DOCSLIB.ORG

On Mathematics As Sense- Making: an Informalattack on the Unfortunate Divorce of Formal and Informal Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Mathematics As Sense- Making: an Informalattack on the Unfortunate Divorce of Formal and Informal Mathematics On Mathematics as Sense­ Making: An InformalAttack On the Unfortunate Divorce of Formal and Informal 16 Mathematics Alan H. Schoenfeld The University of California-Berkeley This chapter explores the ways that mathematics is understood and used in our culture and the role that schooling plays in shaping those mathematical under­ standings. One of its goals is to blur the boundaries between forma] and informal mathematics: to indicate that, in real mathematical thinking, formal and informal reasoning are deeply intertwined. I begin, however, by briefly putting on the formalist's hat and defining formal reasoning. As any formalist will tell you, it helps to know what the boundaries are before you try to blur them. PROLOGUE: ON THE LIMITS AND PURITY OF FORMAL REASONING QUA FORMAL REASONING To get to the heart of the matter, formal systems do not denote; that is, formal systems in mathematics are not about anything. Formal systems consist of sets of symbols and rules for manipulating them. As long as you play by the rules, the results are valid within the system. But if you try to apply these systems to something from the real world (or something else), you no longer engage in formal reasoning. You are applying the mathematics at that point. Caveat emp­ tor: the end result is only as good as the application process. Perhaps the clearest example is geometry. Insofar as most people are con­ cerned, the plane or solid (Euclidean) geometry studied in secondary school provides a mathematical description of the world they live in and of the way the world must be. For example, it appears patently clear that there is always one and only one line through any given point that is parallel to a given line, that parallels 311 16. MATHEMATICS AS SENSE-MAKING 313 312 SCHOENFELD never intersect, and that the sum of the angles of a triangle is always 180 degrees. (2) Manipulations are performed within the It appears so clear, in fact, that for 2,000 years the world's finest mathematicians A formal ) The Formal System formal system System tried to prove these results. The most intensively explored problem in the history of mathematics was the attempt to show that Euclid's parallel postulate was a logical consequence of the other Euclidean axioms. Yet, as we know, that is not (1) Aspects 01 the (3) The results 01 Real-World the formal 1 the only possibility. There are other perfectly consistent geometries in which Situation are manipulatons are there are either infinitely many parallel lines to a given line through a given point represented in the interpreted in the or none at all. And in these geometries, the measures of the angles of a triangle formal system real-world situation do not add up to 180 degrees. Such non-Euclidean geometries are not simply r mathematical curiosities, bearing no relation to reality. The various geometries are, in fact, differentially useful for characterizing different aspects of our phys­ A Real-World The Real-World Situation -------) ical world. Euclidean geometry serves perfectly well for building bookshelves or Situation laying out a garden on a flat piece of land, but it does not work, for example, for astrophysics; on a celestial scale, space is curved, and non-Euclidean geometries Steps 1, 2, and 3 combined yield an analysis of the real-world situauon. are more useful. That is, neither Euclidean nor non-Euclidean geometry is right Note that the quality of the analysis depends on the mappmgs to and from the formal system. If either one of those mappings is lIawed, the or wrong. Both are logically consistent, and it just happens that each seems to fit analysis IS not valid. certain circumstances better than the other. In the use of any formal mathematical system, the mathematics is valid if it is FIG. 16.1. Using formal systems to interpret real-world situations. internally consistent; its applications to the real world, however, are something else altogether. In a statistical model, for example, if the conditions of an experi­ Following are a series of vignettes that illustrate our culture's general (mis)un­ ~f ment fail to match the assumptions of the statistical tests that are used to analyze derstanding the role. and uses of mathematics. I then speculate, in Essay I, on the data, then, regardless of the correctness of the statistical manipulations used why these rmsconcepnons are fostered by certain common teaching practices in to analyze the data, the statistical analysis is i6'v~Hd. Or in formal logic, consider the typical classroom setting. In a second set of vignettes, I describe some cases the following syllogism, offered by Ray Nickerson (chapter 15, this volume): of successful classroom instruction, and, in concluding Essay 2, surmise why these appro~ches work and how they suggest lines of investigation in developing more effective mathematics learning. Nothing is better than eternal happiness. A ham sandwich is better than nothing. A ham sandwich is better than eternal happiness. FOUR VIGNETTES ON THE MAKINGS OF MATHEMATICAL NONSENSE This reductio ad absurdum shows that the mere use of formally correct syllogistic reasoning is no guarantee of the validity of the conclusion. The whole of the Vignette 1: On the Authoritarian/Totalitarian Nature reasoning process is no better than any of its parts, which include the appropri­ and Uses ofJl.i1athematical Formalism, or Formalismus ateness of the syllogistic form in the first place and the accurateness of the Uber Alles ·W__ translation of the argument into syllogistic form. In general, the quality of reasoning using formal systems depends on the Like many rulers of great nations, Catherine the Great amused herself by inviting quality of the maps from the situation being explored to the formal system, on the people of intellectual stature to visit her court. Among those who visited correctness of the reasoning within the formal system, and on the interpretation Catherine's court was Denis Diderot, French philosopher and author of the famed of results in the formal system that is transferred to the situation being analyzed Encyclopedia. E. T. Bell (1937) described the visit: (see Fig. 16.1). All of these must be valid for the reasoning using formal systems to be valid. The mere use (or abuse) of formal systems as one component of the Much to her dismay, Diderot earned his keep by trying to convert the courtiers to reasoning process guarantees nothing about correctness, just as the mere trap­ atheism. Fed up, Catherine commissioned [the great mathematician Leonhard] pings of the scientific method (i.e" scientism rather than science) guarantee Euler to muzzle the windy philosopher. This was easy because all mathematics was nothing about the scientific correctness of a piece of research. Chinese [!] to Diderot. (p. 146) 16. MATHEMATICS AS SENSE-MAKING 314 SCHOENFELD 315 I~, tiseme~t Quoting from De Morgan's Budget of Paradoxes, Bell continued: are sufficiently intimidating that people will question no further. The ad campaign f?r Extra ~trength Brandnarrie was orieof the Iongesfrillinirig and most Diderot was informed that a learned mathematician was in possession of an al­ s~5c~s~ful In Am:nc~n television history. It could only be successful if people gebraical demonstration of the existence of God, and would give it before all the a~1i1cated responsibility for thinking, took the ad's assertions at face value, and Court, if he desired to hear it. Diderot gladly consented .. did not try to make sense of the situation being described. Euler advanced toward Diderot, and said gravely. and in a tone of perfect I must add that such .abuse of mathematics is hardly limited to the packaging conviction: of objects. It IS used, with comparable effectiveness, for the packaging of politi­ "Sir, (a + bn)/n = x, hence God exists; reply!" cal candl~ates .. Overspending and the budget deficit were major issues in the It sounded like sense to Diderot. Humiliated by the unrestrained laughter which 1980 presidential race. Facts and figures were thrown around in "white papers," greeted his embarrassed silence, the poor man asked Catherine's permission to but the whole matter came to a head in the debates themselves. The Republican return at once to France. She graciously gave it. (pp. 146-147) candld~te, ~om~unicating at his best, responded to a budget question with something like this: Now the economists tell me it works like this. There's a line Commentary. Mathematicsisa weapon of great potential power. For those who that goes. like this (moves his arm in an upward direction, from left to right) and understand the mathematics', ofc()urse:E;Ter's assertion is laughable. FOf th~~~ ano~her line that goes like this (moves his arm in a downward direction, from left who believe that mathematics is coherent and bound by logic, Euler's pronounce­ to nght). When those two Jines cross, we'll have a balanced budget. ment is an assertion to be questioned (and probably to be found laughable). But That put the matter to rest. This completely meaningless assertion was not for those who believe mathematics to be a mystical domain-a source of in, followed up with probing questions, either during the debate or in the media after controvertible wisdom that is accessible only to a' few-ge'niuses-mathematics the d:bate. Unmitigated mathematical gobbledygook went unquestioned, and can be a battering ram that knocks them senseless. Amenca bought the product. (I bought pain reliever.) . ~ere, as in Euler's humiliation of Diderot, is another example of the total­ Vignette 2: Euler Comes to Madison Avenue, or With . Itan.an power".of mathematics.
Load more

Recommended publications
  • Ethnomathematics and Education in Africa
    Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example
    [Show full text]
  • Mathematics in Informal Learning Environments: a Summary of the Literature
    MATHEMATICS IN INFORMAL LEARNING ENVIRONMENTS: A SUMMARY OF THE LITERATURE Scott Pattison, Andee Rubin, Tracey Wright Updated March 2017 Research on mathematical reasoning and learning has long been a central part of the classroom and formal education literature (e.g., National Research Council, 2001, 2005). However, much less attention has been paid to how children and adults engage with and learn about math outside of school, including everyday settings and designed informal learning environments, such as interactive math exhibits in science centers. With the growing recognition of the importance of informal STEM education (National Research Council, 2009, 2015), researchers, educators, and policymakers are paying more attention to how these experiences might support mathematical thinking and learning and contribute to the broader goal of ensuring healthy, sustainable, economically vibrant communities in this increasingly STEM-rich world. To support these efforts, we conducted a literature review of research on mathematical thinking and learning outside the classroom, in everyday settings and designed informal learning environments. This work was part of the NSF-funded Math in the Making project, led by TERC and the Institute for Learning Innovation and designed to advance researchers’ and educators’ understanding of how to highlight and enhance the mathematics in making experiences.1 Recognizing that the successful integration of mathematics and making requires an understanding of how individuals engage with math in these informal learning environments, we gathered and synthesized the informal mathematics education literature, with the hope that findings would support the Math in the Making project and inform the work of mathematics researchers and educators more broadly.
    [Show full text]
  • Effective Strategies for Promoting Informal Mathematics Learning
    Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning Prepared for by Lynn Dierking Michael Dalton Jennifer Bachman Oregon State University 2008 with the generous support of This material is based upon work supported by the National Science Foundation under grant Number DRL-0714634. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the Oregon Museum of Science and Industry. © OREGON MUSEUM OF SCIENCE AND INDUSTRY, September 2011 Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning Prepared for the Oregon Museum of Science and Industry Oregon State University Lynn Dierking * Michael Dalton * Jennifer Bachman Executive Summary The Oregon Museum of Science and Industry (OMSI) received a five- year grant from the National Science Foundation (NSF) to develop the Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning project. Access Algebra will include a traveling exhibition and a comprehensive professional development program for educational staff at host museums. Access Algebra will target middle- school age youth and engage visitors in design activities that promote creativity and innovation and build mathematics literacy by taking a non-traditional, experiential approach to mathematics learning. Oregon State University (OSU) faculty from the College of Education and the College of Science with expertise in mathematics teaching, administrative experience at K–20 levels and within the Oregon Department of Education, and in free-choice learning settings and research are collaborating on research and development for the exhibit prototypes and professional development components of the project.
    [Show full text]
  • The Philosophy of Mathematics Education
    The Philosophy of Mathematics Education The Philosophy of Mathematics Education Paul Ernest © Paul Ernest 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without permission in writing from the copyright holder and the Publisher. First published 1991 This edition published in the Taylor & Francis e-Library, 2004. RoutledgeFalmer is an imprint of the Taylor & Francis Group British Library Cataloguing in Publication Data Ernest, Paul The philosophy of mathematics education. 1. Education. Curriculum subjects: Mathematics. Teaching. I. Title 510.7 ISBN 0-203-49701-5 Master e-book ISBN ISBN 0-203-55790-5 (Adobe eReader Format) ISBN 1-85000-666-0 (Print Edition) ISBN 1-85000-667-9 pbk Library of Congress Cataloging-in-Publication Data is available on request Contents List of Tables and Figures viii Acknowledgments ix Introduction xi Rationale xi The Philosophy of Mathematics Education xii This Book xiv Part 1 The Philosophy of Mathematics 1 1 A Critique of Absolutist Philosophies of Mathematics 3 Introduction 3 The Philosophy of Mathematics 3 The Nature of Mathematical Knowledge 4 The Absolutist View of Mathematical Knowledge 7 The Fallacy of Absolutism 13 The Fallibilist Critique of Absolutism 15 The Fallibilist View 18 Conclusion 20 2 The Philosophy of Mathematics Reconceptualized 23 The Scope of the Philosophy of Mathematics 23 A Further Examination of Philosophical Schools 27 Quasi-empiricism
    [Show full text]
  • Secondary Intermediate Math Endorsement (Algebra I, Geometry, Algebra Ii)
    SECONDARY INTERMEDIATE MATH ENDORSEMENT (ALGEBRA I, GEOMETRY, ALGEBRA II) Preparations that can add this Endorsement: Preparation Pedagogy Major Test Coursework Early Childhood Preparation X AND X OR X Elementary Preparation X AND X OR X Secondary Preparation X OR X CTE Preparation X OR X K-12 Preparation X OR X Early Childhood SPED Preparation X AND X OR X K-12 SPED Preparation X OR X State-Designated Pedagogy Requirements: Test # Test Name Cut Score Effective Date 5624 Principals of Learning and Teaching (7-12 PLT) 157 07/01/2012 Waiver of Pedagogy Test 7-12 PLT The PLT test may be waived if verification of two or more years of state-certified Experience in a 5- teaching experience in the grade span of the endorsement is documented. 12 grade span Major in Content Requirements (27 credits with a minimum GPA of 2.7 in the content): 27 credits with a minimum GPA of 2.7 in the content • 27 credits in Math State-Designated Test Requirements (passage of one test): Test # Test Name Cut Score Effective Date 5169 Middle School Math 165 09/01/2013 5161 Mathematics: Content Knowledge 160 09/01/2013 Coursework Requirements: None July 2019 Intermediate Math Endorsement Page 2 Assignment Codes: Code Assignment Description 02001 Informal Mathematics 02002 General Math 02035 Math, Grade 5 02036 Math, Grade 6 02037 Math, Grade 7 02038 Math, Grade 8 02039 Grades 5-8 Math 02040 Grades 5-8 Pre-Algebra 02051 Pre-Algebra I 02052 Algebra I 02053 Algebra I - Part 1 02054 Algebra I - Part 2 02055 Transition Algebra 02056 Algebra II 02062 Integrated Mathematics
    [Show full text]
  • DOING and SHOWING 1. Introduction 1 Part 1. Euclid's Way of Building Mathematical Theories 3 2. Demonstration
    DOING AND SHOWING ANDREI RODIN 1. Introduction 1 Part 1. Euclid's Way of Building Mathematical Theories 3 2. Demonstration and \Monstration" 3 3. Are Euclid's Proofs Logical? 8 4. Instantiation and Objectivity 11 5. Logical Deduction and Geometrical Production 15 Part 2. Modern Axiomatic Method 25 6. Euclid and Modern Mathematics 25 7. Formalization 27 8. Objectivity and Objecthood in Modern Mathematics 33 9. Formal and Informal Bourbaki 37 10. The Role of Symbolism 41 11. Mathematical Constructivism 47 12. Galilean Science and \Unreasonable Effectiveness of Mathematics" 50 References 53 1. Introduction An axiomatic theory consists of a distinguished set of propositions called axioms and an- other set of propositions called theorems, which are deduced from the axioms according to certain rules of inference. Such rules are supposed to be truth-preserving in the following sense: as far as the axioms are true the theorems derived from these axioms are also true. Further, these rules are supposed to be not specific for any given theory: one assumes that the same set of rules of inference applies in all axiomatic theories (logical monism) or at least that any complete set of such rules applies in some large class of theories (logical pluralism). These basic features of the rules of inference can be briefly expressed by saying that those rules are the rules of logical inference. A logical inference of a given theorem from the axioms (possibly through a number of intermediate propositions) is called a proof of this theorem. Here is how this core notion of axiomatic theory is described by Hilbert in his famous address \Axiomatic Thought" delivered before the Swiss Mathematical Society in Zurich in 1917: 1 2 ANDREI RODIN If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construc- tion of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework.
    [Show full text]
  • A Philosophical Examination of Proofs in Mathematics Eric Almeida
    Undergraduate Review Volume 3 Article 13 2007 A Philosophical Examination of Proofs in Mathematics Eric Almeida Follow this and additional works at: http://vc.bridgew.edu/undergrad_rev Part of the Logic and Foundations Commons, and the Logic and foundations of mathematics Commons Recommended Citation Almeida, Eric (2007). A Philosophical Examination of Proofs in Mathematics. Undergraduate Review, 3, 80-84. Available at: http://vc.bridgew.edu/undergrad_rev/vol3/iss1/13 This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. Copyright © 2007 Eric Almeida 0 A Philosophical Examination of Proofs in Mathematics Eric Almeida Eric Almeida is a Philosophy Major. This “The purpose of a proof is to understand, not verify”-- Arnold Ross. project was mentored by Dr. Catherine n mathematics, a proof is a demonstration that, given certain axioms, some Womack. statement of interest is necessarily true. Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. The distinction has led to much Iexamination of current and historical mathematical practice, quasi-empiricism in mathematics. One of the concerns with the philosophy of mathematics is the role of language and logic in proofs, and mathematics as a language. Regardless of one’s attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final word, and the complete proof shows how it follows from the axioms alone.
    [Show full text]
  • Benacerraf's Dilemma and Informal Mathematics
    THE REVIEW OF SYMBOLIC LOGIC Volume 2, Number 4, December 2009 BENACERRAF’S DILEMMA AND INFORMAL MATHEMATICS GREGORY LAVERS Department of Philosophy, Concordia University Abstract. This paper puts forward and defends an account of mathematical truth, and in par- ticular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerraf’s dilemma. The account builds upon Georg Kreisel’s work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. §1. Introduction. Benacerraf’s (1973) paper ‘Mathematical truth’ presents a problem with which any position in the philosophy of mathematics must come to terms. Benacerraf’s paper is often seen as presenting a dilemma where common sense seems to pull in opposite directions. Common sense with respect to the truth and the syntactical form of mathematical statements leads us to conclude that mathematical propositions concern abstract objects. At the same time, common sense with respect to epistemology seems to imply that mathematical propositions cannot concern abstract objects. It is quite generally accepted that no existing position on the philosophy of mathematics is completely adequate in its handling of the Benacerraf dilemma. It is the goal of this paper to show that a position on the philosophy of mathematics that will strike most practicing mathematicians as very natural can aptly handle the Benacerraf challenge.
    [Show full text]
  • Exploring Growth Trajectories of Informal and Formal Mathematics
    LDQXXX10.1177/0731948718786030Learning Disability QuarterlyNamkung et al. 786030research-article2018 Article Learning Disability Quarterly 2019, Vol. 42(2) 80 –91 Exploring Growth Trajectories of Informal © Hammill Institute on Disabilities 2018 Article reuse guidelines: sagepub.com/journals-permissions and Formal Mathematics Skills Among DOI: 10.1177/0731948718786030 Prekindergarten Children Struggling journals.sagepub.com/home/ldq With Mathematics Jessica M. Namkung, PhD1 , Peng Peng, PhD1, J. Marc Goodrich, PhD1, and Victoria Molfese, PhD2 Abstract Growth in two subscales, Informal and Formal Mathematics Skills, of the Test of Early Mathematics Abilitity–3 (TEMA-3) was explored in a sample of 281 children. Children were identified as either typically developing (TYP; n = 205) or having mathematics difficulties (MD; n = 76) based on their total TEMA-3 score at the end of prekindergarten. Their average level of informal and formal mathematics skills, growth rate over time, and rate of acceleration of growth were estimated using conventional growth modeling while controlling for the effects of gender. Results indicated that children with MD had significantly lower informal and formal mathematics knowledge than did TYP children at the end of kindergarten. However, for informal mathematics skills, children with MD grew at a significantly faster rate than did TYP children, and the rate of acceleration was also significantly faster for children with MD. In contrast, both the rate of growth and acceleration of growth in formal mathematics
    [Show full text]
  • Redalyc.Ethnomathematical Research and Drama in Education Techniques
    Revista Latinoamericana de Etnomatemática E-ISSN: 2011-5474 [email protected] Red Latinoamericana de Etnomatemática Colombia Stathopoulou, Charoula; Kotarinou, Panagiota; Appelbaum, Peter Ethnomathematical research and drama in education techniques: developing a dialogue in a geometry class of 10th grade students Revista Latinoamericana de Etnomatemática, vol. 8, núm. 2, junio-septiembre, 2015, pp. 105-135 Red Latinoamericana de Etnomatemática San Juan de Pasto, Colombia Available in: http://www.redalyc.org/articulo.oa?id=274041586006 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Stathopoulou, C., Kotarinou, P., & Appelbaum, P. (2015). Ethnomathematical research and drama in education techniques: developing a dialogue in a geometry class of 10th grade students. Revista Latinoamericana de Etnomatemática, 8(2), 105-135. Artículo recibido el 28 de noviembre 2014; Aceptado para publicación el 12 de mayo de 2015 Ethnomathematical research and drama in education techniques: developing a dialogue in a geometry class of 10th grade students Εθνομαθημαηική έπεςνα και Τεσνικέρ Δπαμαηικήρ Τέσνηρ: αναπηύζζονηαρ ένα διάλογο ζε μια ηάξη Λςκείος ζηο μάθημα ηηρ Γευμεηπίαρ Charoula Stathopoulou1 Panagiota Kotarinou2 Peter Appelbaum3 Abstract Ethnomathematical research, together with digital technologies (WebQuest)
    [Show full text]
  • Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator
    Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator Senfeng Liang University of Wisconsin-Stevens Point, U.S.A., [email protected] www.ijres.net To cite this article: Liang, S. (2016). Teaching the concept of limit by using conceptual conflict strategy and Desmos graphing calculator. International Journal of Research in Education and Science (IJRES), 2(1), 35-48. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Authors alone are responsible for the contents of their articles. The journal owns the copyright of the articles. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of the research material. International Journal of Research in Education and Science Volume 2, Issue 1, Winter 2016 ISSN: 2148-9955 Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator Senfeng Liang* University of Wisconsin-Stevens Point, U.S.A. Abstract Although the mathematics community has long accepted the concept of limit as the foundation of modern Calculus, the concept of limit itself has been marginalized in undergraduate Calculus education. In this paper, I analyze the strategy of conceptual conflict to teach the concept of limit with the aid of an online tool – Desmos graphing calculator. I also provide examples of how to use the strategy of conceptual conflict.
    [Show full text]
  • A Formal System for Euclid's Elements
    THE REVIEW OF SYMBOLIC LOGIC Volume 2, Number 4, December 2009 A FORMAL SYSTEM FOR EUCLID’S ELEMENTS JEREMY AVIGAD Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University EDWARD DEAN Department of Philosophy, Carnegie Mellon University JOHN MUMMA Division of Logic, Methodology, and Philosophy of Science, Suppes Center for History and Philosophy of Science Abstract. We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. §1. Introduction. For more than two millennia, Euclid’s Elements was viewed by mathematicians and philosophers alike as a paradigm of rigorous argumentation. But the work lost some of its lofty status in the nineteenth century, amidst concerns related to the use of diagrams in its proofs. Recognizing the correctness of Euclid’s inferences was thought to require an “intuitive” use of these diagrams, whereas, in a proper mathemat- ical argument, every assumption should be spelled out explicitly. Moreover, there is the question as to how an argument that relies on a single diagram can serve to justify a general mathematical claim: any triangle one draws will, for example, be either acute, right, or obtuse, leaving the same intuitive faculty burdened with the task of ensuring that the argument is equally valid for all triangles.1 Such a reliance on intuition was therefore felt to fall short of delivering mathematical certainty. Without denying the importance of the Elements, by the end of the nineteenth century the common attitude among mathematicians and philosophers was that the appropriate logical analysis of geometric inference should be cast in terms of axioms and rules of inference.
    [Show full text]