The Transition from Comparison of Finite to the Comparison of Infinite Sets: Teaching Prospective Teachers

Tsamir, Pessia

doi:10.1007/978-94-017-1584-3_10

Cited by 19 publications

(33 citation statements)

References 11 publications

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“…The students-future teachers-should also be faced with problems that lead to contradictory solutions when relying on intuition to solve them (for example, compare natural and even numbers) (see Tirosh, 1991;Tsamir, 1999Tsamir, , 2001. This could lead to the realization that Cantor's Set Theory is impossible to avoid when tackling the concept of infinity.…”

Section: Discussionmentioning

confidence: 99%

“…Research in the field of infinity suggests that students' and teacher students' intuitions often interfere with their performance on problems on infinity Kattou et al, 2009;Manfreda Kolar & Hodnik Čadež, 2010;Monaghan, 2001;Pehkonen et al, 2006;Tirosh, 1991;Tirosh & Tsamir, 1996;Tsamir, 1999;Tsamir, 2001;Tsamir & Tirosh, 1999;…).…”

Section: Understanding Of the Concept Of Infinity In Students Primarmentioning

confidence: 94%

“…Similarly to Tirosh (1991), Tsamir (1999) also conducted planned, non-traditional teaching of Cantorian Set Theory, the so-called enrichment or intuitive-based course, the difference being that the course was meant for mathematics teacher students. Its aim was to promote the need to use a consistent method in comparing infinite sets, to become aware of the fact that applying generalizations about finite sets to infinite sets leads to contradiction and to realize that there is one method of comparing infinite sets which is always applicable, that is, one-to-one correspondence.…”

Section: Representation Dependence In the Understanding Of Infinitymentioning

confidence: 99%

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Analysis of factors influencing the understanding of the concept of infinity

Kolar

Čadež

2011

Educ Stud Math

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The article discusses the understanding of infinity in children, teachers and primary teacher students. It focuses on a number of difficulties that people cope with when dealing with problems related to infinity such as its abstract nature, understanding of infinity as an ongoing process which never ends, understanding of infinity as a set of an infinite number of elements and understanding of well-known paradoxes. In the empirical section of the article, a study is described that was conducted at the Faculty of Education, University of Ljubljana, Slovenia. It encompassed 93 third-year students of the Primary Teacher Education study programme with the aim of researching their understanding of the concept of infinity. The focus was on finding out how primary teacher students who received no in-depth instruction on abstract mathematical content understand different types of infinity: infinitely large, infinitely many and infinitely close, what argumentation they provide for their answers to problems on infinity and what their basic misunderstandings about infinity are. The results show that the respondents' understanding of infinity depends on the type of the task and on the context of the task. The respondents' justifications for the solutions are based both on actual and on potential infinity. When solving tasks of the types 'infinitely large' and 'infinitely many', they provide justifications based on actual infinity. When solving tasks of the type 'infinitely close', they use arguments based on potential infinity. We conclude that when they feel unsure of themselves, they resort to their primary method of dealing with infinity, that is, to potential infinity.Keywords Infinity . Concept of infinity . Potential infinity . Actual infinity . Primary teacher students Potential and actual infinityIn the literature, the understanding of the notion of infinity is associated with two different concepts-the concept of potential infinity and the concept of actual infinity (Dubinsky, Educ Stud Math (2012) 80:389-412

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Section: Discussionmentioning

confidence: 99%

Section: Understanding Of the Concept Of Infinity In Students Primarmentioning

confidence: 94%

Section: Representation Dependence In the Understanding Of Infinitymentioning

confidence: 99%

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Analysis of factors influencing the understanding of the concept of infinity

Kolar

Čadež

2011

Educ Stud Math

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“…The researcher explicitly suggested the use of a different approach, but the student tried to use a visual register for a second time. In fact, visual registers are concrete and suitable, particularly with reference to the example considered (this evokes sectorialization: Schoenfeld, 1986): the subject deals with geometrical objects and so with a sector usually expressed by visual registers (concerning the influence of visual representations, see Tsamir andTirosh, 1994, 1999;Tsamir, 1999;Tsamir and Dreyfus, 2002). Symbolic registers are more general (although some symbols are sometimes used with implicit meanings, e.g.…”

Section: An Exercisementioning

confidence: 99%

Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory

Bagni

2006

Educ Stud Math

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The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory -the concepts of inclusion and belonging.To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of mathematical objects draws its meaning from different semiotic systems framed by their own cultural context. Our results suggest that the successful accomplishment of knowledge attainment seems to be linked to the students' ability to suitably distinguish and coordinate the meanings and symbols of the various semiotic systems (e.g. verbal, diagrammatic and symbolic) that encompass their mathematical experience.

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“…She demonstrates how research-based knowledge about students' inconsistent responses to different representations of infinite sets can be used to raise their awareness of contradictions in their own reasoning and to guide them toward using the one-to-one correspondence as the unique criterion for the comparison of infinite quantities. She begins by briefly summarizing the results of previous studies on students' responses to different representations of the comparison-of-infinite-sets tasks (e.g., Tsamir, 1999;Tsamir & Tirosh, 1999). The main result is that a student's decision as to whether two infinite sets have the same number of elements largely depends on the specific representation of the given infinite sets in the problem.…”

Section: Teaching Approachesmentioning

confidence: 99%