%0 Journal Article %1 Sfard91 %A Sfard, Anna %D 1991 %J Educational Studies in Mathematics %K ch2 ijtme2006 limit mythesis process-product reification scalar-functional sequences %P 1--36 %T On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin %U http://www.springerlink.com/openurl.asp?genre=article&issn=0013-1954&volume=22&issue=1&spage=1 %V 22 %X This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions. On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: interiorization, condensation, and reification. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.

@article{Sfard91, abstract = {This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions. On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: interiorization, condensation, and reification. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.}, added-at = {2006-03-07T17:56:12.000+0100}, author = {Sfard, Anna}, biburl = {https://www.bibsonomy.org/bibtex/2d9fce41cb9772f390f5220c0e85fa2d3/yish}, citeulike-article-id = {378466}, comment = {Kidron: "Abstract notions such as the limit concept could be conceived operationally as processes and structurally as objects. The dual character of mathematical concepts that have both a procedural and a structural aspect was investigated by many researchers. Sfard (1991) used the word reification to describe the gradual development of a process becoming an object." The scalar - functional incision stems from a developmental outlook on mathematical concepts building on the work of Piaget (1952). The functional form is seen as the correct, rigorous mathematical concept of a sequence: a function f:N -> R. Children's concept of sequences emerges out of their counting schemes (Steffe, 1988). Once they establish their concept of an explicitly nested number sequence they can progress from counting to counting in twos and threes. "Just as the iterable one was the abstraction of the repeated application of the "one more item" operation when double counting by ones, the iterable composite unit is the result of the abstraction of the repeated application of the "one more four" (say) when double counting by fours." (Olive, 2001). What we see as a multiplicative structure, or a function of the natural numbers (an = k*n), children perceive as a repetitive action. Ana Sfard's (1991) view is much less hierarchical. She also starts from Piaget's observation regarding the trajectory from an operational to a structural view of number. However, she notes that the transition between these views (both epistemologically and phylogenically) is spiral rather than linear; "again and again, processes performed on already accepted abstract objects have been converted into compact wholes, or reified" (ibid.).}, interhash = {508a634f5360d41b15be43eb0ffe6394}, intrahash = {d9fce41cb9772f390f5220c0e85fa2d3}, journal = {Educational Studies in Mathematics}, keywords = {ch2 ijtme2006 limit mythesis process-product reification scalar-functional sequences}, pages = {1--36}, priority = {2}, timestamp = {2008-04-26T23:37:31.000+0200}, title = {On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin}, url = {http://www.springerlink.com/openurl.asp?genre=article\&issn=0013-1954\&volume=22\&issue=1\&spage=1}, volume = 22, year = 1991 }