DOCSLIB.ORG

Mathematical Thinking: from Cacophony to Consensus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Thinking: from Cacophony to Consensus MATHEMATICAL THINKING: FROM CACOPHONY TO CONSENSUS A dissertation submitted to the Kent State University College and Graduate School of Education, Health, and Human Services in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Sean F. Argyle August 2012 2012 by Sean F. Argyle Some Rights Reserved http://creativecommons.org/licenses/by-nc-nd/3.0/ A dissertation written by Sean F. Argyle B.S., Slippery Rock University, 2005 M.Ed., Slippery Rock University, 2006 Ph.D., Kent State University, 2012 Approved by _________________________, Co-director, Doctoral Dissertation Committee Michael G. Mikusa _________________________, Co-director, Doctoral Dissertation Committee Lisa A. Donnelly _________________________, Member, Doctoral Dissertation Committee Natasha Levinson Accepted by _________________________, Director, School of Teaching, Learning Alexa Sandmann and Curriculum Studies _________________________, Dean, College of Education, Health and Human Services Daniel F. Mahony ARGYLE, SEAN F., Ph.D., August 2012 Curriculum and Instruction MATHEMATICAL THINKING: FROM CACOPHONY TO CONSENSUS (199 pp.) Co-Directors of Dissertation: Michael G. Mikusa, Ph.D. Lisa A. Donnelly, Ph.D. Various standards have demanded that teachers improve “mathematical thinking,” but definitions are vague – if present at all. What little research on the subject exists is disjointed and dissenting, leading some researchers to lament the possibility of ever coming to an agreement on how to define “mathematical thinking” as a viable construct. Rather than add one more voice into the cacophony of competing definitions, this dissertation seeks to discuss the results of a conceptual meta-analysis of the term’s use in an appropriately titled journal – Mathematical Thinking and Learning. ACKNOWLEDGEMENTS I am grateful to Valerie, who proofread so much of this paper. Surely, it would have been far muddier without her advice. I am, furthermore, indebted to my mother for her continued modeling of how to proceed through adversity. iv TABLE OF CONTENTS Heading Pg. ACKNOWLEDGEMENTS. iv LIST OF FIGURES. vii CHAPTER I: INTRODUCTION. 1 Guiding Research Questions. 8 Methodology. 9 Conceptual Analysis. 12 Data Analysis. 16 Procedure. 21 CHAPTER II: THE MODEL. 22 “Mathematical Thinking” or “Thinking About Mathematics” . 22 Four Views of Mathematical Thinking. 24 Think about mathematics. 24 Think when doing mathematics. 25 Habits of mind. 26 Think mathematically. 27 Synthesis of the Four Views. 27 1. Mathematical thinking could potentially apply to almost anything. 28 2. Mathematical thinking is implicitly tied to the process of mathematics. 30 3. Mathematical thinking requires habituation. 34 4. Mathematical thinking operates on and outside generalized thinking. 37 An Overview of the Model. 39 CHAPTER III: EVERYDAY EXPERIENCE. 46 Abstraction. 51 Summary. 54 Generalization. 55 Overgeneralization. 60 Summary. 65 Reification. 66 Summary. 68 Chapter Summary. 69 v TABLE OF CONTENTS (CONT.) CHAPTER IV: MATHEMATICAL WORLD. 70 Mathematization. 72 Summary. 81 Justification. 81 Summary. 87 Chapter Summary. 88 CHAPTER V: MATHEMATICAL COMMUNITY. 89 Normalization. 95 Summary. 108 Contribution. 109 Summary. 114 Chapter Summary. 115 CHAPTER VI: MATHEMATICAL DISPOSITION. 116 Internalization. 119 Summary. 123 Intuition. 124 Summary. 127 Motivation. 128 Summary. 132 Aesthetic Evaluation. 133 Summary. 135 Chapter Summary. 136 CHAPTER VII: SENSE-MAKING. 137 Interpretation. 141 Representation. 145 Organization. 149 Chapter Summary. 152 CHAPTER VIII: PRACTICAL IMPLICATIONS. 153 Limitations. 159 Discussion. 162 Statistical Thinking. 163 Curriculum & Instruction. 169 Conclusion. 179 REFERENCES. 180 vi LIST OF FIGURES Figure Pg. 1. Word Cloud. 11 2. Model Research Methodologies. 12 3. Article Category Percentages. 20 4. Annual Category Counts. 20 5. A Meta-analytic Model of Mathematical Thinking. 40 6. Everyday Experience. 46 7..
Load more

Recommended publications
  • A Dissertation Entitled Exploring Common Antecedents of Three Related Decision Biases by Jonathan E. Westfall Submitted As Parti
    A Dissertation Entitled Exploring Common Antecedents of Three Related Decision Biases by Jonathan E. Westfall Submitted as partial fulfillment of the requirements for the Doctor of Philosophy in Psychology __________________________ Advisor: Dr. J. D. Jasper __________________________ Dr. S. D. Christman __________________________ Dr. R. E. Heffner __________________________ Dr. K. L. London __________________________ Dr. M. E. Doherty __________________________ Graduate School The University of Toledo August 2009 Exploring Common Antecedents ii Copyright © 2009 – Jonathan E. Westfall This document is copyrighted material. Under copyright law, no parts of this document may be reproduced in any manner without the expressed written permission of the author. Exploring Common Antecedents iii An Abstract of Exploring Common Antecedents of Three Related Decision Biases Jonathan E. Westfall Submitted as partial fulfillment of the requirements for The Doctor of Philosophy in Psychology The University of Toledo August 2009 “Decision making inertia” is a term loosely used to describe the similar nature of a variety of decision making biases that predominantly favor a decision to maintain one course of action over switching to a new course. Three of these biases, the sunk cost effect, status-quo bias, and inaction inertia are discussed here. Combining earlier work on strength of handedness and the sunk cost effect along with new findings regarding counterfactual thought, this work principally seeks to determine if counterfactual thought may drive the three decision biases of note while also analyzing common relationships between the biases, strength of handedness, and the variables of regret and loss aversion. Over a series of experiments, it was found that handedness differences did exist in the three biases discussed, that amount and type of counterfactuals generated did not predict choice within the status-quo bias, and that the remaining variables potentially thought to drive the biases presented did not link causally to them.
    [Show full text]
  • Weakness of Will, Akrasia, and the Neuropsychiatry of Decision Making: an Interdisciplinary Perspective
    Cognitive, Affective, & Behavioral Neuroscience 2008, 8 (4), 402-417 doi:10.3758/CABN.8.4.402 Weakness of will, akrasia, and the neuropsychiatry of decision making: An interdisciplinary perspective ANNEMARIE KALIS Universíteit Utrech, Utrecht, The Netherlands ANDREAS MOJZISCH Georg-August-Universität Göttingen, Göttingen, Germany T. SOPHIE SCHWEIZER Vrije Universiteit Amsterdam, Amsterdam, The Netherlands AND STEFAN KAISER Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany This article focuses on both daily forms of weakness of will as discussed in the philosophical debate (usually referred to as akrasia) and psychopathological phenomena as impairments of decision making. We argue that bothboth descriptionsdescriptions ofof dysfunctionaldysfunctional decisiondecision makingmaking can bebe organorganizedized wwithinithin a common ttheoreticalheoretical fframeworkramework that divides the decision making process in three different stages: option generation, option selection, and action initiation. We first discuss our theoretical framework (building on existing models of decision-making stages), focusing on option generation as an aspect that has been neglected by previous models. In the main body of this article, we review how both philosophy and neuropsychiatry have provided accounts of dysfunction in each decision-making stage, as well as where these accounts can be integrated. Also, the neural underpinnings of dysfunction in the three different stages are discussed. We conclude by discussing advantages and limitations of our integppgrative approach. Most of us do not always do what, in our own eyes, Such phenomena are often attributed to a weakness of will. we judge to be best. Imagine Bessie, who is enjoying a This term already suggests a certain explanation: Our will holiday after a period of hard work.
    [Show full text]
  • Libertarian Nudges
    Missouri Law Review Volume 82 Issue 3 Summer 2017 - Symposium Article 9 Summer 2017 Libertarian Nudges Gregory Mitchell Follow this and additional works at: https://scholarship.law.missouri.edu/mlr Part of the Law Commons Recommended Citation Gregory Mitchell, Libertarian Nudges, 82 MO. L. REV. (2017) Available at: https://scholarship.law.missouri.edu/mlr/vol82/iss3/9 This Article is brought to you for free and open access by the Law Journals at University of Missouri School of Law Scholarship Repository. It has been accepted for inclusion in Missouri Law Review by an authorized editor of University of Missouri School of Law Scholarship Repository. For more information, please contact [email protected]. Mitchell: Libertarian Nudges Libertarian Nudges Gregory Mitchell* ABSTRACT We can properly call a number of nudges libertarian nudges, but the ter- ritory of libertarian nudging is smaller than is often realized. The domain of libertarian nudges is populated by choice-independent nudges, or nudges that only assist the decision process and do not push choosers toward any particu- lar choice. Some choice-dependent nudges pose no great concern from a lib- ertarian perspective for rational choosers so long as there is a low-cost way to avoid the nudger’s favored choice. However, choice-dependent nudges will interfere with the autonomy of irrational choosers, because the opt-out option will be meaningless for this group. Choice-independent nudges should be of no concern with respect to irrational actors and in fact should be welcomed because they can promote the decision competence fundamental to libertari- anism, but choice-dependent nudges can never truly be libertarian nudges.
    [Show full text]
  • Negative Symptoms and Behavioral Alterations Associated with Dorsolateral Prefrontal Syndrome in Patients with Schizophrenia
    Journal of Clinical Medicine Article Negative Symptoms and Behavioral Alterations Associated with Dorsolateral Prefrontal Syndrome in Patients with Schizophrenia Pamela Ruiz-Castañeda 1,2, María Teresa Daza-González 1,2,* and Encarnación Santiago-Molina 3 1 Neuropsychological Evaluation and Rehabilitation Center (CERNEP), University of Almeria, Carretera de Sacramento, s/n, La Cañada de San Urbano, 04120 Almeria, Spain; [email protected] 2 Department of Psychology, University of Almeria, Carretera de Sacramento, s/n, La Cañada de San Urbano, 04120 Almeria, Spain 3 Mental Health Hospitalization Unit of Torrecárdenas Hospital, Calle Hermandad de Donantes de Sangre, s/n, 04009 Almería, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-950214623 Abstract: The present study had three main aims: (1) to explore the possible relationships between the two dimensions of negative symptoms (NS) with the three frontal behavioral syndromes (dorsolateral, orbitofrontal and the anterior or mesial cingulate circuit) in patients with schizophrenia; (2) to determine the influence of sociodemographic and clinical variables on the severity of the two dimensions of NS (expressive deficits and disordered relationships/avolition); and (3) to explore the possible relationships between the two dimensions of NS and social functioning. We evaluated a group of 33 patients with schizophrenia with a predominance of NS using the self-reported version Citation: Ruiz-Castañeda, P.; of the Frontal System Behavior scale. To quantify the severity of NS, the Assessment of Negative Daza-González, M.T.; Symptoms (SANS) scale was used. The results revealed that the two dimensions of NS correlate Santiago-Molina, E. Negative positively with the behavioral syndrome of dorsolateral prefrontal origin.
    [Show full text]
  • Developing a New Apathy Measurement Scale
    Edinburgh Research Explorer Developing a new apathy measurement scale Citation for published version: Radakovic, R & Abrahams, S 2014, 'Developing a new apathy measurement scale: Dimensional apathy scale', Psychiatry Research, vol. 219, no. 3, pp. 658-663. https://doi.org/10.1016/j.psychres.2014.06.010 Digital Object Identifier (DOI): 10.1016/j.psychres.2014.06.010 Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: Psychiatry Research General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 28. Sep. 2021 Developing a new apathy measurement scale: dimensional apathy scale * a, b, c, d a, c, d Ratko Radakovic and Sharon Abrahams a Human Cognitive Neuroscience-Psychology, School of Philosophy, Psychology & Language Sciences, University of Edinburgh, UK b Alzheimer Scotland Dementia Research Centre, University of Edinburgh, UK c Anne Rowling Regenerative Neurology Clinic, University of Edinburgh, UK d Euan MacDonald Centre for MND Research, University of Edinburgh, UK Abstract Apathy is both a symptom and syndrome prevalent in neurodegenerative disease, including motor system disorders, that affects motivation to display goal directed functions.
    [Show full text]
  • The Cognitive and Neural Basis for Apathy in Frontotemporal Degeneration
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2014 The Cognitive and Neural Basis for Apathy in Frontotemporal Degeneration Lauren M. Massimo University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Neuroscience and Neurobiology Commons, and the Nursing Commons Recommended Citation Massimo, Lauren M., "The Cognitive and Neural Basis for Apathy in Frontotemporal Degeneration" (2014). Publicly Accessible Penn Dissertations. 1360. https://repository.upenn.edu/edissertations/1360 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1360 For more information, please contact [email protected]. The Cognitive and Neural Basis for Apathy in Frontotemporal Degeneration Abstract The syndrome of apathy, defined as a eductionr in goal-directed behavior (GDB), has profound consequences for morbidity and mortality in the patient and for family-caregiver burden. Apathy is one of the primary neuropsychiatric syndromes associated with the disruption of the frontal-striatal system, but the behavioral and biological mechanisms underlying apathy are not well understood. Apathy is especially prevalent in behavioral variant frontotemporal degeneration (bvFTD). In a sample of 20 apathetic adults with bvFTD and 17 normal controls (NC), impairments in three components of GDB--initiation, planning and motivation--were examined using a novel computerized reaction time test. Employing structural neuroimaging techniques, I then examined the neural basis of GDB in these apathetic bvFTD participants. I found evidence that apathy is associated with an impairment in any of the three GDB components. Initiation, planning, and motivation each map onto three distinct brain regions in the frontal lobe that work together in a large-scale neural network.
    [Show full text]
  • Cognitive Hygiene and the Fountains of Human Ignorance
    Cognitive Hygiene and the Fountains of Human Ignorance by Bradford Hatcher © 2019 Bradford Hatcher ISBN: 978-0-9824191-5-1 Download at: https://www.hermetica.info/CH.htm or: https://www.hermetica.info/CH.pdf Statue on cover: Cain, by Henri Vidal, Tuileries Garden, Paris, 1896. Table of Contents Part One: A General Survey of the Issues 1.0 - Preface and Introduction 7 1.1 - Truth Words, Taxa, and False Dichotomies 11 Amathia - the Deliberate Kind of Stupid True as a Verb, and Being or Holding True False Dichotomies and Delusional Schisms Nature and Nurture, Affect and Cognition 1.2 - Would a Rational Being Think Man a Rational Being? 18 Human Nature and Reality's Nature The Nature of Mind and Cognitive Science Emergence, Qualia, and Consciousness Pros and Cons of Ignorance, Delusion, & Self-Deception A Duty to Culture and Paying Our Rent 1.3 - Why Critical Thinking Hurts Our Stupid Feelings 34 Contradiction and Cognitive Dizziness Critical Thinking and Cognitive Hygiene Stupid Feelings and Emotional Intelligence Denial, Relativism, and Limits to Tolerance 1.4 - Science, and Some Other Ways to Be True 43 Scientific Naturalism, Mathematics, and Formal Logic Informal Logic, Statistics, and Skepticism Taxonomies, Categories, Scales, and Matrices Analogies, Thought Experiments, and Parables Exposition, Narrative, Anecdote, and Anecdata Introspection, Phenomenology, and Vipassana Bhavana 1.5 - Framing Issues and Far Horizons 61 Framing and Perspective Narrow-mindedness, Points of View and Perspective Nearsightedness, Spatial Framing and Orders
    [Show full text]
  • Overcoming Organizational Lock-In in Decision-Making Construction Clients Facing Innovation
    ISSN: 1402-1544 ISBN 978-91-7439-XXX-X Se i listan och fyll i siffror där kryssen är DOCTORAL T H E SIS Erika Hedgren Overcoming Organizational Lock-In in Decision-Making Erika Hedgren Overcoming Department of Civil, Environmental and Natural Resources Engineering Division of Structural and Construction Engineering ISSN: 1402-1544 Overcoming Organizational Lock-In ISBN 978-91-7439-572-3 (tryckt) ISBN 978-91-7439-573-0 (pdf) in Decision-Making Luleå University of Technology 2013 Construction Clients Facing Innovation Erika Hedgren Construction Clients Facing Innovation Construction Clients Facing Overcoming Organizational Lock-In in Decision-Making Construction Clients Facing Innovation Erika Hedgren April, 2013 Luleå University of Technology Department of Civil, Environmental and Natural Resources Engineering Division of Structural and Construction Engineering – Timber Structures Printed by Universitetstryckeriet, Luleå 2013 ISSN: 1402-1544 ISBN 978-91-7439-572-3 (tryckt) ISBN 978-91-7439-573-0 (pdf) Luleå 2013 www.ltu.se Overcoming Organizational Lock-In in Decision-Making Construction Clients Facing Innovation Erika Hedgren Luleå University of Technology Department of Civil, Environmental and Natural Resources Engineering Division of Structural and Construction Engineering – Timber Structures Dissertation for the degree of Doctor of Philosophy (Ph.D.) in Timber Structures which, with the permission of the Faculty Board at Luleå University of Technology, will be defended in public in F1031, Friday 26th April, at 10.00 am. Akademisk avhandling
    [Show full text]
  • The Formation of Organizational RAVASI Published9july2018 GREEN
    King’s Research Portal DOI: 10.5465/annals.2016.0124 Document Version Peer reviewed version Link to publication record in King's Research Portal Citation for published version (APA): Ravasi, D., Rindova, V., Etter, M. A., & Cornelissen, J. (2018). The Formation of Organizational Reputation. Academy of management annals, 12(2), 574–599. https://doi.org/10.5465/annals.2016.0124 Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. •Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. •You may not further distribute the material or use it for any profit-making activity or commercial gain •You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim.
    [Show full text]
  • Cognition and Management Managerial Cognition and Organisational Performance
    Cognition and Management Managerial Cognition and Organisational Performance Geoffrey Goodhew December 1998 Submitted to the University of Canterbury for the degree of Doctor of Philosophy. HF lowe so much to so many people who have contributed to this thesis - both directly and indirectly. First and foremost are my family - Cara, Bill, and Claire. I really can't think of anything to say that can express my gratitude to them. Without their love and almost unquestioning support, this would have been impossible. Thank you. My supervisors at University - Peter Cammack, Bob Hamilton and Stephen Dakin have all given me invaluable advice and support. On the many times when I felt I was in over my head, they provided the clarity I needed. All of the staff in the department of management both academic and secretarial- have been supportive and helpful. In particular I have to mention Ian Brooks who has always been willing to stop and have a chat. I have learned so much from all of you. I have also had the fortune of sharing an office with two other Ph.D stUdents - Mark Fox and Jo Berg. Aside from the fact that they have both finished before me, they made the daily grind bearable. I consider all of the other Ph.D students in the department to be friends who have helped me: Mike, Callum, Gavin, Oily, Thomas, Andrew, Steve and Ashley. The two names missing from that list are Tristram and Toby. They have both gone out of their way to help me by either reading drafts or figuring out how to make this computer work or just talking to me.
    [Show full text]
  • Inertial Behavior and Generalized Partition"
    Inertial Behavior and Generalized Partition David Dillenbergery Philipp Sadowskiz May 2016 Abstract Behavior is inertial if it does not react to the apparent arrival of relevant informa- tion. In a context where information is subjective, we formulate an axiom that captures inertial behavior, and provide a representation that explains such behavior as that of a rational decision maker who perceives a particular type of information structure, we call a generalized partition. We characterize the learning processes that can be described by a generalized partition. We then investigate behavioral protocols that may lead individuals to perceive a generalized partition (and thus to display inertial behavior) even when facing a di¤erent type of information structure: A cognitive bias referred to as cognitive inertia and a bound on rationality, which we term shortsightedness. Key words: Inertial behavior, subjective learning, generalized partition, uniform cover, cognitive inertia, shortsighted rationality 1. Introduction Individuals often seem reluctant to change their behavior in response to new information, unless that information is conclusive. Examples include managers who do not adapt their strategy to changes in the business environment (Hodgkinson 1997 and reference therein), a bias in favor of renewing the status quo health plan when reviewing other options after a change in health status, or a “one advice suits all”approach among consultants, such as medical doctors or …nancial advisors, dealing with a heterogeneous population (clients with di¤erent symptoms or di¤erent levels of risk attitude). We refer to this apparent bias as inertial behavior. We contend that in most applied situations, the analyst may suspect inertial behavior even without being aware of the exact information available to each agent; behavior does Some of the results in this paper previously appeared in Dillenberger, D., and P.
    [Show full text]
  • Biography of a Quest(Ion)
    THE DENIAL OF RELEVANCE: BIOGRAPHY OF A QUEST(ION) AMIDST THE MIN(D)FIELDS—GROPING AND STUMBLING Marion Turner VanBebber, JD, MS, BS Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS August 2014 APPROVED: Brian C. O’Connor, Major Professor James Duban, Committee Member Jodi L. Kearns, Committee Member Suliman Hawamdeh, Chair of the Department of Library and Information Science Herman Totten, Dean of the College of Information Mark Wardell, Dean of the Toulouse Graduate School VanBebber, Marion Turner. The Denial of Relevance: Biography of a Quest(ion) Amidst the Min(d)fields—Groping and Stumbling. Doctor of Philosophy (Information Science), August 2014, 251 pp., 15 figures, references, 226 titles. Early research on just why it might be the case that “the mass of men lead lives of quiet desperation” suggested that denial of relevance was a significant factor. Asking why denial of relevance would be significant and how it might be resolved began to raise issues of the very nature of questions. Pursuing the nature of questions, in light of denial of relevance and Thoreau’s “quiet desperation” provoked a journey of modeling questions and constructing a biography of the initial question of this research and its evolution. Engaging literature from philosophy, neuroscience, and retrieval then combined with deep interviews of successful lawyers to render a thick, biographical model of questioning. Copyright 2014 by Marion Turner VanBebber ii ACKNOWLEDGMENTS I am immensely grateful to so many mentors who have
    [Show full text]