Reading and Writing: An Interdisciplinary Journal, 2019
Abstract
Although the Toulmin model (1958) has dominated argumentation research, it does not pro... more Abstract Although the Toulmin model (1958) has dominated argumentation research, it does not provide many tools for evaluating argument quality. Towards that end, we draw on work in philosophy on argument schemes, and critical questions for evaluating those schemes. In our approach, we integrate the teaching of critical questions with argumentation vee diagrams (AVDs) and with oral and written discourse. AVDs are graphic organizers that prompt students to write arguments and counterarguments on different sides of the vee, and at the bottom of the vee, an integrating paragraph supporting a final conclusion. The present study was conducted in three sections of an undergraduate general education seminar. Two sections, comprising the experimental group, used AVDs containing a critical questions box reflecting questions for arguments from consequences (Walton, 1996). One section used AVDs without the critical question box. Students completed AVDs prior and during class discussions on social issues (e.g., drug legalization). Over time, students in the experimental group included increasingly more refutations related to the critical questions compared to the control group. The effect transferred to an in-class essay where no question prompts were provided, but not to a course paper written on whistleblowing. However, students in the experimental condition did include in their papers more explicit discussion of moral principles. We explain these effects in relation to argument schema theory, in particular the development and automation of a weighing schema. The critical questions appeared to provide students with a structure for evaluating arguments and counterarguments.
In this paper, I assume, perhaps controversially, that translation into a language of formal logi... more In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deduc... more Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deductive Logic in Mathematics: The Probability of Conjectures; James Franklin.- Chapter 2. Arguments, Proofs, and Dialogues; Erik C. W. Krabbe.- Chapter 3. Argumentation in Mathematics; Jesús Alcolea Banegas.- Chapter 4. Arguing Around Mathematical Proofs; Michel Dufour.- Part II. Argumentation as a Methodology for Studying Mathematical Practice.- Chapter 5. An Argumentative Approach to Ideal Elements in Mathematics; Paola Cantù.- Chapter 6. How Persuaded Are You? A Typology of Responses; Matthew Inglis and Juan Pablo Mejía-Ramos.- Chapter 7. Revealing Structures of Argumentations in Classroom Proving Processes; Christine Knipping and David Reid.- Chapter 8. Checking Proofs; Jesse Alama and Reinhard Kahle.- Part III. Mathematics as a Testbed for Argumentation Theory.- Chapter 9. Dividing by Zero—and Other Mathematical Fallacies; Lawrence H. Powers.- Chapter 10. Strategic Maneuvering in Mathematical Proofs; Erik C. W. Krabbe.- Chapter. 11 Analogical Arguments in Mathematics; Paul Bartha.- Chapter 12. What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures; Brendan Larvor.- Part IV. An Argumentational Turn in the Philosophy of Mathematics.- Chapter 13. Mathematics as the Art of Abstraction; Richard L. Epstein.- Chapter 14. Towards a Theory of Mathematical Argument; Ian J. Dove.- Chapter 15. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics; Alison Pease, Alan Smaill, Simon Colton and John Lee.- Chapter 16. Mathematical Arguments and Distributed Knowledge; Patrick Allo, Jean Paul Van Bendegem and Bart Van Kerkhove.- Chapter 17. The Parallel Structure of Mathematical Reasoning; Andrew Aberdein.- Index.
Reading and Writing: An Interdisciplinary Journal, 2019
Abstract
Although the Toulmin model (1958) has dominated argumentation research, it does not pro... more Abstract Although the Toulmin model (1958) has dominated argumentation research, it does not provide many tools for evaluating argument quality. Towards that end, we draw on work in philosophy on argument schemes, and critical questions for evaluating those schemes. In our approach, we integrate the teaching of critical questions with argumentation vee diagrams (AVDs) and with oral and written discourse. AVDs are graphic organizers that prompt students to write arguments and counterarguments on different sides of the vee, and at the bottom of the vee, an integrating paragraph supporting a final conclusion. The present study was conducted in three sections of an undergraduate general education seminar. Two sections, comprising the experimental group, used AVDs containing a critical questions box reflecting questions for arguments from consequences (Walton, 1996). One section used AVDs without the critical question box. Students completed AVDs prior and during class discussions on social issues (e.g., drug legalization). Over time, students in the experimental group included increasingly more refutations related to the critical questions compared to the control group. The effect transferred to an in-class essay where no question prompts were provided, but not to a course paper written on whistleblowing. However, students in the experimental condition did include in their papers more explicit discussion of moral principles. We explain these effects in relation to argument schema theory, in particular the development and automation of a weighing schema. The critical questions appeared to provide students with a structure for evaluating arguments and counterarguments.
In this paper, I assume, perhaps controversially, that translation into a language of formal logi... more In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deduc... more Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deductive Logic in Mathematics: The Probability of Conjectures; James Franklin.- Chapter 2. Arguments, Proofs, and Dialogues; Erik C. W. Krabbe.- Chapter 3. Argumentation in Mathematics; Jesús Alcolea Banegas.- Chapter 4. Arguing Around Mathematical Proofs; Michel Dufour.- Part II. Argumentation as a Methodology for Studying Mathematical Practice.- Chapter 5. An Argumentative Approach to Ideal Elements in Mathematics; Paola Cantù.- Chapter 6. How Persuaded Are You? A Typology of Responses; Matthew Inglis and Juan Pablo Mejía-Ramos.- Chapter 7. Revealing Structures of Argumentations in Classroom Proving Processes; Christine Knipping and David Reid.- Chapter 8. Checking Proofs; Jesse Alama and Reinhard Kahle.- Part III. Mathematics as a Testbed for Argumentation Theory.- Chapter 9. Dividing by Zero—and Other Mathematical Fallacies; Lawrence H. Powers.- Chapter 10. Strategic Maneuvering in Mathematical Proofs; Erik C. W. Krabbe.- Chapter. 11 Analogical Arguments in Mathematics; Paul Bartha.- Chapter 12. What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures; Brendan Larvor.- Part IV. An Argumentational Turn in the Philosophy of Mathematics.- Chapter 13. Mathematics as the Art of Abstraction; Richard L. Epstein.- Chapter 14. Towards a Theory of Mathematical Argument; Ian J. Dove.- Chapter 15. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics; Alison Pease, Alan Smaill, Simon Colton and John Lee.- Chapter 16. Mathematical Arguments and Distributed Knowledge; Patrick Allo, Jean Paul Van Bendegem and Bart Van Kerkhove.- Chapter 17. The Parallel Structure of Mathematical Reasoning; Andrew Aberdein.- Index.
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Papers by Ian Dove
Although the Toulmin model (1958) has dominated argumentation research, it does not provide many tools for evaluating argument quality. Towards that end, we draw on work in philosophy on argument schemes, and critical questions for evaluating those schemes. In our approach, we integrate the teaching of critical questions with argumentation vee diagrams (AVDs) and with oral and written discourse. AVDs are graphic organizers that prompt students to write arguments and counterarguments on different sides of the vee, and at the bottom of the vee, an integrating paragraph supporting a final conclusion. The present study was conducted in three sections of an undergraduate general education seminar. Two sections, comprising the experimental group, used AVDs containing a critical questions box reflecting questions for arguments from consequences (Walton, 1996). One section used AVDs without the critical question box. Students completed AVDs prior and during class discussions on social issues (e.g., drug legalization). Over time, students in the experimental group included increasingly more refutations related to the critical questions compared to the control group. The effect transferred to an in-class essay where no question prompts were provided, but not to a course paper written on whistleblowing. However, students in the experimental condition did include in their papers more explicit discussion of moral principles. We explain these effects in relation to argument schema theory, in particular the development and automation of a weighing schema. The critical questions appeared to provide students with a structure for evaluating arguments and counterarguments.
Books by Ian Dove
Although the Toulmin model (1958) has dominated argumentation research, it does not provide many tools for evaluating argument quality. Towards that end, we draw on work in philosophy on argument schemes, and critical questions for evaluating those schemes. In our approach, we integrate the teaching of critical questions with argumentation vee diagrams (AVDs) and with oral and written discourse. AVDs are graphic organizers that prompt students to write arguments and counterarguments on different sides of the vee, and at the bottom of the vee, an integrating paragraph supporting a final conclusion. The present study was conducted in three sections of an undergraduate general education seminar. Two sections, comprising the experimental group, used AVDs containing a critical questions box reflecting questions for arguments from consequences (Walton, 1996). One section used AVDs without the critical question box. Students completed AVDs prior and during class discussions on social issues (e.g., drug legalization). Over time, students in the experimental group included increasingly more refutations related to the critical questions compared to the control group. The effect transferred to an in-class essay where no question prompts were provided, but not to a course paper written on whistleblowing. However, students in the experimental condition did include in their papers more explicit discussion of moral principles. We explain these effects in relation to argument schema theory, in particular the development and automation of a weighing schema. The critical questions appeared to provide students with a structure for evaluating arguments and counterarguments.