What Is the Object of the Encapsulation of a Process?

Tall, David; Thomas, Michael; Davis, George K.; Gray, Eddie; Simpson, Adrian P.

doi:10.1016/s0732-3123(99)00029-2

Cited by 66 publications

(40 citation statements)

References 8 publications

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“…We need to understand the reasons behind this failure in order to establish an effective communication with students in the class. In this sense, exploring how various mathematical concepts exist in students' minds is considered to be an important phase (Vinner & Dreyfus, 1989;Tall et al, 2000). In accordance with given explanation, the purpose of this study is to explore the nature and causes of the "gap" between parallelogram's formal concept and the concept of parallelogram in individuals' minds by revealing the figures and definitions constructed by the individuals regarding the concept.…”

Section: Introductionmentioning

confidence: 99%

Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms

Ülger¹

2017

EUROPEAN J ED RES

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Abstract:This study attempts to reveal pre-service teachers' conceptions, definitions, and understanding of quadrilaterals and their internal relationships in terms of personal and formal figural concepts via case of the parallelograms. To collect data, an open-ended question was addressed to 27 pre-service mathematics teachers, and clinical interviews were conducted with them. The factors influential on pre-service teachers' definitions of parallelograms and conceptions regarding internal relationships between quadrilaterals were analyzed. The strongest result involved definitions based on prototype figures and partially seeing internal relationships between quadrilaterals via these definitions. As a different result from what is reported in the literature, it was found that the fact that rectangle remains as a special case of parallelogram in pre-service teachers' figural concepts leads them not to adopt the hierarchical relationship. The findings suggested that learners were likely to recognize quadrilaterals by a special case of them and prototypical figures, even though they knew the formal definition in general. This led learners to have difficulty in understanding the inclusion relations of quadrilaterals.

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Section: Introductionmentioning

confidence: 99%

Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms

Ülger¹

2017

EUROPEAN J ED RES

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“…Importantly, the extent to which students engaged functions as objects in non-ordered pair strategies may not plumb the full depth of the structural conception posed in the theory of reification. The distinct manifestations of objects underlying non-ordered pair and multiple-ordered pair strategies, respectively, may reflect the differing notions of object that Tall, Thomas, Davis, Gray & Simpson (2000) argue are embedded within Sfard's characterization of structural perspectives on function. While "her description of a function as a 'set of ordered pairs' as structural...agrees with the formal Bourbaki approach," she also "considers the visual imagery of a graph of a function to be structural" (p. 234).…”

Section: Discussionmentioning

confidence: 99%

Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

White

2009

Educ Stud Math

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This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups' developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.

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“…They noted students tended to focus on surface representational features (such as commas between terms in a sequence) and had difficulties with reconciling a list-based view of sequences with a functional view and seeing a sequence as a cognitive whole. Such problems also manifest themselves in students making sense of sequences defined in unfamiliar ways: for example, the sequence (a n ) where a n = 1 − ( 1 n ) for n odd and a n = 1 n for n even, may be seen as two sequences, with two limits (Tall, Thomas, Davis, Gray, & Simpson, 1999). Sierpínska (1987) focussed on students' understandings of limits of sequences (most notably in considering infinite decimals like 0.999 .…”

Section: Limits Of Sequencesmentioning

confidence: 99%

Three concepts or one? Students’ understanding of basic limit concepts

Fernández-Plaza

Simpson

2016

Educ Stud Math

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Citation for published item:pern¡ ndezElzD tFeF nd impsonD eF @PHITA 9hree onepts or onec tudents9 understnding of si limit oneptsF9D idutionl studies in mthemtisFD WQ @QAF ppF QISEQQPF Further information on publisher's website:The nal publication is available at Springer via https://doi.org/10.1007/s10649-016-9707-6Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of limit connect them. This paper examines the nature of connections made by students exposed to this fragmented curriculum. The study adopted a phenomenographic approach and used card sorting and comparison tasks to expose students to symbols representing these different types of limit. The findings suggest that, while some students treat limit cases as separate, some can draw connections, but often do so in ways which are at odds with the formal mathematics. In particular, while there are occasional, implicit uses of neighbourhood notions, no student in the study appeared to possess a unifying organisational framework for all three basic uses of limit.

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What Is the Object of the Encapsulation of a Process?

Cited by 66 publications

References 8 publications

Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms

Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms

Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

Three concepts or one? Students’ understanding of basic limit concepts

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