caa a22 4500 445881682 CHVBK 20180317145545.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s10649-011-9313-6 doi (NATIONALLICENCE)springer-10.1007/s10649-011-9313-6 Kidron Ivy Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, POB 16031, 91160, Jerusalem, Israel aut Tacit models, treasured intuitions and the discrete—continuous interplay [Elektronische Daten] [Ivy Kidron] We explore conditions for productive synthesis between formal reasoning and intuitive representations through analysis of college students' understanding of the limit concept in the definition of the derivative. In particular, we compare and contrast cognitive processes that accompany different manifestations of persistence of intuitions and tacit models that coexist with students' logical reasoning. The students are highly trained in mathematics. We encounter expressions of the persistence and impact of intuitions and tacit pictorial models as described by Fischbein. But we also observe some new characterization of persistence of tacit models in which the tacit pictorial model continues to interfere in the student's reasoning process, coexists with a logical reasoning but does not prevent the student from reaching a feeling of logical consistency. The empirical analysis and the theoretical discussion offered in the present paper permit us to highlight this very special integration of the formal and the intuitive components of the reasoning process. Springer Science+Business Media B.V., 2011 Actual infinity and potential infinity nationallicence Discrete and continuous nationallicence Infinitely small nationallicence Intuition nationallicence Limit nationallicence Tacit model nationallicence Educational Studies in Mathematics Springer Netherlands 78/1(2011-09-01), 109-126 0013-1954 78:1<109 2011 78 10649 https://doi.org/10.1007/s10649-011-9313-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10649-011-9313-6 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Kidron Ivy Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, POB 16031, 91160, Jerusalem, Israel aut NATIONALLICENCE 773 0- Educational Studies in Mathematics Springer Netherlands 78/1(2011-09-01), 109-126 0013-1954 78:1<109 2011 78 10649 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer