston. (Original work published 1972)
Dreyfus T, Kidron I (2006) Interacting parallel constructions. A solitary learner and the bifurcation diagram. Rech Didact Math 26:295–336
Dreyfus T, Tsamir P (2004) Ben’s consolidation of knowledge structures about infinite sets. J Math Behav 23:271–300
Dreyfus T, Hershkowitz R, Schwarz B (2015) The nested epistemic actions model for abstraction in context – theory as methodological tool and methodological tool as theory. In: Bikner-Ahsbahs A, Knipping C, Presmeg N (eds) Approaches to qualitative research in mathematics education: examples of methodology and methods. Advances in mathematics education series. Springer, Dordrecht, pp 185–217
Gilboa N, Dreyfus T, Kidron I (in press) Constructing a mathematical definition: the case of tangent. Int J Math Educ Sci Technol
Gravemeijer K (1999) How emergent models may foster the constitution of formal mathematics. Math Think Learn 1:155–177
Hershkowitz R (2009) Contour lines between a model as a theoretical framework and the same model as methodological tool. In: Schwarz BB, Dreyfus T, Hershkowitz R (eds) Transformation of knowledge through classroom interaction. Routledge, London, pp 273–280
Hershkowitz R, Schwarz B, Dreyfus T (2001) Abstraction in context: epistemic actions. J Res Math Educ 32:195–222
Hershkowitz R, Hadas N, Dreyfus T, Schwarz B (2007) Processes of abstraction, from individuals’ constructing of knowledge to a group’s “shared knowledge”. Math Educ Res J 19(2):41–68
Hershkowitz R, Tabach M, Dreyfus T (2016) Creativity within shifts of knowledge in the mathematics classroom. In: Csíkos C, Rausch A, Szitányi J (eds) Proceedings of the 40th conference of the international group for the psychology of mathematics education, vol 2. PME, Szeged, pp 385–392
Kidron I (2008) Abstraction and consolidation of the limit procept by means of instrumented schemes: the complementary role of three different frameworks. Educ Stud Math 69:197–216
Kidron I, Dreyfus T (2010a) Justification enlightenment and combining constructions of knowledge. Educ Stud Math 74:75–93
Kidron I, Dreyfus T (2010b) Interacting parallel constructions of knowledge in a CAS context. Int J Comput Math Learn 15:129–149
Kidron I, Monaghan J (2009) Commentary on the chapters on the construction of knowledge. In: Schwarz BB, Dreyfus T, Hershkowitz R (eds) Transformation of knowledge through classroom interaction. Routledge, London, pp 81–90
Kouropatov A, Dreyfus T (2014) Learning the integral concept by constructing knowledge about accumulation. ZDM 46:533–548
Monaghan J, Ozmantar MF (2006) Abstraction and consolidation. Educ Stud Math 62:233–258
Pontecorvo C, Girardet H (1993) Arguing and reasoning in understanding historical topics. Cogn Instr 11:365–395
Rasmussen C, Stephan M (2008) A methodology for documenting collective activity. In: Kelly AE, Lesh RA, Baek JY (eds) Handbook of innovative design research in science, technology, engineering, mathematics (STEM) education. New York: Taylor and Francis, pp 195–215
Ron G, Dreyfus T, Hershkowitz R (2010) Partially correct constructs illuminate students’ inconsistent answers. Educ Stud Math 75:65–87
Ron G, Dreyfus T, Hershkowitz R (2017) Looking back to the roots of partially correct constructs: the case of the area model in probability. J Math Behav 45:15–34
Tabach M, Hershkowitz R, Rasmussen C, Dreyfus T (2014) Knowledge shifts in the classroom – a case study. J Math Behav 33:192–208
Tabach M, Rasmussen C, Dreyfus T, Hershkowitz R (2017) Abstraction in context and documenting collective activity. In: Dooley T, Gueudet G (eds) Proceedings of the tenth conference of the European Society for Research in Mathematics Education (CERME10). Dublin City University and ERME, Dublin, pp 2692–2699
Treffers A, Goffree F (1985) Rational analysis of realistic mathematics education. In: Streefland L (ed) Proceedings of the 9th international conference for the psychology of mathematics education, vol II. OW&OC, Utrecht, pp 97–123
van Oers B (1998) The fallacy of decontextualization. Mind Cult Act 5:143–153
Weiss D (2011) Processes of mathematical knowledge construction with analogical models. Unpublished PhD thesis, Tel Aviv University. (In Hebrew)
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Hershkowitz, R., Dreyfus, T., Schwarz, B.B. (2020). Abstraction in Context. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_100032
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