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Author: Nina Attridge
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
This thesis investigates the Theory of Formal Discipline (TFD): the idea that studying mathematics develops general reasoning skills. This belief has been held since the time of Plato (2003/375B.C), and has been cited in recent policy reports (Smith, 2004; Walport, 2010) as an argument for why mathematics should hold a privileged place in the UK's National Curriculum. However, there is no rigorous research evidence that justifies the claim. The research presented in this thesis aims to address this shortcoming. Two questions are addressed in the investigation of the TFD: is studying advanced mathematics associated with development in reasoning skills, and if so, what might be the mechanism of this development? The primary type of reasoning measured is conditional inference validation (i.e. `if p then q; not p; therefore not q'). In two longitudinal studies it is shown that the conditional reasoning behaviour of mathematics students at AS level and undergraduate level does change over time, but that it does not become straightforwardly more normative. Instead, mathematics students reason more in line with the `defective' interpretation of the conditional, under which they assume p and reason about q. This leads to the assumption that not-p cases are irrelevant, which results in the rejection of two commonly-endorsed invalid inferences, but also in the rejection of the valid modus tollens inference. Mathematics students did not change in their reasoning behaviour on a thematic syllogisms task or a thematic version of the conditional inference task. Next, it is shown that mathematics students reason significantly less in line with a defective interpretation of the conditional when it is phrased `p only if q' compared to when it is phrased `if p then q', despite the two forms being logically equivalent. This suggests that their performance is determined by linguistic features rather than the underlying logic. The final two studies investigated the heuristic and algorithmic levels of Stanovich's (2009a) tri-process model of cognition as potential mechanisms of the change in conditional reasoning skills. It is shown that mathematicians' defective interpretation of the conditional stems in part from heuristic level processing and in part from effortful processing, and that the executive function skills of inhibition and shifting at the algorithmic level are correlated with its adoption. It is suggested that studying mathematics regularly exposes students to implicit `if then' statements where they are expected to assume p and reason about q, and that this encourages them to adopt a defective interpretation of conditionals. It is concluded that the TFD is not supported by the evidence; while mathematics does seem to develop abstract conditional reasoning skills, the result is not more normative reasoning.
Author: Nina Attridge
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
This thesis investigates the Theory of Formal Discipline (TFD): the idea that studying mathematics develops general reasoning skills. This belief has been held since the time of Plato (2003/375B.C), and has been cited in recent policy reports (Smith, 2004; Walport, 2010) as an argument for why mathematics should hold a privileged place in the UK's National Curriculum. However, there is no rigorous research evidence that justifies the claim. The research presented in this thesis aims to address this shortcoming. Two questions are addressed in the investigation of the TFD: is studying advanced mathematics associated with development in reasoning skills, and if so, what might be the mechanism of this development? The primary type of reasoning measured is conditional inference validation (i.e. `if p then q; not p; therefore not q'). In two longitudinal studies it is shown that the conditional reasoning behaviour of mathematics students at AS level and undergraduate level does change over time, but that it does not become straightforwardly more normative. Instead, mathematics students reason more in line with the `defective' interpretation of the conditional, under which they assume p and reason about q. This leads to the assumption that not-p cases are irrelevant, which results in the rejection of two commonly-endorsed invalid inferences, but also in the rejection of the valid modus tollens inference. Mathematics students did not change in their reasoning behaviour on a thematic syllogisms task or a thematic version of the conditional inference task. Next, it is shown that mathematics students reason significantly less in line with a defective interpretation of the conditional when it is phrased `p only if q' compared to when it is phrased `if p then q', despite the two forms being logically equivalent. This suggests that their performance is determined by linguistic features rather than the underlying logic. The final two studies investigated the heuristic and algorithmic levels of Stanovich's (2009a) tri-process model of cognition as potential mechanisms of the change in conditional reasoning skills. It is shown that mathematicians' defective interpretation of the conditional stems in part from heuristic level processing and in part from effortful processing, and that the executive function skills of inhibition and shifting at the algorithmic level are correlated with its adoption. It is suggested that studying mathematics regularly exposes students to implicit `if then' statements where they are expected to assume p and reason about q, and that this encourages them to adopt a defective interpretation of conditionals. It is concluded that the TFD is not supported by the evidence; while mathematics does seem to develop abstract conditional reasoning skills, the result is not more normative reasoning.
Author: William Johnston
Publisher: OUP USA
ISBN: 0195310764
Category : Mathematics
Languages : en
Pages : 766
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Book Description
Preface 1. Mathematical Logic 2. Abstract Algebra 3. Number Theory 4. Real Analysis 5. Probability and Statistics 6. Graph Theory 7. Complex Analysis Answers to Questions Answers to Odd Numbered Questions Index of Online Resources Bibliography Index.
Author: Danilo R. Diedrichs
Publisher: CRC Press
ISBN: 1000581667
Category : Mathematics
Languages : en
Pages : 552
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Book Description
This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics. The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide, that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline. Part I offers: An introduction to logic and set theory. Proof methods as a vehicle leading to topics useful for analysis, topology, algebra, and probability. Many illustrated examples, often drawing on what students already know, that minimize conversation about "doing proofs." An appendix that provides an annotated rubric with feedback codes for assessing proof writing. Part II presents the context and culture aspects of the transition experience, including: 21st century mathematics, including the current mathematical culture, vocations, and careers. History and philosophical issues in mathematics. Approaching, reading, and learning from journal articles and other primary sources. Mathematical writing and typesetting in LaTeX. Together, these Parts provide a complete introduction to modern mathematics, both in content and practice. Table of Contents Part I - Introduction to Proofs Logic and Sets Arguments and Proofs Functions Properties of the Integers Counting and Combinatorial Arguments Relations Part II - Culture, History, Reading, and Writing Mathematical Culture, Vocation, and Careers History and Philosophy of Mathematics Reading and Researching Mathematics Writing and Presenting Mathematics Appendix A. Rubric for Assessing Proofs Appendix B. Index of Theorems and Definitions from Calculus and Linear Algebra Bibliography Index Biographies Danilo R. Diedrichs is an Associate Professor of Mathematics at Wheaton College in Illinois. Raised and educated in Switzerland, he holds a PhD in applied mathematical and computational sciences from the University of Iowa, as well as a master’s degree in civil engineering from the Ecole Polytechnique Fédérale in Lausanne, Switzerland. His research interests are in dynamical systems modeling applied to biology, ecology, and epidemiology. Stephen Lovett is a Professor of Mathematics at Wheaton College in Illinois. He holds a PhD in representation theory from Northeastern University. His other books include Abstract Algebra: Structures and Applications (2015), Differential Geometry of Curves and Surfaces, with Tom Banchoff (2016), and Differential Geometry of Manifolds (2019).
Author: Stanley J. Farlow
Publisher: John Wiley & Sons
ISBN: 1119563518
Category : Mathematics
Languages : en
Pages : 480
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Book Description
Provides a smooth and pleasant transition from first-year calculus to upper-level mathematics courses in real analysis, abstract algebra and number theory Most universities require students majoring in mathematics to take a “transition to higher math” course that introduces mathematical proofs and more rigorous thinking. Such courses help students be prepared for higher-level mathematics course from their onset. Advanced Mathematics: A Transitional Reference provides a “crash course” in beginning pure mathematics, offering instruction on a blendof inductive and deductive reasoning. By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive setting. Clear and concise chapters cover all the essential topics students need to transition from the "rote-orientated" courses of calculus to the more rigorous "proof-orientated” advanced mathematics courses. Topics include sentential and predicate calculus, mathematical induction, sets and counting, complex numbers, point-set topology, and symmetries, abstract groups, rings, and fields. Each section contains numerous problems for students of various interests and abilities. Ideally suited for a one-semester course, this book: Introduces students to mathematical proofs and rigorous thinking Provides thoroughly class-tested material from the authors own course in transitioning to higher math Strengthens the mathematical thought process of the reader Includes informative sidebars, historical notes, and plentiful graphics Offers a companion website to access a supplemental solutions manual for instructors Advanced Mathematics: A Transitional Reference is a valuable guide for undergraduate students who have taken courses in calculus, differential equations, or linear algebra, but may not be prepared for the more advanced courses of real analysis, abstract algebra, and number theory that await them. This text is also useful for scientists, engineers, and others seeking to refresh their skills in advanced math.
Author: Ron Larson
Publisher: National Geographic Learning
ISBN: 9781680336467
Category :
Languages : en
Pages :
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Author: Scott A. Chamberlin
Publisher: Pieces of Learning
ISBN: 1937113159
Category : Gifted children
Languages : en
Pages : 130
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Author: Annie Selden
Publisher: Routledge
ISBN: 1135478457
Category : Education
Languages : en
Pages : 112
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Book Description
This is Volume 7, Issue 1 2005, a Special Issue of 'Mathematical Thinking and Learning' which looks at Advanced Mathematical Thinking. Opening with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. The articles follow the recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices.
Author: Matthew Inglis
Publisher: World Scientific
ISBN: 1786340704
Category : Mathematics
Languages : en
Pages : 204
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Book Description
For centuries, educational policymakers have believed that studying mathematics is important, in part because it develops general thinking skills that are useful throughout life. This 'Theory of Formal Discipline' (TFD) has been used as a justification for mathematics education globally. Despite this, few empirical studies have directly investigated the issue, and those which have showed mixed results.Does Mathematical Study Develop Logical Thinking? describes a rigorous investigation of the TFD. It reviews the theory's history and prior research on the topic, followed by reports on a series of recent empirical studies. It argues that, contrary to the position held by sceptics, advanced mathematical study does develop certain general thinking skills, however these are much more restricted than those typically claimed by TFD proponents.Perfect for students, researchers and policymakers in education, further education and mathematics, this book provides much needed insight into the theory and practice of the foundations of modern educational policy.
Author: Nancy Mather
Publisher: John Wiley & Sons
ISBN: 1118860748
Category : Psychology
Languages : en
Pages : 617
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Book Description
Includes online access to new, customizable WJ IV score tables, graphs, and forms for clinicians Woodcock-Johnson IV: Reports, Recommendations, and Strategies offers psychologists, clinicians, and educators an essential resource for preparing and writing psychological and educational reports after administering the Woodcock-Johnson IV. Written by Drs. Nancy Mather and Lynne E. Jaffe, this text enhances comprehension and use of this instrument and its many interpretive features. This book offers helpful information for understanding and using the WJ IV scores, provides tips to facilitate interpretation of test results, and includes sample diagnostic reports of students with various educational needs from kindergarten to the postsecondary level. The book also provides a wide variety of recommendations for cognitive abilities; oral language; and the achievement areas of reading, written language, and mathematics. It also provides guidelines for evaluators and recommendations focused on special populations, such as sensory impairments, autism, English Language Learners, and gifted and twice exceptional students, as well as recommendations for the use of assistive technology. The final section provides descriptions of the academic and behavioral strategies mentioned in the reports and recommendations. The unique access code included with each book allows access to downloadable, easy-to-customize score tables, graphs, and forms. This essential guide Facilitates the use and interpretation of the WJ IV Tests of Cognitive Abilities, Tests of Oral Language, and Tests of Achievement Explains scores and various interpretive features Offers a variety of types of diagnostic reports Provides a wide variety of educational recommendations and evidence-based strategies
Author: William Johnston
Publisher: Oxford University Press
ISBN: 0199718660
Category : Mathematics
Languages : en
Pages : 766
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Book Description
A Transition to Advanced Mathematics: A Survey Course promotes the goals of a "bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical calculations) to theoretical upper-level mathematics courses (where they will have to prove theorems and grapple with mathematical abstractions). The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis. The main objective is "to bring about a deep change in the mathematical character of students -- how they think and their fundamental perspectives on the world of mathematics." This text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language, to use mathematically sound reasoning, and to ask probing questions about mathematics. In short, we hope that working through A Transition to Advanced Mathematics encourages students to become mathematicians in the fullest sense of the word. A Transition to Advanced Mathematics has a number of distinctive features that enable this transformational experience. Embedded Questions and Reading Questions illustrate and explain fundamental concepts, allowing students to test their understanding of ideas independent of the exercise sets. The text has extensive, diverse Exercises Sets; with an average of 70 exercises at the end of section, as well as almost 3,000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. We have also interwoven embedded reflections on the history, culture, and philosophy of mathematics throughout the text.
