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Understanding Mathematical Discourse
 DIALOGUE. AMSTERDAM UNIVERSITY
, 1999
"... Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers ..."
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Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers a welldefined set of discourse relations and forces/allows us to apply mathematical reasoning. We give a brief discussion on selected linguistic phenomena of mathematical discourse, and an analysis from the mathematician’s point of view. Requirements for a theory of discourse representation are given, followed by a discussion of proofs plans that provide necessary context and structure. A large part of semantics construction is defined in terms of proof plan recognition and instantiation by matching and attaching.
Secondarylevel Student Teachers ’ Conceptions of Mathematical Proof
"... Recent reforms in mathematics education have led to an increased emphasis on proof and reasoning in mathematics curricula. The National Council of Teachers of Mathematics highlights the important role that teachers ’ knowledge and beliefs play in shaping students ’ understanding of mathematics, thei ..."
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Recent reforms in mathematics education have led to an increased emphasis on proof and reasoning in mathematics curricula. The National Council of Teachers of Mathematics highlights the important role that teachers ’ knowledge and beliefs play in shaping students ’ understanding of mathematics, their confidence in and outlook on mathematics education, and their ability to use math to solve fundamental problems. It is crucial that teachers, especially the uninitiated, understand on a deep level the mathematical concepts that they are expected to teach to adolescents. Thus, it becomes critical for teacher educators to assess the understanding and abilities of student teachers in constructing mathematical proof. The analysis in this study is based on three factors: 1) meaning of proof, 2) ideas about teaching methods on proof, and 3) ideas about the usefullness of proof in a mathematics classroom. An analysis of the data collected from this study indicates that current student teachers ’ conceptions of mathematical proof are limited. The uneasiness expressed by student teachers about mathematical proof may suggest an examination of students ’ experiences with the mathematical proof in both secondary and post secondary classrooms.
Mathematically Gifted Students' Geometrical Reasoning and Informal Proof
 In Helen
, 2005
"... This paper provides an analysis of a teaching experiment designed to foster students' geometrical reasoning and verification in small group. The purposes of the teaching experiment in this paper were to characterize gifted students ' proof constructions and to contribute to the theoretical ..."
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This paper provides an analysis of a teaching experiment designed to foster students' geometrical reasoning and verification in small group. The purposes of the teaching experiment in this paper were to characterize gifted students ' proof constructions and to contribute to the theoretical body of knowledge about gifted students' mathematical thinking. The experiment was conducted as part of education for gifted sixthgraders (12 years of age). The analysis of the students ' responses in this paper documents the evolution of the students ' proving ability as they participated in activities from an instructional sequence designed to support geometrical reasoning. Three types of reasoning (pragmatic, semantic, intellectual) and creative informal proofs were identified in the analysis. In order for mathematically gifted students to develop their proving ability, teachers need to draw explicit attention to the value of informal proofs. Likewise, in order for students to develop their sense of geometrical reasoning, they need a lot of experience in conjecturing, testing, and then verifying in a mathematical way.
9 Informal prerequisites for informal proofs
"... Abstract: Reasoning and proof play an important role in the mathematics classroom. However, prerequisites for the learning of mathematical reasoning and proof, such as logical competence or the understanding of concepts and proofs, are rarely taught explicitly. In an empirical survey with 106 stude ..."
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Abstract: Reasoning and proof play an important role in the mathematics classroom. However, prerequisites for the learning of mathematical reasoning and proof, such as logical competence or the understanding of concepts and proofs, are rarely taught explicitly. In an empirical survey with 106 students in grade 8 we investigated students ’ declarative and methodological knowledge related to some of these prerequisites. The results show that there are certain deficits which make it difficult for students to learn reasoning and proof. Kurzreferat: Beweisen und Begründen sind wichtige Ziele des Mathematikunterrichts. Das Erlernen von mathematischem
Proof and Its Classroom Role: A Survey
"... There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and 1999 the leading journals of mathematics education published over one hundred research papers on this topic. This in itself is an indication that proof remains a prominent issue in mathematics education ..."
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There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and 1999 the leading journals of mathematics education published over one hundred research papers on this topic. This in itself is an indication that proof remains a prominent issue in mathematics education.
ARGUMENTATION, PROOF AND THE UNDERSTANDING OF PROOF
"... Argumentation and proof play an important role in mathematics. In recent years several empirical studies have revealed deficits in students ’ abilities in logical argumentation and in their understanding of mathematical proofs. In an empirical survey with upper secondary students we investigated dif ..."
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Argumentation and proof play an important role in mathematics. In recent years several empirical studies have revealed deficits in students ’ abilities in logical argumentation and in their understanding of mathematical proofs. In an empirical survey with upper secondary students we investigated different components of both mathematical, and general, competence like declarative knowledge, methodological knowledge, metacognition and spatial abilities in relation to students ’ performance for geometrical proof items. The results indicate that these components explain a comparatively high proportion of interindividual differences. 1 Theoretical framework 1.1 Argumentation and proof in the mathematics classroom Logical argumentation, reasoning and proof may be regarded as highly important topics in mathematics. Despite the fact that mathematics may even be regarded as a proving science the role of proof in the school curriculum did not always reflect that importance. In the last few years there was a significant change in the teaching of reasoning and proof in the mathematics classroom. During the lively discussions of the 70s and 80s as to whether proofs should be part of the mathematics curriculum in secondary schools, mathematics educators argued that proving in the classroom had developed into a topic that particularly emphasised formal aspects but disregarded mathematical understanding (Hanna, 1983). In the 90s the situation changed: proof is again regarded as an important topic in the mathematics curriculum (NCTM, 2000) and is an essential aspect of mathematical competence. However proof was not necessarily used as a synonym for formal proof. Several authors like Hanna and Jahnke
TEACHERS ’ CONCEPTIONS OF PROOF IN THE CONTEXT OF
"... ABSTRACT. Current reform efforts in the United States are calling for substantial changes in the nature and role of proof in secondary school mathematics – changes designed to provide all students with rich opportunities and experiences with proof throughout the entire secondary school mathematics c ..."
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ABSTRACT. Current reform efforts in the United States are calling for substantial changes in the nature and role of proof in secondary school mathematics – changes designed to provide all students with rich opportunities and experiences with proof throughout the entire secondary school mathematics curriculum. This study examined 17 experienced secondary school mathematics teachers ’ conceptions of proof from their perspectives as teachers of school mathematics. The results suggest that implementing “proof for all” may be difficult for teachers; teachers viewed proof as appropriate for the mathematics education of a minority of students. The results further suggest that teachers tended to view proof in a pedagogically limited way, namely, as a topic of study rather than as a tool for communicating and studying mathematics. Implications for mathematics teacher education are discussed in light of these findings. KEY WORDS: proof, reform, secondary mathematics, teacher conceptions
ORIGINAL ARTICLE Visualisation and proof: a brief survey of philosophical
, 2006
"... Abstract The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhau ..."
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Abstract The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification. 1 Foreword It is a great honour to be asked to contribute to this special issue in memory of Hans Georg Steiner. He was a friend and mentor who had an enormous impact on the field of mathematics education, as acknowledged in the volume dedicated to him, Didactics of mathematics as a scientific discipline, which appeared in 1994 to mark both his 65th birthday and 20 years of work at the Institut für Didaktik der Mathematik (IDM) in Bielefeld. I had the great pleasure and privilege of meeting