Basics of qualitative research: Grounded theory procedures and techniques

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Using Multivariate Statistics

Emotion and Adaptation

미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

The Social Psychology of Groups. J. W Thibaut & H. H. Kelley. New York: alley, 1959. The team of Thibaut and Kelley goes back to 1946 when, after serving in different units of the armed services psychology program, the authors joined the Research Center for Group Dynamics, first at M.LT and then at the University of Michigan. Their continued association eventuated in appointments as fellows at the Center for Advanced Study in the Behavioral Sciences, 19561957. It is during these years that their collaboration resulted in the publication of The Social Psychology of Groups. The book was designed to "bring order and coherence to present-day research in interpersonal relations and group functioning." To accomplish this aim, the authors introduced, defined, and illustrated basic concepts in an effort to explain the simplest of social phenomena, the two-person relationship. These basic principles and concepts were then employed to illuminate larger problems and more complex social relationships and to examine the significance of such concepts as roles, norm, power, group cohesiveness, and status. The lasting legacy of this book is derived from the fact that the concepts and principles discussed therein serve as a foundation for one of the dominant conceptual frameworks in the field of family studies today-the social exchange framework. Specifically, much of our contemporary thinking about the process of interpersonal attraction and about how individuals evaluate their close relationships has been influenced by the theory and concepts introduced in The Social Psychology of Groups. Today, as a result of Thibaut and Kelley, we think of interpersonal attraction as resulting from the unique valence of driving and restraining forces, rewards and costs, subjectively thought to be available from a specific relationship and its competing alternatives. We understand, as well, that relationships are evaluated through complex and subjectively based comparative processes. As a result, when we think about assessing the degree to which individuals are satisfied with their relationships, we take into consideration the fact that individuals differ in terms of the importance they attribute to different aspects of a relationship (e.g., financial security, sexual fulfillment, companionship). We also take into consideration the fact that individuals differ in terms of the levels of rewards and costs that they believe are realistically obtainable and deserved from a relationship. In addition, as a result of Thibaut and Kelley's theoretical focus on the concept of dependence and the interrelationship between attraction and dependence, there has evolved within the field of family studies a deeper appreciation for the complexities and variability found within relationships. Individuals are dependent on their relationships, according to Thibaut and Kelley, when the outcomes derived from the existing relationship exceed those perceived to be available in competing alternatives. Individuals who are highly dependent on their relationships are less likely to act to end their relationships. This dependence and the stability it engenders may or may not be voluntary, depending on the degree to which individuals are attracted to and satisfied with their relationships. When individuals are both attracted to and dependent on their relationships, they can be thought of as voluntarily participating in their relationship. That is, they are likely to commit themselves to the partner and relationship and actively work for its continuance. Thibaut and Kelley termed those relationships characterized by low levels of satisfaction and high levels of dependence "nonvoluntary relationships. …

The Social Psychology of Groups

The ability to construct proofs is an important skill for all mathematicians. Despite its importance, students have great difficulty with this task. In this paper, I first demonstrate that undergraduates often are aware of and able to apply the facts required to prove a statement but still fail to prove it. They thus fail to construct a proof because they could not use the syntactic knowledge that they had. By comparing doctoral students and undergraduates constructing proofs in abstract algebra, I have hypothesized four types of `strategic knowledge' – knowledge of how to choose which facts and theorems to apply – which the doctoral students appeared to possess and undergraduates did not. The doctoral students appeared to know the powerful proof techniques in abstract algebra, which theorems are most important, when particular facts and theorems are likely to be useful, and when one should or should not try and prove theorems using symbol manipulation.

Student difficulty in constructing proofs: The need for strategic knowledge

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.

Semantic and Syntactic Proof Productions

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

Series Editor's Foreword: The Soul of Mathematics, Alan H Schoenfeld Preface List of Contributors Introduction Section I: Theoretical Considerations on the Teaching and Learning of Proof 1 What I Would Like My Students to Already Know About Proof, Reuben Hersh 2 Exploring Relationships Between Disciplinary Knowledge and School Mathematics: Implications For Understanding the Place of Reasoning And Proof in School Mathematics, Daniel Chazan and H Michael Lueke 3 Proving and Knowing In Public: The Nature of Proof in A Classroom, Patricio Herbst and Nicolas Balacheff Section II: Teaching and Learning of Proof in the Elementary Grades 4 Representation-based Proof in the Elementary Grades, Deborah Schifter 5 Representations that Enable Children To Engage in Deductive Argument, Anne K Morris 6 Young Mathematicians At Work: The Role of Contexts And Models in the Emergence of Proof, Catherine Twomey Fosnot and Bill Jacob 7 Children's Reasoning: Discovering the Idea of Mathematical Proof, Carolyn A Maher 8 Aspects of Teaching Proving In Upper Elementary School, David A Reid and Vicki Zack Section III: Teaching and Learning of Proof in Middle Grades and High School 9 Middle School Students' Production of Mathematical Justifications, Eric J Knuth, Jeffrey M Choppin and Kristen N Bieda 10 From Empirical to Structural Reasoning in Mathematics: Tracking Changes Over Time, Dietmar Kuchemann and Celia Hoyles 11 Developing Argumentation and Proof Competencies in the Mathematics Classroom, Aiso Heinze and Kristina Reiss 12 Formal Proof in High School Geometry: Student Perceptions of Structure, Validity And Purpose, Sharon M Soucy McCrone and Tami S Martin 13 When is an Argument Just An Argument? The Refinement of Mathematical Argumentation, Kay McClain 14 Reasoning-and-Proving in School Mathematics: The Case of Pattern Identification, Gabriel J Stylianides and Edward A Silver 15 "Doing Proofs" in Geometry Classrooms, Patricio Herbst, Chialing Chen, Michael Weiss, and Gloriana Gonzalez, with Talli Nachlieli, Maria Hamlin and Catherine Brach Section IV: Teaching and Learning of Proof in College 16 College Instructors' Views of Students Vis-a-Vis Proof, Guershon Harel and Larry Sowder 17 Understanding Instructional Scaffolding in Classroom Discourse on Proof, Maria L Blanton, Despina A Stylianou and M Manuela David 18 Building a Community of Inquiry in a Problem-Based Undergraduate Number Theory Course: The Role of the Instructor, Jennifer Christian Smith, Stephanie Ryan Nichols, Sera Yoo and Kurt Oehler 19 Proof in Advanced Mathematics Classes: Semantic and Syntactic Reasoning in the Representation System of Proof, Keith Weber and Lara Alcock 20 Teaching Proving by Coordinating Aspects of Proofs with Students' Abilities, John Selden and Annie Selden 21 Current Contributions toward Comprehensive Perspectives on the Learning and Teaching of Proof, Guershon Harel & Evan Fuller References Index

Teaching and learning proof across the grades : a K-16 perspective

The purpose of this article is to investigate the mathematical practice of proof validation�that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and example-based reasoning.

How Mathematicians Determine if an Argument Is a Valid Proof

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