The Evolution of Students' Understanding of Mathematical Induction PDF Download
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Author: Stacy A. Brown
Publisher:
ISBN:
Category : Induction (Mathematics)
Languages : en
Pages : 846
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Book Description
This dissertation examines how students' understandings of proof by mathematical induction evolved during an 8-week teaching experiment. The design of the experiment was informed by a theoretical perspective that is a synthesis of two complementary theories: the Theory of Didactical Situations (Brousseau, 1997) and the Necessity Principle, Harel's (1998) theory of intellectual need. This study provides an account of how the proof schemes and ways of understanding of a cohort of students progressed through three stages: pre-transformational, restrictive transformational, and transformational, as they worked through a series of proof by mathematical induction appropriate tasks. It also reports on the various didactical and epistemological obstacles the students encountered at each stage. Harel's (1998) Dual Assertion and Harel and Sowder's (1998) proof schemes are used to explain the students' ways of acting in terms of two coexisting schemes, the students' ways of thinking and ways of understanding. The results of the study indicate that the students' conceptions of what constitutes a convincing argument changed in response to a series of shifts in the students' understandings of generality.
Author: Stacy A. Brown
Publisher:
ISBN:
Category : Induction (Mathematics)
Languages : en
Pages : 846
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Book Description
This dissertation examines how students' understandings of proof by mathematical induction evolved during an 8-week teaching experiment. The design of the experiment was informed by a theoretical perspective that is a synthesis of two complementary theories: the Theory of Didactical Situations (Brousseau, 1997) and the Necessity Principle, Harel's (1998) theory of intellectual need. This study provides an account of how the proof schemes and ways of understanding of a cohort of students progressed through three stages: pre-transformational, restrictive transformational, and transformational, as they worked through a series of proof by mathematical induction appropriate tasks. It also reports on the various didactical and epistemological obstacles the students encountered at each stage. Harel's (1998) Dual Assertion and Harel and Sowder's (1998) proof schemes are used to explain the students' ways of acting in terms of two coexisting schemes, the students' ways of thinking and ways of understanding. The results of the study indicate that the students' conceptions of what constitutes a convincing argument changed in response to a series of shifts in the students' understandings of generality.
Author: Etta Mae Whitton
Publisher:
ISBN:
Category : Algebra
Languages : en
Pages : 498
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Book Description
Author: YEE-HO GENTHEW. LEUNG
Publisher:
ISBN: 9781361079768
Category :
Languages : en
Pages :
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Book Description
This dissertation, "An Evaluation of a Teaching Approach to Improve Students' Understanding of Mathematical Induction" by Yee-ho, Genthew, Leung, 梁以豪, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: ABSTRACT "Mathematical Induction" is not a new mathematics topic in Hong Kong secondary school Mathematics curriculum. Dubinsky & Lewin (1986), (Dubinsky 1986, 1989) provided a theoretically based instructional method, using computer experiences, in teaching MI. These three papers was a beginning of a theory for teaching MI based on the genetic epistemology of Jean Piaget. Dubinsky & Lewin (1986) worked out a genetic decomposition of MI to describe the required construction process for learning it. Dubinsky (1986,1989) designed and improved an instructional method for teaching MI which based on students' computer experiences. He claimed that the method is well developed and it can be used for teaching. Although this instructional method is remarkable, it entirely depends on students' computer experience. Can we have a similar instructional method which is independent of computer experience? Movshovitz-Hadar (1993) posted ten classroom activities to enhance students' understanding in MI. I found some of these activities are suitable for formulating an instructional method. I decided to take some of these 10 activities and some self-designed activities to form a new instructional method. The main purpose of the study is to do an evaluation on this computer independent teaching approach. The overall performance on the evaluations suggests that this teaching approach was effective in bringing students to the concept of Mathematical Induction. I hope that it could be developed until we can use it in any ordinary classroom even ii where there are no computer facilities. Also, there were various examples of reflective abstraction in the interview. Students built up the concept from one schema to another. It reflected that the genetic decomposition proposed by Dubinsky & Lewin (1986) is a reasonable way to meditate the construct of Mathematical Induction. iii DOI: 10.5353/th_b3551612 Subjects: Induction (Mathematics) - Study and teaching (Secondary) - China - Hong Kong
Author: Marilyn Paula Carlson
Publisher: MAA
ISBN: 9780883851838
Category : Mathematics
Languages : en
Pages : 340
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Book Description
The chapters in this volume convey insights from mathematics education research that have direct implications for anyone interested in improving teaching and learning in undergraduate mathematics. This synthesis of research on learning and teaching mathematics provides relevant information for any math department or individual faculty member who is working to improve introductory proof courses, the longitudinal coherence of precalculus through differential equations, students' mathematical thinking and problem-solving abilities, and students' understanding of fundamental ideas such as variable and rate of change. Other chapters include information about programs that have been successful in supporting students' continued study of mathematics. The authors provide many examples and ideas to help the reader infuse the knowledge from mathematics education research into mathematics teaching practice. University mathematicians and community college faculty spend much of their time engaged in work to improve their teaching. Frequently, they are left to their own experiences and informal conversations with colleagues to develop new approaches to support student learning and their continuation in mathematics. Over the past 30 years, research in undergraduate mathematics education has produced knowledge about the development of mathematical understandings and models for supporting students' mathematical learning. Currently, very little of this knowledge is affecting teaching practice. We hope that this volume will open a meaningful dialogue between researchers and practitioners toward the goal of realizing improvements in undergraduate mathematics curriculum and instruction.
Author: Gila Hanna
Publisher: Springer Science & Business Media
ISBN: 9400721293
Category : Education
Languages : en
Pages : 468
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Book Description
*THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.
Author: Roza Leikin
Publisher: Springer Science & Business Media
ISBN: 9048139902
Category : Science
Languages : en
Pages : 300
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Book Description
The idea of teachers Learning through Teaching (LTT) – when presented to a naïve bystander – appears as an oxymoron. Are we not supposed to learn before we teach? After all, under the usual circumstances, learning is the task for those who are being taught, not of those who teach. However, this book is about the learning of teachers, not the learning of students. It is an ancient wisdom that the best way to “truly learn” something is to teach it to others. Nevertheless, once a teacher has taught a particular topic or concept and, consequently, “truly learned” it, what is left for this teacher to learn? As evident in this book, the experience of teaching presents teachers with an exciting opp- tunity for learning throughout their entire career. This means acquiring a “better” understanding of what is being taught, and, moreover, learning a variety of new things. What these new things may be and how they are learned is addressed in the collection of chapters in this volume. LTT is acknowledged by multiple researchers and mathematics educators. In the rst chapter, Leikin and Zazkis review literature that recognizes this phenomenon and stress that only a small number of studies attend systematically to LTT p- cesses. The authors in this volume purposefully analyze the teaching of mathematics as a source for teachers’ own learning.
Author: Dr. Matthew Yip
Publisher: Mathewmatician
ISBN:
Category : Education
Languages : en
Pages : 4
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Book Description
Author: Theodore A. Sundstrom
Publisher: Prentice Hall
ISBN: 9780131877184
Category : Logic, Symbolic and mathematical
Languages : en
Pages : 0
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Book Description
Focusing on the formal development of mathematics, this book shows readers how to read, understand, write, and construct mathematical proofs.Uses elementary number theory and congruence arithmetic throughout. Focuses on writing in mathematics. Reviews prior mathematical work with “Preview Activities” at the start of each section. Includes “Activities” throughout that relate to the material contained in each section. Focuses on Congruence Notation and Elementary Number Theorythroughout.For professionals in the sciences or engineering who need to brush up on their advanced mathematics skills. Mathematical Reasoning: Writing and Proof, 2/E Theodore Sundstrom
Author: Charles Warren Nelson
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 370
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Book Description
Author: Richard A. Lesh
Publisher: Routledge
ISBN: 1000149501
Category : Education
Languages : en
Pages : 437
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Book Description
The central question addressed in Foundations for the Future in Mathematics Education is this: What kind of understandings and abilities should be emphasized to decrease mismatches between the narrow band of mathematical understandings and abilities that are emphasized in mathematics classrooms and tests, and those that are needed for success beyond school in the 21st century? This is an urgent question. In fields ranging from aeronautical engineering to agriculture, and from biotechnologies to business administration, outside advisors to future-oriented university programs increasingly emphasize the fact that, beyond school, the nature of problem-solving activities has changed dramatically during the past twenty years, as powerful tools for computation, conceptualization, and communication have led to fundamental changes in the levels and types of mathematical understandings and abilities that are needed for success in such fields. For K-12 students and teachers, questions about the changing nature of mathematics (and mathematical thinking beyond school) might be rephrased to ask: If the goal is to create a mathematics curriculum that will be adequate to prepare students for informed citizenship—as well as preparing them for career opportunities in learning organizations, in knowledge economies, in an age of increasing globalization—how should traditional conceptions of the 3Rs be extended or reconceived? Overall, this book suggests that it is not enough to simply make incremental changes in the existing curriculum whose traditions developed out of the needs of industrial societies. The authors, beyond simply stating conclusions from their research, use results from it to describe promising directions for a research agenda related to this question. The volume is organized in three sections: *Part I focuses on naturalistic observations aimed at clarifying what kind of “mathematical thinking” people really do when they are engaged in “real life” problem solving or decision making situations beyond school. *Part II shifts attention toward changes that have occurred in kinds of elementary-but-powerful mathematical concepts, topics, and tools that have evolved recently—and that could replace past notions of “basics” by providing new foundations for the future. This section also initiates discussions about what it means to “understand” the preceding ideas and abilities. *Part III extends these discussions about meaning and understanding—and emphasizes teaching experiments aimed at investigating how instructional activities can be designed to facilitate the development of the preceding ideas and abilities. Foundations for the Future in Mathematics Education is an essential reference for researchers, curriculum developers, assessment experts, and teacher educators across the fields of mathematics and science education.
