# Annie SeldenNew Mexico State University | NMSU · Department of Mathematical Sciences

Annie Selden

PhD Clarkson University

## About

458

Publications

133,461

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2,591

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Introduction

I, along with my husband John Selden, conduct research in undergraduate mathematics education. This research investigates the way undergraduate, and sometimes graduate, students do and do not learn mathematical concepts. In the past, we researched students' learning of calculus, but now we mainly research students' understanding of proofs and proving. We were instrumental in the founding of the MAA's Special Interest Group on Research in Undergraduate Mathematics Education (SIGMAA on RUME).
Through the Mathematical Association of America, we established the Annie and John Selden Award for Research in Undergraduate Mathematics Education awarded biannually. Previous awardees include: Chris Rasmussen, Marilyn Carlson, Keith Weber, Lara Alcock, Matthew Inglis, Pablo Mejia-Ramos.

Additional affiliations

Education

September 1969 - June 1973

September 1960 - June 1963

September 1959 - July 1960

## Publications

Publications (458)

This is a book review of a volume in Springer's History of Mathematics Education Series.

This is a book review of "Doing Research: A New Researcher's Guide" published by Springer as part of the Research in Mathematics Education Series.

This is a chapter for a book edited by Keith Weber and Milos Savic, tentatively titled, "New Directions in University Proving: Honoring the Legacy of John and Annie Selden".
The chapter describes undergraduate mathematics education research on proof/proving from its inception in about the late 1980s.

This is a tribute to the visionary work of Ed Dubinsky, who passed away in May 2022. It concentrates his work on the establishment of the Special Interest Group of the Mathematical Association of America for Research in Undergraduate Mathematics Education (SIGMAA on RUME) and on his preparatory work towards a journal for research in undergraduate m...

This book review defines mathematical encounters, pedagogical detours and disturbance. It also gives a brief overview of Chapter 6 (with an illustration from calculus) that describes the authors' "If not, what yes?" method of getting students to engage in first refuting an incorrect statement by providing a counterexample, then refining the stateme...

Education began as a community of practice. This idea, introduced by the social learning theorist, Etienne Wenger [2], describes communities that come together around a joint enterprise and work not only to develop the practices and ideas that draw them together but also to negotiate "a shared repertoire of communal resources (routines, sensibiliti...

This short article details some of the mathematics education work of Dr. Karen King and of our work together in the MPWR group.

This is a book review for an MAA Online Book Review of this ICME-13 monograph on creativity.

This is a book review for MAA Online on this ICME-13 monograph. The volume's chapters are revised and extended versions of papers presented at the ICME-13 Topic Study Group 18 on Reasoning and Proof in Mathematics Education.

This abstract is for an upcoming special issue of the Journal of Humanistic Mathematics on creativity in mathematics and mathematics education,

This is a book review written for MAA Online Book Reviews.

This is an important, under-research question.

This handout provides information on 5 topics regarding 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/rejecti...

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...

This is a book review. 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 speciali...

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...

This is a book review of an IBL-teaching textbook for an introduction-to-proof course, published by MAA Textbook Series.

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, sta...

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...

This is a book review that gives an overview of the operational definitions of creativity and giftedness used in this volume. It also discusses two chapters as examples of interest.

This is a book review. This ICMI Study Volume has 15 chapters authored by 47 contributors from 22 countries including Brazil, South Africa, Germany, Denmark, Cameroon, Tanzania, Russia, USA, and Canada. Each ICMI Study volume is written after an ICMI Study Conference—this one was in Brazil in 2011. The conference itself had 91 participants from 27...

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 pro...

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...

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...

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...

This presentation mentioned the genre of proof, the structure of proofs, the formal-informal distinction, proofs of Types 0, 1, 2, and 3, and exploration It also discussed psychological aspects of proving such as situation-action links, behavioral schemas, automaticity, S2 and S1 cognition, self-efficacy, and local memory.

This abstract is for a contributed paper session. The presentation will be about proof frameworks and operable interpretations of definitions.

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 wri...

These short abstracts of mathematics education research journal articles were written for the Media Highlights section of the College Mathematics Journal. They cover a variety of topics.

This presentation considers the difficulty that one masters student, Dori, had when unpacking an informally worded mathematical theorem statement into its formal equivalent, in order to prove it.

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...

This is a book review, written for the Mathematical Association of America's (MAA's) Online Book Reviews. The book considered is one of the ICME-13 Topical Surveys, written by the organizers of ICME-13Topic Study Group 2 (TSG2) on Mathematics Education at Tertiary Level.

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...

This is a book 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 math...

This reports the dissertation work of my PhD student, Valeria Aguirre Holguin, on university students' use of definitions in examples, proof, and true-false statements.

This is a 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 [ph...

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.

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 proof...

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...

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 in...

This is a list of specific reading questions that I prepare for each paper that we discuss in the seminar. I have discovered that it is useful to have such a list on hand in case there is a "lull" in the conversation, or in case I want to redirect the discussion in a more useful direction. I update this list from time-to-time over the course of the...

This PowerPoint presentation 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...

This PowerPoint presentation 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 incl...

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...

This book review 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.,...

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 th...

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...

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 th...

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. The paper has been r...

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,...

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 a...

This theoretical presentation 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 pr...

This theoretical paper considers several perspectives for understanding and teaching university students' autonomous proof construction.

In the mathematics education research literature on proof, four concepts have been discussed: proof comprehension, proof construction, proof validation, and proof evaluation.

This completes the list of questions on papers we read in the seminar this semester. I generate these questions so that the discussion can lead to fruitful directions, in case there is a "lull" in the conversation or the conversation needs to be redirected towards the paper.

This is an overview chapter written at the request of the editors.It begins as follows:
In the early 1980s, to complement a previous PME emphasis on elementary mathematical thinking, some PME members, chiefly Gontran Ervynck and David Tall, wanted to consider “mathematics in school that led on to university mathematics and linked … to the thinking...

This gives the list of papers we have read and discussed in our mathematics education research seminar and the questions I compiled to discuss, in case there is a "lull" in the conversation. I find having such questions handy can help focus the conversation in positive directions. This listing is updated from time-to-time until the semester is over...

This c.v. lists my publications (of all kinds), my professional service activities (reviewing, editorial boards, professional organizations with offices held, etc.), my Ph.D. students, and some unpublished works available through www.researchgate.net or www.academia.edu.

USING A THEORETICAL PERSPECTIVETO TEACH A PROVING SUPPLEMENT FOR AN UNDERGRADUATE REAL ANALYSIS COURSE
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 (...

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.

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,...

This is an abstract for a contributed paper session at the Joint Mathematics Meetings in Seattle, Washington, January 2016. We will discuss how we came to the idea of proof frameworks and demonstrate the writing of several proof frameworks.

This is an abstract for a paper to be presented at the Joint Mathematics Meetings, Seattle, WA in January 2016. In it, 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.

This is a foreword for the upcoming MAA Notes volume, “Beyond Lecture: Techniques to Improve Student Proof-Writing Across the Curriculum”, edited by Rachel Schwell,, Aliza Steurer, and Jennifer F. Vasquez. It contains 40 short, approximately six-page, pieces of usable advice, written by college and university mathematics teachers who have tried out...

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...

This is a proposal for a paper on a proving supplement for real analysis that we have taught three times.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...

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 descripti...

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 descripti...

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 descripti...

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, difficulti...

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, difficulti...

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...

This is a list of specific reading questions that I prepare for each paper that we discuss in the seminar. I have discovered that it is useful to have such a list on hand in case there is a "lull" in the conversation, or in case I want to redirect the discussion in a more useful direction. I update this list from time-to-time over the course of the...

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-base...

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-base...

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,...

This should be the final version of this paper.
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 p...

This is a list of specific reading questions that I prepare for each paper that we discuss in the seminar. I have discovered that it is useful to have such a list on hand in case there is a "lull" in the conversation, or in case I want to redirect the discussion in a more useful direction. I update this list from time-to-time over the course of the...

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...

This Powerpoint presentation (which as allowed 20 minutes) is longer and more detailed than the CERME9 WG1 Powerpoint of the same name (which was allowed only 10 minutes). In it we suggest a perspective for understanding undergraduate proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behav...

This 10-minute Powerpoint presentation is a summary of the 10-page working group paper with the same title, which can also be found on Researchgate.

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 p...

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...

These abstracts of mathematics education research articles are written for the Media Highlights section of The College Mathematics Journal, mainly for mathematicians.

## Questions

Questions (8)

The idea of "proof" in mathematics is really a meta-mathematical concept. Students seem to learn what proofs are by "doing proofs" in various mathematics courses. What are they implicitly learning about proof by doing so?

I register for an ORCiD account because the journal, Mathematical Thinking and Learning, indicated that I should if I wanted to continue reviewing for them. But I don't know much about ORCiD.

I am particularly interested in how these terms are used in mathematics education research papers. I was under the impression that an "instructional sequence" as something of longer duration than an "intervention".

I am looking for some case studies of research in mathematics education to read in my mathematics education research seminar. I am especially interested in single case studies of individuals learning mathematics.