An Unexplored Aspect of Following a Rule by Rachael Elizabeth Driver
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On Mathematical Problem Posing*
On Mathematical Problem Posing* EDWARD A. SILVER In mathematics classes at all levels of schooling in all itself an exercise in exploring, conjecturing, examining, and countries of the world, students can be observed solving testing-all aspects of problem solving .. 1 asks should be problems .. The quality and authenticity of these mathemat created and presented that are accessible to students and ics problems has been the subject of many discussions and ~xtend their knowledge of mathematics and problem solv debates in recent years. Much of this attention has resulted ing. Students should be given opportunities to formulate in a rich, more diverse collection of problems being incor problems from given situations and create new problems by porated into school mathematics curricula. Although the modifying the conditions of a given problem [NCIM, problems themselves have received much scrutiny, less 1991, p. 95] attention has been paid to diversifying the sources for the Despite this interest, however, there is no coherent, com problems that students are asked to consider in school Stu prehensive account of problem posing as a prut of mathe dents are almost always asked to solve only the problems matics curriculum and instruction nor has there been sys that have been presented by a teacher or a textbook Stu tematic research on mathematical problem posing [Kil dents are rarely, if ever, given opportunities to pose in patrick, 1987] For the past several years, I have been some public way their own mathematics problems. Tradi working with colleagues and students on a number of tional transmission/reception models of mathematics investigations into various aspects of problem posing . -
Ludwig.Wittgenstein.-.Philosophical.Investigations.Pdf
PHILOSOPHICAL INVESTIGATIONS By LUDWIG WITTGENSTEIN Translated by G. E. M. ANSCOMBE BASIL BLACKWELL TRANSLATOR'S NOTE Copyright © Basil Blackwell Ltd 1958 MY acknowledgments are due to the following, who either checked First published 1953 Second edition 1958 the translation or allowed me to consult them about German and Reprint of English text alone 1963 Austrian usage or read the translation through and helped me to Third edition of English and German text with index 1967 improve the English: Mr. R. Rhees, Professor G. H. von Wright, Reprint of English text with index 1968, 1972, 1974, 1976, 1978, Mr. P. Geach, Mr. G. Kreisel, Miss L. Labowsky, Mr. D. Paul, Miss I. 1981, 1986 Murdoch. Basil Blackwell Ltd 108 Cowley Road, Oxford, OX4 1JF, UK All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of this publication may be NOTE TO SECOND EDITION reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or THE text has been revised for the new edition. A large number of otherwise, without the prior permission of the publisher. small changes have been made in the English text. The following passages have been significantly altered: Except in the United States of America, this book is sold to the In Part I: §§ 108, 109, 116, 189, 193, 251, 284, 352, 360, 393,418, condition that it shall not, by way of trade or otherwise, be lent, re- 426, 442, 456, 493, 520, 556, 582, 591, 644, 690, 692. -
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Advances in Social Science, Education and Humanities Research, volume 104 2nd Annual International Seminar on Transformative Education and Educational Leadership (AISTEEL 2017) Development of Learning Devices Oriented Problem Based Learning to Increase Student’s Combinatorial Thinking in Mathematical Problem Solving Ability Ammamiarihta Department of Mathematics Education Universitas Negeri Medan Medan, Indonesia Corresponding Email: [email protected] Edi Syahputra Department of Mathematics Education Universitas Negeri Medan Medan, Indonesia Edy Surya Department of Mathematics Education Universitas Negeri Medan Medan, Indonesia Abstract–This research study is research and students, and learn the initial knowledge of students, all this development learning devices. This study aimed to describe will unravel its implementation in the learning device[1]. how the validity, practically, and effectiveness of learning Learning devices should not only provide materials devices oriented of problem based learning which is developed instantly, but be able to lead students to the ability to and knowing about increase students’ combinatorial thinking understand learned concepts. It aims to determine the extent in mathematical problem solving ability after using learning to which learning devices have been presented, what devices which is developed. The product that produce in this indicators to be achieved, to how the follow-up will be done study is lesson plan, handbook’s teacher, student’s book, and by the teacher. worksheet. Learning devices development using 4D model which developed by Thiagarajan, Semmel and Semmel with In the development of quality learning devices need four step, that is define, design, develop and disseminate. This an assessment of products developed. In order to make the study was conducted in two trials in two different class . -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Wittgenstein on Rules and Private Language
SAUL A. KRIPKE Wittgenstein on Rules and Private Language An Elementary Exposition Harvard University Press Cambridge, Massachusetts Contents Copyright © 1982 by Saul A. Kripke All rights reserved Preface Vll EIGHTH PRINTING, 1995 1 Introductory 1 Printed in the United States of America 2 The Wittgensteinian Paradox 7 3 The Solution and the 'Private Language' Argument 55 Postscript Wittgenstein and Other Minds Library ofCongress Cataloging in Publication Data 114 Index Kripke, Saul A., 1940- 147 Wittgenstein on rules and private language. Includes bibliographical references and index. Wittgenstein, Ludwig, 1889-1951. I. Title B3376.W564K74 192 81-20070 AACR2 ISBN 0-674-95401-7 (paper) - To my parents Preface The main part ofthis work has been delivered at various places as lectures, series oflectures, or seminars. It constitutes, as I say, 'an elementary exposition' ofwhat I take to be the central thread of Wittgenstein's later work on the philosophy of language and the philosophy of mathematics, including my interpretation of the 'private language argument', which on my view is principally to be explicated in terms ofthe problem of 'following a rule'. A postscript presents another problem Wittgenstein saw in the conception ofprivate language, which leads to a discussion of some aspects of his views on the problem ofother minds. Since I stress the strong connection in Wittgenstein's later philosophy between the philosophy of psychology and the philosophy of mathematics, I had hoped to add a second postscript on the philosophy ofmathematics. Time has not permitted this, so for the moment the basic remarks on philosophy ofmathematics in the main text must suffice. -
Problem Posing As a Means for Developing Mathematical
PROBLEM POSING AS A MEANS FOR DEVELOPING MATHEMATICAL KNOWLEDGE OF PROSPECTIVE TEACHERS Ilana Lavy and Atara Shriki Emek Yezreel Academic College / Oranim Academic College of Education Abstract In the present study we aim at exploring the development of mathematical knowledge and problem solving skills of prospective teachers as result of their engagement in problem posing activity. Data was collected through the prospective teachers’ reflective portfolios and weekly class discussions. Analysis of the data shows that the prospective teachers developed their ability to examine definition and attributes of mathematical objects, connections among mathematical objects, and validity of an argument. However, they tend to focus on common posed problems, being afraid of their inability to prove their findings. This finding suggests that overemphasizing the importance of providing a formal proof prevents the development of inquiry abilities. Introduction Problem posing (PP) is recognized as an important component of mathematics teaching and learning (NCTM, 2000). In order that teachers will gain the knowledge and the required confidence for incorporating PP activities in their classes, they have to experience it first. While experiencing PP they will acknowledge its various benefits. Hence such an experience should start by the time these teachers are being qualified towards their profession. Therefore, while working with prospective teachers (PT) we integrate activities of PP into their method courses. In addition, accompanying the process with reflective writing might make the PT be more aware to the processes they are going through (Campbell et al, 1997), and as a result increase the plausibility that the PT will internalize the effect of the processes that are involved in PP activities. -
Problem Solving Strategies Students Use When Solving
PROBLEM SOLVING STRATEGIES STUDENTS USE WHEN SOLVING COMBINATORIAL PROBLEMS by GARY YUEN A THESIS SUBMITTED iN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Mathematics Education) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2008 © Gary Yuen, 2008 ABSTRACT This research is a case study that examines the strategies that three grade 11 students use to manoeuvre through a series of three combinatorial problems. Grade 11 students were chosen as participants because they have had no formal training in solving this class of math problems. Data includes video recordings of each participant’s problem solving sessions along with each participant’s written work. Through analysis of this data, several themes related to problem solving strategies were identified. First, students tend to rely on algebraic representation and methods as they approach a problem. Second, students use the term “guess and check” to describe any strategy where the steps to a solution are not clearly defined. Thirdly, as students negotiate problems, they tend to search for patterns that will streamline their methods. Fourthly, students approach complicated problems by breaking up the problem into smaller parts. Finally, students who verify their work throughout the problems solving process tend to experience more success than those who do not. From these findings, I suggest that mathematics teachers need to ensure that they are not over-emphasizing algebraic strategies in the classroom. In addition, students need to be given the opportunity to explore various solution strategies to a given problem. Finally, students should be taught how to verify their work, and be encouraged to perform this step throughout the problem solving process. -
Karl Popper and the Philosophy of Mathematics Proceedings of the Conference Held in Klagenfurt, 5 – 7 April, 2018
Symposium Karl Popper and the Philosophy of Mathematics Proceedings of the Conference held in Klagenfurt, 5 – 7 April, 2018 Edited by Reinhard Neck Alpen-Adria-Universität Klagenfurt 2018 Contents Preface ................................................................................................................ v Programme ...................................................................................................... vii Abstracts and Preliminary Papers .................................................................. 1 Schroeder-Heister, Peter Popper on deductive logic and logical education ...................................................... 2 Binder, David A Critical Edition of Popper's Work on Logic .......................................................... 3 Brîncuş, Constantin and Toader, Iulian Non-normal Interpretations of Positive Logic .......................................................... 8 Pimbé, Daniel Popper and “absolute proofs” ................................................................................. 12 Albert, Max Critical Rationalism and Decision Theory .............................................................. 25 Del Santo, Flavio The physical motivations for a propensity interpretation of probability ................. 26 Afisi, Oseni Taiwo Prospensity Probability and Its Applications of Knowledge in Ifa ......................... 32 Miller, David Independence (Probabilistic) and Independence (Logical) ..................................... 36 Burgoyne, Bernard From cosmic paths to psychic -
The Influence of Teaching Mathematical Problem Solving Strategies on Students’ Attitudes in Middle School
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2012 Mathematic Strategies For Teaching Problem Solving: The Influence Of eachingT Mathematical Problem Solving Strategies On Students' Attitudes In Middle School Kelly Lynn Klingler University of Central Florida Part of the Science and Mathematics Education Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Klingler, Kelly Lynn, "Mathematic Strategies For Teaching Problem Solving: The Influence Of eachingT Mathematical Problem Solving Strategies On Students' Attitudes In Middle School" (2012). Electronic Theses and Dissertations, 2004-2019. 2316. https://stars.library.ucf.edu/etd/2316 MATHEMATIC STRATEGIES FOR TEACHING PROBLEM SOLVING: THE INFLUENCE OF TEACHING MATHEMATICAL PROBLEM SOLVING STRATEGIES ON STUDENTS’ ATTITUDES IN MIDDLE SCHOOL by KELLY LYNN KLINGLER B.A. Mount Vernon Nazarene University, 2006 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Education in the School of Teaching, Learning and Leadership in the College of Education at the University of Central Florida Orlando, Florida Spring Term 2012 © 2012 Kelly L. Klingler ii ABSTRACT The purpose of this action research study was to observe the influence of teaching mathematical problem solving strategies on students’ attitudes in middle school. The goal was to teach five problem solving strategies: Drawing Pictures, Making a Chart or Table, Looking for a Pattern, Working Backwards, and Guess and Check, and have students reflect upon the process. -
HISTORY-BASED PROBLEMS in the TEACHING of SENIOR HIGH SCHOOL Mathematics in Mainland China Xiaoqin Wang
HISTORY-BASED PROBLEMS IN THE TEACHING OF SENIOR HIGH SCHOOL Mathematics in Mainland China Xiaoqin Wang To cite this version: Xiaoqin Wang. HISTORY-BASED PROBLEMS IN THE TEACHING OF SENIOR HIGH SCHOOL Mathematics in Mainland China. History and Pedagogy of Mathematics, Jul 2016, Montpellier, France. hal-01349273 HAL Id: hal-01349273 https://hal.archives-ouvertes.fr/hal-01349273 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HISTORY-BASED PROBLEMS IN THE TEACHING OF SENIOR HIGH SCHOOL Mathematics in Mainland China Xiaoqin Wang Department of Mathematics East China Normal University, Shanghai 200241 汪晓勤 [email protected] ABSTRACT This paper deals with the integration of history in mathematics teaching. Four possible approaches are considered: complementation, replication, accommodation, reconstruction. The main focus is on the relation between. mathematical problems posed on the basis of historical information (history-based problems) and mathematics teaching from the HPM perspective. In this paper 20 HPM lessons are analysed. 1 Introduction Integrating the history of mathematics into mathematics teaching is one of the important research fields in HPM. As shown in table 1, four ways are identified in which historical information is used in classroom teaching in Mainland China. -
Philosophy of Mathematics, Mathematics Education, and Philosophy of Mathematics Education Yuxin Zheng Nanjing University
Humanistic Mathematics Network Journal Issue 9 Article 9 2-1-1994 Philosophy of Mathematics, Mathematics Education, and Philosophy of Mathematics Education Yuxin Zheng Nanjing University Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Logic and Foundations of Mathematics Commons, Mathematics Commons, Scholarship of Teaching and Learning Commons, and the Science and Mathematics Education Commons Recommended Citation Zheng, Yuxin (1994) "Philosophy of Mathematics, Mathematics Education, and Philosophy of Mathematics Education," Humanistic Mathematics Network Journal: Iss. 9, Article 9. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss9/9 This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Philosophy of Mathematics, Mathematics Education and Philosophy of Mathematics Education Zheng Yuxin (Y. Zheng) Department of Philosophy Nanling University. China (P. R. C.) As a philosopher of mathematics. I have been the great infl uence of the philosophy of thinking about, or rather, worried about the mathematics on mathematics education. The third follow ing qu esti on: is there any important section discusses the problem of how to develop rel ation ship between the philosophy of the subject 'philosophy of mathematics education , mathematics and actual mathematical activities which in fact can be regarded as an impetus from (including mathematical research, teaching and mathematics education to the further development learning)? Or, does the philosophy of mathematics of philosophy of mathematics and philosophy in have any important infl uence on act ual general. -
Platonism in Modern Mathematics
PLATONISM IN MODERN MATHEMATICS A University Thesis Presented to the Faculty of California State University, East Bay In Partial Fulfillment ofthe Requirements for the Degree Master of Science in Interdisciplinary Studies By Ramal Lamar October, 2012 PLATONISM IN MODERN MATHEMATICS By Ramal Lamar Date: INTRODUCTION In this thesis we investigate the existence of Platonic ideas in the concepts and foundations of modern mathematics. Platonism refers to the viewpoint that the objects and entities constructed and defined in the work of mathematics actually exist independent of our sense preception. In the first chapter, we look at how Platonism itself has been interpreted differently by modern thinkers and how this leads to confusion at times as to what is actually Platonism. Therefore by articulating specific ideas constructed by Plato and showing their analogs in today’s mathematical notation, we can speak of ‘mathematical Platonism.’ For example, numbers are not just figments of the imagination; there is a symbol known as one to refer to the concept of ‘oneness.’ Of course, philosophers disagree as to what constitutes Platonism, whether Platonism, also known as realism, is necessary in order to obtain truth and to solve mathematical problems with implications in the applied areas of science and technology. Also in this thesis, we describe the various schools of mathematical thought. Each school tackles its own philosophical problem and we highlight some of the major contributions associated with each school. In the second chapter, we survey a recent history of mathematics by exploring the five foundations of mathematics. The philosophical program of the logicist school is to ground mathematics proper in the rules of logic.