- Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM, USA
- Tennessee Technological University, Mathematics, Emeritusadd
- Research in Undergraduate Mathematics Education, Proof and Proving in Mathematics Education, Reading Comprehension, Proof and Reasoning, Procedural learning, Conceptual Learning, and 26 moreProblem solving (Cognitive Psychology), History of Reading, Readability, Mathematical Sciences, Calculus, Mathematical reasoning and proof, Logic Teaching, Researches in Mathematics Education, Pure Mathematics, Undergraduate Mathematics Education, Reading Research, Advanced Mathematical Thinking, University Mathematics Education, Tertiary Mathematics Education, Ressearch in Undergraduate Mathematics Education, Psychology, Communication, Discourse Analysis, Higher Education, Logical reasoning, Mathematics Education, Teaching and Learning Mathematics, Research of Mathematics Education, Mathematical Reasoning and Proofs, Deductive reasoning, and Mathematical Problem Solvingedit
- Fellow of the American Association for the Advancement of Science, elected 2003. Recipient of the 12th Annual Asso... moreFellow of the American Association for the Advancement of Science, elected 2003.
Recipient of the 12th Annual Association for Women in Mathematics' Louise Hay Award for Contributions to Mathematics Education, 2002.
Former Coordinator, Special Interest Group on Research in Undergraduate Mathematics Education (SIGMAA on RUME)edit - John Seldenedit
Inspired by Schoenfeld's resources, orientations, and goals (ROG) framework, we present a brief analysis of the goals we have for students in our inquiry-based (IBL) proofs course. These goals are described and divided into those related... more
Inspired by Schoenfeld's resources, orientations, and goals (ROG) framework, we present a brief analysis of the goals we have for students in our inquiry-based (IBL) proofs course. These goals are described and divided into those related to structuring the course notes and those more concerned with the day-today teaching of the course. Some of the goals are more cognitive and mathematical, while others are more affective and psychological, and this is indicated. We briefly relate our goals to our resources and orientations.
Research Interests:
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We... more
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on writing proofs. This is the technique of writing proof frameworks, based on the logical structure of the statements of the theorems and associated definitions. Also, in order to unpack the conclusion and know what is to be proved, students need definitions to become " operable " .
Research Interests:
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.
Research Interests:
Alice Slowly Develops Self-Efficacy with Writing Proof Frameworks, but Her Initial Progress and Sense of Self-Efficacy Evaporates When She Encounters Unfamiliar Concepts: However, It Eventually Returns. Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education.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... 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.
Research Interests:
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:
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
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.
Research Interests:
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.
Research Interests:
This discussion paper was written for the Advanced Mathematical Thinking Working (AMT) at the 14th Annual Conference of the International Group for the Psychology of Mathematics Education (PME). It deals with how undergraduate students... more
This discussion paper was written for the Advanced Mathematical Thinking Working (AMT) at the 14th Annual Conference of the International Group for the Psychology of Mathematics Education (PME). It deals with how undergraduate students learn to read, write, and create proofs.
Research Interests:
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.
Research Interests:
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.
Research Interests:
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.
Research Interests:
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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This paper is a report on a small benchmark study of the question: Can Average Calculus Students Work Nonroutine Problems?
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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.
Research Interests:
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.
Research Interests:
Research Interests:
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.
Research Interests:
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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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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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 is a report on two yearlong experimental courses and an analysis of the technology needed. The research aim of the project is to gather information useful in the development of future courses that produce students who can use their... more
This is a report on two yearlong experimental courses and an analysis of the technology needed. The research aim of the project is to gather information useful in the development of future courses that produce students who can use their knowledge in flexible and creative ...
Research Interests:
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).
Research Interests:
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and prepare for proof-based courses, such as abstract algebra and real analysis. We have... more
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and prepare for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on writing proofs. This is the technique of writing proof frameworks, based on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level proof framework.
Research Interests:
This theoretical paper considers several perspectives for understanding and teaching university students' autonomous proof construction. We describe the logical structure of statements, the formal-rhetorical part of a proof text, and... more
This theoretical paper considers several perspectives for understanding and teaching university students' autonomous proof construction. We describe the logical structure of statements, the formal-rhetorical part of a proof text, and proof frameworks. We view proof construction as a sequence of actions, and consider actions in the proving process, both situation-action pairs and behavioral schemas. We call on several ideas from the psychological literature and introduce the concept of local memory – a subset of memory that is partly activated during prolonged consideration of a proof.
Research Interests: Proof and Proving in Mathematics Education, Mathematical reasoning and proof, Research of Mathematics Education, Research in Mathematics education, Mathematics Education, Qualitative Research, Philosophy of Mathematics and Science, and 2 moreResearch in Undergraduate Mathematics Education and Researches in Mathematics Education
This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the... more
This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the literature, some conjectural. Lastly, it considers some related teaching implications and research.
Research Interests:
This case study documents the progression of one non-traditional individual’s proof-writing through a semester. We analyzed the videotapes of this individual’s one-on-one sessions working through our course notes for an inquiry-based... more
This case study documents the progression of one non-traditional individual’s proof-writing through a semester. We analyzed the videotapes of this individual’s one-on-one sessions working through our course notes for an inquiry-based transition-to-proof course. Our theoretical perspective informed our work with this individual and included the view that proof construction is a sequence of (mental, as well as physical) actions. It also included the use of proof frameworks as a means of initiating a written proof. This individual’s early reluctance to use proof frameworks, after an initial introduction to them, was documented, as well as her later acceptance of, and proficiency with, them. By the end of the first semester, she had developed considerable facility with both the formal-rhetorical and problem-centered parts of proofs and a sense of self-efficacy.
Research Interests:
We will describe a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical... more
We will describe a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical perspective (Selden & Selden, in press-a, in press-b). We briefly mention our theoretical perspective, where it came from, and how we came to teach the supplement/intervention. After that, we describe the actual teaching of the supplement. Finally, we discuss the effectiveness of the supplement and provide some evidence that it “worked”. Since no major reorganization of the real analysis course itself was undertaken, we feel such a supplement could be implemented practically using an advanced mathematics graduate student by many mathematics departments.
Research Interests:
This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the... more
This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the literature, some conjectural. Lastly, it raises some teaching implication questions and suggests a few possible answers.
Research Interests:
This theoretical paper considers several perspectives for understanding and teaching university students’ autonomous proof construction. We describe the logical structure of statements, the formal-rhetorical part of a proof text, and... more
This theoretical paper considers several perspectives for understanding and teaching university students’ autonomous proof construction. We describe the logical structure of statements, the formal-rhetorical part of a proof text, and proof frameworks. We view proof construction as a sequence of actions, and consider actions in the proving process, both situation-action pairs and behavioral schemas. We introduce the concept of local memory – subsets of memory that are partly activated during prolonged consideration of proofs and facilitate success.
Research Interests:
We present results on the proof validation behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course. Participants were interviewed individually towards the end of the course using the same protocol... more
We present results on the proof validation behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course. Participants were interviewed individually towards the end of the course using the same protocol used by Selden and Selden (2003). We describe participants’ observed validation behaviors and provide descriptions of their evaluative comments and their sense-making attempts. We make the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course emphasizing proof construction, in which validation was modelled, may not enhance students’ abilities to judge the correctness of other students’ proof attempts.
Research Interests:
In this paper, we consider proof validation as a type of reading and sense-making of texts within the genre of proof. To illustrate these ideas, we present the results of a study of the proof validation abilities and behaviors of sixteen... more
In this paper, we consider proof validation as a type of reading and sense-making of texts within the genre of proof. To illustrate these ideas, we present the results of a study of the proof validation abilities and behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course that emphasized proof construction and demonstrated proof validation. Participants were interviewed individually towards the end of the course using the same protocol that was used by Selden and Selden (2003) at the beginning of a transition-to-proof course. Results include a description of the participants’ observed validation behaviors, a description of their proffered evaluative comments, a description of their sense-making attempts, and the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course emphasizing proof construction may not enhance students’ abilities to judge the correctness of other students’ proof attempts. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelations.
Research Interests:
This theoretical paper suggests a perspective for understanding university students’ proof construction. It is based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, cognitive feelings and beliefs,... more
This theoretical paper suggests a perspective for understanding university students’ proof construction. It is based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, cognitive feelings and beliefs, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, 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.
This paper is now available online in the RUME 2015 Proceedings. It was given a Meritorious Citation, which means it was in the running for best paper, but was not selected.
This paper is now available online in the RUME 2015 Proceedings. It was given a Meritorious Citation, which means it was in the running for best paper, but was not selected.
Research Interests:
This should be the final revised version. Abstract: This theoretical paper suggests a perspective for understanding university students’ proof construction based on the ideas of conceptual and procedural knowledge, explicit and... more
This should be the final revised version.
Abstract: 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.
Keywords: university level, proving actions, behavioural schemas, System 1 and System 2 cognition, proof framework.
Abstract: 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.
Keywords: university level, proving actions, behavioural schemas, System 1 and System 2 cognition, proof framework.
Research Interests:
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.
Research Interests:
This theoretical paper suggests a perspective for understanding undergraduate proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory,... more
This theoretical paper suggests a perspective for understanding undergraduate 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. This is a shorter, and slightly different version, of our CERME9 WG1 draft paper.
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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 is my presentation of my PhD student's (Valeria Algeria Holguin's) work. It describes her various findings, especially it has conjecture stages of coming to know and use mathematical definitions that are new to students.
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University students often do not know there is a difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004), yet in order to succeed in their university mathematics courses, they must often construct... more
University students often do not know there is a difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004), yet in order to succeed in their university mathematics courses, they must often construct original (to them) mathematical proofs. Only a little research has been conducted to discover how university students handle definitions new to them (e.g., Dahlberg & Housman, 1997). Our specific research question was: How do university students use definitions to evaluate and justify examples and non-examples, in proving, and to evaluate and justify true/false statements
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This is an expanded version of our proposal of the same name. It has been submitted for the Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education (RUME-17). It has a textual analysis of four sample... more
This is an expanded version of our proposal of the same name. It has been submitted for the Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education (RUME-17). It has a textual analysis of four sample correct proofs and four sample incorrect student proof attempts. The abstract is: The purpose of this study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included drawing inferences about students’ sometimes automated links between situations and mental, as well as physical, actions. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework, not unpacking the conclusion, and not using definitions correctly. The ultimate goal is to contribute to an understanding of some of these kinds of difficulties as pedagogical content knowledge with which to teach or redesign transition-to-proof courses.
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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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The purpose of the reported study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. We have tentatively identified... more
The purpose of the reported study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. We have tentatively identified categories of difficulties such as nonstandard language/notation, insufficient warrants, and extraneous statements. The ultimate goal is to use these categories as pedagogical content knowledge with which to redesign an existing transition-to-proof course to alleviate the difficulties for future students.
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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 first discuss a theoretical perspective on proof, proof construction, and proving actions, as well as the need for persistence. Persistence is important for successful proving because it allows one to “explore”, including making... more
We first discuss a theoretical perspective on proof, proof construction, and proving actions, as well as the need for persistence. Persistence is important for successful proving because it allows one to “explore”, including making arguments in directions of unknown value, until one ultimately makes progress. Persistence can be 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 next examine actions needed for a successful proof construction of a theorem given to mid-level U.S. undergraduates in a transition-to-proof course. We contrast those actions with the actual actions of a mathematician proving the same theorem. Finally, we give some teaching implications.
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We first discuss our perspective on, and three useful actions in, proving that depend on persistence. Persistence is important for successful proving because it allows one to “explore”, including making arguments in directions of unknown... more
We first discuss our perspective on, and three useful actions in, proving that depend on persistence. Persistence is important for successful proving because it allows one to “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 actions needed for a successful proof construction of a theorem given to mid-level undergraduates in a transition-to-proof course. We contrast those actions with the actual actions of a mathematician proving the same theorem. Then, after a brief discussion, we end with some teaching implications.
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In this article, we argue that: (1) there has been little research on how students read their mathematics textbooks; (2) there has been research on opportunities to learn from textbooks, on how teachers use textbooks, and on how textbooks... more
In this article, we argue that: (1) there has been little research on how students read their mathematics textbooks; (2) there has been research on opportunities to learn from textbooks, on how teachers use textbooks, and on how textbooks are selected; (3) reading comprehension research, while valuable in general, does not sufficiently inform one about good reading strategies for mathematical text; (3) text relevance research on the kinds of goals students have when reading their mathematics textbooks may be a useful direction; and (4) more generally, that research on what parts of their mathematics textbooks students read and use, and why, would greatly inform both research and practice.
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This study describes eight undergraduates -- four preservice secondary mathematics majors and four regular mathematics majors – early in a sophomore transition-to-proof course. They were asked to check and reflect on four... more
This study describes eight undergraduates -- four preservice secondary mathematics majors and four regular mathematics majors – early in a sophomore transition-to-proof course. They were asked to check and reflect on four student-generated arguments purported to be proofs of a single elementary number theory theorem. At first, the students were essentially at chance level in judging which arguments were proofs and which were not. However, they improved as they continued to reflect on the arguments. The students tended to focus on surface, rather than structural, features of the arguments. We concluded that, without additional instruction, those that became secondary mathematics teachers would have limited ability to determine the correctness of their own students’ proofs. Additional studies have confirmed and extended these results.
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We first discuss our perspective and three useful actions in proof construction that depend on persistence. Persistence is important for successful proving because it allows one to “explore”, including making arguments in directions of... more
We first discuss our perspective and three useful actions in proof construction that depend on persistence. Persistence is important for successful proving because it allows one to “explore”, including making arguments in directions of unknown value, until one ultimately makes progress. Persistence can be 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 discuss a study of U.K. undergraduates’ perceived sense of self-efficacy with regard to proving (Iannone & Inglis, 2010). We then examine actions needed for a successful proof construction of a theorem given to mid-level U.S. undergraduates in a transition-to-proof course. We contrast those actions with the actual actions of a mathematician proving the same theorem.
Research Interests:
This paper is concerned with undergraduate and graduate students’ problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for... more
This paper is concerned with undergraduate and graduate students’ problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld’s (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task.
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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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I briefly describe how my husband and I, who have PhDs in mathematics, got into research in mathematics education. We taught university first in the U.S. and then for 11 years, overseas in Turkey and Nigeria. During this time, we... more
I briefly describe how my husband and I, who have PhDs in mathematics, got into research in mathematics education. We taught university first in the U.S. and then for 11 years, overseas in Turkey and Nigeria. During this time, we published our first mathematics education paper. In it, we analyzed university students’ errors in logical reasoning for a Turkish journal (Selden & Selden, 1978). This was later “recast” in terms of misconceptions for the 1987 Cornell Misconceptions Conference (Selden & Selden, 1987).
In 1988, we attended the Calculus for a New Century Symposium, held at the National Academy of Sciences, and shortly thereafter, we did a sequence of three small empirical studies on university students’ ability to solve non-routine first calculus problems (Selden, Mason, & Selden, 1989; Selden, Selden, & Mason, 1994; Selden, Selden, Hauk, & Mason, 2000). These will be described.
Subsequently, because we had seen many university students’ proving difficulties during our teaching, we switched our research area from university students’ difficulties with calcluls to their difficulties with proof and proving. The bulk of the talk will be devoted to this work including a description of our “unpacking” and “validation” papers (Selden & Selden, 1995, 2003), which will lead up to a discussion of our more recent theoretical work (Selden & Selden, 2015), including our consideration of the structure of proof texts, as well as our consideration of concepts from the psychological literature.
In 1988, we attended the Calculus for a New Century Symposium, held at the National Academy of Sciences, and shortly thereafter, we did a sequence of three small empirical studies on university students’ ability to solve non-routine first calculus problems (Selden, Mason, & Selden, 1989; Selden, Selden, & Mason, 1994; Selden, Selden, Hauk, & Mason, 2000). These will be described.
Subsequently, because we had seen many university students’ proving difficulties during our teaching, we switched our research area from university students’ difficulties with calcluls to their difficulties with proof and proving. The bulk of the talk will be devoted to this work including a description of our “unpacking” and “validation” papers (Selden & Selden, 1995, 2003), which will lead up to a discussion of our more recent theoretical work (Selden & Selden, 2015), including our consideration of the structure of proof texts, as well as our consideration of concepts from the psychological literature.
Research Interests:
I briefly describe how my husband and I, who have PhDs in mathematics, got into research in mathematics education. We taught university first in the U.S. and then for 11 years, overseas in Turkey and Nigeria. During this time, we... more
I briefly describe how my husband and I, who have PhDs in mathematics, got into research in mathematics education. We taught university first in the U.S. and then for 11 years, overseas in Turkey and Nigeria. During this time, we published our first mathematics education paper. In it, we analyzed university students’ errors in logical reasoning for a Turkish journal (Selden & Selden, 1978). This was later “recast” in terms of misconceptions for the 1987 Cornell Misconceptions Conference (Selden & Selden, 1987).
In 1988, we attended the Calculus for a New Century Symposium, held at the National Academy of Sciences, and shortly thereafter, we did a sequence of three small empirical studies on university students’ ability to solve non-routine first calculus problems (Selden, Mason, & Selden, 1989; Selden, Selden, & Mason, 1994; Selden, Selden, Hauk, & Mason, 2000). These will be described.
Subsequently, because we had seen many university students’ proving difficulties during our teaching, we switched our research area from university students’ difficulties with calcluls to their difficulties with proof and proving. The bulk of the talk will be devoted to this work including a description of our “unpacking” and “validation” papers (Selden & Selden, 1995, 2003), which will lead up to a discussion of our more recent theoretical work (Selden & Selden, 2015), including our consideration of the structure of proof texts, as well as our consideration of concepts from the psychological literature.
In 1988, we attended the Calculus for a New Century Symposium, held at the National Academy of Sciences, and shortly thereafter, we did a sequence of three small empirical studies on university students’ ability to solve non-routine first calculus problems (Selden, Mason, & Selden, 1989; Selden, Selden, & Mason, 1994; Selden, Selden, Hauk, & Mason, 2000). These will be described.
Subsequently, because we had seen many university students’ proving difficulties during our teaching, we switched our research area from university students’ difficulties with calcluls to their difficulties with proof and proving. The bulk of the talk will be devoted to this work including a description of our “unpacking” and “validation” papers (Selden & Selden, 1995, 2003), which will lead up to a discussion of our more recent theoretical work (Selden & Selden, 2015), including our consideration of the structure of proof texts, as well as our consideration of concepts from the psychological literature.
Research Interests:
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.
Research Interests:
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.
Research Interests:
We describe a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical... more
We describe a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical perspective (Selden & Selden, in press-a, in press-b). We briefly mention our theoretical perspective, where it came from, and how we came to teach the supplement/intervention. After that, we describe the actual teaching of the supplement. Finally, we discuss the effectiveness of the supplement and provide some evidence that it “worked”. Since no major reorganization of the real analysis course itself was undertaken, we feel such a supplement could be implemented practically using an advanced mathematics graduate student by many mathematics departments.
Research Interests:
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.
Research Interests:
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.
Research Interests:
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.
Research Interests:
This is the small size version of our poster. In it, we examine: • an example of the usefulness/necessity of consciousness, and • a suggestion of its role in human evolution. The kind of consciousness we are referring to is often called... more
This is the small size version of our poster. In it, we examine:
• an example of the usefulness/necessity of consciousness, and
• a suggestion of its role in human evolution.
The kind of consciousness we are referring to is often called phenomenal consciousness, that is, a kind of awareness that everyone experiences. “We have P-conscious states [phenomenal consciousness] when we see, hear, smell, taste and have pains.” (Block, 1995, p. 230)
In order to say something about the contribution of a human characteristic, such as consciousness, to the evolution of homo sapiens, that characteristic should at least contribute to survival value, in an essential way.
• an example of the usefulness/necessity of consciousness, and
• a suggestion of its role in human evolution.
The kind of consciousness we are referring to is often called phenomenal consciousness, that is, a kind of awareness that everyone experiences. “We have P-conscious states [phenomenal consciousness] when we see, hear, smell, taste and have pains.” (Block, 1995, p. 230)
In order to say something about the contribution of a human characteristic, such as consciousness, to the evolution of homo sapiens, that characteristic should at least contribute to survival value, in an essential way.
Research Interests: Consciousness (Psychology), Deductive reasoning, Proof and Reasoning, Consciousness Studies, Procedural Knowledge, and 5 moreResearch of Mathematics Education, Procedural learning, Procedural and Declarative Knowledge, Research in Undergraduate Mathematics Education, and Researches in Mathematics Education
In the mathematics education research literature on proof, four concepts have been discussed: proof comprehension, proof construction, proof validation, and proof evaluation. This presentation examined their interrelationship, the... more
In the mathematics education research literature on proof, four concepts have been discussed: proof comprehension, proof construction, proof validation, and proof evaluation. This presentation examined their interrelationship, the research conducted to date, and speculate on how these ideas could be conveyed and made helpful to students.
Research Interests:
This presentation documents the progression of one non-traditional student’s proof-writing through a semester. Videotapes of this individual’s one-on-one sessions working through the transition-to-proof course notes were analyzed. Proof... more
This presentation documents the progression of one non-traditional student’s proof-writing through a semester. Videotapes of this individual’s one-on-one sessions working through the transition-to-proof course notes were analyzed. Proof construction was viewed a sequence of (mental, as well as physical) actions, and proof frameworks were used to initiate the writing of proofs. This student’s early reluctance to use proof frameworks was documented, as well as her later acceptance of, and proficiency with, them. By the end of the first semester, she had developed considerable facility with the construction of moderately difficult proofs and had developed a sense of self-efficacy.
Research Interests:
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.
Research Interests:
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 upper-division proof-based courses. Many students in such courses... 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 upper-division proof-based courses. Many students in such courses do not know how to start writing proofs. We first have students write proof frameworks to structure their proofs. This can be done based only on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level proof framework. We discuss how we came to the idea of proof framework and demonstrate how they are written in elementary number theory, elementary set theory, and beginning real analysis. We also show how we use our theoretical perspective on proving actions to analyze an incorrect student proof attempt. We go on to discuss a "harder" proof of a theorem on semigroups, where proof frameworks are not enough and a significant amount of "exploration", self-efficacy, and persistence are needed.
Research Interests:
This is the PowerPoint of a paper that considers the difficulty that university students’ may have when unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure, and hence,... more
This is the PowerPoint of a paper that considers the difficulty that university students’ may have when unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure, and hence, construct 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. But later, students were presented with some informally worded theorem statements to prove. We go on to consider the question of when, and how, to enculturate students into the often informal way that theorem statements are normally written, while still enabling them to progress in their proof construction abilities.
Research Interests: Deductive reasoning, Negation (Logic), Proof and Proving in Mathematics Education, Negation, Mathematical reasoning and proof, and 8 moreLogical reasoning, Negation (pragmatics), Research of Mathematics Education, Research in Mathematics education, Educational Research in Mathematics, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, and Logical and Reasoning
This PowerPoint details a case study that documents the progression of one non-traditional individual’s proof-writing through a semester. We analyzed the videotapes of this individual’s one-on-one sessions working through our course notes... more
This PowerPoint details a case study that documents the progression of one non-traditional individual’s proof-writing through a semester. We analyzed the videotapes of this individual’s one-on-one sessions working through our course notes for an inquiry-based transition-to-proof course. Our theoretical perspective informed our work with this individual and included the view that proof construction is a sequence of (mental, as well as physical) actions. It also included the use of proof frameworks as a means of initiating a written proof. This individual’s early reluctance to use proof frameworks, after an initial introduction to them, was documented, as well as her later acceptance of, and proficiency with, them. By the end of the first semester, she had developed considerable facility with both the formal-rhetorical and problem-centered parts of proofs and a sense of self-efficacy.
Research Interests: Proof and Reasoning, Proof and Proving in Mathematics Education, Undergraduate Mathematics Education, Mathematical reasoning and proof, Logical reasoning, and 6 moreResearch of Mathematics Education, Research in Mathematics education, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, Mathematical Reasoning and Proofs, and Proofs and Mathematical Reasoning
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 now many community colleges do not offer such courses, but may want to begin doing so. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on proof writing. This is the technique of writing proof frameworks, based on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level of a proof framework. We discuss how we came to the idea of proof frameworks and demonstrate the writing of several proof frameworks.
Research Interests: Proof and Reasoning, Teaching Mathematics, Mathematics Teaching, Mathematical reasoning and proof, Research of Mathematics Education, and 4 moreTeaching and Learning Mathematics, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, and Mathematical Reasoning and Proofs
We describe an intervention in the form of a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were... more
We describe an intervention in the form of a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical perspective (Selden & Selden, in press-a, in press-b). Since no major reorganization of the real analysis course itself was undertaken, we feel such a supplement could be implemented practically by many mathematics departments. We will briefly mention relevant parts of our theoretical perspective, where it came from, and how we came to teach the supplement/intervention. After that, we will describe our teaching actions as facilitators in preparing for, and leading, what we came to call co-construction (McKee, Savic, Selden, & Selden, 2010). After describing a sample supplement session, we will discuss the usefulness, advantages and disadvantages of this kind of supplement/intervention, some evidence that it “worked”, who benefited, what sorts of things the participating students learned, and what kinds of questions they asked during the supplement.
Research Interests:
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 now many community colleges do not offer such courses, but may want to begin doing so. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on proof writing. This is the technique of writing proof frameworks, based on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level of a proof framework. We will discuss how we came to the idea of proof frameworks and demonstrate the writing of several proof frameworks.
Research Interests:
This presentation considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the... more
This presentation considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the literature, some conjectural. Lastly, it raises some teaching implication questions and suggests a few possible answers.
Research Interests: Proof and Proving in Mathematics Education, Mathematical reasoning and proof, Research of Mathematics Education, Research in Mathematics education, Educational Research in Mathematics, and 3 moreResearch in Undergraduate Mathematics Education, Researches in Mathematics Education, and Mathematical Reasoing and Proof
We present results on the proof validation behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course. Participants were interviewed individually towards the end of the course using the same protocol... more
We present results on the proof validation behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course. Participants were interviewed individually towards the end of the course using the same protocol used by Selden and Selden (2003). We describe participants’ observed validation behaviors and provide descriptions of their evaluative comments and their sense-making attempts. We make the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course emphasizing proof construction, in which validation was modelled, may not enhance students’ abilities to judge the correctness of other students’ proof attempts.
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This presentation discussed the course we have developed to help beginning graduate students with proving.
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We present the results of an analysis of undergraduate students’ examination papers from an IBL transition-to-proof course. Students’ papers were considered from the point of view of their actions (mental, as well as physical), instead of... more
We present the results of an analysis of undergraduate students’ examination papers from an IBL transition-to-proof course. Students’ papers were considered from the point of view of their actions (mental, as well as physical), instead of their possible misconceptions. In doing so, we identified process, rather than mathematical content, difficulties, and this has resulted in the detection of both beneficial actions and detrimental actions that students often take.
Thus far, we have identified the following categories of difficulties: omitting beneficial actions; taking detrimental actions; inadequate proof framework (e.g., not unpacking the conclusion); mathematical syntax errors; wrong or improperly used definitions; misuse of logic; insufficient warrant; assumption of all or part of the conclusion; extraneous statements; assumption of the negation of a previously established fact; difficulties with proof by contradiction; inappropriately mimicking a prior proof; mathematical syntax errors, failure to use cases when appropriate; incorrect deduction; assertion of an untrue result; and computational errors. Examples of some of these difficulties will be presented
Thus far, we have identified the following categories of difficulties: omitting beneficial actions; taking detrimental actions; inadequate proof framework (e.g., not unpacking the conclusion); mathematical syntax errors; wrong or improperly used definitions; misuse of logic; insufficient warrant; assumption of all or part of the conclusion; extraneous statements; assumption of the negation of a previously established fact; difficulties with proof by contradiction; inappropriately mimicking a prior proof; mathematical syntax errors, failure to use cases when appropriate; incorrect deduction; assertion of an untrue result; and computational errors. Examples of some of these difficulties will be presented
Research Interests:
Creating proofs is a vital part of succeeding in courses such as abstract algebra and real analysis during a student’s junior and senior years. Many mathematics departments have instituted transition-to-proof courses for second semester... more
Creating proofs is a vital part of succeeding in courses such as abstract algebra and real analysis during a student’s junior and senior years. Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs. We have taught transition-to-proof courses using inquiry-based learning techniques and our own notes, in order to provide students many opportunities to experience proof writing.
In order to understand where students are “coming from” and to help them learn to construct proofs. We have analyzed students’ examination papers from several such courses. We have identified process, rather than mathematical content, difficulties such as not unpacking the conclusion, and not using definitions correctly. Examples of these difficulties will be presented.
In order to understand where students are “coming from” and to help them learn to construct proofs. We have analyzed students’ examination papers from several such courses. We have identified process, rather than mathematical content, difficulties such as not unpacking the conclusion, and not using definitions correctly. Examples of these difficulties will be presented.
Research Interests: Deductive reasoning, Proof and Proving in Mathematics Education, Undergraduate Mathematics Education, Logical reasoning, Research of Mathematics Education, and 3 moreResearch in Undergraduate Mathematics Education, Researches in Mathematics Education, and Ressearch in Undergraduate Mathematics Education
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...
Research Interests:
In this paper, we consider proof validation as a type of reading and sense-making of texts within the genre of proof. To illustrate these ideas, we present the results of a study of the proof validation abilities and behaviors of sixteen... more
In this paper, we consider proof validation as a type of reading and sense-making of texts within the genre of proof. To illustrate these ideas, we present the results of a study of the proof validation abilities and behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course that emphasized proof construction and demonstrated proof validation. Participants were interviewed individually towards the end of the course using the same protocol that was used by Selden and Selden (2003) at the beginning of a transition-to-proof course. Results include a description of the participants’ observed validation behaviors, a description of their proffered evaluative comments, a description of their sense-making attempts, and the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course emphasizing proof construction may not enhance students’ abilities to judge the correctness of other students’ proof attempts.
Research Interests:
This Powerpoint is longer than the CERME9 WG1 Powerpoint (of the same name) and contains more details. We were allowed 20 minutes for this presentation and only 10 minutes for the CERFME9 WG1 presentation. In it, we suggest a perspective... more
This Powerpoint is longer than the CERME9 WG1 Powerpoint (of the same name) and contains more details. We were allowed 20 minutes for this presentation and only 10 minutes for the CERFME9 WG1 presentation. In it, we suggest a perspective for understanding undergraduate 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 10-minute Powerpoint summarizes our 10-page paper for the CERME9, WG1 on Argumentation and Proof (available elsewhere on my www.academia.edu page).
Research Interests:
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.
Research Interests:
This presentation detailed two teaching experiments: (1) a special topics course to help advanced undergraduate and beginning mathematics graduate students with proving; and (2) a proving supplement to an undergraduate real analysis... more
This presentation detailed two teaching experiments: (1) a special topics course to help advanced undergraduate and beginning mathematics graduate students with proving; and (2) a proving supplement to an undergraduate real analysis course.
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 problem-centered part.
Second, we discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the undergraduate real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process, by our division of proofs into their formal-rhetorical and problem-centered parts, and by observations of students’ proving difficulties.
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 problem-centered part.
Second, we discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the undergraduate real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process, by our division of proofs into their formal-rhetorical and problem-centered parts, and by observations of students’ proving difficulties.
Research Interests:
This is the Powerpoint presentation. University students often do not know there is a difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004), yet in order to succeed in their university mathematics... more
This is the Powerpoint presentation. University students often do not know there is a difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004), yet in order to succeed in their university mathematics courses, they must often construct original (to them) mathematical proofs.
Our specific research question was: How do university students use definitions to evaluate and justify examples and non-examples, in proving, and to evaluate and justify true/false statements? Data were collected through individual task-based interviews with volunteers from a transition-to-proof course. Each student was provided with a particular definition and asked to consider examples and non-examples, construct a proof, and consider true/false statements, in that order. Altogether there were five definitions: function, continuity, ideal, isomorphism, and group, but each student was asked to consider only one of the five.
Our specific research question was: How do university students use definitions to evaluate and justify examples and non-examples, in proving, and to evaluate and justify true/false statements? Data were collected through individual task-based interviews with volunteers from a transition-to-proof course. Each student was provided with a particular definition and asked to consider examples and non-examples, construct a proof, and consider true/false statements, in that order. Altogether there were five definitions: function, continuity, ideal, isomorphism, and group, but each student was asked to consider only one of the five.
Research Interests:
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.
Research Interests:
This paper presents the results of a study of the proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. It is an expansion of our RUME Conference paper of the same... more
This paper presents the results of a study of the proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. It is an expansion of our RUME Conference paper of the same title to include audience participatnion. 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 interrelation.
Research Interests:
This paper presents 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... more
This paper presents 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 validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelation.
Research Interests:
The purpose of the study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included the sometimes... more
The purpose of the study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included the sometimes automated links between situations and (mental, as well as physical) actions. We have tentatively identified process, rather than content, categories of difficulties such as nonstandard language/notation, insufficient warrants, and extraneous statements. The ultimate goal is to use an understanding of some of these categories as pedagogical content knowledge with which to redesign an existing transition-to-proof course.
Research Interests:
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.
This "after dinner" talk at the MPWR Seminar considered 40 "lessons learned" from a long career in research in undergraduate mathematics educaton (RUME).
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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.
Research Interests:
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.
Research Interests:
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.
Have you ever wondered how to pick a research question, which methodology to use, how to get started on writing a mathematics education research journal article, which mathematics education research journals are ranked the highest, how... more
Have you ever wondered how to pick a research question, which methodology to use, how to get started on writing a mathematics education research journal article, which mathematics education research journals are ranked the highest, how mathematics education research papers are reviewed, what you need to do go get tenure as a mathematics education researcher in a mathematics department? This talk will discuss these topics and more. There will be no definitive or categorical answers, just observations from a long career in the field.
Research Interests:
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."
Research Interests:
First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 and each time we are... more
First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 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 the remainder of the proof the problem-centered part.
Second, I will discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process and by our division of proofs into their formal-rhetorical and problem-centered parts.
Second, I will discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process and by our division of proofs into their formal-rhetorical and problem-centered parts.
Research Interests:
In this presentation, we discussed some characteristics of the genre of proof and discussed how we teach proving.
Research Interests:
This question was asked of the presenters of CERME8, WG1. This PowerPoint gives our answer.
Research Interests:
We first discuss three useful actions in proof construction that depend on persistence. Persistence is important for successful proving because it allows one to explore and to continue making arguments in unknown directions until one... more
We first discuss three useful actions in proof construction that depend on persistence. Persistence is important for successful proving because it allows one to explore and to continue making arguments in unknown directions until one ultimately makes progress. Persistence can be 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 discuss a study (Iannone & Inglis, 2010) of U.K. undergraduates’ perceived sense of self-efficacy with regard to proving. We then examine actions needed for a successful proof construction of a theorem given to mid-level U.S. undergraduates in a transition-to-proof course. We contrast those actions with the actual actions of a mathematician proving the same theorem.
Research Interests:
We consider the role of text relevance in formulating an explanation for why undergraduate students do not read large parts of their beginning mathematics textbooks. In a previous paper (Shepherd, Selden, & Selden, in press), we asked why... more
We consider the role of text relevance in formulating an explanation for why undergraduate students do not read large parts of their beginning mathematics textbooks. In a previous paper (Shepherd, Selden, & Selden, in press), we asked why it is that good readers, who were also good at mathematics, did not read large parts of their beginning mathematics textbooks effectively, that is, why they could not work straightforward tasks based directly on that reading. Here, we reanalyze that data in terms of text relevance to consider the role that students’ personal implicit or explicit goals may play.
Keywords: Post-secondary education, instructional activities and practices, curriculum
Keywords: Post-secondary education, instructional activities and practices, curriculum
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This is a book review submitted to MAA Online Book Reviews. The book's main chapters are revised and extended versions of papers presented at ICME-13 Topic Study Group 18 on Reasoning and Proof in Mathematics Education.
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This is a brief book review written for MAA Online Book Reviews.
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This is a book review prepared for MAA Online Book Reviews. It begins "This book, although indicated to be about improving mathematics and science education, appears in the Springer Series, Studies in Computational Intelligence, rather... more
This is a book review prepared for MAA Online Book Reviews. It begins "This book, although indicated to be about improving mathematics and science education, appears in the Springer Series, Studies in Computational Intelligence, rather than in the Springer Series on Mathematics Education. It is unusual in that it is not an edited “chapter book” with many different authors who specialize in mathematics education research. Indeed, the more I perused the book, the more I came to the conclusion that it is really about the possibilities of applying interval and fuzzy techniques to education-related problems, some more closely connected to teaching than others, than it is about traditional mathematics education research."
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This is a book review. It begins, " What would proving, that is, age appropriate mathematical reasoning, look like for 8- and 9-year olds? Through actual classroom observations and interventions, initially gathered for research, the... more
This is a book review. It begins, "
What would proving, that is, age appropriate mathematical reasoning, look like for 8- and 9-year olds? Through actual classroom observations and interventions, initially gathered for research, the author describes “what one should look for and expect of young children engaged in proving activities” (Forward). The eight main classroom episodes come from a Year 4 class in England (8- and 9-year olds) taught by Mrs. Howard (a pseudonym) or from a third-grade class in the U.S. taught by Deborah Ball. The proving activities are mainly arithmetical or combinatorial, but one can imagine similar geometrical tasks. A key question that the author asks, and attempts to answer, is: What would it take for elementary teachers to productively engage their students in proving? (p. 36)."
What would proving, that is, age appropriate mathematical reasoning, look like for 8- and 9-year olds? Through actual classroom observations and interventions, initially gathered for research, the author describes “what one should look for and expect of young children engaged in proving activities” (Forward). The eight main classroom episodes come from a Year 4 class in England (8- and 9-year olds) taught by Mrs. Howard (a pseudonym) or from a third-grade class in the U.S. taught by Deborah Ball. The proving activities are mainly arithmetical or combinatorial, but one can imagine similar geometrical tasks. A key question that the author asks, and attempts to answer, is: What would it take for elementary teachers to productively engage their students in proving? (p. 36)."
Research Interests:
This is a book review of this transition-to-proof course textbook, published by MAA Press for its Textbook Series.
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This commentary describes the chapters in and the flavor of this book, which covers various kinds of algebra including college algebra and linear algebra.
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This is a review of a book that reports the development of a teaching method, called the TR/NYCity Model. The model was developed over time by combining prior mathematics education theory and research results with the editors' own... more
This is a review of a book that reports the development of a teaching method, called the TR/NYCity Model. The model was developed over time by combining prior mathematics education theory and research results with the editors' own teaching observations.
Research Interests: Mathematics Education, 2-Year Developmental Mathematics Programs, Developmental Mathematics, Developmental Mathematics Education, Research of Mathematics Education, and 3 moreResearch in Undergraduate Mathematics Education, Researches in Mathematics Education, and Developmental Education Programs in Mathematics
This is a book review that describes the overall structure of the book, noting various operational definitions of creativity and giftedness, and gives more detail on a couple of chapters.
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This is a book review that gives an idea of what is in the volume, particularly Chapters 2 and 13.
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This is a book review of the ICME-13 Topical Survey titled, "Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-Art and Looking Ahead," written by the five organizers of ICME-13 TSG2 on Mathematics Education... more
This is a book review of the ICME-13 Topical Survey titled, "Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-Art and Looking Ahead," written by the five organizers of ICME-13 TSG2 on Mathematics Education at Tertiary Level.
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This is a review of this edited, 6-chapter, 80-page paperback volume discusses the current school level mathematics education situation in East Africa. The various chapters are devoted to issues of quality mathematics education in East... more
This is a review of this edited, 6-chapter, 80-page paperback volume discusses the current school level mathematics education situation in East Africa. The various chapters are devoted to issues of quality mathematics education in East Africa; harmonization of curricula across Kenya, Rwanda, Tanzania, and Uganda; a comparative analysis of mathematics achievement across East Africa, insights from East African classrooms; teacher training in East Africa; and integration of ICT in East Africa.
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This tome, 680 pages plus introductory material, published in 1973, is a classic in the mathematics education literature. Many of the ideas expressed in it are current today, especially those of Dutch Realistic Mathematics Education... more
This tome, 680 pages plus introductory material, published in 1973, is a classic in the mathematics education literature. Many of the ideas expressed in it are current today, especially those of Dutch Realistic Mathematics Education (RME). 1 The ideas of reinvention (now often referred to as guided reinvention) and mathematization are discussed in detail in Chapters 6 and 7. For guided reinvention to work, the teacher, or the curriculum developer, should ensure that the learner will regard the knowledge as his/her own. Freudenthal saw guided reinvention as an elaboration of the Socratic Method, but one in which students are much more active. That said, how can one adequately summarize, or even indicate, the contents of this huge volume? There are 19 chapters plus two appendices. They range over the aim of mathematics instruction, the Socratic Method, mathematical rigor, logic, number, sets and functions, geometry, analysis, probability and statistics, and more, with the first of the...
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This book stems from the 13th International Conference on the Teaching of Mathematical Modeling and Applications (ICTMA 13). The first part focuses on research into what it means for students to understand models and modeling processes (6... more
This book stems from the 13th International Conference on the Teaching of Mathematical Modeling and Applications (ICTMA 13). The first part focuses on research into what it means for students to understand models and modeling processes (6 sections, comprising 23 chapters), while the second part considers what is needed for modeling activities to be productive in classrooms (5 sections, comprising 30 chapters). The authors come from many countries: South Africa, Taiwan, Japan, Mexico, Germany, Israel, Brazil, Argentina, Sweden, Italy, Australia, Canada, Cyprus, U.K., Spain, Switzerland, Denmark, and the U.S. This book is a veritable fount of information about modeling in classrooms at all levels from elementary through university. In addition, it has chapters dealing with how teachers develop models of modeling (Chapters 30-46) and how new technologies influence modeling in classrooms (Chapters 47-51).
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This is a book review. According to the editors, this handbook provides a “comprehensive overview of the most recent theoretical and practical developments in the field … spanning established and emerging topics, diverse workplace and... more
This is a book review. According to the editors, this handbook provides a “comprehensive overview of the most recent theoretical and practical developments in the field … spanning established and emerging topics, diverse workplace and school environments, and globally representative priorities”. Packed into this 726-page paperback third edition are 12 revised and 17 entirely new chapters spread across five sections: (1) priorities in international research in mathematics education -- 4 chapters; (2) democratic access to mathematics learning -- 7 chapters; (3) transformative learning contexts -- 6 chapters; (4) advances in research methodologies -- 5 chapters; and (5) influence of advanced technologies -- 6 chapters, with one final commentary. There are new perspectives on such topics as embodied learning, educating future mathematics education professors, multi-modal technologies, and e-textbooks. The authors, of these sometimes multi-authored chapters, stem from a variety of countries including Australia, Cypress, Canada, United States, Mexico, Greece, Israel, Portugal, Denmark, United Kingdom, Italy, Norway, Netherlands, and France. The rest of the review describes the content of various chapters.
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This is a book review written for MAA Online Reviews. It begins as follows: This edited book of 26 chapters is divided into four parts: defining the field; mathematical problem posing in the school curriculum, mathematical problem posing... more
This is a book review written for MAA Online Reviews. It begins as follows: This edited book of 26 chapters is divided into four parts: defining the field; mathematical problem posing in the school curriculum, mathematical problem posing in teacher education, and concluding remarks. It is not a slim book—there are 569 pages contributed by 52 authors from 16 countries, such as the U.S., Canada, Australia, Israel, Japan, Norway, Czech Republic, Singapore, Serbia, Romania, Belgium, Sweden, Italy, and the Netherlands. Despite these many and varied contributions, one gets the distinct impression that problem-posing research is still in its infancy.
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This is a review of a book about Hans Freudenthal’s ideas on the didactics of mathematics. As the author states, this dissertation study tried to answer the question, “What was Freudenthal’s role in mathematics education?” (p. 3). To... more
This is a review of a book about Hans Freudenthal’s ideas on the didactics of mathematics. As the author states, this dissertation study tried to answer the question, “What was Freudenthal’s role in mathematics education?” (p. 3). To answer this question, the author offers a reconstruction of the development of Freudenthal’s ideas, based primarily on documents found in Freudenthal’s 16 meter long personal archive. It is not meant to be a biography of his life, but rather a historical analysis of the development of his didactical ideas.
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This is a book review of this classic 1973 book by Hans Freudenthal. In addition, it has a lot of information on how he went from being a topologist to becoming interested in mathematics education and devoting the rest of his life to it.
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First, a disclaimer—I am not a logician. Indeed, I have never taken a course in mathematical logic, but I was interested in whether this book would provide some background information for teaching the logic portion of a traditional... more
First, a disclaimer—I am not a logician. Indeed, I have never taken a course in mathematical logic, but I was interested in whether this book would provide some background information for teaching the logic portion of a traditional transition-to-proof course. The answer is no, but I will do my best to give some sense of what is in this book. The goal of this paperback textbook, written for a course in logic, is to prove Gödel’s completeness and incompleteness theorems.
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How much mathematics is used in various occupations? What kind and in what ways? Are there any implications for teaching or learning? Answers to these questions will vary with the jobs -- auto workers use mathematics differently from... more
How much mathematics is used in various occupations? What kind and in what ways? Are there any implications for teaching or learning? Answers to these questions will vary with the jobs -- auto workers use mathematics differently from biologists -- and with the perspectives of those who do the looking. In the past few years, researchers (mainly in mathematics education) have observed auto workers, nurses, bankers, biologists, ecologists, and others, as they go about their day-to-day activities.
While all such studies have gathered empirical data on the mathematics used in various workplaces, they have also investigated such things as the nature of modeling and abstraction, the role of representations, and various associated learning difficulties. Below is a description of several such studies conducted in the U.S., U.K., and Canada, progressing from mainly empirical to more theoretical. We do not here discuss any of the equally interesting studies of Brazilian street sellers or South African carpenters that are often classified as ethnomathematics.
While all such studies have gathered empirical data on the mathematics used in various workplaces, they have also investigated such things as the nature of modeling and abstraction, the role of representations, and various associated learning difficulties. Below is a description of several such studies conducted in the U.S., U.K., and Canada, progressing from mainly empirical to more theoretical. We do not here discuss any of the equally interesting studies of Brazilian street sellers or South African carpenters that are often classified as ethnomathematics.
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We discuss and give background for the following questions: Question I. Do many mathematicians commonly end validations as we have described? Question II. Assuming mathematicians conclude their validations with an unbroken train of... more
We discuss and give background for the following questions: Question I. Do many mathematicians commonly end validations as we have described? Question II. Assuming mathematicians conclude their validations with an unbroken train of thought, do students at various levels do something similar? Question III. Does concluding a validation by reviewing the entire proof in an unbroken train of thought provide more reliability? Does a mathematician or student who concludes a validation in this way find more errors?
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This article has four parts: 1. In mathematics education, especially research in mathematics education. A. Constructivism in its moderate (also called simple or trivial) form is a philosophical position characterized by the idea that... more
This article has four parts:
1. In mathematics education, especially research in mathematics education.
A. Constructivism in its moderate (also called simple or trivial) form is a philosophical position characterized by the idea that knowledge is not passively received, but rather actively constructed by an individual. 2. In popular usage, especially in the press and amongst some teachers and mathematicians.
B. Constructivism in its radical form can be characterized by two additional ideas.
(1) The function of cognition is adaptive in the biological sense of tending toward "fit" or viability. (Trees in a forest have more leaves where there is more light and runners have strong legs. Both have adapted to fit a situation.)
(2) Cognition organizes the experiential world, but does not allow discovery of objective reality.
2. In popular usage, especially in the press and amongst some teachers and mathematicians. Constructivism is often used to refer to a teaching method, or the advocacy of a teaching method, in which students construct (invent, discover) their own mathematics.
3. In the history and philosophy of science and mathematics.
Constructivism (also called social constructivism) is a philosophy of science in which scientists are viewed as socially constructing knowledge of the external world rather than uncovering it directly from nature.
4. In mathematics. Constructivism in mathematics (also sometimes called intuitionism) refers to a philosophical position (with several variations) that seeks to place mathematics on a firm logical foundation by basing it only on what advocates regard as intuitively clear insights. This includes theorems that can be proved constructively based on integers and finite sets, but avoids the logical laws of double negation and excluded middle, as well as proof by contradiction.
1. In mathematics education, especially research in mathematics education.
A. Constructivism in its moderate (also called simple or trivial) form is a philosophical position characterized by the idea that knowledge is not passively received, but rather actively constructed by an individual. 2. In popular usage, especially in the press and amongst some teachers and mathematicians.
B. Constructivism in its radical form can be characterized by two additional ideas.
(1) The function of cognition is adaptive in the biological sense of tending toward "fit" or viability. (Trees in a forest have more leaves where there is more light and runners have strong legs. Both have adapted to fit a situation.)
(2) Cognition organizes the experiential world, but does not allow discovery of objective reality.
2. In popular usage, especially in the press and amongst some teachers and mathematicians. Constructivism is often used to refer to a teaching method, or the advocacy of a teaching method, in which students construct (invent, discover) their own mathematics.
3. In the history and philosophy of science and mathematics.
Constructivism (also called social constructivism) is a philosophy of science in which scientists are viewed as socially constructing knowledge of the external world rather than uncovering it directly from nature.
4. In mathematics. Constructivism in mathematics (also sometimes called intuitionism) refers to a philosophical position (with several variations) that seeks to place mathematics on a firm logical foundation by basing it only on what advocates regard as intuitively clear insights. This includes theorems that can be proved constructively based on integers and finite sets, but avoids the logical laws of double negation and excluded middle, as well as proof by contradiction.
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Examining examples and non-examples can help students understand definitions. While a square may be defined as a quadrilateral with four equal sides and one right angle, seeing concrete examples of squares of various sizes, as well as... more
Examining examples and non-examples can help students understand definitions. While a square may be defined as a quadrilateral with four equal sides and one right angle, seeing concrete examples of squares of various sizes, as well as considering rectangular non-examples, can help children clarify the notion of square. When we teach linear algebra and introduce the concept of subspace, we often provide examples and non-examples for students. We may point out that the polynomials of degree less than or equal to two form a subspace of the space of all polynomials, whereas the polynomials of degree two do not. Is the provision of such examples always desirable? Would it perhaps be better to ask undergraduate students to provide their own examples and non-examples? Would they be able to? Given a false conjecture, would students be able to come up with counterexamples? Several studies shed light on these questions.
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Does thinking of using things one already knows (facts, procedures, etc.) play a major role when solving mathematical problems? Focusing this question somewhat more narrowly, does "accessing" one's "knowledge base" play a major role in... more
Does thinking of using things one already knows (facts, procedures, etc.) play a major role when solving mathematical problems?
Focusing this question somewhat more narrowly, does "accessing" one's "knowledge base" play a major role in helping one solve novel problems? On the one hand, success in solving novel problems might depend mainly on one's reasoning skills and the quality of one's knowledge base, and accessing it might be relatively routine, automatic, and unproblematic. On the other hand, it might be that one can know quite a lot, but often fail to solve a problem through not accessing the appropriate part of one's knowledge.
In I., we discuss the meanings of the terms we use and place them in a somewhat larger setting, i.e., we sketch a theoretical framework. In II., we discuss some related literature that suggests the answer to our initial question might sometimes be yes. In III., we narrow the question to what might be called a directly researchable question, and in IV., we suggest a way to partially answer that question.
Focusing this question somewhat more narrowly, does "accessing" one's "knowledge base" play a major role in helping one solve novel problems? On the one hand, success in solving novel problems might depend mainly on one's reasoning skills and the quality of one's knowledge base, and accessing it might be relatively routine, automatic, and unproblematic. On the other hand, it might be that one can know quite a lot, but often fail to solve a problem through not accessing the appropriate part of one's knowledge.
In I., we discuss the meanings of the terms we use and place them in a somewhat larger setting, i.e., we sketch a theoretical framework. In II., we discuss some related literature that suggests the answer to our initial question might sometimes be yes. In III., we narrow the question to what might be called a directly researchable question, and in IV., we suggest a way to partially answer that question.
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The answer to this question depends both on how you define problem and on how you define expert. Problems can be routine textbook exercises or they can be difficult mathematical tasks which take weeks, months, or even years to solve. An... more
The answer to this question depends both on how you define problem and on how you define expert. Problems can be routine textbook exercises or they can be difficult mathematical tasks which take weeks, months, or even years to solve. An expert may mean someone who knows the domain thoroughly and can solve problems in a nearly automatic manner, or it can mean someone who can think of things to do even when no clear solution method suggests itself, marshaling strategies, heuristics, analogies, alternative representations, etc. While both types of "experts" often possess an extensive content knowledge base, the latter are more successful at solving nonroutine problems.
To get a handle on expertise, cognitive psychologists, who want to understand it, and knowledge engineers, who want to use it in AI programs, have extensively examined both general problem-solving heuristics and expertise in particular, often narrow, domains. They have designed artificial intelligence programs which duplicate, and sometimes even exceed, human expertise. Two early examples from the 70's are Newell and Simon's General Problem Solver and MYCIN, a rule-based deduction system for diagnosing bacterial infections. [Cf. Newell & Simon, Human Problem Solving, 1972; Winston, Artificial Intelligence, 3rd ed., 1992, pp.130-132.]
To get a handle on expertise, cognitive psychologists, who want to understand it, and knowledge engineers, who want to use it in AI programs, have extensively examined both general problem-solving heuristics and expertise in particular, often narrow, domains. They have designed artificial intelligence programs which duplicate, and sometimes even exceed, human expertise. Two early examples from the 70's are Newell and Simon's General Problem Solver and MYCIN, a rule-based deduction system for diagnosing bacterial infections. [Cf. Newell & Simon, Human Problem Solving, 1972; Winston, Artificial Intelligence, 3rd ed., 1992, pp.130-132.]
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The way teachers explain mathematics depends, to a large extent, on the conceptual grasp they acquire in their college classes. In addition, they often teach as they were taught, modeling themselves after their college mathematics... more
The way teachers explain mathematics depends, to a large extent, on the conceptual grasp they acquire in their college classes. In addition, they often teach as they were taught, modeling themselves after their college mathematics teachers as much as their K-12 teachers. Mathematics departments share in the preparation of all preservice teachers, and in particular, secondary education mathematics majors take many of the same courses as regular mathematics majors.
What views of mathematics and its teaching do preservice teachers get while in college? A lot of research effort has gone into probing their understandings of both mathematics and how it is taught. Somewhat curiously perhaps, getting at students' understandings of mathematics seems to be a very different enterprise from assessing students' work for the purposes of assigning grades. Students can often carry out algorithms well (i.e., have sufficient procedural knowledge to pass a course), yet are unable to explain why the algorithms work (i.e., have little underlying conceptual knowledge). Thus, it can be hard to gauge college students', including preservice teachers', conceptual grasp through normal testing alone -- hence, the need for in-depth studies of their knowledge of mathematics and how to teach it.
What views of mathematics and its teaching do preservice teachers get while in college? A lot of research effort has gone into probing their understandings of both mathematics and how it is taught. Somewhat curiously perhaps, getting at students' understandings of mathematics seems to be a very different enterprise from assessing students' work for the purposes of assigning grades. Students can often carry out algorithms well (i.e., have sufficient procedural knowledge to pass a course), yet are unable to explain why the algorithms work (i.e., have little underlying conceptual knowledge). Thus, it can be hard to gauge college students', including preservice teachers', conceptual grasp through normal testing alone -- hence, the need for in-depth studies of their knowledge of mathematics and how to teach it.
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Mathematicians, mathematics educators, and psychologists often seem to be addressing different questions. While creativity may interest mathematicians and mathematics educators seek to understand mathematical learning with the aim of... more
Mathematicians, mathematics educators, and psychologists often seem to be addressing different questions. While creativity may interest mathematicians and mathematics educators seek to understand mathematical learning with the aim of improving it, psychologists tend to use mathematical tasks to study aspects of general cognition such as problem solving. Even so, couldn't general learning theories tell us something about learning mathematics? Many mathematics educators would say "not much" -- knowledge acquisition is largely domain specific. Learning mathematics has features unlike learning, say, biology. Solving mathematics problems is not the same as employing a heuristic search to solve syntactic reasoning puzzles, like the Tower of Hanoi. Rather, what counts is a rich, organized set of connections between concepts, together with imagery and reasoning.
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Does it consist of competencies, such as being able to factor the difference of cubes, in knowing why certain results such as the Mean Value Theorem hold, or in seeing connections between mathematical concepts and results, or between... more
Does it consist of competencies, such as being able to factor the difference of cubes, in knowing why certain results such as the Mean Value Theorem hold, or in seeing connections between mathematical concepts and results, or between various representations of a single concept like function? Often teachers' views of what it means to know mathematics are not explicitly expressed. Rather, how they teach, and especially how they test, provides clues to their mostly tacit views.
In this, our first Research Sampler column for MAA Online, we consider various answers to the above epistemological question. We take as our starting point the deliberations of the Working Group, Forms of Mathematical Knowledge, at this year's International Congress on Mathematical Education (ICME-8) in Seville. Organized by Dina Tirosh of Israel, these sessions brought together a variety of mathematics education researchers from around the world -- John Mason of the Open University, Paul Ernest of Exeter University, Eddie Gray and David Tall of Warwick University, Tommy Dreyfus of the Weizmann Institute, Anna Graeber of University of Maryland, Tom Cooney of University of Georgia, and Michele Artigue of France. Although normally this column will report results of published research, here we describe various perspectives researchers are taking as they seek to clarify underlying ideas which may guide their future investigations.
In this, our first Research Sampler column for MAA Online, we consider various answers to the above epistemological question. We take as our starting point the deliberations of the Working Group, Forms of Mathematical Knowledge, at this year's International Congress on Mathematical Education (ICME-8) in Seville. Organized by Dina Tirosh of Israel, these sessions brought together a variety of mathematics education researchers from around the world -- John Mason of the Open University, Paul Ernest of Exeter University, Eddie Gray and David Tall of Warwick University, Tommy Dreyfus of the Weizmann Institute, Anna Graeber of University of Maryland, Tom Cooney of University of Georgia, and Michele Artigue of France. Although normally this column will report results of published research, here we describe various perspectives researchers are taking as they seek to clarify underlying ideas which may guide their future investigations.
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As UME Trends completes its seventh year of publication with a possible change of format ahead, we depart from our normal routine to engage in some retrospection. Instead of describing others’ work, we explain how we came to write this... more
As UME Trends completes its seventh year of publication with a possible change of format ahead, we depart from our normal routine to engage in some retrospection. Instead of describing others’ work, we explain how we came to write this column, recount what we have tried to do with it, and finally describe
something of our own research on calculus and proof.
something of our own research on calculus and proof.
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This Research Sampler column discussed much of Jere Confrey's work on covariation of function up to the date of its publication. It also contains a separate section titled, "Irrelevant Illustrations in U.S. Math Texts".
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This column first discusses research on learning styles. According to Samuel Messick of Educational Testing Service in an invited address to the American Psychological Association, the literature is “peppered with unstable and... more
This column first discusses research on learning styles. According to Samuel Messick of Educational Testing Service in an invited address to the American Psychological Association, the literature is “peppered with unstable and inconsistent findings” with a major source of difficulty being that different researchers use different measures for the same construct or similar measures for different constructs. ["The Matter of Style: Manifestations of Personality in Cognition, Learning, and Teaching," Educational Psychologist, (1994) 29(3), 121-136.] This is followed by a discussion of how traditional students “know” what mathematics is and how it should be taught: Mathematics problems have precisely one correct solution method which teachers should explain, ordinary students can't expect to understand mathematics, and proof is irrelevant. It goes on to consider the NCTM Standards and speculate on whether these will change our incoming students' views. Next it discusses research on how students come to terms with abstract mathematical concepts. In particular, it begin to look at research to answer this question--namely, it considers undergraduate students' evolving understandings of group, subgroup, coset, normality, and quotient group, as described by Ed Dubinsky and colleagues in terms of APOS theory. Finally, it discusses the work of Robert C. Moore on the difficulties that transition-to-proof course students have with proofs and proving.
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This column discusses the first of a projected series of annual volumes titled Research in Collegiate Mathematics Education, published by CBMS Series, Issues in Mathematics Education. The volumes are a project of the AMS/MAA Committee for... more
This column discusses the first of a projected series of annual volumes titled Research in Collegiate Mathematics Education, published by CBMS Series, Issues in Mathematics Education. The volumes are a project of the AMS/MAA Committee for Research in Undergraduate Mathematics Education (CRUME). The editors, Ed Dubinsky, Alan Schoenfeld, and Jim Kaput, intend to serve two audiences -- researchers in collegiate mathematics education and mathematicians who may be looking for applications to instruction or are just plain curious.
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A mathematics learning problem, with potentially widespread implications, came to our attention recently. Writing in the journal of the Research Council for Diagnostic and Prescriptive Mathematics (RCDPM), two special education... more
A mathematics learning problem, with potentially widespread implications, came to our attention recently. Writing in the journal of the Research Council for Diagnostic and Prescriptive Mathematics (RCDPM), two special education professors at the University of Nevada at Las Vegas (UNLV) describe their case study of Kathy, a 39-year-old returning preservice teacher. They suggest " . . . some teacher candidates experience misconceptions and processing difficulties which go beyond those commonly identified in the literature." [Babbitt and Van Vactor, Focus on Learning Problems in Mathematics, Winter 1993.]
This column discusses other research under the title, "Dealing with Change from an Early Age". In it, we consider a somewhat novel idea, based upon research with
children--calculus reform should not be limited to redesigning current courses. Rather, the study of change should become a fundamental longitudinal strand in the learning of school mathematics. Ricardo Nemirovsky and colleagues at TERC, a nonprofit education research and development organization, have asked questions like: How is change experienced at different ages? Can such information be used to gain a new perspective on calculus and its role in mathematics education?
This column discusses other research under the title, "Dealing with Change from an Early Age". In it, we consider a somewhat novel idea, based upon research with
children--calculus reform should not be limited to redesigning current courses. Rather, the study of change should become a fundamental longitudinal strand in the learning of school mathematics. Ricardo Nemirovsky and colleagues at TERC, a nonprofit education research and development organization, have asked questions like: How is change experienced at different ages? Can such information be used to gain a new perspective on calculus and its role in mathematics education?
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This column discusses a recent issue of Focus on Learning Problems in Mathematics [15, 2 \& 3, 1993], which was devoted to Vygotskian psychology and mathematics education. A growing number of mathematics education researchers are taking... more
This column discusses a recent issue of Focus on Learning Problems in Mathematics [15, 2 \& 3, 1993], which was devoted to Vygotskian psychology and mathematics education. A growing number of mathematics education researchers are taking theoretical inspiration from his work. First, it discusses why cooperative groups might be effective. It goes on to discuss how Vygotsky, like Piaget, held that cognitive development proceeds from action to thought. While Piaget saw the development naturalistically, Vygotsky saw it resulting from the internalization of socially meaningful activity. Next, it discusses how Vygotsky's ideas were extended by followers such as V. V. Davydov. Finally, it discusses how Vygotsky's experimental studies with children and adolescents led him to posit the idea of the
individual's ZPD (zone of proximal development).
individual's ZPD (zone of proximal development).
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This column discusses research on non-verbal learning disabilities. Next, it discusses research of Ricardo Nemirovsky and colleagues at TERC, a nonprofit education research and development organization, who have asked questions like: How... more
This column discusses research on non-verbal learning disabilities. Next, it discusses research of Ricardo Nemirovsky and colleagues at TERC, a nonprofit education research and development organization, who have asked questions like: How
is change experienced at different ages? Can such information be used to gain a new perspective on calculus and its role in mathematics education?
is change experienced at different ages? Can such information be used to gain a new perspective on calculus and its role in mathematics education?
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This entire column is devoted to research on how individuals view probability. For example, it discusses the 1991 overview research volume, Chance Encounters: Probability in Education, edited by Kapadia and Borovcnik. It also discuss the... more
This entire column is devoted to research on how individuals view probability. For example, it discusses the 1991 overview research volume, Chance Encounters: Probability in Education, edited by Kapadia and Borovcnik. It also discuss the work of cognitive psychologists Kahneman, Slovic and Tversky, who explain people's intuitive ideas of probability in terms of judgemental heuristics such as representativeness. In addition, it discuss Fischbein's and others' work on probability. Fischbein distinquishes between primary and secondary intuitions--the former are naive, unschooled ones, whereas the latter have been informed by
instructional interventions
instructional interventions
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We first update the rather extensive research on the Students-and Professors Problem: Write an equation using the variables S and P to represent the following statement: “There are six times as many students as professors at this... more
We first update the rather extensive research on the Students-and Professors Problem: Write an equation using the variables S and P to represent the following statement: “There are six times as many students as professors at this university.” Use S for the number of students and P for the number of professors. We next describe the delightfully told, true tale of how the Russian mathematician Alexander Zvonkin organized a once-a-week “mathematics circle” for his four-year-old son, Dima, and three playmates.[ See “Mathematics for Little Ones”, Journal of Mathematical Behavior, 11(2), 207-219, 1992]. We next discuss research on college students' problem-solving strategies and the persistence of their naive ideas about inverses for composite functions and matrices.
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Thia column first discusses the diversity of knowing, in particular, the work of Sherry Turkle and Seymour Papert, who make a case for epistemological pluralism. It next discusses a recent survey of research on K-12 teachers' beliefs, in... more
Thia column first discusses the diversity of knowing, in particular, the work of Sherry Turkle and Seymour Papert, who make a case for epistemological pluralism. It next discusses a recent survey of research on K-12 teachers' beliefs, in which Dona Kagan of the University of Alabama suggests that strongly held, highly personal pedagogies develop because there are many alternative explanations of the nature of learning and cognitive growth, of why students behave the way they do, and of the best way to teach a topic. As a result, preservice teachers' beliefs are highly resistant to change and they tend to leave college with their beliefs about teaching unaffected by the research they read, their education courses and their student teaching.
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In this column we note that reading to learn mathematics, just like writing to learn mathematics, is currently being rethought in a similar constructivist way. The work of Raffaella Borasi, a mathematics education researcher, together... more
In this column we note that reading to learn mathematics, just like writing to learn mathematics, is currently being rethought in a similar constructivist way. The work of Raffaella Borasi, a mathematics education researcher, together with Marjorie Siegel, a reading researcher, is discussed. Next, we discuss a large-scale study of 751 students conducted at a mid-sized state university, whose results show that a full year of high-school calculus, whether AP or not, improved traditionally-taught first semester college calculus students’ grades one notch, from C- to B-. Finally, we note a study of how novice and expert views of “random” and “non-random” differ.
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In this column, we first discuss beliefs research of Alan Schoenfeld and others, specially as it effects mathematical problem solving. Next, we discuss a meta-analysis of Noreen Webb, who concludes, among other things, that the giving of... more
In this column, we first discuss beliefs research of Alan Schoenfeld and others, specially as it effects mathematical problem solving. Next, we discuss a meta-analysis of Noreen Webb, who concludes, among other things, that the giving of elaborate content-related explanations is positively related to one's own achievement
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Brief description of the 1991 study, "What is Mathematics to Children", published in Journal of Mathematical Behavior, Volume 10, pp 105-113.
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This column first discusses microworlds for mathematics learning, especially Seymore Paper's Logo and Sharon Dugdale's Green Globs. It goes on to discuss a dissertation by B. Lynn Bodner on diagram drawing competencies.
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First, this column discusses effective mathematical problem solving and the importance of having students understand the importance of metacognition, that is, “thinking about their own thinking.” In this regard, the work of Alan... more
First, this column discusses effective mathematical problem solving and the importance of having students understand the importance of metacognition, that is, “thinking about their own thinking.” In this regard, the work of Alan Schoenfeld and others is discussed. Next, the idea of using concept maps in mathematics, and other, learning is discussed. Concept maps resemble directed graphs with concept names at the vertices and relational links labeling the edges, although, to our knowledge, no one has yet used any graph theory. Concept maps are meant to help students integrate non-linear knowledge, which textbooks, of necessity, must present linearly.
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This entire column is devoted to reporting details of a conference on what is known about students' understandings of the function concept. It was held at Purdue University in October 1990. Many well-known mathematics education... more
This entire column is devoted to reporting details of a conference on what is known about students' understandings of the function concept. It was held at Purdue University in October 1990. Many well-known mathematics education researchers participated, for example, Michele Artigue, Anna Sierpinska, Judah Schwatrz, Steve Monk, and Anna Sfard. This eventually resulted in an 1992 MAA Notes volume, edited by G. Harel and E. Dubinsky, titled, The Concept of Function: Aspects of Epistemology and Pedagogy. We wrote the introductory chapter for this volume.
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One prequisite for becoming a mathematician is the ability to recognize a proof when you see one, and presenting proofs clearly is how you teach them. Or is it? For some in mathematics education and the philosophy of science, the... more
One prequisite for becoming a mathematician is the ability to recognize a proof when you see one, and presenting proofs clearly is how you teach them. Or is it? For some in mathematics education and the philosophy of science, the acceptance of new mathematics, including the proof of a new theorem, is a process of social negotiation. Pedagogical consequences are drawn from this view, and almost everyone writing on the subject reference, quotes, interprets, of misinterprets Imre Lakatos’ seminal work, Proofs and Refutations: The Logic of Mathematical Discovery. Using a closely reasoned Socratic dialogue based on the historical evolution of the proof of the Descartes=Euler conjecture that for polyhedral, V – 3 +F = 2. Lakatos develops the method of “proofs and refutations.” It has four basic stages: (1) a primitive conjecture, (2) a rough proof, (3) emergence of counterexamples, and (4) re-examination of the proof and improvement of the conjecture by redefining the concepts involved. He indicates that the teaching of mathematics should better reflect its practice. There seems to be more to say about creating mathematics, and we wonder how Lakatos’s work might have evolved had it not been truncated by his untimely death.
The second piece is about incorporating technology in teaching mathematics. In particular, Bernard Cornu of the University of Grenoble surveys this topic in “Didactical Aspects of the Use of Computers for Teaching and Learning Mathematics” Educational Computing in Mathematics, North-Holland, 1988. A computer can be used in many ways from classroom demonstrations where it is essentially an “improved blackboard” to labs where it is a resource for homework or projects. However, hardware and software are not enough; there needs to be a pedagogical strategy of a “didactical approach” for their use.
The second piece is about incorporating technology in teaching mathematics. In particular, Bernard Cornu of the University of Grenoble surveys this topic in “Didactical Aspects of the Use of Computers for Teaching and Learning Mathematics” Educational Computing in Mathematics, North-Holland, 1988. A computer can be used in many ways from classroom demonstrations where it is essentially an “improved blackboard” to labs where it is a resource for homework or projects. However, hardware and software are not enough; there needs to be a pedagogical strategy of a “didactical approach” for their use.
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The first part of this column is devoted to work of David Tall and Bernard Cornu on students' understanding of the derivative and limits, more generally. It then goes on to discuss J. Michael Shaughnessy's article, “Misconceptions of... more
The first part of this column is devoted to work of David Tall and Bernard Cornu on students' understanding of the derivative and limits, more generally. It then goes on to discuss J. Michael Shaughnessy's article, “Misconceptions of Probability: An Experiment with a Small-Group, Activity-Based, Model Building Approach to Introductory Probability at the College Level." After that, it discusses the article of Hall, Kibler, Wenger, and Truxaw, “Exploring the episodic structure of algebra story problem solving,” on algebra students word problem solving.
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First, we discuss several articles, and a book, about people's ability to make judgments under uncertainty. Next we discuss what is known about how experts solve problems, as opposed to how novices solve problems. Lastly, we discuss... more
First, we discuss several articles, and a book, about people's ability to make judgments under uncertainty. Next we discuss what is known about how experts solve problems, as opposed to how novices solve problems. Lastly, we discuss misconceptions and what can be done about them.
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First, we discuss both research and informed opinion that support the promising view that the focus in calculus can be shifted away from mechanics to central ideas using technology (Richard Shumway in "Symbolic Computer Systems and the... more
First, we discuss both research and informed opinion that support the promising view that the focus in calculus can be shifted away from mechanics to central ideas using technology (Richard Shumway in "Symbolic Computer Systems and the Calculus," AMATYC Review, September 1989). Next, we consider how Daniel Alibert and his colleagues at the University of Grenoble view their first-year university calculus students and discuss why they decided to design the method of "scientific debate" in which students construct their own the need for mathematics. Lastly, we discuss what prospective mathematics teachers may be learning from their university instructors.
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The first part of this column discusses the classic Students-and-Professors Problem. The original research was done by Peter Rosnick and John Clement under an NSF grant at the Cognitive Development Project, University of Massachusetts,... more
The first part of this column discusses the classic Students-and-Professors Problem. The original research was done by Peter Rosnick and John Clement under an NSF grant at the Cognitive Development Project, University of Massachusetts, Amherst. The problem is: Write an equation using the variables S and P to represent the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. The second part of this column discusses Alan Schoenfeld's short intellectual autobiography (“Confessions of an Accidental Theorist,” For the Learning of Mathematics, 8, 1). The third part of this column discusses John Sweller's article, “Cognitive Load During Problem Solving: Effects on Learning” (Cognitive Science, 12 (1988), 257-285). The fourth, and last part of this column briefly discusses Ole Skovsmose's critical pedagogy in which students are encouraged to reflect on the uses, misuses, and societal consequences of particular models [“Mathematics as a Part of Technology”, Educational Studies in Mathematics, 19(1988), 23-41].
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In July the International Group for the Psychology of Mathematics Education (PME) met in Recife, where participants were treated to miles of tropical beach, unpredictable short rain showers, and sumptuous lunches ending with petite cups... more
In July the International Group for the Psychology of Mathematics Education (PME) met in Recife, where participants were treated to miles of tropical beach, unpredictable short rain showers, and sumptuous lunches ending with petite cups of thick, sweet coffee. As happens when PME meets in unusual places, attendance was somewhat lower, with some "regulars" missing. Travel arrangements were occasionally awkward with Australian attendees flying over Antarctica, while those from the U.S. and Canada accumulated frequent flyer miles journeying via Sao Paulo and backtracking a thousand miles.
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This is how Carole B. Lacampagne, a mathematician working at U.S. Department of Education, concluded her recent AMS-MAA invited address, titled “Reform in mathematics education: New or simply a variation on an old theme?”, at the... more
This is how Carole B. Lacampagne, a mathematician working at U.S. Department of Education, concluded her recent AMS-MAA invited address, titled “Reform in mathematics education: New or simply a variation on an old theme?”, at the Minneapolis Mathfest. She outlined the current reform effort beginning with the 1983 publication of A Nation at Risk, continuing
with the 1989 publication of the NCTM Standards and the 1990 National Education Goals (a.k.a. the governors' goals for the year 2000), and ending with the 1994 Goals 2000: Educate America Act.
with the 1989 publication of the NCTM Standards and the 1990 National Education Goals (a.k.a. the governors' goals for the year 2000), and ending with the 1994 Goals 2000: Educate America Act.
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A subdued pink cover, instead of the customary red, distinguishes this first special issue of Educational Studies in Mathematics (Vol. 25, Nos. 1-2, 1993), which is dedicated to its founder and editor for ten years. H.F., as colleagues... more
A subdued pink cover, instead of the customary red, distinguishes this first special issue of Educational Studies in Mathematics (Vol. 25, Nos. 1-2, 1993), which is dedicated to its founder and editor for ten years. H.F., as colleagues referred to him, helped propel the scientific discipline of mathematics education forward, as well as being the inspiration behind the Dutch realistic mathematics education movement.
Compiled by Guest Editor Leen Streefland of the (now) Freudenthal Institute at Utrecht University, this issue contains nine articles, all but one in English. Devoted mainly to aspects of Freudenthal's work in mathematics education, there is also a short sampling from his research in homotopy theory, Lie groups, and geometry.
Compiled by Guest Editor Leen Streefland of the (now) Freudenthal Institute at Utrecht University, this issue contains nine articles, all but one in English. Devoted mainly to aspects of Freudenthal's work in mathematics education, there is also a short sampling from his research in homotopy theory, Lie groups, and geometry.
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With these words Frank Demana, founder and honorary co-chair with Bert Waits, opened the Sixth Annual International Conference on Technology in Collegiate Mathematics (ICTCM), hosted by Montclair State College in Parsippany, New Jersey... more
With these words Frank Demana, founder and honorary
co-chair with Bert Waits, opened the Sixth Annual International
Conference on Technology in Collegiate Mathematics (ICTCM),
hosted by Montclair State College in Parsippany, New Jersey last
November. Just under fifteen hundred participants came from a
large mix of universities, colleges, junior colleges, and high
schools. Among them were over three hundred presenters from
such far-flung places as Japan, Australia, England, Holland, and
France, as well as the U.S. ICTCM has no sustaining
professional organization. It is the interest and enthusiasm of
volunteers, some forty of whom were on the program committee,
as well as the financial support of Addison-Wesley, Casio,
Hewlett-Packard, Sharp and Texas Instruments, that keep it
going.
co-chair with Bert Waits, opened the Sixth Annual International
Conference on Technology in Collegiate Mathematics (ICTCM),
hosted by Montclair State College in Parsippany, New Jersey last
November. Just under fifteen hundred participants came from a
large mix of universities, colleges, junior colleges, and high
schools. Among them were over three hundred presenters from
such far-flung places as Japan, Australia, England, Holland, and
France, as well as the U.S. ICTCM has no sustaining
professional organization. It is the interest and enthusiasm of
volunteers, some forty of whom were on the program committee,
as well as the financial support of Addison-Wesley, Casio,
Hewlett-Packard, Sharp and Texas Instruments, that keep it
going.
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These were the themes of the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held at the Asilomar Conference Center on the Monterey Peninsula of California... more
These were the themes of the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held at the Asilomar Conference
Center on the Monterey Peninsula of California in October 1993.
Center on the Monterey Peninsula of California in October 1993.
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The AMS-MAA Summer Meetings were held in conjunction with the Canadian Mathematical Society (CMS) on the beautifully situated campus of the University of British Columbia. Dorm accommodations were widely acclaimed as more than adequate... more
The AMS-MAA Summer Meetings were held in conjunction with the Canadian Mathematical Society (CMS) on the beautifully situated campus of the University of British Columbia. Dorm accommodations were widely acclaimed as more than adequate and included spacious apartments, student union food was moderately priced and varied, meeting rooms were sizable, e-mail was obtainable through Internet, a daily newsletter kept one up-to-date on schedule changes, and the weather was excellent with mostly sunny days and 75 degree(Fahrenheit) highs.
Mathematics education was well represented at the meetings. At the Opening Banquet, recipients of the MAA Award for Distinguished College or University Teaching of Mathematics, V. Frederick Rickey, Bowling Green State, Doris W. Schattschneider, Moravian College, and Philip D. Straffin, Jr., Beloit College, gave personal insights regarding teaching. Minicourses, which are becoming very popular, ranged from environmental modelling and earth algebra to implementing the Harvard calculus curriculum and teaching abstract algebra with ISETL.
Mathematics education was well represented at the meetings. At the Opening Banquet, recipients of the MAA Award for Distinguished College or University Teaching of Mathematics, V. Frederick Rickey, Bowling Green State, Doris W. Schattschneider, Moravian College, and Philip D. Straffin, Jr., Beloit College, gave personal insights regarding teaching. Minicourses, which are becoming very popular, ranged from environmental modelling and earth algebra to implementing the Harvard calculus curriculum and teaching abstract algebra with ISETL.
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Held about a decade after the first such conference, about 500 participants from around the world met on the Cornell campus in early August for the Third International Seminar on Misconceptions and Educational Strategies in Science and... more
Held about a decade after the first such conference, about 500 participants from around the world met on the Cornell campus in early August for the Third International Seminar on Misconceptions and Educational Strategies in Science and Mathematics. The original seminar idea is due to two Cornell education professors, D. Bob Gowin and Joseph D. Novak, whose numerous masters and Ph.D. students have investigated both meaningful learning and students' misconceptions in science and mathematics using Gowan's Vee and Novak's concept maps. The Vee can help a researcher identify and clarify methodological elements (the collecting, analyzing, and interpreting of data, as well as the value of a study) and conceptual elements (the constructs, principles, theories, and perspectives that cause one to ask certain questions). Concept maps, which resemble graphs with concepts at the nodes and connecting phrases along the edges and are also hierarchically-structured, are tools for analyzing either one's own or students' knowledge structures. [See “Concept Maps on the Computer," UME Trends, May 1991, 6.]
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The March AMS meeting in Knoxville featured a special session, “Interventions to Assure Success: Calculus Through Junior Faculty.” Sponsored by the AMS-MAA-SIAM Committee on Preparation for College Teaching, this event was the first ever... more
The March AMS meeting in Knoxville featured a special session, “Interventions to Assure Success: Calculus Through Junior Faculty.” Sponsored by the AMS-MAA-SIAM
Committee on Preparation for College Teaching, this event was the first ever education session at a regional meeting. Though the contents should have been of broad general interest and the session did attract the occasional drop-in, the audience was composed mainly of people working in the area, much like other special sessions devoted to such topics as continua and optimal control.
Committee on Preparation for College Teaching, this event was the first ever education session at a regional meeting. Though the contents should have been of broad general interest and the session did attract the occasional drop-in, the audience was composed mainly of people working in the area, much like other special sessions devoted to such topics as continua and optimal control.
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Professional standards were a big concern for the approximately one thousand attendees at the eighteenth annual conference of the American Mathematical Association of Two-Year Colleges (AMATYC) in Indianapolis last November. Currently a... more
Professional standards were a big concern for the approximately one thousand attendees at the eighteenth annual conference of the American Mathematical Association of Two-Year Colleges (AMATYC) in Indianapolis last November.
Currently a volunteer organization with no headquarters or permanent staff, AMATYC is well aware of the fact that two-year colleges teach over half the students taking lower-division mathematics and thinks the time has come for sweeping organizational change. Supported by a grant from the Exxon Educational Foundation, AMATYC engaged in a year-long planning effort resulting in a strategic plan with twelve main goals and numerous lesser objectives. This was approved by its contentious delegate assembly, whose energetic debate used different microphones for those speaking for or against an issue.
Currently a volunteer organization with no headquarters or permanent staff, AMATYC is well aware of the fact that two-year colleges teach over half the students taking lower-division mathematics and thinks the time has come for sweeping organizational change. Supported by a grant from the Exxon Educational Foundation, AMATYC engaged in a year-long planning effort resulting in a strategic plan with twelve main goals and numerous lesser objectives. This was approved by its contentious delegate assembly, whose energetic debate used different microphones for those speaking for or against an issue.
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As an "add-on" to the usual Advanced Placement (AP) Calculus Reading at Clemson University this past June, 150 readers sponsored by NSF, 50 AP consultants sponsored by College Board, and assorted "specials" stayed over for a three-day... more
As an "add-on" to the usual Advanced Placement (AP) Calculus Reading at Clemson University this past June, 150 readers sponsored by NSF, 50 AP consultants sponsored by College Board, and assorted "specials" stayed over for a three-day graphing calculator workshop, Technology Intensive Calculus with Advanced Placement. TICAP, pronounced "tie-cap," is a program for future workshop leaders to provide them with materials and expertise for inservice to AP calculus teachers, so College Board can implement its announced interest in requiring the use of graphing
calculators on the May 1995 AP calculus exam. Although the final decison has yet to be made by College Board, much preparatory work is being undertaken.
calculators on the May 1995 AP calculus exam. Although the final decison has yet to be made by College Board, much preparatory work is being undertaken.
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“Research on learning affects us all.” This is what MAA President Lida Barrett said when she introduced the CTUM panel on research in learning undergraduate mathematics at the Summer Meeting in Columbus. James Kaput began the panel of... more
“Research on learning affects us all.” This is what MAA President Lida Barrett said when she introduced the CTUM panel on research in learning undergraduate mathematics at the Summer Meeting in Columbus.
James Kaput began the panel of three by giving an overview of different kinds research: (1) surveys giving descriptive information on large populations, e.g., NAEP and international comparison studies; (2) the “agricultural-botany” type of controlled experiments, which dominated educational research until the late 1970's, where one studied the effect of some small change in what one did--with statistical significance being a main result; (3) historical or philosophical research looking at longer term events; (4) formative or summative research evaluating the outcome of a change in curriculum materials; (5) case studies or clinical research involving careful studies of individuals in complex learning situations with the aim of providing detailed models of cognitive
processes. The latter, which Kaput sees at the most promising, aims at descriptions of what’s going on in the minds of individuals as they learn mathematics.
James Kaput began the panel of three by giving an overview of different kinds research: (1) surveys giving descriptive information on large populations, e.g., NAEP and international comparison studies; (2) the “agricultural-botany” type of controlled experiments, which dominated educational research until the late 1970's, where one studied the effect of some small change in what one did--with statistical significance being a main result; (3) historical or philosophical research looking at longer term events; (4) formative or summative research evaluating the outcome of a change in curriculum materials; (5) case studies or clinical research involving careful studies of individuals in complex learning situations with the aim of providing detailed models of cognitive
processes. The latter, which Kaput sees at the most promising, aims at descriptions of what’s going on in the minds of individuals as they learn mathematics.
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There are at least four ways students might learn mathematics: spontaneously, inductively, constructively, and pragmatically. Each view has implications for teaching. If students learn mathematics spontaneously, there is little one can do... more
There are at least four ways students might learn mathematics: spontaneously, inductively, constructively, and pragmatically. Each view has implications for teaching. If students learn mathematics spontaneously, there is little one can do directly to help them, except perhaps provide them a good environment, that is, good explanations and materials. If students learn inductively by working examples and extracting common features and important ideas, then one might present them many carefully structured examples to facilitate this. If students learn by making mental constructions to handle mathematical ideas, then one might want to study these mental constructions to discover how to help students make them. If students learn pragmatically as a response to real world problems, one might search for interesting applications. In regard to calculus, for example, most mathematicians say they believe students learn inductively through examples, whereas they often teach as if calculus were learned spontaneously by listening and watching. (Ed Dubinsky, paper presented at the October St. Olaf Conference.) Like most researchers in mathematics education today, Dubinsky takes the constructivist view.
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At the November 17-18 meeting of NSF Calculus Principal Investigators, there was a poster session of all projects, plus software demonstrations from CALCULUS T/L to uses of MathCAD. Cross fertilization of ideas occurred in many informal... more
At the November 17-18 meeting of NSF Calculus Principal Investigators, there was a poster session of all projects, plus software demonstrations from CALCULUS T/L to uses of MathCAD. Cross fertilization of ideas occurred in many informal conversations.
Wayne Roberts reported on what he's been seeing on campus visits for CRAFTY. Technology is being widely introduced, but not always imaginatively. Everyone seems to be writing a new, but not necessarily “lean” book. Delivery is being changed more than course content, with everything from longer projects with a writing component to group learning. Portability of some experimental projects to other campuses is a concern.
There were several panels; the one on evaluation offered many good ideas on what one might learn from calculus projects if one asked the right questions. The Duke project aims to get a clearer picture of what both their new course and the old course do and do not accomplish with regard to writing, concepts, skills, problem-solving, and attitudes. Kathleen Heid remarked that a project should at least be tested according to its own goals. Too often a new course is merely tested against the standards of the old course. She suggested many kinds of data one might collect.
Wayne Roberts reported on what he's been seeing on campus visits for CRAFTY. Technology is being widely introduced, but not always imaginatively. Everyone seems to be writing a new, but not necessarily “lean” book. Delivery is being changed more than course content, with everything from longer projects with a writing component to group learning. Portability of some experimental projects to other campuses is a concern.
There were several panels; the one on evaluation offered many good ideas on what one might learn from calculus projects if one asked the right questions. The Duke project aims to get a clearer picture of what both their new course and the old course do and do not accomplish with regard to writing, concepts, skills, problem-solving, and attitudes. Kathleen Heid remarked that a project should at least be tested according to its own goals. Too often a new course is merely tested against the standards of the old course. She suggested many kinds of data one might collect.
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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."
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The Thirteenth Conference of the International Group for the Psychology of Mathematics Education (PME 13) met at the Sorbonne in Paris, July 9 - 13, with the celebration of the bicentennial of the French Revolution as backdrop.... more
The Thirteenth Conference of the International Group for the Psychology of Mathematics Education (PME 13) met at the Sorbonne in Paris, July 9 - 13, with the celebration of the bicentennial of the French Revolution as backdrop. Appropriately, the opening plenary address by Jean Dhombres concentrated on the sweeping scientific accomplishments of the Revolution from the introduction of the metric system to the espousal of the analytic method.
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Spreading the word about the state of the art in existing calculus projects was the intent of the one day conference on “The Future of Calculus," held on the campus of Ithaca College April 15. Organizers Steve Hilbert and John Maceli,... more
Spreading the word about the state of the art in existing calculus projects was the intent
of the one day conference on “The Future of Calculus," held on the campus of Ithaca College April 15. Organizers Steve Hilbert and John Maceli, with support obtained under
their NSF Planning Grant, wanted to inform faculty unable to attend national meetings.
of the one day conference on “The Future of Calculus," held on the campus of Ithaca College April 15. Organizers Steve Hilbert and John Maceli, with support obtained under
their NSF Planning Grant, wanted to inform faculty unable to attend national meetings.
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This was a short report on the very first ICTCM(International Conference on Technology in Collegiate Mathematics) conference held at Ohio State University.
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This article gives information on the Prize. The awardee is determined by a committee of the MAA (Mathematical Association of America). Awardees have been: Chris Rasmussen, 2006; Marilyn Carlson, 2008; Keith Weber, 2010; Lara Alcock,... more
This article gives information on the Prize. The awardee is determined by a committee of the MAA (Mathematical Association of America). Awardees have been: Chris Rasmussen, 2006; Marilyn Carlson, 2008; Keith Weber, 2010; Lara Alcock, 2012; Matthew Inglis, 2014. Nomination information can be found on the MAA Website.
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Here is the introduction: At the top left corner of almost every page of MAA Online are three buttons: Home, Search, and Contact Us. When you click on Contact Us, you find a number of people you might e-mail. If you had a question about... more
Here is the introduction:
At the top left corner of almost every page of MAA Online are three buttons: Home, Search, and Contact Us. When you click on Contact Us, you find a number of people you might e-mail. If you had a question about mathematics or mathematics learning, which of these people would you ask? The Editor? Maybe. I know Fernando Gouvea gets a lot of e-mail about broken links, as well as requests to publish the latest angle trisection. But you probably would not ask one of the associate editors for MAA programs, professional development, awards, or sections. And, unless you want to know about subscriptions, book orders, donations, or committees, that leaves me -- the Associate Editor for Teaching and Learning -- which seems to be the catchall, grab bag category.
Until assuming this volunteer position, I had no idea what it might feel like to be a mathematical Ann Landers. While no one at MAA asked me to do so, I decided that, in the interests of good public relations, every sincere request for help should get answered, even if it's only: Perhaps you could "Ask Dr. Math" at the Math Forum [http://mathforum.org]. Alternatively, if a query poses a mathematical problem, I sometimes suggest asking the MathNerds [www.mathnerds.com/texan], a site willing to provide "free, discovery-based, mathematical guidance via an international, volunteer network of mathematicians," one of whom is W. Ted Mahavier of Lamar University . Just recently, I found a site called "The Math Doctor is In!" run by Dan McQuillan of the University of Western Ontario, which I might recommend in the future [http://homepages.msn.com/LibraryLawn/learnmath/].
At the top left corner of almost every page of MAA Online are three buttons: Home, Search, and Contact Us. When you click on Contact Us, you find a number of people you might e-mail. If you had a question about mathematics or mathematics learning, which of these people would you ask? The Editor? Maybe. I know Fernando Gouvea gets a lot of e-mail about broken links, as well as requests to publish the latest angle trisection. But you probably would not ask one of the associate editors for MAA programs, professional development, awards, or sections. And, unless you want to know about subscriptions, book orders, donations, or committees, that leaves me -- the Associate Editor for Teaching and Learning -- which seems to be the catchall, grab bag category.
Until assuming this volunteer position, I had no idea what it might feel like to be a mathematical Ann Landers. While no one at MAA asked me to do so, I decided that, in the interests of good public relations, every sincere request for help should get answered, even if it's only: Perhaps you could "Ask Dr. Math" at the Math Forum [http://mathforum.org]. Alternatively, if a query poses a mathematical problem, I sometimes suggest asking the MathNerds [www.mathnerds.com/texan], a site willing to provide "free, discovery-based, mathematical guidance via an international, volunteer network of mathematicians," one of whom is W. Ted Mahavier of Lamar University . Just recently, I found a site called "The Math Doctor is In!" run by Dan McQuillan of the University of Western Ontario, which I might recommend in the future [http://homepages.msn.com/LibraryLawn/learnmath/].
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This short article giving an update on the SIGMAA on RUME and its activities was written for MAA Focus.
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This news article tells of the formation of ARUME (Association for Research in Undergraduate Mathematics Education), which became the SIGMAA on RUME (Special Interest Group of the Mathematical Association of America on Research in... more
This news article tells of the formation of ARUME (Association for Research in Undergraduate Mathematics Education), which became the SIGMAA on RUME (Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education), somewhat later. It was the first special interest group of the MAA.
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I plan to take you on a journey through how my husband John and I, who have PhDs in mathematics and spent our early academic years in pure mathematics, got into research in mathematics education. Along the way, I will discuss the kinds of... more
I plan to take you on a journey through how my husband John and I, who have PhDs in mathematics and spent our early academic years in pure mathematics, got into research in mathematics education. Along the way, I will discuss the kinds of research we have done and where we are in our thinking today. We have been mainly concerned with university students’ learning of mathematical ideas and concepts, especially with proof and proving.
Before we got interested in mathematics education as a research subject, we were mathematicians teaching at least some upper-division and graduate mathematics courses using the Moore Method (Mahavier, 1999). Doing so gave us a lot of exposure to students’ proving difficulties in courses like abstract algebra and topology. However, in other courses, like calculus, we lectured. The problem, as we saw it, was that despite our seemingly well-constructed and thoughtful lectures, students often had misconceptions and didn’t perform as we wished. We wondered why. This was the germ of our interest in mathematics education research at the university level.
Before we got interested in mathematics education as a research subject, we were mathematicians teaching at least some upper-division and graduate mathematics courses using the Moore Method (Mahavier, 1999). Doing so gave us a lot of exposure to students’ proving difficulties in courses like abstract algebra and topology. However, in other courses, like calculus, we lectured. The problem, as we saw it, was that despite our seemingly well-constructed and thoughtful lectures, students often had misconceptions and didn’t perform as we wished. We wondered why. This was the germ of our interest in mathematics education research at the university level.
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This foreword was written at the request of the editors of this volume. It contains 40 short, approximately six-page, pieces of usable advice, written by college and university mathematics teachers who have tried out the ideas themselves... more
This foreword was written at the request of the editors of this volume. It contains 40 short, approximately six-page, pieces of usable advice, written by college and university mathematics teachers who have tried out the ideas themselves and found them worthy of presenting to others. In short, it’s a handbook of practical suggestions—some large, some small—of “what works” for helping students develop proof-writing skills.
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This paper recounts the prehistory and history of the Mathematical Association of American’s Special Interest Group on Research in Mathematics Education (SIGMAA on RUME). It relates the many events leading up to the formation ARUME, the... more
This paper recounts the prehistory and history of the Mathematical Association of American’s Special Interest Group on Research in Mathematics Education (SIGMAA on RUME). It relates the many events leading up to the formation ARUME, the Association for Research in Mathematics Education, which was the predecessor of the SIGMAA on RUME. It continues with
the establishment of SIGMAA on RUME and its growth during the first ten years of its existence.
the establishment of SIGMAA on RUME and its growth during the first ten years of its existence.
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This preface gives an overview of the research contained in this volume. The major themes of the reported studies include curriculum, teaching and learning calculus and analysis.
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This is a short article written for the AWM Newsletter. It reports on my invited address, titled "Two Research Traditions Separated by a Common Subject: Mathematics and Mathematics Education," given at MathFest in Burlington, VT. In the... more
This is a short article written for the AWM Newsletter. It reports on my invited address, titled "Two Research Traditions Separated by a Common Subject: Mathematics and Mathematics Education," given at MathFest in Burlington, VT. In the address itself, I discussed the nature of mathematics education research, the kinds of claims made, and the evidence provided. I also contrasted the way research is done and evaluated in mathematics and mathematics education.
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This article notes that even with reasonable assumptions (see accompanying table) that one cannot reasonable expect the percentage of women administrators to get even close to 50% in 10 years time.
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This short article describes our experimental calculus course in which we used our own notes, had students present their original solutions to problems at the board, and critiqued them. This was done in a very Modified Moore Method way,... more
This short article describes our experimental calculus course in which we used our own notes, had students present their original solutions to problems at the board, and critiqued them. This was done in a very Modified Moore Method way, except there were no theorems to prove. The theorems and definitions were provided in the notes, but there were no sample worked problems.
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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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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 omega-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. In addition, most of our arguments depend on joint continuity. Recall that in a topological inverse semigroup both the multiplication and the inverse are continuous.
Some of our results apply to the continuous version of the extended bicyclic semigroup, that is, R x R with the usual topology of the plane and multiplication as in the extended bicyclic semigroup. However, the variablilty in the attachment of the group of units cannot occur. We will comment when our arguments carry over and leave substantially different arguments for a later investigation.
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. In addition, most of our arguments depend on joint continuity. Recall that in a topological inverse semigroup both the multiplication and the inverse are continuous.
Some of our results apply to the continuous version of the extended bicyclic semigroup, that is, R x R with the usual topology of the plane and multiplication as in the extended bicyclic semigroup. However, the variablilty in the attachment of the group of units cannot occur. We will comment when our arguments carry over and leave substantially different arguments for a later investigation.
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Eberhart and Selden have shown that the only Hausdorff topology on the bicyclic semigroup making it a topological semigroup is the discrete topology. J. W. Hogan has shown that the only locally compact topology on the alpha-bicyclic... more
Eberhart and Selden have shown that the only Hausdorff topology on the bicyclic semigroup making it a topological semigroup is the discrete topology. J. W. Hogan has shown that the only locally compact topology on the alpha-bicyclic semigroup making it a topological inverse semigroup is the discrete topology. As Hogan's paper is a generalization of that of Eberhart and Selden, it seems natural to ask whether the assumption of local compactness is necessary. The following example shows that it is.
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A bisimple ω-semigroup is determined by its group of units G and an endomorphism of G. Suppose this bisimple ω-semigroup is contained in T a locally compact topological inverse semigroup. A characterization of its closure has been given... more
A bisimple ω-semigroup is determined by its group of units G and an endomorphism of G. Suppose this bisimple ω-semigroup is contained in T a locally compact topological inverse semigroup. A characterization of its closure has been given when G is compact and the endomorphism is onto. In this paper, we investigate the kernel of that endomorphism, as well as the kernels of related maps. We prove that the component of the identity of each is compact. In particular, if the kernel of the endomorphism of G is connected, it is compact. From this, it follows that certain bisimple ω-semigroups are closed in every containing T.
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We obtain a pair of theorems. The first shows that given any bisimple omega-semigroup B with G compact and the determining endomorphism onto, a certain construction produces a locally compact topological inverse semigroup which is the... more
We obtain a pair of theorems. The first shows that given any bisimple omega-semigroup B with G compact and the determining endomorphism onto, a certain construction produces a locally compact topological inverse semigroup which is the closure of B. The second shows that the locally compact closure of every bisimple omega-semigroup with G compact and the determining endomorphism onto is algebraically and topologically isomorphic to one of these constructions.
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Reilly has characterized bisimple omega-semigroups in terms of their group of units and a natural endomorphism of that group. The closures of bisimple omega-semigroups, which are contained in locally compact topological inverse... more
Reilly has characterized bisimple omega-semigroups in terms of their group of units and a natural endomorphism of that group. The closures of bisimple omega-semigroups, which are contained in locally compact topological inverse semigroups, are studied. A characterization theorem is obtained for the closure in the case where the group of units is compact and the natural endomorphism is onto. N.B. Because of the length of this paper, it had to be scanned into my computer in two parts. The first part contains pages 15-53 and the second part contains pages 53-77.
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"Reilly [10] has characterized bisimple omega-semigroups in terms of their group of units and a natural endomorphism of that group. We study the closures of bisimple omega-semigroups which are contained in locally compact topological... more
"Reilly [10] has characterized bisimple omega-semigroups in terms of their group of units and a natural endomorphism of that group. We study the closures of bisimple omega-semigroups which are contained in locally compact topological inverse semigroups. A characterization theorem is obtained for the closure in the case where the group of units is compact and the natural endomorphism is onto. Several other cases are studied. The contents are as follows:
Introduction
Chapter 1 Preliminaries.
Chapter 2 General Results.
Chapter 3 Three Examples.
Chapter 4 A Characterization Theorem for the Case G Compact and the map alpha onto.
Chapter 5 A Partial Description of the Topological Structure of B*, under Certain Natural Conditions, Given the Topology of B and the Topology of Eo*.
Chapter 6 The structure of B* - B.
Chapter 7 The Kernel of the map alpha.
Introduction
Chapter 1 Preliminaries.
Chapter 2 General Results.
Chapter 3 Three Examples.
Chapter 4 A Characterization Theorem for the Case G Compact and the map alpha onto.
Chapter 5 A Partial Description of the Topological Structure of B*, under Certain Natural Conditions, Given the Topology of B and the Topology of Eo*.
Chapter 6 The structure of B* - B.
Chapter 7 The Kernel of the map alpha.
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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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A list of collected references.
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A list of references prepared for our talk on the rhetoric of proofs at the ARUME Conference in 1999.
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A list of references prepared for the colloquium talk.
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This annotated list provides information on what (mathematics) education journal editors and reviewers look for in research papers. It accompanied our talk at the ARUME Conference.
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This annotated bibliography provides information on what mathematics education journal editors and reviewers (i.e., referees) look for in research papers. It was prepared as a supplement to our talk "Where's the Theorem? Where's the... more
This annotated bibliography provides information on what mathematics education journal editors and reviewers (i.e., referees) look for in research papers. It was prepared as a supplement to our talk "Where's the Theorem? Where's the Proof? An Analysis of Why Math Ed Research Papers Get Rejected" given at the RUMEC Conference on Research in Mathematics Education, South Bend, Indiana, September 1998.
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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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A list of references to accompany my talk to the Tennessee Collaborative Academy.
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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 handout covers the following 5 topics on publishing mathematics education research: 1. Finding time/space/support for the actual writing 2. Choosing a journal 3. What to do, or not do, to improve your chances of acceptance 4. An... more
This handout covers the following 5 topics on publishing mathematics education research:
1. Finding time/space/support for the actual writing
2. Choosing a journal
3. What to do, or not do, to improve your chances of acceptance
4. An overview of the submission/publication process
5. What to expect and do when you get criticism/rejection.
1. Finding time/space/support for the actual writing
2. Choosing a journal
3. What to do, or not do, to improve your chances of acceptance
4. An overview of the submission/publication process
5. What to expect and do when you get criticism/rejection.
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This is the protocol used for the study of the same name.
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These questions are handed out to students in our graduate level mathematics education research seminar, which concentrates on reading and analyzing mostly qualitative research articles in undergraduate mathematics education (RUME). They... more
These questions are handed out to students in our graduate level mathematics education research seminar, which concentrates on reading and analyzing mostly qualitative research articles in undergraduate mathematics education (RUME). They were collected over a long time and have just been updated and rearranged. They are meant to help focus class discussions in a productive way.
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For this working group, participants were asked to bring examples of habits of mind for discussion. John Selden and I brought this handout to the Working Group with examples of behavioral schemas for proving, which can be thought of as... more
For this working group, participants were asked to bring examples of habits of mind for discussion. John Selden and I brought this handout to the Working Group with examples of behavioral schemas for proving, which can be thought of as small habits of mind. There were four examples of behavioral schemas that we observed with students, dubbed Edward, Mary, Willy, and Sofia.
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This handout was given to the students who attended the voluntary proving supplement. The students co-constructed a proof similar (in actions) to an assigned homework proof problem. At the end of the supplement class, they each received... more
This handout was given to the students who attended the voluntary proving supplement. The students co-constructed a proof similar (in actions) to an assigned homework proof problem. At the end of the supplement class, they each received one of these handouts, which could be considered a co-construction proof trajectory. While not identical to what they would have produced, it was sufficiently similar so that they would have an account of what had been done. It is a step-by-step account of the proving process beginning the first- and second-level proof frameworks.
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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.
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We begin by reporting on students’ work in a design experiment – an unusual course whose whole purpose is to teach beginning graduate and advanced undergraduate students to prove theorems. We then discuss an emerging theoretical... more
We begin by reporting on students’ work in a design experiment – an unusual course whose whole purpose is to teach beginning graduate and advanced undergraduate students to prove theorems. We then discuss an emerging theoretical framework, describing kinds and aspects of proofs, that we have found useful for monitoring course breadth and individual student progress. Next we describe two persistent student difficulties, “starting in the wrong place” and “reluctance to introduce a fixed, but arbitrary object.” Finally, we suggest a theoretical approach to a kind of procedural knowledge that we see as playing a large role in constructing proofs.
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People construct their own understandings based on prior knowledge and past experiences, whether just listening, as in a lecture, or actively engaged in some project. They engage in sense-making, attempting to fit new knowledge into what... more
People construct their own understandings based on prior knowledge and past experiences, whether just listening, as in a lecture, or actively engaged in some project. They engage in sense-making, attempting to fit new knowledge into what they already know. Sometimes this can lead to misconceptions. For example, when children having a good grasp of whole numbers are introduced to decimals, they sometimes say that 3.214 >3.8 because 214>8. Such misunderstandings are a consequence of attempting to integrate new material with prior knowledge and, at least initially, cannot be entirely avoided. However, active engagement with appropriate activities, along with deep reflection, can lead to more meaningful, and less errorful, learning.
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In this paper, we will discuss the way various features of consciousness interact with each other and with cognition, specifically, the cognition of mathematical reasoning and problem solving. Thus we are interested in how consciousness... more
In this paper, we will discuss the way various features of consciousness interact with each other and with cognition, specifically, the cognition of mathematical reasoning and problem solving. Thus we are interested in how consciousness and cognition "work," in a somewhat mechanistic way, rather than in larger philosophical questions about consciousness. Our goal is ultimately to answer questions like: Where does one's "next" idea come from? Answers to such smaller questions may eventually help in understanding the nature of consciousness itself.
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This is a handout for my transition-to-proof course given at Tennessee Technological University, Spring Semester, 2003.
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This handout contains the agenda for the initial organizational meeting to form the Association for Research in Mathematics Education (ARUME), which later became the Special Interest Group of the Mathematical Association of America for... more
This handout contains the agenda for the initial organizational meeting to form the Association for Research in Mathematics Education (ARUME), which later became the Special Interest Group of the Mathematical Association of America for Research in Undergraduate Mathematics Education (SIGMAA on RUME). It also contains the initial by-laws and a slate of officers. I (with others) was in charge of preparing the agenda and the by-laws.
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This is an important, under-research question.
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This paper begins: "Just as with the reading of any text, one's reading of mathematical proofs depends upon one's purposes. Does one just want to get the gist of an argument so one is well-informed about current mathematical discoveries... more
This paper begins: "Just as with the reading of any text, one's reading of mathematical proofs depends upon one's purposes. Does one just want to get the gist of an argument so one is well-informed about current mathematical discoveries generally, or in one's subfield, such as number theory or differential equations? Does one want to find out whether a result, and its proof, are relevant to one's own research? Does one think one can actually use a result in one's own research? Has one been asked to blind referee a manuscript for publication in a journal? Is one looking over a dissertation as a member of a PhD student's committee? Is one grading a graduate course examination, consisting entirely of proofs and counterexamples, in a course such as real analysis? Is one grading an examination, consisting of proofs, in undergraduate transition-to proof course?"
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This paper begins "Today in a RUME class, we were asked, after mentioning epistemological, cognitive, and didactical obstacles, whether there were affective obstacles to mathematics learning. We answered a bit about the well-studied... more
This paper begins "Today in a RUME class, we were asked, after mentioning epistemological, cognitive, and didactical obstacles, whether there were affective obstacles to mathematics learning. We answered a bit about the well-studied concept of mathematics anxiety. However, there should be a lot of other affective obstacles, which could be studied. "
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After reading Robert Moore's IJRUME paper on professors' grading of transition-to-proof course students' proof attempts, the following research questions came to mind.
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Why is it that one sometimes finds oneself (unintentionally and automatically) interchanging some mathematical symbols like epsilon and delta when discussing with (students or others) what is to be done in a proof of, say, continuity of a... more
Why is it that one sometimes finds oneself (unintentionally and automatically) interchanging some mathematical symbols like epsilon and delta when discussing with (students or others) what is to be done in a proof of, say, continuity of a function at a point?
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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.
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This is a proposal for a research paper on a theoretically-based proving intervention. We will describe an intervention in the form of a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we... more
This is a proposal for a research paper on a theoretically-based proving intervention. We will describe an intervention in the form of a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical perspective. Since no major reorganization of the real analysis course itself was undertaken, we feel such a supplement could be implemented practically by many mathematics departments.
Research Interests: Real Analysis, Real Analysis (Mathematics), Proof and Reasoning, Real and Complex Analysis, Proof and Proving in Mathematics Education, and 6 moreResearch of Mathematics Education, Educational Research in Mathematics, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, Reasoning and Proof, and Mathematical Reasoning and Proofs
This unpublished paper contains some ideas on knowing how, and when, to check an equation, a rule, or a theorem in the moment to gain confidence that one has done it correctly or remembered it correctly. There are now (as of March 5,... more
This unpublished paper contains some ideas on knowing how, and when, to check an equation, a rule, or a theorem in the moment to gain confidence that one has done it correctly or remembered it correctly. There are now (as of March 5, 2015) four example situations and a question about whether knowing-to-check is something that one learns implicitly and something which causes S2 cognition to override S1 cognition.
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Here is a list of questions about misconceptions (whether that is always an appropriate term), bugs (perhaps a better term sometimes), mistakes (something done wrongly, perhaps because of working memory overload), and flawed actions (the... more
Here is a list of questions about misconceptions (whether that is always an appropriate term), bugs (perhaps a better term sometimes), mistakes (something done wrongly, perhaps because of working memory overload), and flawed actions (the result of a flawed behavior, or habit, that has been automated).
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This short unpublished note is a musing on why university students have so much trouble distinguishing between applying a definition to prove a result and proving that a mathematical object satisfies a definition.
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We discuss the following question: In a moment-to-moment way, what are the (immediate) origins of students’ (physical or mental) actions in the proving process?
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This is an unpublished, and unfinished, paper, considering the question, "What do it mean to have a proof?" Does it mean one has an idea for a proof? Or, does it mean that it should be written up in an acceptable mathematical style, so... more
This is an unpublished, and unfinished, paper, considering the question, "What do it mean to have a proof?" Does it mean one has an idea for a proof? Or, does it mean that it should be written up in an acceptable mathematical style, so readers can tell whether it is correct? These topics are considered. However, the rest of the paper, giving the questionable student proof, was never finished.
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This unpublished paper contains further thoughts on actions in the proving process, collected at various times in 2009. Some of these thoughts have been published in our iJMEST paper, "Affect, Behavioural Schemas and the Proving Process",... more
This unpublished paper contains further thoughts on actions in the proving process, collected at various times in 2009. Some of these thoughts have been published in our iJMEST paper, "Affect, Behavioural Schemas and the Proving Process", but not all.
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This is a description of how we used to teach Moore Method mathematics courses, followed by our more recent views of studying actions in the proving process.
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This is a proposed explanation of how procedural knowledge [or procedural memory, as described, on pages 636-7, in The Oxford Handbook of Memory (2000), edited by Tulving and Craik] is often used in doing mathematics. The explanation... more
This is a proposed explanation of how procedural knowledge [or procedural memory, as described, on pages 636-7, in The Oxford Handbook of Memory (2000), edited by Tulving and Craik] is often used in doing mathematics. The explanation takes the form of six characteristics of procedural knowledge that are not currently directly observable, except as reports of subjective experience. This (unpublished) paper is an explanation of some of our ideas that later appeared in our paper, "Affect, Behavioural Schemas, and the Proving Process" in the International Journal for Mathematical Education in Science and Technology", Vol. 41(2), pp. 199-215.
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Often a propitious choice of symbols, including various mathematical notations and diagrams, can facilitate thought positively when solving problems and constructing arguments. However, there are also ways that symbol use can lead to... more
Often a propitious choice of symbols, including various mathematical notations and diagrams, can facilitate thought positively when solving problems and constructing arguments. However, there are also ways that symbol use can lead to errors in reasoning. Symbols are sometimes so strongly associated with the objects in a particular mathematical domain, such as the real numbers, that they become strongly associated with the properties and manipulations of the objects in that mathematical domain that they are carried over automatically, even when this is inappropriate, thus introducing errors. We provide numerous examples.
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This compilation of themes was done in preparation for writing a chapter for the Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future, edited by A. Gutierrez and P. Boero, and published by Sense... more
This compilation of themes was done in preparation for writing a chapter for the Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future, edited by A. Gutierrez and P. Boero, and published by Sense Publishers in 2006. However, much of it was never used in writing the chapter.
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"How much logic does one need to know in order to succeed as a mathematics major? What part of that logic is natural? That is, to what extent do entering university students already have the ability to reason deductively? Are they able... more
"How much logic does one need to know in order to succeed as a mathematics major? What part of that logic is natural? That is, to what extent do entering university students already have the ability to reason deductively? Are they able to use some parts of logic, such as modus ponens, with ease whereas they find other aspects, such as modus tollens, difficult? For those parts that undergraduates use easily, is that logic known in a tacit, almost automatic way? For mathematics majors, how does one best build on the logic they know to help them learn the parts they find difficult? Not all of these questions have been answered or even partially answered. This article contains a potpourri of what is known.
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This unpublished paper considers 'negative' definitions in mathematics, like those of prime number and irrational number, that lack representations to work with, and why they are hard for students.
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This small example shows how the instructor could have a number of things come to mind in response to a student's confusion over a proof.
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A lot of cognition seems to depend on the interaction of procedural and conceptual knowledge. Here we will discuss only procedural knowledge and we will focus mainly on how it is executed or used, as opposed to how it is obtained. More... more
A lot of cognition seems to depend on the interaction of procedural and conceptual knowledge. Here we will discuss only procedural knowledge and we will focus mainly on how it is executed or used, as opposed to how it is obtained. More specifically, we will examine how linear equations are solved in a moment-to-moment way and will restrict our comments to how ordinary knowledgeable people solve such equations in an ordinary way, that is, by writing several lines on paper.
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This short conjectural paper is meant to provoke further thought, as well as research, on the topic of various ways of reading proofs
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This unpublished paper, originally sent in an email to a colleague, contains some thoughts on how automating certain (as yet to be determined) algebra facts might facilitate the learning of calculus.
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This article describes the activities of the MAA project, Preparing Mathematicians to Educate Teachers (PMET), which began in Summer 2002 and lasted at least through 2005. Strengthening the mathematical education of America's... more
This article describes the activities of the MAA project, Preparing Mathematicians to Educate Teachers (PMET), which began in Summer 2002 and lasted at least through 2005.
Strengthening the mathematical education of America's teachers is the immediate goal of the NSF-funded MAA project Preparing Mathematicians to Educate Teachers (PMET). The primary lever for PMET in achieving this goal is to assist college and university mathematics faculty in providing better courses for future K-12 teachers. During the first eight months of PMET's four-year term, 105 faculty have participated in PMET workshops, eighteen have attended a PMET minicourse, and numerous others have been encouraged to join PMET's effort.
Strengthening the mathematical education of America's teachers is the immediate goal of the NSF-funded MAA project Preparing Mathematicians to Educate Teachers (PMET). The primary lever for PMET in achieving this goal is to assist college and university mathematics faculty in providing better courses for future K-12 teachers. During the first eight months of PMET's four-year term, 105 faculty have participated in PMET workshops, eighteen have attended a PMET minicourse, and numerous others have been encouraged to join PMET's effort.
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This unpublished paper discusses and illustrates what we call the hierarchical structure of proofs, as well as the linear structure of proofs. It also discusses what we have come to call the formal-rhetorical part of a proof and the... more
This unpublished paper discusses and illustrates what we call the hierarchical structure of proofs, as well as the linear structure of proofs. It also discusses what we have come to call the formal-rhetorical part of a proof and the problem-solving part of a proof. Lastly, it discusses the idea of "logic in action".
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This unpublished paper contains ideas about: (I.) the structure of proofs and (II.) about the way logic is used,.
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Alice Mason was a coauthor on: "Can average calculus students solve nonroutine problems?", "Even good calculus students can't solve nonroutine problems", and "Why can't calculus students access their knowledge to solve nonroutine... more
Alice Mason was a coauthor on: "Can average calculus students solve nonroutine problems?", "Even good calculus students can't solve nonroutine problems", and "Why can't calculus students access their knowledge to solve nonroutine problems?" A memorial was held for her at Tennessee Technological University and I was asked to speak.
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This short piece gives an example from a Calculus I class of "thinking one thing, but writing another" when dealing with notation in a problem about limits.
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This unfinished paper is about how mathematcial proofs are written. It was to include the results of an interview study of mathematicians' views of seven features of mathematical proofs. The mathematicians were presented with the seven... more
This unfinished paper is about how mathematcial proofs are written. It was to include the results of an interview study of mathematicians' views of seven features of mathematical proofs. The mathematicians were presented with the seven features (Appendix A) and asked about them. They were asked to bring one of their own unpublished papers to the interview and to asked questions (Appendix B) about the style in which it was written. Appendix C suggests some questions that one should ask when one is reading a proof for understanding.
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We demonstrate a prototype of a supplementary tool for helping undergraduates understand, and participate in reading and writing, proofs. It consists of hypertext presentations of actual proofs with links to explanations regarding... more
We demonstrate a prototype of a supplementary tool for helping undergraduates understand, and participate in reading and writing, proofs. It consists of hypertext presentations of actual proofs with links to explanations regarding questions one should ask and answer in reading proofs, the "proofs within proofs" structure, the application of theorems and definitions, the special usage of words like "let," etc. The structure of these hypertext documents is both "deep," in having several layers of explanations within explanations, and "wide," in that relevant explanations are linked to parallel information, e.g., when "or" is explained, there is also a link to "and." Such hypertext presentations are to be explored, rather than read, and may help students form mental links that can later bring information to mind when it is needed in constructing proofs. The explanatory material, e.g., about sets and logic, is thus provided in context rather than in an abstract decontextualized way, and consists of precisely what is used in student proofs -- in contrast to how it is presented in most "transition" courses.
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We point out seven features that appear to partly constitute a distinctive rhetorical style in which mathematical proofs are written. As evidence for this, we report on a mathematics journal survey and on interviews with eight... more
We point out seven features that appear to partly constitute a distinctive rhetorical style in which mathematical proofs are written. As evidence for this, we report on a mathematics journal survey and on interviews with eight mathematicians. We also report on a survey of U.S. undergraduate "transition course" textbooks that suggest this style is part of the implicit, but not the explicit, curriculum. We conjecture that this style is useful in minimizing certain kinds of validation errors, i.e., errors in checking the correctness of proofs, and is thus important in learning to construct proofs. Finally, we suggest that it might be useful to treat this style explicitly when introducing undergraduate students to writing proofs.
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This lists some difficulties (often interacting) that I have observed with college algebra students, along with some ideas about how one might investigate them.
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This unpublished paper considers two ideas. The first idea is just that mathematics deals with more types of entities than object concepts (e.g., groups). It also concerns property concepts (e.g., compact) and activity concepts (e.g.,... more
This unpublished paper considers two ideas. The first idea is just that mathematics deals with more types of entities than object concepts (e.g., groups). It also concerns property concepts (e.g., compact) and activity concepts (e.g., factoring). Our second idea comes for our interest in consciousness. We think a key feature of a conception is its association with a unique, unitary, rememberable, mental feature that can be conscious and that one might call a quality/feeling (= qualia ?). We think such qualities/feelings are idiosyncratic and cannot be communicated from one person to another
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Why is learning to write mathematics hard? First of all, unlike natural languages such as English or Chinese which are written in alphabets or ideograms, written mathematics comes in at least four forms: logograms, pictograms,... more
Why is learning to write mathematics hard? First of all, unlike natural languages such as English or Chinese which are written in alphabets or ideograms, written mathematics comes in at least four forms: logograms, pictograms, punctuation symbols, and alphabetic symbols.
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This course, which did not get enough registrations, was not offered. The description began as follows: The first third of this survey course will emphasize the concepts, perspectives, and vocabulary arising from research in undergraduate... more
This course, which did not get enough registrations, was not offered. The description began as follows: 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.
programming as an aid to conceptual understanding.
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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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This short paper was never finished. It deals with situated cognition and transfer in mathematical problem solving.
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Groups seem to have some social benefits, as well as the usual cognitive ones of getting people to monitor and question each other's work, be reflective, etc. Somehow, students do not seem to be as afraid to speak out when they are in a... more
Groups seem to have some social benefits, as well as the usual cognitive ones of getting people to monitor and question each other's work, be reflective, etc. Somehow, students do not seem to be as afraid to speak out when they are in a group. American university students, especially, as compared, for example, to Nigerian university students, whom I taught for seven years, do not like to be "wrong". Often they would rather not answer at all. But, if they can 'shout out' an answer for the group, and probably not be recognized as an individual, they seem more willing to try. There seems to be less fear of embarrassment when speaking out for the group. This unpublished paper contains two observations from courses I taught; one was an Honors Calculus course, the other was an Abstract Algebra course. These observations are very suggestive. Someone could possibly take these ideas and do research on them.
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This is a collection of thoughts and references, mainly on what is know in the psychological literature on human (deductive) reasoning, as of 1996. This raises a number of questions about how students reason logically in mathematics,... more
This is a collection of thoughts and references, mainly on what is know in the psychological literature on human (deductive) reasoning, as of 1996. This raises a number of questions about how students reason logically in mathematics, especially when constructing proofs.
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This describes an incident from one of my transition-to-proof classes concerning students difficulty understanding a particular exam problem in which they were to prove a certain function was onto, along with a conjecture about why it... more
This describes an incident from one of my transition-to-proof classes concerning students difficulty understanding a particular exam problem in which they were to prove a certain function was onto, along with a conjecture about why it happened.
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This unpublished, and unfinished, paper contains some ideas about how the concept of mathematical definitions might be taught and how students' understandings of mathematical definitions might be researched.
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What makes for college teacher change? This is a topic we are studying now. What makes a workshop effective? There are no simple "how to" manuals, but we have some ideas*.
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This paper discusses constructivism in mathematics education, which is the most influential and widely accepted philosophical perspective in mathematics education today. This view, which holds that individuals construct their own... more
This paper discusses constructivism in mathematics education, which is the most influential and widely accepted philosophical perspective in mathematics education today. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget and beyond. While it takes many forms, at its simplest, it sees the learner as an active participant, not as a blank slate upon which we write or as an empty vessel which we fill. In this view, 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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These observations were a result of a dinner discussion with John Selden when we were observing one of Ed Dubinsky's workshops for teaching college mathematics teachers to use his and Leron's abstract algebra textbook. The workshops, and... more
These observations were a result of a dinner discussion with John Selden when we were observing one of Ed Dubinsky's workshops for teaching college mathematics teachers to use his and Leron's abstract algebra textbook. The workshops, and the textbooks, were based on his APOS (action, process, object, schema) view of concept development.
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Here are some thoughts about how mathematical definitions might develop, with particular emphasis on the idea of covariation.
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Some musings about the concept of proof.
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This unpublished paper was probably written for a presentation to the Advanced Mathematical Thinking Working Group (AMT) at the annual conference of the International Group for the Psychology of Mathematics Education (PME). It deals with... more
This unpublished paper was probably written for a presentation to the Advanced Mathematical Thinking Working Group (AMT) at the annual conference of the International Group for the Psychology of Mathematics Education (PME). It deals with mathematical concepts at the undergraduate level and beyond.
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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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This is an abstract for a paper describing a voluntary proving supplement to an undergraduate real analysis course.
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This in an abstract for a contributed paper on proof frameworks.
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We describe a perspective for examining the enactment of a common kind of procedural knowledge in mathematics and how that enactment relates to consciousness. Here, we view procedural knowledge in a very fine-grained way, for example,... more
We describe a perspective for examining the enactment of a common kind of procedural knowledge in mathematics and how that enactment relates to consciousness. Here, we view procedural knowledge in a very fine-grained way, for example, considering a single step in a mathematical procedure, and discuss knowledge that includes, not only how to, but also to, or when to, physically or mentally act. We call the mental structure that links information allowing one to recognize that an act is to be performed, to what is to be done and how to do it, a behavioral schema.
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This is the abstract for a presentation in which we will discuss how we came to the idea of proof frameworks and demonstrate how they are written.
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This presentation will compare proof comprehension, proof construction, proof validation, and proof evaluation.
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This theoretical paper considers several perspectives for understanding and teaching university students’ autonomous proof construction.
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We present the results of an analysis of undergraduate students’ examination papers from an IBL transition-to-proof course. Students’ papers were considered from the point of view of their actions (mental, as well as physical), instead of... more
We present the results of an analysis of undergraduate students’ examination papers from an IBL transition-to-proof course. Students’ papers were considered from the point of view of their actions (mental, as well as physical), instead of their possible misconceptions. In doing so, we identified process, rather than mathematical content, difficulties, and this has resulted in the detection of both beneficial actions and detrimental actions that students often take.
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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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First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 and each time we are learning... more
First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 and each time we are learning something more about students' proving capabilities.
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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 understand where students are “coming from” and to help them learn to construct... 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 understand where students are “coming from” and to help them learn to construct proofs. We have analyzed students’ examination papers from several such courses. We have identified process, rather than mathematical content, difficulties such as not unpacking the conclusion, and not using definitions correctly.
Research Interests: Proof and Proving in Mathematics Education, Undergraduate Mathematics Education, Mathematical reasoning and proof, Logical reasoning, Research of Mathematics Education, and 4 moreEducational Research in Mathematics, Research in Undergraduate Mathematics Education, Researches in Mathematics Education, and Undergraduate Mathematics Education Research
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.
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Have you ever wondered how to pick a research question, which methodology to use, how to get started on writing a mathematics education research journal article, which mathematics education research journals are ranked the highest, how... more
Have you ever wondered how to pick a research question, which methodology to use, how to get started on writing a mathematics education research journal article, which mathematics education research journals are ranked the highest, how mathematics education research papers are reviewed, what you need to do go get tenure as a mathematics education researcher in a mathematics department?
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This is an abstract for my presentation as part of the panel on proof, held at the Symposium in honor of Ted Eisenburg's retirement at Ben Gurion University in Israel in May 2012.
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First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 and each time we are learning... more
First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 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 the remainder of the proof the problem-centered part. Second, I will discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times to since Fall 2009. Each week, one proof problem was selected or created to " resemble in construction " an assigned homework proof problem that the real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and " on the surface " would often not resemble the assigned proof problem. The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process and by our division of proofs into their formal-rhetorical and problem-centered parts.
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Abstract for a talk to the math ed group in the Mathematics Department at the University of Oklahoma.
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First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs.
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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 paper in which we proposed to describe our perspective on the structure of deductive reasoning, that is, we describe the principal psychological components of that reasoning and how they interact. The components... more
This is an abstract for a paper in which we proposed to describe our perspective on the structure of deductive reasoning, that is, we describe the principal psychological components of that reasoning and how they interact. The components include: consciousness, the automated guiding of actions, stimulus-independent thought, and what we call local memory, a temporary, easily accessed, part of memory. In discussing deductive reasoning, we restrict our attention to the construction of mathematical arguments, and in particular, to proof construction, but no prior knowledge of mathematics will be required to understand our examples.
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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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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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In designing and studying a Modified Moore Method course, we are learning about the relationship between affect, behavioral schemas, and the proving process. Behavioral schemas are enduring mental structures linking situations to... more
In designing and studying a Modified Moore Method course, we are learning about the relationship between affect, behavioral schemas, and the proving process. Behavioral schemas are enduring mental structures linking situations to actions; they appear to drive many mental actions in the proving process.
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I will discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate students to construct proofs. This course has been taught at least six times to date and each time... more
I will discuss a design experiment, sometimes also called a teaching experiment, for helping advanced undergraduate and beginning graduate students to construct proofs. This course has been taught at least six times to date and each time we are learning something more about students' proving capabilities.
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There are certain aspects of proving that mathematicians do automatically, but that students are often unaware of.
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This submission concerns both the teaching of and the cognitive aspects of 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... more
This submission concerns both the teaching of and the cognitive aspects of 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.
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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... 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.
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This is an abstract for a paper in which we point out a way consciousness is necessary for a particular kind of mathematical reasoning.
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This is an abstract for a presentation on our beginning graduate transition-to-proof course.
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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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We will present a brief look at some mathematics education research results on students’ difficulties in learning to understand and construct proofs. We will also describe a course we are currently developing that is meant to help... more
We will present a brief look at some mathematics education research results on students’ difficulties in learning to understand and construct proofs. We will also describe a course we are currently developing that is meant to help beginning graduate students and prospective graduate students become more skilled at constructing proofs.
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People construct their own understandings based on prior knowledge and past experiences, whether just listening, as in a lecture, or actively engaged in some project.
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This describes a paper we presented at this conference. In the paper, we discussed the nature of proofs that mathematics undergraduate or beginning graduate students read or construct. In particular, we will focus mainly on three... more
This describes a paper we presented at this conference. In the paper, we discussed the nature of proofs that mathematics undergraduate or beginning graduate students read or construct. In particular, we will focus mainly on three structures of proofs: a hierarchical structure, a linear path through which a proof could have been constructed by an idealized prover, and a division into the "rhetorical" and problem-solving parts. These three structures appear to play a role in the remarkable reliability and stability of mathematical results, and in particular, in mathematicians' ability to discern, and agree on, the correctness of proofs. The three structures, once articulated, can be combined and reduced to a single three-dimensional diagram that visually reveals the complexity of a proof. Both the articulation of the three structures and the diagram are likely to be useful in mathematics education research and teaching.
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This is an abstract for a presentation on students' understanding and construction of proofs. It was intended for a general AAAS (American Association for the Advancement of Science) audience.
Research Interests: Deductive reasoning, Mathematical reasoning and proof, Logical reasoning, Definitions, Research of Mathematics Education, and 5 moreResearch in Undergraduate Mathematics Education, Researches in Mathematics Education, Perform Abstract Reasoning, Universal quantifiers, and Undergraduate Mathematics Education Research
"There are no proofs in mathematics education." While this is true, claims are made in mathematics education research and evidence is provided for them. Perhaps as a result, there are significant differences between the ways research is... more
"There are no proofs in mathematics education." While this is true, claims are made in mathematics education research and evidence is provided for them. Perhaps as a result, there are significant differences between the ways research is done and evaluated in the two fields.
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This paper 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 paper 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.
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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.
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We describe hypertext presentations of proofs and say we will demonstrate a prototype. That is, we will explore explanations in a hypertext presentation of a particular proof while commenting on both the structure of the hypertext... more
We describe hypertext presentations of proofs and say we will demonstrate a prototype. That is, we will explore explanations in a hypertext presentation of a particular proof while commenting on both the structure of the hypertext document and the nature of the explanations contained in it.
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Questions regarding the teaching and learning of undergraduate mathematics are beginning to be investigated by a small but growing number of researchers.
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This is an abstract for our poster on non-routine problem solving in calculus at a poster session sponsored by the Association for Research in Undergraduate Mathematics Education (ARUME). held at the January Joint Mathematics Meetings... more
This is an abstract for our poster on non-routine problem solving in calculus at a poster session sponsored by the Association for Research in Undergraduate Mathematics Education (ARUME). held at the January Joint Mathematics Meetings (JMM) of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA).
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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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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 for a talk given at the DIMACS Symposium on Teaching Logic and Reasoning, held at Rutgers University, July 25--26, 1996.
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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 was for a talk that discussed interesting tasks that mathematics education researchers have posed to college students.
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This paper was presented to a conference of the Mathematical Association of Nigeria in 1981.
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These short abstracts give information about mathematics education research journal articles.
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These abstracts of mathematics education research articles were written for college and university teachers of mathematics and cover a variety of topics.
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These are the abstracts of mathematics education research articles written for the College Mathematics Journal between January 2014 and January 2015, mainly for mathematicians.
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These abstracts of (mainly) mathematics education research articles were written for mathematicians.
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These abstracts of (mainly) mathematics education research articles were written for mathematicians.
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These abstracts of (mainly) mathematics education research articles were written for mathematicians.
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This is an abstract written for the College Mathematics Journal's Media Highlights section.
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These abstracts of (mainly) mathematics education research articles were written for mathematicians.
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The abstracts of (mainly) mathematics education research articles were written for mathematicians.
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These abstracts of (mainly) mathematics education research articles were written for mathematicians.
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This is the narrative of a Type I TUES research and development proposal that involves cooperation between a mathematician and two mathematics education researchers. It seeks to produce a transformational and fundamentally new way of... more
This is the narrative of a Type I TUES research and development proposal that involves cooperation between a mathematician and two mathematics education researchers. It seeks to produce a transformational and fundamentally new way of helping undergraduate introductory real analysis students learn to construct mathematical proofs and to gain confidence in their own proving ability.
This will be achieved through a weekly one-hour supplement to an introductory real analysis course. The supplement will be based on a set of paired real analysis theorems, along with detailed descriptions of the actions involved in proving them. One pair is to be selected each week. The proof of one of these will be assigned as homework in the real analysis class. The proof of the other will be co-constructed by the students in the supplement before the homework is due.
This will be achieved through a weekly one-hour supplement to an introductory real analysis course. The supplement will be based on a set of paired real analysis theorems, along with detailed descriptions of the actions involved in proving them. One pair is to be selected each week. The proof of one of these will be assigned as homework in the real analysis class. The proof of the other will be co-constructed by the students in the supplement before the homework is due.
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The project summary for this grant proposal was as follows: This empirical research proposal is in the STEM learning strand. It will produce transformational, fundamental knowledge about: How can advanced university students learn, and... more
The project summary for this grant proposal was as follows: This empirical research proposal is in the STEM learning strand. It will produce transformational, fundamental knowledge about: How can advanced university students learn, and be taught, to construct mathematical proofs autonomously? The emphasis will be on what students are capable of, in contrast to the literature’s focus on their difficulties. The proposal is also partly in the cognitive underpinnings strand because it will develop a new theoretical view of how the mind controls its own deductive reasoning.
The above question (and more tractable subquestions) will be answered through a naturalistic study (Lincoln & Guba, 1985) using a generative/integrative analysis (Clement, 2000) of data arising from an unusual experimental “proofs” course. This course is for beginning graduate and some advanced undergraduate mathematics students who need help with proving. Its sole purpose is to provide that help, and classes consist very largely of students presenting their proofs and receiving critiques and advice. The course is consistent with a constructivist point of view, in that the teachers attempt to help students reflect on, and learn from, their own proof writing experiences. It is also somewhat Vygotskian (1978) in that the teachers represent to the students how the mathematics community writes proofs. That is, they are instruments in the cultural mediation of community norms and practices.
Data will be collected mainly in four ways: (1) videoing and analyzing all class meetings; (2) videoing extensive one-on-one tutoring of students constructing proofs; (3) task-based interviews of former students; and (4) for the first time, using tablet computers to record and observe students independently constructing proofs.
The theoretical framework used in a data analysis of a naturalistic study emerges from the data itself. From pilot project data a framework has already started emerging including: (1) kinds of proofs – needed to keep track of student progress; and (2) aspects of a proof – mathematicians fail to prove a theorem because of its “problem-oriented aspect”, but students (mainly) fail to prove a theorem because they cannot write its “formal-rhetorical aspect” (which is teachable).
The pilot project data have also yielded a view of much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed proof construction. Similar situations tend to yield similar actions that can become permanently linked with those situations. The smallest of such linked, automated situation-action pairs are regarded as persistent mental structures, called behavioral schemas. An example of a behavioral schema, that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action]. Such schemas are stored in procedural memory and can be activated without requiring conscious recollection, thus reducing the demands on working memory. The idea of behavioral schemas in this project is not derived from other schema theories in psychology. However, it is reminiscent of social psychologists’ view of automaticity in everyday life, except that the linking of situations to actions is reified into mental objects.
Intellectual merit: (1) The idea of a course and a teaching experiment devoted to teaching proving at this level is novel, and the project will produce much detailed information on what students can do, rather than focusing mainly on their difficulties. (2) One of the proposed data collection methods is new and will provide insights into students’ actions and thinking when alone. (3) The emerging framework will allow one to see that an aspect of a proof that blocks a mathematician’s progress is different in nature from the aspect that blocks most students. Thus teaching students to mimic mathematicians may not help. (4) The emerging view of behavioral schemas should prove very helpful in teaching, and elucidate how procedural knowledge can guide the use of conceptual knowledge. Such schemas may also illustrate the utility of converting System 2 cognition to System 1 cognition (Stanovich & West, 2000).
Broader impact: (1)This project will provide information for designing courses for teaching students to prove theorems on a variety of topics and for students at a variety of levels – beginning graduate, mid-level undergraduate, and even high school. (2) It is common knowledge among teachers of beginning graduate students that significantly many need help constructing proofs, and some, who do not improve, drop out of mathematics. The experimental course in this project could be conveniently adapted to other universities to alleviate this situation. (3) When students first learn to construct proofs, they often find the experience empowering, and raise their career expectations and their interest in mathematics. This can be particularly helpful for minority populations whose expectations are not high.
The above question (and more tractable subquestions) will be answered through a naturalistic study (Lincoln & Guba, 1985) using a generative/integrative analysis (Clement, 2000) of data arising from an unusual experimental “proofs” course. This course is for beginning graduate and some advanced undergraduate mathematics students who need help with proving. Its sole purpose is to provide that help, and classes consist very largely of students presenting their proofs and receiving critiques and advice. The course is consistent with a constructivist point of view, in that the teachers attempt to help students reflect on, and learn from, their own proof writing experiences. It is also somewhat Vygotskian (1978) in that the teachers represent to the students how the mathematics community writes proofs. That is, they are instruments in the cultural mediation of community norms and practices.
Data will be collected mainly in four ways: (1) videoing and analyzing all class meetings; (2) videoing extensive one-on-one tutoring of students constructing proofs; (3) task-based interviews of former students; and (4) for the first time, using tablet computers to record and observe students independently constructing proofs.
The theoretical framework used in a data analysis of a naturalistic study emerges from the data itself. From pilot project data a framework has already started emerging including: (1) kinds of proofs – needed to keep track of student progress; and (2) aspects of a proof – mathematicians fail to prove a theorem because of its “problem-oriented aspect”, but students (mainly) fail to prove a theorem because they cannot write its “formal-rhetorical aspect” (which is teachable).
The pilot project data have also yielded a view of much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed proof construction. Similar situations tend to yield similar actions that can become permanently linked with those situations. The smallest of such linked, automated situation-action pairs are regarded as persistent mental structures, called behavioral schemas. An example of a behavioral schema, that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action]. Such schemas are stored in procedural memory and can be activated without requiring conscious recollection, thus reducing the demands on working memory. The idea of behavioral schemas in this project is not derived from other schema theories in psychology. However, it is reminiscent of social psychologists’ view of automaticity in everyday life, except that the linking of situations to actions is reified into mental objects.
Intellectual merit: (1) The idea of a course and a teaching experiment devoted to teaching proving at this level is novel, and the project will produce much detailed information on what students can do, rather than focusing mainly on their difficulties. (2) One of the proposed data collection methods is new and will provide insights into students’ actions and thinking when alone. (3) The emerging framework will allow one to see that an aspect of a proof that blocks a mathematician’s progress is different in nature from the aspect that blocks most students. Thus teaching students to mimic mathematicians may not help. (4) The emerging view of behavioral schemas should prove very helpful in teaching, and elucidate how procedural knowledge can guide the use of conceptual knowledge. Such schemas may also illustrate the utility of converting System 2 cognition to System 1 cognition (Stanovich & West, 2000).
Broader impact: (1)This project will provide information for designing courses for teaching students to prove theorems on a variety of topics and for students at a variety of levels – beginning graduate, mid-level undergraduate, and even high school. (2) It is common knowledge among teachers of beginning graduate students that significantly many need help constructing proofs, and some, who do not improve, drop out of mathematics. The experimental course in this project could be conveniently adapted to other universities to alleviate this situation. (3) When students first learn to construct proofs, they often find the experience empowering, and raise their career expectations and their interest in mathematics. This can be particularly helpful for minority populations whose expectations are not high.
Research Interests: Theorem Proving, Mathematics Teaching, Mathematics Education, with a Particular Emphasis on the Teaching and Learning of Mathematics at University Level and the Psychology of Mathematical Learning – Cognitive, Social and Affective Issues of Engagement with Mathematics, Teaching and Learning Mathematics, Research in Undergraduate Mathematics Education, and 2 moreResearches in Mathematics Education and Teacing and Learning Mathematics
The Project Summary was as follows: Intellectual Merit. This proposed contextual research – empirical research project will use a new theoretical perspective on the learning of proof construction. We view much of the proof construction... more
The Project Summary was as follows: Intellectual Merit. This proposed contextual research – empirical research project will use a new theoretical perspective on the learning of proof construction. We view much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed process. Similar situations tend to yield similar actions that can become, more or less, permanently linked with those situations. We regard the smallest of such linked, automated situation-action pairs as persistent mental structures, and call them behavioral schemas.
One such behavioral schema, that we have observed and that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action].
The idea of behavioral schemas is similar to ideas that some social psychologists have been investigating about the automaticity of much of everyday life, except that we are reifying the linking of a situation to an action into a mental object (a behavioral schema) that can be altered with the aid of a teacher. We have developed a six-point theoretical sketch of the genesis and enactment of behavioral schemas (given in the proposal). One point is that behavioral schemas are acquired through practice. That is, to acquire a beneficial schema a person should actually carry out the appropriate action correctly a number of times – not just understand its appropriateness. Changing a detrimental behavioral schema, which may have been tacitly acquired, requires similar, perhaps longer, practice.
This research project will produce immediately usable outcomes. We will find and categorize, ways to promote beneficial behavioral proving schemas, as well as uncover and find ways to change detrimental ones. We will also find and categorize kinds and aspects of proofs at a level of detail that will be practical for teachers. The overarching research method is the design experiment. As with most design experiments, our intent is to develop theories about the process of learning (to construct proofs), and about the teaching and teaching materials that support learning. The research results, however, are intended to be paradigmatic, that is, to apply more widely to situations and students other than our own.
We are proposing six interconnected subprojects: (1) a design experiment studying the development of a “proofs” course for advanced undergraduate and beginning graduate mathematics students; (2) a design experiment developing a “proving skills” supplement for undergraduate real analysis; (3) a design experiment developing a nontraditional transition-to-proof course; (4) conducting and studying sustained one-on-one tutoring; (5) using tablet computers to record and observe students independently constructing proofs; and (6) developing a new kind of electronic book/course notes that is adjustable by teachers to suit differing classes and specific content.
Broader impact. It is common knowledge, among many who teach beginning mathematics graduate students in the U.S., that significantly many such students have difficulty constructing proofs and that this difficulty sometimes persists for a long time. Sometimes the inability to prove theorems causes students to fail because content course assessments are based primarily on that ability. The remaining students, who cannot significantly improve their proving skills, are likely to leave mathematics voluntarily for some other field. A similar problem occurs at the upper undergraduate level for students, such as preservice secondary mathematics teachers, in proof-based content courses such as abstract algebra and real analysis. This is a time in their studies when some undergraduates realize (perhaps wrongly) that mathematics is going to require a skill that they cannot do well and that may seem impossible to learn. Such students are likely to start seriously considering a non-mathematical career.
The current “solution” for graduate students is often to put them in “remedial” courses, such as undergraduate real analysis or abstract algebra, but by and large, this attempted “solution” is not successful. Perhaps this is because such undergraduate content courses concentrate on extending students’ knowledge of mathematical topics, and can rarely devote much class time to improving proving skills. Because beginning mathematics graduate students are the survivors of a long sequence of younger students turning away from mathematics, sometimes referred to as “the leaky pipeline,” they represent a considerable national investment. A Challenge of Numbers noted that the attrition rate from mathematics from ninth grade on, is roughly 50% per year, and the attrition rate for Latinos and African Americans is significantly larger. This is a waste of the nation’s resources and does not bode well for the future of the U.S. STEM workforce. The information to be provided by this project is meant to solve both the above graduate and undergraduate problems.
One such behavioral schema, that we have observed and that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action].
The idea of behavioral schemas is similar to ideas that some social psychologists have been investigating about the automaticity of much of everyday life, except that we are reifying the linking of a situation to an action into a mental object (a behavioral schema) that can be altered with the aid of a teacher. We have developed a six-point theoretical sketch of the genesis and enactment of behavioral schemas (given in the proposal). One point is that behavioral schemas are acquired through practice. That is, to acquire a beneficial schema a person should actually carry out the appropriate action correctly a number of times – not just understand its appropriateness. Changing a detrimental behavioral schema, which may have been tacitly acquired, requires similar, perhaps longer, practice.
This research project will produce immediately usable outcomes. We will find and categorize, ways to promote beneficial behavioral proving schemas, as well as uncover and find ways to change detrimental ones. We will also find and categorize kinds and aspects of proofs at a level of detail that will be practical for teachers. The overarching research method is the design experiment. As with most design experiments, our intent is to develop theories about the process of learning (to construct proofs), and about the teaching and teaching materials that support learning. The research results, however, are intended to be paradigmatic, that is, to apply more widely to situations and students other than our own.
We are proposing six interconnected subprojects: (1) a design experiment studying the development of a “proofs” course for advanced undergraduate and beginning graduate mathematics students; (2) a design experiment developing a “proving skills” supplement for undergraduate real analysis; (3) a design experiment developing a nontraditional transition-to-proof course; (4) conducting and studying sustained one-on-one tutoring; (5) using tablet computers to record and observe students independently constructing proofs; and (6) developing a new kind of electronic book/course notes that is adjustable by teachers to suit differing classes and specific content.
Broader impact. It is common knowledge, among many who teach beginning mathematics graduate students in the U.S., that significantly many such students have difficulty constructing proofs and that this difficulty sometimes persists for a long time. Sometimes the inability to prove theorems causes students to fail because content course assessments are based primarily on that ability. The remaining students, who cannot significantly improve their proving skills, are likely to leave mathematics voluntarily for some other field. A similar problem occurs at the upper undergraduate level for students, such as preservice secondary mathematics teachers, in proof-based content courses such as abstract algebra and real analysis. This is a time in their studies when some undergraduates realize (perhaps wrongly) that mathematics is going to require a skill that they cannot do well and that may seem impossible to learn. Such students are likely to start seriously considering a non-mathematical career.
The current “solution” for graduate students is often to put them in “remedial” courses, such as undergraduate real analysis or abstract algebra, but by and large, this attempted “solution” is not successful. Perhaps this is because such undergraduate content courses concentrate on extending students’ knowledge of mathematical topics, and can rarely devote much class time to improving proving skills. Because beginning mathematics graduate students are the survivors of a long sequence of younger students turning away from mathematics, sometimes referred to as “the leaky pipeline,” they represent a considerable national investment. A Challenge of Numbers noted that the attrition rate from mathematics from ninth grade on, is roughly 50% per year, and the attrition rate for Latinos and African Americans is significantly larger. This is a waste of the nation’s resources and does not bode well for the future of the U.S. STEM workforce. The information to be provided by this project is meant to solve both the above graduate and undergraduate problems.
Research Interests:
Informal observations of students in first-year college mathematics courses indicate that students rarely read a mathematics textbook and that students find reading mathematics textbook material hard. Our intention in this research is... more
Informal observations of students in first-year college mathematics courses indicate that students rarely read a mathematics textbook and that students find reading mathematics textbook material hard. Our intention in this research is to explore how first-year students approach reading mathematical text with the hopes that this research project will lead first to an interview script/protocol to be used with students in an exploratory study, and later, to developing and testing strategies to address problems that are noted. We plan to use grounded theory and expect that relevant constructs will emerge in the course of the analysis.
Research Interests:
This is a grant proposal written for release time when I was at Tennessee Technological University. Proofs are specialized deductive arguments used in mathematics to establish the correctness of theorems (mathematical results).... more
This is a grant proposal written for release time when I was at Tennessee Technological University. Proofs are specialized deductive arguments used in mathematics to establish the correctness of theorems (mathematical results). Upper-level undergraduate mathematics majors are expected to validate (check the correctness of) proofs and to construct their own sound and well-written proofs -- something that, in addition to subject matter knowledge, requires considerable facility with deductive reasoning. Normally, students learn, or do not learn, proof and validation skills through an extended trial-and-error process, mainly by handing in original proofs that their mathematics professors critique. Because proofs are deductive arguments, some mathematics departments require majors to study a modest amount of rather abstract logic in the form of truth tables, propositional and predicate calculus, and valid arguments -- usually in a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and beginning differential equations) to more proof-based upper-division courses (e.g., abstract algebra and real analysis). However, the abstract study of logic does not seem in practice to improve general deductive reasoning (Nisbett, Gong, Lehman, & Cheng, 1987).
Research Interests:
This proposal was written for non-instructional assignment (a.k.a., sabbatical leave) for Spring 2002, during which time I wanted to conduct research and visit at University of Northern Colorado with a colleague. At the time I wrote,... more
This proposal was written for non-instructional assignment (a.k.a., sabbatical leave) for Spring 2002, during which time I wanted to conduct research and visit at University of Northern Colorado with a colleague.
At the time I wrote, "I am applying for full time non-instructional assignment for either Fall Semester 2001 or Spring Semester 2002, depending which is more convenient for the TTU Mathematics Department. I plan to spend the time on research, writing, and the building of a new national research organization. Currently in the U.S. there is need for much more research into the teaching and learning of undergraduate mathematics. All three of my goals relate to filling that need."
At the time I wrote, "I am applying for full time non-instructional assignment for either Fall Semester 2001 or Spring Semester 2002, depending which is more convenient for the TTU Mathematics Department. I plan to spend the time on research, writing, and the building of a new national research organization. Currently in the U.S. there is need for much more research into the teaching and learning of undergraduate mathematics. All three of my goals relate to filling that need."
Research Interests:
Proofs are specialized deductive arguments used in mathematics to establish the correctness of theorems (mathematical results). Upper-level undergraduate mathematics majors are expected to construct proofs that are both mathematically... more
Proofs are specialized deductive arguments used in mathematics to establish the correctness of theorems (mathematical results). Upper-level undergraduate mathematics majors are expected to construct proofs that are both mathematically sound and well written -- something that, in addition to subject matter knowledge, requires considerable facility with the style in which proofs are written. Normally, students learn, or do not learn, this ability through an extended trial-and-error process, by handing in proofs which their mathematics professors critique. There have been few attempts at isolating elements of this style in order to make them explicit, and hence, more readily accessible to novices. However, if one were to isolate features of this style and teach them explicitly, the learning process might be greatly enhanced and accelerated.
Research Interests:
This grant proposal was written while I was a Tennessee Technological University to apply for full time non-instructional assignment for Fall Semester 2000. The plan was to spend the time as a visiting professor in the Mathematics... more
This grant proposal was written while I was a Tennessee Technological University to apply for full time non-instructional assignment for Fall Semester 2000. The plan was to spend the time as a visiting professor in the Mathematics Department of Arizona State University undertaking a variety of projects. Goal I. Working with graduate students and colleagues in mathematics education research. Goal II. Furthering the Development of the Association for Research in Undergraduate Mathematics Education (ARUME). Goal III. Working on the Teaching and Learning portion of MAA Online. Goal IV. Writing up my research results for publication.
Research Interests:
While a certain amount of research on the learning of algebra by high school students exists, not much work has been done on college students' difficulties with algebra. From my recent experiences teaching college algebra, in which I... more
While a certain amount of research on the learning of algebra by high school students exists, not much work has been done on college students' difficulties with algebra. From my recent experiences teaching college algebra, in which I used small group work and thus had ample opportunity to observe students closely, I noticed a variety of apparently interacting student difficulties, both cognitive and affective, that are not widely discussed in the research literature. Because this college course has a high failure rate nationally as well as locally, it would be valuable to catalogue and understand these difficulties, and the ways they interact -- that is the goal of this project.
In this grant proposal for release time, written when I was in the Mathematics Department of Tennessee Technological University, I included some difficulties I had observed which are not examined in the literature.
In this grant proposal for release time, written when I was in the Mathematics Department of Tennessee Technological University, I included some difficulties I had observed which are not examined in the literature.
Research Interests:
This project aims to answer the following questions: How do mathematicians and students go about validating proofs, i.e., deciding whether they are correct? What kind of logic is involved? Is it often invoked subconsciously?... more
This project aims to answer the following questions: How do mathematicians and students go about validating proofs, i.e., deciding whether they are correct? What kind of logic is involved? Is it often invoked subconsciously?
Mathematics departments rarely require students to study much logic before working with proofs. Normally, the most they require (or 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 proof?
Mathematics departments rarely require students to study much logic before working with proofs. Normally, the most they require (or 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 proof?
Research Interests:
This grant proposal, written to obtain release time when I was in the Mathematics Department at Tennessee Technological University, contains observations and ideas for a study in undergraduate mathematics education, regarding the elusive... more
This grant proposal, written to obtain release time when I was in the Mathematics Department at Tennessee Technological University, contains observations and ideas for a study in undergraduate mathematics education, regarding the elusive concept of "mathematical maturity".
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:
This is an (updated) grant proposal written to gain release time when I was in the Department of Mathematics at Tennessee Technological University. It uses the concept of statement image, which is a generalization of the idea of concept... more
This is an (updated) grant proposal written to gain release time when I was in the Department of Mathematics at Tennessee Technological University. It uses the concept of statement image, which is a generalization of the idea of concept image. The idea of statement image was introduced in my paper, "Unpacking the logic of mathematical statements" (with J. Selden), Educational Studies in Mathematics 29 (1995), 123-151.
Research Interests:
This is an (updated) version of a grant proposal I wrote to obtain released time from my university in order to investigate U.S. lower-division university students conceptions of proof. I believe that it was funded and (much later)... more
This is an (updated) version of a grant proposal I wrote to obtain released time from my university in order to investigate U.S. lower-division university students conceptions of proof. I believe that it was funded and (much later) resulted in the article, "Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?", (with J. Selden). Journal for Research in Mathematics Education, 34(1), 2003, 4-36.
Research Interests:
We proposed 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... more
We proposed 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:
This proposal was written to apply for full time non-instructional assignment for Fall Semester 1993. In addition, I planned to ask for leave of absence and outside funding for Spring Semester 1994. If external funding was not... more
This proposal was written to apply for full time non-instructional assignment for Fall Semester 1993. In addition, I planned to ask for leave of absence and outside funding for Spring Semester 1994. If external funding was not available, I would seek a modest Spring Semester teaching assignment at Berkeley or one of the nearby colleges where I had professional contacts. I was also prepared to provide some funding myself, if needed. Goal I. Research in Students' Understandings of Function. Goal II. A Research Volume on Student Proofs.
Research Interests:
This completes the readings for Fall 2015 and the specific reading questions I generated for them. I find such questions useful in case there is a "lull" in the discussion or in case the discussion should get "off track" so I can bring it... more
This completes the readings for Fall 2015 and the specific reading questions I generated for them. I find such questions useful in case there is a "lull" in the discussion or in case the discussion should get "off track" so I can bring it back to the paper at hand.
Research Interests:
This is a list of the papers, with discussion questions, that we are reading and discussing in our math ed research seminar this semester. I find it useful to have such questions prepared ahead of time in case there is a "lull" in the... more
This is a list of the papers, with discussion questions, that we are reading and discussing in our math ed research seminar this semester. I find it useful to have such questions prepared ahead of time in case there is a "lull" in the conversation, or if the conversation goes in unfruitful directions, so it can be redirected to discussing the paper at hand.
Research Interests:
This is a list of discussion questions that I am using in my math ed research seminar this fall. The discussion goes better when I have some prepared questions, especially when the participants have little to offer. It will be updated... more
This is a list of discussion questions that I am using in my math ed research seminar this fall. The discussion goes better when I have some prepared questions, especially when the participants have little to offer. It will be updated from time-to-time.
Research Interests:
This is a list of specific reading questions that I have used in my ongoing seminar to help the discussion along and to check whether the students have read each paper carefully (and gotten some of the most important points/concepts). I... more
This is a list of specific reading questions that I have used in my ongoing seminar to help the discussion along and to check whether the students have read each paper carefully (and gotten some of the most important points/concepts). I have discovered that having a list of specific reading questions available for when there is a "lull" in the conversation or for when the conversation gets "off track" is useful for refocusing the discussion in a useful direction. This list is updated from time-to-time. This completes the questions for Spring 2015. At the end, is my take-home exam, emphasizing certain concepts that were used in the papers.
Research Interests:
This list of questions is now complete. The questions cover all the papers we read in our Fall 2014 mathematics education research seminar. I have discovered that having a list of specific reading questions available for when there is a... more
This list of questions is now complete. The questions cover all the papers we read in our Fall 2014 mathematics education research seminar. I have discovered that having a list of specific reading questions available for when there is a "lull" in the conversation or for when the conversation gets "off track" is useful for refocusing the discussion in a useful direction.
Research Interests:
This is a "work-in-progress". I discovered that my reading/discussion questions for research articles were not appropriate for chapters and survey articles, so produced this one.
Research Interests:
In order that discussions be productive, these are some general questions I provided students to guide their reading and help them get more out of reading mathematics education research articles. This is an updated version as of Oct. 8,... more
In order that discussions be productive, these are some general questions I provided students to guide their reading and help them get more out of reading mathematics education research articles. This is an updated version as of Oct. 8, 2014.
Research Interests:
This is an example of a detailed proof framework giving both Level 1 and Level 2 of the framework.
Research Interests:
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.
Research Interests:
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.
Research Interests:
This questionnaire/survey was handed out, and collected, at an R. L. Moore Legacy Conference.
Research Interests:
This interview script contains 7 questions about the style in which proofs are written. We asked these questions of some mathematicians at the Park City Mathematics Institute one summer. The results of that study, while not fully... more
This interview script contains 7 questions about the style in which proofs are written. We asked these questions of some mathematicians at the Park City Mathematics Institute one summer. The results of that study, while not fully analyzed, were used in writing "The Genre of Proof", which was published in 2013 in the book edited by M. N. Fried and T. Dreyfus, "Mathematics and Mathematics Education: Searching for Common Ground" (pp. 248-251). Advances in Mathematics Education series. New York: Springer.
Research Interests:
This interview script was devised to elicit undergraduate students' views of proof.
Research Interests:
This questionnaire was devised to, and used to, interview mathematicians. It has three parts: I. Mathematicians' views of mathematics, II. Mathematicians' views of learning and teaching mathematics, and III. The way mathematicians... more
This questionnaire was devised to, and used to, interview mathematicians. It has three parts: I. Mathematicians' views of mathematics, II. Mathematicians' views of learning and teaching mathematics, and III. The way mathematicians actually teach specific lower-division and upper-division mathematics courses
Research Interests:
There is an initial written questionnaire about which mathematics courses the interviewee has taken, followed by the actual interview script. This questionnaire and interview script was devised, and used, to interview undergraduate... more
There is an initial written questionnaire about which mathematics courses the interviewee has taken, followed by the actual interview script. This questionnaire and interview script was devised, and used, to interview undergraduate mathematics majors, mainly at the transition-to-proof course (sophomore) level.
Research Interests:
This is a task-based interview script. There is one task on equivalent or nonequivalent restatements of a theorem. There is one task on evaluating a hypothetical student's proof. There is one task on deciding whether a given mathematical... more
This is a task-based interview script. There is one task on equivalent or nonequivalent restatements of a theorem. There is one task on evaluating a hypothetical student's proof. There is one task on deciding whether a given mathematical statement is reasonable.
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:
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:
The notes were typed up by our former graduate student, Milos Savic, who assisted in this course.
Research Interests:
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.
Research Interests:
This is a report to the University of Texas at El Paso on what I talked about in my colloquium. It was written to get expenses reimbursed. It given a better indication of what was in that talk than the abstract. It begins as follows: On... more
This is a report to the University of Texas at El Paso on what I talked about in my colloquium. It was written to get expenses reimbursed. It given a better indication of what was in that talk than the abstract. It begins as follows:
On Oct. 23, 2009, I delivered a colloquium address to the UTEP Mathematics Department. The title of my talk was, " Understanding and Constructing Proofs: A Design Experiment. " I described the design experiment, a course my husband, John Selden, and I have been developing for helping advanced undergraduate and beginning graduate mathematics students construct proofs. The course has been taught at least eight times to date and each time we learned 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 have defined 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.
On Oct. 23, 2009, I delivered a colloquium address to the UTEP Mathematics Department. The title of my talk was, " Understanding and Constructing Proofs: A Design Experiment. " I described the design experiment, a course my husband, John Selden, and I have been developing for helping advanced undergraduate and beginning graduate mathematics students construct proofs. The course has been taught at least eight times to date and each time we learned 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 have defined 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.