Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

Zandieh, Michelle; Rasmussen, Chris

doi:10.1016/j.jmathb.2010.01.001

Cited by 94 publications

(46 citation statements)

References 19 publications

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“…Zandieh & Rasmussen, 2010) and understanding of quadrilateral definitions (e.g. Govender & De Villiers, 2004;Levenson, Tirosh & Tsamir, 2011).…”

Section: Theoretical Opinions Regarding Concept Image and Concept Defmentioning

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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“…Zandieh & Rasmussen, 2010) and understanding of quadrilateral definitions (e.g. Govender & De Villiers, 2004;Levenson, Tirosh & Tsamir, 2011).…”

Section: Theoretical Opinions Regarding Concept Image and Concept Defmentioning

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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“…Van Dormolen & Zaslavsky, 2003), students' beliefs about them (Zaslavsky & Shir, 2005), and students' guided collaborative development of definitions in contexts new to them (Larsen & Zandieh, 2008;Zandieh & Rasmussen, 2010). Studies have suggested that students who spontaneously generate examples in response to new definitions later perform better on tasks involving the defined concepts (Dahlberg & Housman, 1997), and there has been much recent discussion of the way in which example generation tasks might help students to modify their example spaces so that these more closely coincide with the extensions of the defined concepts (Mason, 2002;Watson & Mason, 2005;Zazkis, Liljedahl & Chernoff, 2008;Zazkis & Leikin, 2008).…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…Zandieh and Rasmussen (2010) and Zaslavsky and Shir (2005) have shown that college students and high-achieving upper high school students can engage in discussions about desirable properties of definitions and do seem to understand that the goal is to decide on a definition that covers all possible cases. However, in both cases this work was conducted in a collaborative learning situation, so it does not speak to the behaviour of individuals when asked to classify objects without discussion, nor to the issue of whether classification is consistent in the absence of a definition.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Classification and Concept Consistency

Alcock

Simpson

2011

Canadian Journal of Science, Mathematics and Technology Educati

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This paper investigates the extent to which undergraduates consistently use a single mechanism as a basis for classifying mathematical objects. We argue that the concept image/concept definition distinction focuses on whether students use an accepted definition, but does not necessarily capture the more basic notion that there should be a fixed basis for classification. We examine students' classification of real sequences before and after exposure to definitions of 'increasing' and 'decreasing'; we develop an abductive 'plausible explanations' method to estimate the consistency within the participants' responses and suggest that this provides evidence that many students may lack what we call 'concept consistency'.Running head: Classification and Concept Consistency

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“…A descriptive approach to defining a circle with 12 to 13-year-old students via modelling using Play-Doh was reported by Neel-Romine, Paul and Shafer (2012), involving students in writing and critiquing possible definitions for a circle. Zandieh and Rasmussen (2010) effectively used the ideas of concept image and concept definition in a basically descriptive (a posteriori) way to explore the meaning and definition of a triangle with undergraduate students by considering extreme cases (e.g., vertices in straight line) as well as contrasting planar triangles with spherical ones. The comparative approach to teaching geometry developed by István Lénárt similarly involves the simultaneous, initial introduction of geometric concepts such as lines, triangles, angle, etc.…”

Section: Understanding the Process Of Defining And The Need For Definmentioning

confidence: 99%

Recent research on geometry education: an ICME-13 survey team report

Sinclair

Bartolini

Villiers

et al. 2016

ZDM Mathematics Education

125

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This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

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Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

Cited by 94 publications

References 19 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

Classification and Concept Consistency

Recent research on geometry education: an ICME-13 survey team report

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