- I received my Ph.D. in topological semigroups from the University of Georgia in 1963. I am a 5th generation Moore st... moreI received my Ph.D. in topological semigroups from the University of Georgia in 1963. I am a 5th generation Moore student [R. L. Moore, Gordon Whyburn, Alexander Donaphin Wallace, Robert Koch, Robert P. Hunter, John Selden]. I have taught at various universities in the U.S., Canada, Turkey, and Nigeria, and have directed 9 Ph.D.'s in mathematics. I served as Dean of Science, Bayero University in Nigeria, and upon returning to the U.S., took up research in undergraduate mathematics education. I was a member of the AMS/MAA Joint Committee on Research in Undergraduate Mathematics Education (CRUME) and various other MAA Committees. I was a Visiting Scholar at Peabody College; Vanderbilt, at the Division of Education in Mathematics, Science, and Technology, University of California at Berkeley; at the Center for Research in Mathematics and Science Education, San Diego State University; and the Department of Mathematics, Arizona State University. I am currently Adjunct Professor of Mathematics, New Mexico State University.edit
This paper is based on my Topic Session presentation to the Canadian Mathematics Education Study Group in Montreal in June 2017.
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We document Alice's progression with proof-writing over two semesters. We analyzed videotapes of her one-on-one sessions working through the course notes for our inquiry-based transition-to-proof course. Our theoretical perspective... more
We document Alice's progression with proof-writing over two semesters. We analyzed videotapes of her one-on-one sessions working through the course notes for our inquiry-based transition-to-proof course. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental and physical, actions. It also includes the use of proof frameworks as a means of getting started. Alice's early reluctance to use proof frameworks, after an initial introduction to them, is documented, as well as her subsequent acceptance of and proficiency with them by the end of the real analysis section of the course notes, along with a sense of self-efficacy. However, during the second semester, upon first encountering semigroups, with which she had no prior experience, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and regained a sense of self-efficacy. This case study focuses on how one non-traditional mature individual, Alice, in one-on-one sessions, progressed from an initial reluctance to use the technique of proof frameworks (Selden & Selden, 1995; Selden, Benkhalti, & Selden, 2014) to a gradual acceptance of, and eventual proficiency with, both writing proof frameworks and completing many entire proofs with familiar content.
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This case study continues the story of the development of Alice's proof-writing skills into the second semester. We analyzed the videotapes of her one-on-one sessions working through our inquiry-based transition-to-proof course notes. Our... more
This case study continues the story of the development of Alice's proof-writing skills into the second semester. We analyzed the videotapes of her one-on-one sessions working through our inquiry-based transition-to-proof course notes. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental, as well as physical, actions. It also includes the use of proof frameworks as a means of initiating a written proof. Previously, we documented Alice's early reluctance to use proof frameworks, followed by her subsequent seeming acceptance of, and proficiency with, them by the end of the first semester (Benkhalti, Selden, & Selden, 2016). However, upon first encountering semigroups, with which she had no prior experience, during the second semester, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and seemed to regain a sense of self-efficacy.
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This paper presents a theoretical perspective for understanding and teaching university students' proof construction. It includes features of proof texts with which students may be unfamiliar. It considers psychological aspects of proving... more
This paper presents a theoretical perspective for understanding and teaching university students' proof construction. It includes features of proof texts with which students may be unfamiliar. It considers psychological aspects of proving such as behavioral schemas, automaticity, working memory, consciousness, cognitive feelings, and local memory. We discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
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This case study elucidates the difficulty that university students' may have in unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure and facilitate constructing a proof.... more
This case study elucidates the difficulty that university students' may have in unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure and facilitate constructing a proof. This situation is illustrated with the case of Dori who encountered just such a difficulty with a hidden double negative. She was taking a transition-to-proof course that began by having students first prove formally worded " if-then " theorem statements that enabled them to construct proof frameworks, and thereby make initial progress on constructing proofs. However, later, students, such as Dori, were presented with more informally worded theorem statements to prove. We discuss what additional linguistic difficulties students might have when interpreting informally worded theorem statements and structuring their proofs. This paper sits at the border between linguistics and mathematics education. It considers linguistic obstacles that university students often have when unpacking informally worded mathematical statements into their formal equivalents. This can become especially apparent when students are attempting to prove such statements. We illustrate this with an example from Dori, who was taking a transition-to-proof course that began by having students construct proofs for formally worded " if, then " theorem statements. Early on, she was introduced to the idea of constructing proof frameworks (Selden & Selden, 1995, 2015) and was successful. Later, she encountered difficulty when attempting to interpret and prove an informally worded statement with a hidden double negative. Theoretical Perspective We adopt the theoretical perspective of Selden and Selden (2015) and consider a proof construction to be a sequence of mental or physical actions, some of which do not appear in the final written proof text. Each action is driven by a situation in the partly completed proof construction and its interpretation. For example, suppose that in a partly completed proof, there is an " or " in the hypothesis of a statement yet to be proved: If A or B, then C. Here, the situation is having to prove this statement. The interpretation is realizing that C can be proved by cases. The action is constructing two independent sub-proofs; one in which one supposes A and proves C, the other in which one supposes B and proves C. A proof can also be divided into a formal-rhetorical part and a problem-centered part. The formal-rhetorical part is the part of a proof that depends only on unpacking and using the logical structure of the statement of the theorem, associated definitions, and earlier results. In general, this part does not depend on a deep understanding of, or intuition about, the concepts involved or on genuine problem solving in the sense of Schoenfeld (1985, p. 74). The remaining part of a proof has been called the problem-centered part. It is the part that does depend on genuine problem solving, intuition, heuristics, and a deeper understanding of the concepts involved (Selden & Selden, 2013).
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This theoretical paper suggests a perspective for understanding university students’ proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working... more
This theoretical paper suggests a perspective for understanding university students’ proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we will discuss proving actions, such as the construction of proof frameworks, that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
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This paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students' reading, we... more
This paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students' reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. These students had high ACT mathematics and high ACT reading comprehension test scores and used many of the helpful metacognitive strategies developed in reading comprehension research. However, they were not effective readers of their mathematics textbooks. In discussing this, we draw on the psychology literature to suggest that cognitive gaps, that is, periods of lapsed or diminished focus, during reading may explain some of the ineffectiveness of the students' reading. Finally, we suggest some implications for teaching and pose questions for future research.
We describe the practices of a team of U.S. university teacher/researchers who were invited to attempt to alleviate students’ proving difficulties in an undergraduate real analysis course by offering a voluntary “proving skills... more
We describe the practices of a team of U.S. university teacher/researchers who were invited to attempt to alleviate students’ proving difficulties in an undergraduate real analysis course by offering a voluntary “proving skills supplement.” We analyze what happened in the supplement and why it happened in terms of our theoretical perspective concerning actions in the proving process. This perspective includes that the proving process is a sequence of actions, some of which are not visible or are difficult to recall, and that understanding the justification for an action differs from a tendency to execute it autonomously. Also, the real analysis course and that teacher’s somewhat traditional style of teaching are briefly described, and a comparison is made between proofs co-constructed in the supplement and proofs assigned in the real analysis course. Finally, some student difficulties, views of the supplement given by three students, and the effect of the supplement are briefly discussed.
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In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our... more
In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to express. Next we introduce the idea of behavioral schemas as enduring mental structures that link situations to actions, in short, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment – a course intended to help advanced undergraduate and beginning graduate students to improve their proving abilities.
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This is the abstract for a largely theoretical paper in which we planned to discuss how one kind of affect, specifically feelings of rightness, appropriateness, caution, etc., can both arise from and contribute to reasoning " moves... more
This is the abstract for a largely theoretical paper in which we planned to discuss how one kind of affect, specifically feelings of rightness, appropriateness, caution, etc., can both arise from and contribute to reasoning " moves " in the process of proving theorems. This will be illustrated by the cases of Katherine, a graduate student, and Edward, an advanced undergraduate student. But first we will describe a design experiment, from which the data arose, and discuss the nature of feelings. The course The design experiment consisted of a modified R. L. Moore method course for beginning graduate and advanced undergraduate students, the purpose of which was to improve proving ability. Video recordings and field notes were made and a theoretical framework started to emerge, only two ideas of which will be useful here. First, proofs have a problem-oriented part in which conceptual understanding is used to solve a problem and a formal-rhetorical part that can be based only ...
In this paper, we describe preliminary results arising from the development of a modified R. L. Moore Method course devoted entirely to helping advanced university mathematics students improve their proving abilities. The paper describes... more
In this paper, we describe preliminary results arising from the development of a modified R. L. Moore Method course devoted entirely to helping advanced university mathematics students improve their proving abilities. The paper describes the course and why it might be needed. We also discuss kinds and aspects of proofs in a way that may be useful in gauging student progress, and in particular, introduce the idea of the formal-rhetorical and problem-centered parts of a proof. We go on to propose a theoretical perspective suggesting that much of proving depends on procedural knowledge in the form of small habits of mind, or behavioral schemas, and give three examples. This paper reports on results arising from an ongoing design experiment. The purpose of the experiment is to examine ways advanced undergraduate and beginning graduate mathematics students construct, and learn to construct, proofs and to design a course to facilitate that learning. The paper has a research orientation, b...
In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our... more
In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to express. Next we introduce the idea of behavioral schemas as enduring mental structures that link situations to actions, in short, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment – a course intended to help advanced undergraduate and beginning graduate students to improve their proving abilities. Finally, drawing on data from the course, along with several interviews, we illustrate how these perspectives on affect and on behavioral schemas appear to explain, and are consistent with, our students’ actions.
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In most mathematics courses students learn the subject matter of the course and also improve their skill at reasoning. Usually the subject matter is emphasized with the improvement in reasoning ability occurring in a natural but unplanned... more
In most mathematics courses students learn the subject matter of the course and also improve their skill at reasoning. Usually the subject matter is emphasized with the improvement in reasoning ability occurring in a natural but unplanned way. In this paper we give a brief description of a course in which we have reversed the traditional emphasis, placing more stress on improvement of reasoning ability. We then give a classification of student reasoning errors with examples. The purpose of this classification, and indeed of the course described here, is to find a way to remove these errors as early and efficiently as possible thereby accelerating students' development.
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This article sets the stage for the following 3 articles. It opens with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of... more
This article sets the stage for the following 3 articles. It opens 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 ...
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ABSTRACT
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We describe a perspective and a framework for understanding the role of nonemotional cognitive feelings in proving theorems. We begin with a brief discussion of the nature of affect, emotions, and nonemotional cognitive feelings. We see... more
We describe a perspective and a framework for understanding the role of nonemotional cognitive feelings in proving theorems. We begin with a brief discussion of the nature of affect, emotions, and nonemotional cognitive feelings. We see kinds of situations as mentally linked to kinds of feelings that then participate in enacting behavioral schemas to yield actions.
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Abstract: This exploratory study examined the experiences and difficulties certain first-year university students displayed in reading new passages from their mathematics textbooks. We interviewed eleven precalculus and calculus students.
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This partly empirical, partly theoretical paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness... more
This partly empirical, partly theoretical paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students’ reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. These students had high ACT mathematics and high ACT reading comprehension test scores and used many of the helpful metacognitive strategies developed in reading comprehension research. However, they were not effective readers of their mathematics textbooks. In discussing this, we draw on the psychology literature to suggest that cognitive gaps, that is, periods of lapsed or diminished focus, during reading may explain some of the ineffectiveness of the students’ reading. Finally, we suggest some implications for teaching and pose questions for future research.
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In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our... more
In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to express. Next we introduce the idea of behavioral schemas as enduring mental structures that link situations to actions, in short, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment – a course intended to help advanced undergraduate and beginning
graduate students to improve their proving abilities. Finally, drawing on data from the course, along with several interviews, we illustrate how these perspectives on affect and on behavioral schemas appear to explain, and are consistent with, our students’ actions.
graduate students to improve their proving abilities. Finally, drawing on data from the course, along with several interviews, we illustrate how these perspectives on affect and on behavioral schemas appear to explain, and are consistent with, our students’ actions.
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In this paper we describe a number of types of errors and underlying misconceptions that arise in mathematical reasoning. Other types of mathematical reasoning errors, not associated with specific misconceptions, are also discussed. We... more
In this paper we describe a number of types of errors and underlying misconceptions that arise in mathematical reasoning. Other types of mathematical reasoning errors, not associated with specific misconceptions, are also discussed. We hope the characterization and cataloging of common reasoning errors will be useful in studying the teaching of reasoning in mathematics.
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We first discuss our perspective on, and three useful actions in, proof constructions that depend on persistence. Persistence is often important for successful proving because it helps one “explore,” including making arguments in... more
We first discuss our perspective on, and three useful actions in, proof constructions that depend on persistence. Persistence is often important for successful proving because it helps one “explore,” including making arguments in directions of unknown value, until one ultimately makes progress. Persistence can be supported by a sense of self-efficacy, which is “a person’s belief in his or her ability to succeed in a particular situation” (Bandura, 1995). We then examine the actions of one mathematician when proving a theorem that had previously been given to mid-level undergraduates in a transition-to-proof course. We end with some teaching implications.
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In this chapter, we introduce some concepts for analyzing proofs, including various structures, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We then discuss how the coordination of these... more
In this chapter, we introduce some concepts for analyzing proofs, including
various structures, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We then discuss how the coordination of these two analyses might be used to improve students’ ability to construct proofs.
For this purpose, we need a richer framework for keeping track of students’
progress than the everyday one. We need to know more than that a particular student can, or cannot, prove theorems by induction or contradiction or can, or cannot, prove certain theorems in beginning set theory or analysis. It will be more useful to describe a student's work in terms of a finer-grained framework including various smaller abilities that contribute to proving and that may be learned in differing ways and at differing periods of a student’s development.
Developing a fine-grained framework for analyzing students’ abilities is not an
especially novel idea. In working with higher primary and secondary students, Gutiérrez and Jaime (1998) developed a fine-grained framework of reasoning processes in order to more accurately and easily assess student van Hiele levels.
various structures, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We then discuss how the coordination of these two analyses might be used to improve students’ ability to construct proofs.
For this purpose, we need a richer framework for keeping track of students’
progress than the everyday one. We need to know more than that a particular student can, or cannot, prove theorems by induction or contradiction or can, or cannot, prove certain theorems in beginning set theory or analysis. It will be more useful to describe a student's work in terms of a finer-grained framework including various smaller abilities that contribute to proving and that may be learned in differing ways and at differing periods of a student’s development.
Developing a fine-grained framework for analyzing students’ abilities is not an
especially novel idea. In working with higher primary and secondary students, Gutiérrez and Jaime (1998) developed a fine-grained framework of reasoning processes in order to more accurately and easily assess student van Hiele levels.
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This paper presents the results of an empirical study of the proof validation behaviors of sixteen undergraduates after taking a transition-to-proof course that emphasized proof construction. Students were interviewed individually towards... more
This paper presents the results of an empirical study of the proof validation behaviors of sixteen undergraduates after taking a transition-to-proof course that emphasized proof construction. Students were interviewed individually towards the end of the course using the same protocol used by Selden and Selden (2003) at the beginning of a similar course. Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the suggestion that taking a transition-to-proof course does not seem to enhance students’ validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and point out the need for future research on how these concepts are related.
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We present the results of a study of the observed proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. Students were interviewed individually towards the end of the... more
We present the results of a study of the observed proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. Students were interviewed individually towards the end of the course using the same protocol that we had used earlier at the beginning of a similar course (Selden and Selden, 2003). Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course does not seem to enhance validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelations.
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Tertiary mathematics education research is disciplined inquiry into the learning and teaching of mathematics at the university level. It can be conducted from an individual cognitive perspective or from a social perspective of the... more
Tertiary mathematics education research is disciplined inquiry into the learning and teaching of mathematics at the university level. It can be conducted from an individual cognitive perspective or from a social perspective of the classroom or broader community. It can also coordinate the two, providing insight into how the psychological and social perspectives relate to and affect one another.
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Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more... more
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations)
to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our
name for the process by which proofs are read and checked.
First, we will discuss the style in which mathematical proofs are traditionally written and its apparentutility for reducing validation errors. We will then examine the relationship between the need for logic invalidating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses donot seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we
will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning.
Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will Þnd it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reßection, unpacking the meaning of statements,
and the distinction between procedural and conceptual knowledge.
to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our
name for the process by which proofs are read and checked.
First, we will discuss the style in which mathematical proofs are traditionally written and its apparentutility for reducing validation errors. We will then examine the relationship between the need for logic invalidating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses donot seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we
will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning.
Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will Þnd it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reßection, unpacking the meaning of statements,
and the distinction between procedural and conceptual knowledge.
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This dissertation consists of an Introduction; Chapter 1 titled "Rings, Groups, Semigroups"; Chapter 2 titled "Semirings"; Chapter 3 titled "Gamma-Compact Semirings", and a Bibliography.
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A clan is a compact connected semigroup with unit. We show the question "Which clans are groups?" has a purely topological answer.
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Introduction. Suppose that 5 is a topological semigroup which contains the bicyclic semigroup B as a subsemigroup. Let T denote the closure of B in 5. We investigate the structure of the semigroup T and the extent to which B determines... more
Introduction. Suppose that 5 is a topological semigroup which contains the bicyclic semigroup B as a subsemigroup. Let T denote the closure of B in 5. We investigate the structure of the semigroup T and the extent to which B determines this structure. In §1, two properties of F are established which hold for arbitrary 5; namely, that B is a discrete open subspace of F and T\B is an ideal of T if it is nonvoid. In §11, we introduce the notion of a topological inverse semigroup and establish several properties of such objects. Some questions are posed. In §111, it is shown that if 5 is a topological inverse semigroup, then T\B is a group with a dense cyclic subgroup. §IV contains a description of three examples of a topological semigroup which contains B as a dense proper subsemigroup. Finally, in §V, we assume that 5 is a locally compact topological inverse semigroup and show that either B is closed in 5 or F is isomorphic with the last of the examples described in §IV. A corollary about homomorphisms from B into a locally compact topo-logical inverse semigroup is obtained which generalizes a result due to A. Weil [1, p. 96] concerning homomorphisms from the integers into a locally compact group. All spaces are topological Hausdorff in this paper.
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Several papers have described the structure of topological semigroups containing a dense copy of the bicyclic semigroup, or variants thereof. Eberhart and J. Selden [2] showed that only one nontrivial structure is possible when the... more
Several papers have described the structure of topological semigroups containing a dense copy of the bicyclic semigroup, or variants thereof. Eberhart and J. Selden [2] showed that only one nontrivial structure is possible when the bicyclic semigroup is densely embedded in a locally compact topological inverse semigroup. When A. Selden [5] studied the case of bisimple -semigroups densely embedded in locally compact topological inverse semigroups, a variety of structures proved possible and a complete characterization was obtained only when additional hypotheses were considered. Ahre [1] examined possible closures of the continuous version of the bicyclic semigroup. In the present paper we give a complete description of all possible structures for locally compact topological inverse semigroups T containing a dense copy X of the extended bicyclic semigroup. In such semigroups, X is an open and dense subsemigroup, endowed with the discrete topology; to this subsemigroup can be attached a minimal ideal K and a group of units H. If they exist H and K are both closed, K is isomorphic to the additive group of integers Z and H is isomorphic to a subgroup of Z. Although K can be attached to X in precisely one way, H can be attached in a variety of ways. Our main result is a theorem characterizing T as one of several kinds of examples. Amongst these, two are isomorphic as algebraic semigroups and homeomorphic as topological spaces but are not isomorphic as topological semigroups. This is a consequence of the subtle interaction of algebra and topology near the group of units. Ruppert [4, I, 4.14] described compact semitopological semigroups containing a dense copy of the bicyclic semigroup. This theorem suggests that a result analogous to the one presented here might be obtained in the semitopological setting. However, any such description would be somewhat different from ours because the extended bicyclic semigroup is not compactifiable as a topological semigroup.
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A topological semiring is a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, x(y+z) =xy+xz and (x+y)z = xz+yz for all... more
A topological semiring is a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, x(y+z) =xy+xz and (x+y)z = xz+yz for all x, y, and z in s . Note that, in contrast to the purely algebraic situation [l], [2], we do not postulate the existence of an additive identity which is a multiplicative zero. In this note we characterize compact additively commutative semirings which are multiplicatively left zero simple. This is accomplished by first examining semirings which are multiplicatively groups with zero and then proceeding to the general situation. For other remarks on compact semirings the reader is referred to [3], [4]. The notation follows closely that of topological semigroups [5].
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By a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that x(y + z) = xy +... more
By a topological semiring we mean a Hausdorff space S together with two
continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that
x(y + z) = xy + xz and (x + y) z = xz + yz for all x,y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.
In this note we point out conditions under which the existence of such an
element is equivalent to the double simplicity of the semiring. We also discuss
maximal and minimal double ideals together with several examples.
continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that
x(y + z) = xy + xz and (x + y) z = xz + yz for all x,y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.
In this note we point out conditions under which the existence of such an
element is equivalent to the double simplicity of the semiring. We also discuss
maximal and minimal double ideals together with several examples.
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By a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that x(y+z)=xy+xz and... more
By a topological semiring we mean a Hausdorff space S together
with two continuous associative operations on S such that one (called
multiplication) distributes across the other (called addition). That
is, we insist that x(y+z)=xy+xz and (x+y)z = xz+yz for all x, y,
and z in S. Note that, in contrast to the purely algebraic situation,
we do not postulate the existence of an additive identity which is a
multiplicative zero.
In this note we point out a rather weak multiplicative condition
under which each additive subgroup of a compact semiring is totally
disconnected. We also give several corollaries and examples.
with two continuous associative operations on S such that one (called
multiplication) distributes across the other (called addition). That
is, we insist that x(y+z)=xy+xz and (x+y)z = xz+yz for all x, y,
and z in S. Note that, in contrast to the purely algebraic situation,
we do not postulate the existence of an additive identity which is a
multiplicative zero.
In this note we point out a rather weak multiplicative condition
under which each additive subgroup of a compact semiring is totally
disconnected. We also give several corollaries and examples.
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A topological semiring is a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is,x(y+z) =xy+xz and (x+y)z = xz+yz for all... more
A topological semiring is a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is,x(y+z) =xy+xz and (x+y)z = xz+yz for all x, y, and z in S. Note that, in contrast to the purely algebraic situation [l], [2], we do not
postulate the existence of an additive identity which is a multiplicative
zero.
In this note we characterize compact additively commutative semirings which are multiplicatively left zero simple. This is accomplished by first examining semirings which are multiplicatively groups with zero and then proceeding to the general situation. For other remarks on compact semirings the reader is referred to [3], [4].
The notation follows closely that of topological semigroups [5].
postulate the existence of an additive identity which is a multiplicative
zero.
In this note we characterize compact additively commutative semirings which are multiplicatively left zero simple. This is accomplished by first examining semirings which are multiplicatively groups with zero and then proceeding to the general situation. For other remarks on compact semirings the reader is referred to [3], [4].
The notation follows closely that of topological semigroups [5].
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We answer the question, "Which clans are groups?"
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This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroupin a locally compact topological inverse semigroup. In it we characterize all... more
This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroupin a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free
one-parameter inverse semigroups and then describe their congruences.
one-parameter inverse semigroups and then describe their congruences.
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These notes were typed up by our former graduate student, Milos Savic, at the time. They are rather "bare bones", as not much explanation is given. There are just definitions, statements of theorems, and questions for the students to work... more
These notes were typed up by our former graduate student, Milos Savic, at the time. They are rather "bare bones", as not much explanation is given. There are just definitions, statements of theorems, and questions for the students to work on. The students present their proofs in class and we critique them rather extensively. The student who has presented a proof then writes it up, using the critique, hands it in, and we duplicate copies for the students. That way, each student has at least one correct proof for each theorem in the notes by the end of the semester.
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This submission for ICME-11, Topic Study Group 18 on reasoning, proof, and proving, concerns both the teaching of and the cognitive aspects of reasoning, proof, and proving (RPP). We have two aims. First, we aim to describe a new and... more
This submission for ICME-11, Topic Study Group 18 on reasoning, proof, and proving, concerns both the teaching of and the cognitive aspects of reasoning, proof, and proving (RPP). We have two aims. First, we aim to describe a new and unusual mathematics course for graduate and advanced undergraduate students. The entire purpose of this course is to help students improve their proving ability. This contrasts with other graduate and upper division university courses, whose purpose is normally mostly to inform students about some mathematical topic, such as real analysis or abstract algebra. Second, treating the development of the course as a design experiment, we aim to report on a theoretical framework emerging from the data and several other preliminary findings. In the following synopsis, we omit numerous references, examples, and excerpts of transcripts which would be included in a longer submission.
This is a small size version of the poster we presented. In this poster, we point out a way consciousness is necessary for a particular common kind of mathematical reasoning. To describe such mathematical reasoning we need to first give... more
This is a small size version of the poster we presented. In this poster, we point out a way consciousness is necessary for a particular common kind of mathematical reasoning. To describe such mathematical reasoning we need to first give the perspective in which we will describe it.
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We argue that: Mathematical proofs are written in a distinctive style that is part of the implicit curriculum and that is probably useful in avoiding certain validation errors.
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For more than ten years, Annie Selden and I have co-taught a small experimental course for beginning mathematics graduate students who felt they needed help with proof construction. In the course, students are provided a variety of... more
For more than ten years, Annie Selden and I have co-taught a small experimental course for beginning mathematics graduate students who felt they needed help with proof construction. In the course, students are provided a variety of definitions and theorems, and with some advice, construct their proofs. I describe some student proving difficulties that we have observed and do so from an easily understood psychological perspective that we are finding useful.
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Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. We... more
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. We understand that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof framework. Generating a first-level proof framework is often easy, provided the theorem is stated in the standard “If …, then …” form. However, formulating a second-level proof framework requires knowing how to use the relevant mathematical definitions, that is, being able to put them in an operable form. For example, the definition of the inverse image of a set D under a function f:X→Y is usually given as f -1 (D) = {x∊X| f(x)∊D}. However, in constructing a proof, one needs to be able to use this in an operable way: If a∊ f -1 (D), then one can say f(a)∊D, and conversely, if f(a)∊D, then a∊ f -1(D). This may seem obvious, but it not obvious for some beginning students.
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Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our... more
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our understanding that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof framework. Generating a first-level proof framework is often easy, provided the theorem is stated in the standard “If …, then …” form. However, formulating a second-level proof framework requires knowing how to use the relevant mathematical definitions, that is, being able to put them in an operable form. For example, the definition of the inverse image of a set D under a function f:X→Y is usually given as f -1 (D) = {x∊X| f(x)∊D}. However, in constructing a proof, one needs to be able to use this in an operable way: If a∊ f -1 (D), then one can say f(a)∊D, and conversely, if f(a)∊D, then a∊ f -1(D). This may seem obvious for us, but it not obvious for some beginning students.
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This case study continues the story of the development of Alice’s proof-writing skills into the second semester. We analyzed the videotapes of her one-on-one sessions working through our inquiry-based transition-to-proof course notes. Our... more
This case study continues the story of the development of Alice’s proof-writing skills into the second semester. We analyzed the videotapes of her one-on-one sessions working through our inquiry-based transition-to-proof course notes. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental, as well as physical, actions. It also includes the use of proof frameworks as a means of initiating a written proof. Previously, we documented Alice’s early reluctance to use proof frameworks, followed by her subsequent seeming acceptance of, and proficiency with, them by the end of the first semester (Benkhalti, Selden, & Selden, 2016). However, upon first encountering semigroups, with which she had no prior experience, during the second semester, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and seemed to regain a sense of self-efficacy.
Research Interests: Proof and Reasoning, Proof and Proving in Mathematics Education, Mathematical reasoning and proof, Research of Mathematics Education, Research in Mathematics education, and 3 moreEducational Research in Mathematics, Research in Undergraduate Mathematics Education, and Researches in Mathematics Education
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our... more
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our understanding that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof framework. Generating a first-level proof framework is often easy, provided the theorem is stated in the standard “If …, then …” form. However, formulating a second-level proof framework requires knowing how to use the relevant mathematical definitions, that is, being able to put them in an operable form. For example, the definition of the inverse image of a set D under a function f:X→Y is usually given as f -1 (D) = {x∊X| f(x)∊D}. However, in constructing a proof, one needs to be able to use this in an operable way: If a∊ f -1 (D), then one can say f(a)∊D, and conversely, if f(a)∊D, then a∊ f -1(D). This may seem obvious for us, but it not obvious for some beginning students.
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This presentation considers the difficulty that university students may have when unpacking an informally worded statement into its formal equivalent when attempting a proof. This difficulty is illustrated with the case of Dori who... more
This presentation considers the difficulty that university students may have when unpacking an informally worded statement into its formal equivalent when attempting a proof. This difficulty is illustrated with the case of Dori who encountered just such a difficulty with a hidden double negative.
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In this largely theoretical paper we discuss how one kind of affect, specifically feelings of rightness, appropriateness, caution, etc., can both arise from and contribute to reasoning “moves” in the process of proving theorems. This will... more
In this largely theoretical paper we discuss how one kind of affect, specifically feelings of rightness, appropriateness, caution, etc., can both arise from and contribute to reasoning “moves” in the process of proving theorems. This will be illustrated by the cases of Katherine, a graduate student, and Edward, an advanced undergraduate student. But first we will describe a design experiment, from which the data arose, and discuss the nature of feelings.
This Powerpoint presentation discusses university students’ mathematical and non-mathematical proving difficulties. A total of over forty proving difficulties have been observed and organized into nine categories. Of these difficulties,... more
This Powerpoint presentation discusses university students’ mathematical and non-mathematical proving difficulties. A total of over forty proving difficulties have been observed and organized into nine categories. Of these difficulties, more than twenty will be described. These observations come from several years of teaching an experimental proving course to beginning graduate and advanced undergraduate mathematics students and from teaching an experimental voluntary proving supplement to an undergraduate real analysis course. We believe that discussing and categorizing these difficulties will lead to a greater understanding of students’ thinking with regard to proof and potentially to better teaching.
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Persistence is important to successful non-routine problem solving and proving. It allows one to continue despite making arguments in seemingly unhelpful directions until one ultimately makes progress and comes to a successful result.... more
Persistence is important to successful non-routine problem solving and proving. It allows one to continue despite making arguments in seemingly unhelpful directions until one ultimately makes progress and comes to a successful result. Persistence is supported by a self-efficacy belief, which is“a person’s belief in his or her ability to succeed in a particular situation” (Bandura, 1995). We present three observations spanning university level mathematics from beginning undergraduates solving just explained, straightforward exercises, to actions needed in proof construction in mid-level U.S. university transition-to-proof courses, to the actual proof construction processes of mathematicians.
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We analyzed undergraduate students’ examination papers from a transition-to-proof course. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework first, not unpacking the... more
We analyzed undergraduate students’ examination papers from a transition-to-proof course. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework first, not unpacking the conclusion, and not using definitions correctly. Examples of these difficulties were presented.
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We analyzed undergraduate students’ examination papers from a transition-to-proof course. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework first, not unpacking the... more
We analyzed undergraduate students’ examination papers from a transition-to-proof course. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework first, not unpacking the conclusion, and not using definitions correctly. Examples of these difficulties will be presented.
First, we discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times since Fall 2007 and each time we are... more
First, we discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times since Fall 2007 and each time we are learning something more about students’ proving capabilities and difficulties. For example, there are certain aspects of proving that mathematicians do automatically, but that students are often unaware of. We define the formal-rhetorical part of a proof to be those aspects of a proof that can be written by examining the logical structure of the statement of a theorem and by unpacking associated definitions. This is something we have called a proof framework. Examples include writing the first and last lines, “unpacking” the meaning of the last line, and considering what that means for the structure of a proof. Writing the formal-rhetorical part of a proof can expose "the real problem(s)" to be solved. We call the remainder of the proof the p...
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We report the results of a study of the proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. Students were interviewed individually towards the end of the course... more
We report the results of a study of the proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. Students were interviewed individually towards the end of the course using the same protocol that we had used earlier at the beginning of a similar course (Selden and Selden, 2003). Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course does not seem to enhance students’ validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelation.
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Students find many aspects of mathematical proof confusing. One area that they find especially perplexing is the manner in which proofs are written, which is often at variance with other genres of writing. Nardi and Iannone (2006) claimed... more
Students find many aspects of mathematical proof confusing. One area that they find especially perplexing is the manner in which proofs are written, which is often at variance with other genres of writing. Nardi and Iannone (2006) claimed that mastering proof involves acquiring a different genre of communication.
In the mathematics education literature, a variety of genres of mathematical writing have been considered. For example, in discussing mathematical writing at the school level, Marks and Mousley (1990, p. 119) distinguished narrative genre, procedural genre, description and report genres, exploratory genre, and expository genre. While all these have a place when examining mathematical writing more generally, in this short contribution, we restrict our considerations to the genre of proof, and in particular, to how mathematicians write proofs for publication. As Konior (1993) argued, studying the genre of mathematical proof is particularly important for mathematics educators, as this can inform how students should read and write proofs.
Mathematics educators, mathematicians, and philosophers have written about the genre of mathematical proof, emphasizing it special nature, its long evolution, and the impossibility of making it entirely explicit. For example, Ernest (1998, p. 169) stated, “Mathematical proof is a special form of text, which since the time of the ancient Greeks, has been presented in monological [rather than dialogical] form.” Jaffe (1990, p. 146) asserted that “The standards of what constitutes a proof have evolved over hundreds of years; there is no doubt in the minds of traditional mathematicians what a proof means.” Furthermore, according to Kitcher (1984, p. 163), in mathematical practice both tacit knowledge, or “know how,” and meta-mathematical views (including standards for proof) are important, and it is not possible for those standards to be made fully explicit. However, it may be possible to identify some significant features that generally occur in the genre of proofs.
In the mathematics education literature, a variety of genres of mathematical writing have been considered. For example, in discussing mathematical writing at the school level, Marks and Mousley (1990, p. 119) distinguished narrative genre, procedural genre, description and report genres, exploratory genre, and expository genre. While all these have a place when examining mathematical writing more generally, in this short contribution, we restrict our considerations to the genre of proof, and in particular, to how mathematicians write proofs for publication. As Konior (1993) argued, studying the genre of mathematical proof is particularly important for mathematics educators, as this can inform how students should read and write proofs.
Mathematics educators, mathematicians, and philosophers have written about the genre of mathematical proof, emphasizing it special nature, its long evolution, and the impossibility of making it entirely explicit. For example, Ernest (1998, p. 169) stated, “Mathematical proof is a special form of text, which since the time of the ancient Greeks, has been presented in monological [rather than dialogical] form.” Jaffe (1990, p. 146) asserted that “The standards of what constitutes a proof have evolved over hundreds of years; there is no doubt in the minds of traditional mathematicians what a proof means.” Furthermore, according to Kitcher (1984, p. 163), in mathematical practice both tacit knowledge, or “know how,” and meta-mathematical views (including standards for proof) are important, and it is not possible for those standards to be made fully explicit. However, it may be possible to identify some significant features that generally occur in the genre of proofs.
We discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times to date and... more
We discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times to date and each time we are learning something more about students’ proving capabilities. For example, there are certain aspects of proving that mathematicians do automatically, but that students are often unaware of. We define the formal-rhetorical part of a proof to be those aspects of a proof that can be written by examining the logical structure of the statement of a theorem and by unpacking associated definitions. Examples include writing the first and last lines, “unpacking” the meaning of the last line, and considering what strategy one might invoke to prove that. Writing the formal-rhetorical part of a proof can expose "the real problem(s)" to be solved. We call remainder of the proof the problem-centered part.
Students need to make writing the formal-rhetorical part of a proof automatic, that is, doing so must become a “habit of mind.” For example, beginning students often fail to examine the conclusion to see where they are going; instead, they begin with the hypotheses and forge ahead rather blindly. This is a "bad habit" that needs to be replaced by the "good habit" of examining and unpacking the conclusion. We will discuss what we have been learning about students’ proving capabilities, present a theoretical perspective that we have been developing, and indicate what we have been doing to help students succeed.
Students need to make writing the formal-rhetorical part of a proof automatic, that is, doing so must become a “habit of mind.” For example, beginning students often fail to examine the conclusion to see where they are going; instead, they begin with the hypotheses and forge ahead rather blindly. This is a "bad habit" that needs to be replaced by the "good habit" of examining and unpacking the conclusion. We will discuss what we have been learning about students’ proving capabilities, present a theoretical perspective that we have been developing, and indicate what we have been doing to help students succeed.
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ABSTRACT. We describe the practices of a team of US university teacher/researchers who were invited to attempt to alleviate students' proving difficulties in an undergraduate real analysis course by offering a voluntary "proving skills... more
ABSTRACT. We describe the practices of a team of US university teacher/researchers who were invited to attempt to alleviate students' proving difficulties in an undergraduate real analysis course by offering a voluntary "proving skills supplement."
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This presentation discussed the course we have developed to help beginning graduate students with proving.
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This presentation reports on the relationship between the logic taught and the logic used in students' proofs in one transition-to-proof course. In the U.S.A, such courses are often taken by second year mathematics majors and preservice... more
This presentation reports on the relationship between the logic taught and the logic used in students' proofs in one transition-to-proof course. In the U.S.A, such courses are often taken by second year mathematics majors and preservice secondary teachers. The focus is on teaching students to construct proofs of theorems. Typically logic is presented early in a somewhat abstract, decontextualized way and the students' experiences with theorem proving come later.
Having taught a variety of Moore Method mathematics courses -- topology, algebraic topology, abstract algebra, geometry, topological semigroups -- over many years, we have often wondered why they worked. Is there a theory, or a... more
Having taught a variety of Moore Method mathematics courses -- topology, algebraic topology, abstract algebra, geometry, topological semigroups -- over many years, we have often wondered why they worked. Is there a theory, or a mechanism, that might explain their effectiveness? The best of these courses seemed not only to teach students mathematics, but also about proving theorems, about themselves, and about what we might call the culture of mathematics.
When behaviorist psychology was in its heyday, one of us viewed this in terms of reinforcement. When a student found a proof and it was accepted, this amounted to a very positive reward. Having to admit that one did not have a proof, or presenting a proof with a mistake in it, was negative. The way conditioning works suggests, for example, emphasizing the positive and minimizing the negative experiences, even if negative experience might serve as a source of motivation.
While such analyses were of some use, behaviorist psychology limited itself to analyzing externally observable events. Consideration of what goes on inside the mind is more promising and no longer taboo. There are now more cognitively-based ways of looking at the teaching/learning of mathematics and a constructivist perspective may be helpful. For example, Vygotsky's Zone of Proximal Development (ZPD) may have some explanatory power relative to the Moore Method. Vygotsky was a contemporary of Piaget and the ZPD refers to what a student can do in the presence of a "teacher" but not alone. We will discuss how this and a number of other ideas from research in mathematics education might help to understand Moore Method teaching and the teaching of "transition" courses to mid-level undergraduate mathematics students.
When behaviorist psychology was in its heyday, one of us viewed this in terms of reinforcement. When a student found a proof and it was accepted, this amounted to a very positive reward. Having to admit that one did not have a proof, or presenting a proof with a mistake in it, was negative. The way conditioning works suggests, for example, emphasizing the positive and minimizing the negative experiences, even if negative experience might serve as a source of motivation.
While such analyses were of some use, behaviorist psychology limited itself to analyzing externally observable events. Consideration of what goes on inside the mind is more promising and no longer taboo. There are now more cognitively-based ways of looking at the teaching/learning of mathematics and a constructivist perspective may be helpful. For example, Vygotsky's Zone of Proximal Development (ZPD) may have some explanatory power relative to the Moore Method. Vygotsky was a contemporary of Piaget and the ZPD refers to what a student can do in the presence of a "teacher" but not alone. We will discuss how this and a number of other ideas from research in mathematics education might help to understand Moore Method teaching and the teaching of "transition" courses to mid-level undergraduate mathematics students.
We present the results of a study of eight mid-level undergraduates' validations (reading and checking) of four student-generated short arguments, all purported to be proofs of the same simple number theory theorem. The results suggest... more
We present the results of a study of eight mid-level undergraduates' validations (reading and checking) of four student-generated short arguments, all purported to be proofs of the same simple number theory theorem. The results suggest that mid-level undergraduates' ability to determine whether arguments are proofs is very limited -- perhaps more so than either they or their teachers recognize.
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Some surprising student results on tasks used in mathematics education research were presented and discussed, and it was proposed that tasks other than "Find" and "Solve" be asked of one's own students.
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We will discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate mathematics students construct proofs.
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The abstract begins: The most influential and widely accepted philosophical perspective in mathematics education today is constructivism. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget... more
The abstract begins: The most influential and widely accepted philosophical perspective in mathematics education today is constructivism. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget and beyond. It sees the learner as an active participant, not as a blank slate upon which we write. Cognition is considered adaptive, in the sense that it tends to organize experiences so they "fit" with a person's previously constructed knowledge. As a consequence, both researchers and teachers ask, "What is going on in students' minds when . . . ?", rather than speaking of behavioral outcomes and asking, "Which stimulus will elicit a desired response?"
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This abstract is for a talk that describes a course we have been developing for beginning mathematics graduate students.
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First, we will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs.
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We will discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times to date... more
We will discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times to date and each time we are learning something more about students' proving capabilities.
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Responses were collected during 1995 and 1996 from a broad range of academic mathematicians regarding their views of mathematics. This allowed for the formation of several small "group portraits," e.g., leaders in their (research)... more
Responses were collected during 1995 and 1996 from a broad range of academic mathematicians regarding their views of mathematics. This allowed for the formation of several small "group portraits," e.g., leaders in their (research) fields, state college faculty, junior college teachers, etc. -- it is these group portraits that we intend to describe.
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This is an abstract written for the College Mathematics Journal's Media Highlights section.
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We will describe an example of a supplement to a proof-based course, such as real analysis, in which students are having difficulties constructing proofs or getting started constructing proofs. The supplemented course can itself be taught... more
We will describe an example of a supplement to a proof-based course, such as real analysis, in which students are having difficulties constructing proofs or getting started constructing proofs. The supplemented course can itself be taught in any way, provided it requires students to construct proofs. In the example we will describe, we were invited by an NMSU teacher of first undergraduate real analysis to help her students with their proving skills. We provided a voluntary 75-minute supplement each week. Its sole purpose was to help students improve their proving skills.
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This is an abstract for a presentation on our beginning graduate transition-to-proof course.
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for JMM Atlanta, January 2017 Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior... more
for JMM Atlanta, January 2017 Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our understanding that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof frame
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We present the results of a survey of, and interviews with, practicing mathematicians on mathematics, whose views on the fallibility of mathematics differ from those of Paul Ernest (1991).
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This is an example of a proof framework for a simple set theory theory. It has both a first level and a second level framework. One gets the second level framework (which can be seen inside the box) by unpacking the conclusion.
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This was done by a student on the blackboard. The levels were indicated by her after the proof had been completed. She wanted to remind herself which was the Level 1 and which was the Level 2 proof framework. She indicated on the bottom... more
This was done by a student on the blackboard. The levels were indicated by her after the proof had been completed. She wanted to remind herself which was the Level 1 and which was the Level 2 proof framework. She indicated on the bottom that the Level 2 proof framework was gotten by unpacking the conclusion, which appears at the bottom of the Level 1 proof framework.
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Topics include: teachers, elementary level, secondary level, undergraduate and graduate level, workplace mathematics, experts' perspective and practice, theory in mathematics education, research techniques in mathematics education.
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This bibliography supplements the paper, “Acquainting graduate students with research in undergraduate mathematics education," delivered to the AMS Special Session on Preparing for Future College Teaching at the 1992 Annual Meeting of the... more
This bibliography supplements the paper, “Acquainting graduate students with research in undergraduate mathematics education," delivered to the AMS Special Session on Preparing for Future College Teaching at the 1992 Annual Meeting of the American Mathematical Society and the Mathematical Association of America in Baltimore, Maryland. It consists of those research articles used up to that date in the University of Kentucky seminar that we taught to acquaint mathematics graduate students with this literature
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This bibliography was prepared for the talk, "Constructivism in Mathematics Education -- What Does It Mean?" at the RCME Meeting, Central Michigan University, Sept. 5-8, 1996
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We think having a description of the structure of already written proofs might be useful in analyzing students' attempts to read proofs or construct their own proofs. We consider what we call a hierarchical structure based on... more
We think having a description of the structure of already written proofs might be useful in analyzing students' attempts to read proofs or construct their own proofs.
We consider what we call a hierarchical structure based on subproofs, subconstructions, and parallel arguments (such as case arguments). This is an extension of our proof framework idea and is illustrated by a proof of the theorem: is continuous at a point provided and are, which appears below in the Appendix A.
We consider what we call a hierarchical structure based on subproofs, subconstructions, and parallel arguments (such as case arguments). This is an extension of our proof framework idea and is illustrated by a proof of the theorem: is continuous at a point provided and are, which appears below in the Appendix A.
Here are some thoughts about how mathematical definitions might develop, with particular emphasis on the idea of covariation.
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This paper points out a number of features of the distinctive style in which proofs are written and links them to minimizing validation errors due to working memory overload, rather than the enhancement of insight or conceptual... more
This paper points out a number of features of the distinctive style in which proofs are written and links them to minimizing validation errors due to working memory overload, rather than the enhancement of insight or conceptual understanding. Although a theoretical paper, it suggests several empirical questions concerning the mathematics research literature and the role of working memory in validation errors.
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Some time ago we came across an e-preprint of David Tall and colleagues' paper, "What is the object of the encapsulation of a process?" 2. We found this paper interesting as we have been trying to get a more unified, coherent... more
Some time ago we came across an e-preprint of David Tall and colleagues' paper, "What is the object of the encapsulation of a process?" 2. We found this paper interesting as we have been trying to get a more unified, coherent understanding of some theoretical ideas in mathematics education. We describe two related ideas.
Research Interests:
In order to understand how summer workshops might affect college mathematics teaching, we would like to know something about your teaching, as well as about your views on teaching, students, and mathematics. As we cannot be sure of the... more
In order to understand how summer workshops might affect college mathematics teaching, we would like to know something about your teaching, as well as about your views on teaching, students, and mathematics. As we cannot be sure of the right questions to ask, we hope you will add comments. Anything you say may be helpful to us. In attempting to sort out college teachers' views on teaching mathematics, we will make no value judgements and, of course, everything you tell us will be strictly confidential.
Research Interests:
This draft version of our theoretical paper suggests a perspective for understanding university students' proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral... more
This draft version of our theoretical paper suggests a perspective for understanding university students' proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we will discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
Research Interests: Deductive reasoning, Theorem Proving, Proof and Reasoning, Proof and Proving in Mathematics Education, Teaching Mathematics, and 11 moreMathematics Teaching, Mathematical reasoning and proof, Research of Mathematics Education, Teaching and Learning Mathematics, Educational Research in Mathematics, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, Reasoning and Proof, Mathematical Reasoning and Proofs, Ressearch in Undergraduate Mathematics Education, and Proofs and Mathematical Reasoning
Curriculum reform projects seek to implement change in teaching approaches, curricula, etc. The ultimate objective is broad and permanent improvement of collegiate mathematics instruction, and consequently, an enlarged and improved... more
Curriculum reform projects seek to implement change in teaching approaches, curricula, etc. The ultimate objective is broad and permanent improvement of collegiate mathematics instruction, and consequently, an enlarged and improved technological work force (including women and minorities).
We propose to study the effects of one curriculum project workshop and subsequent implementation attempts on its college mathematics teacher-participants. ["Learning Abstract Algebra: A Research Based Laboratory and Cooperative Learning Approach,"
funded through NSF Research in Learning and Teaching Program, Ed Dubinsky, Purdue University, PI.] Specifically, we propose to study the teaching practices and personal pedagogies of the participants, i.e., their pedagogical and mathematical beliefs, attitudes, priorities, and behaviors. We plan to observe the workshop and evaluate how its college teacher-participants' personal pedagogies and behaviors are changed by it. We will gather interviews and self-reported data and evaluate participants' subsequent teaching practices, as well as their teaching environment and backgrounds in mathematics.
We propose to study the effects of one curriculum project workshop and subsequent implementation attempts on its college mathematics teacher-participants. ["Learning Abstract Algebra: A Research Based Laboratory and Cooperative Learning Approach,"
funded through NSF Research in Learning and Teaching Program, Ed Dubinsky, Purdue University, PI.] Specifically, we propose to study the teaching practices and personal pedagogies of the participants, i.e., their pedagogical and mathematical beliefs, attitudes, priorities, and behaviors. We plan to observe the workshop and evaluate how its college teacher-participants' personal pedagogies and behaviors are changed by it. We will gather interviews and self-reported data and evaluate participants' subsequent teaching practices, as well as their teaching environment and backgrounds in mathematics.
Research Interests:
Course description: The first third of this survey course will emphasize the concepts, perspectives, and vocabulary arising from research in undergraduate mathematics education. These will then be used during the remainder of the... more
Course description:
The first third of this survey course will emphasize the
concepts, perspectives, and vocabulary arising from
research in undergraduate mathematics education.
These will then be used during the remainder of the
course to examine and compare various forms of teaching
(and assessment) including: lecture, whole-class discussion,
group and collaborative work, problem solving, distance
learning, discovery learning, and the Moore method.
The course will also examine the use of writing, computers
and calculators, various kinds of software (including CAS's
and aids to visualization), student projects and labs, and
programming as an aid to conceptual understanding.
The course will meet for the equivalent of a 15-week
3-credit hour course and involve discussions (equivalent
to three hours per week), conducted over the internet, partly
asynchronously and partly synchronously. Where practical,
participants will work in small groups and may be asked to
discuss their own and each other's teaching. The course will
require at least six hours' reading per week and at its conclusion
each participant will write a brief (individually produced) paper.
The first third of this survey course will emphasize the
concepts, perspectives, and vocabulary arising from
research in undergraduate mathematics education.
These will then be used during the remainder of the
course to examine and compare various forms of teaching
(and assessment) including: lecture, whole-class discussion,
group and collaborative work, problem solving, distance
learning, discovery learning, and the Moore method.
The course will also examine the use of writing, computers
and calculators, various kinds of software (including CAS's
and aids to visualization), student projects and labs, and
programming as an aid to conceptual understanding.
The course will meet for the equivalent of a 15-week
3-credit hour course and involve discussions (equivalent
to three hours per week), conducted over the internet, partly
asynchronously and partly synchronously. Where practical,
participants will work in small groups and may be asked to
discuss their own and each other's teaching. The course will
require at least six hours' reading per week and at its conclusion
each participant will write a brief (individually produced) paper.
Research Interests:
Brief description of the 1991 study, "What is Mathematics to Children", published in Journal of Mathematical Behavior, Volume 10, pp 105-113.
Research Interests:
This short piece begins "For some in mathematics education research and the philosophy of mathematics, mathematics is a social process in which meaning is negotiated. Gila Hanna of the Ontario Institute for Studies in Education considers... more
This short piece begins "For some in mathematics education research and the philosophy of mathematics, mathematics is a social process in which meaning is negotiated. Gila Hanna of the Ontario Institute for Studies in Education considers rigorous proof as only one element in the acceptance of a theorem, and not the most important. More important for her are: (1) understanding the theorem and its implications, (2) significance of the theorem in its relation to various branches of mathematics, (3) compatibility of the theorem with other accepted mathematical results, (4) the reputation of the author, and (5) a convincing argument of a type encountered before, whether rigorous or not."