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This paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students' reading, we 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. Appended to this document are: (1) Precalculus Reading Passages; (2) Calculus Reading Passages with Interruptions; and (3) Debriefing Questions. (Contains 2 tables and 13 endnotes.)

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 who were considered to be good at mathematics, as indicated by high ACT mathematics scores. These students were also good general readers, as indicated by their high ACT reading comprehension scores and by their use of many of the metacognitive strategies developed in reading comprehension research. To gauge the effectiveness of students' reading of passages from their mathematics textbooks, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. Our students demonstrated enough difficulties with these tasks, that it appears they do not benefit from reading their textbooks as much as their teachers or textbook authors would hope. Analysis of the data suggests that the reading strategies used by these students were not sufficient for them to complete many of the tasks. Instruction or guidance in strategies that are specifically related to mathematics reading may be needed to help students deal with mathematical text. Appended to this document are: (1) Appendix A: Precalculus Reading Passages; (2) Appendix B: Calculus Reading Passages with Interruptions; and (3) Appendix C: Debriefing Questions. (Contains 3 tables and 11 footnotes.) [Partial support for this research was provided by the Mathematical Association of America's Special Interest Group on Research in Undergraduate Mathematics Education.]

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 is 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. (Contains 7 figures and 9 footnotes.)

This paper discusses the curriculum and students' and teachers' conceptions of proof. It goes on to discuss university students' difficulties with proving related to understanding and using definitions and theorems, understanding the structure of a proof, knowing how to read and check proofs, knowing and using relevant concepts, bringing appropriate knowledge to mind, and knowing what's important and useful. Finally, there are suggestions for teaching proof and proving at the university level. (Contains 22 footnotes.)

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. We will discuss the relationship between consciousness and cognition in terms of two illustrations, and recall and extend some features of consciousness pointed out by Mangan (1993, 2001), following William James. The resulting framework will then be used do analyze three situations from mathematics: students (I) evaluating proofs, (II) writing equations, and (III) failing to use "adequate" knowledge to solve problems. A bibliography is included.

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. (Contains 9 endnotes.) [This paper is published in: Novak, Joseph D., Ed. "Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics" Volume III, July 1987, Cornell University, p457-470.]

The authors discuss the question, "What is mathematics education research at the university level?" They go on to consider "Making progress in tertiary mathematics education research: pushing the field forward." This includes a discussion of basic concepts and theoretical frameworks used. Further, they consider who does such research and how one gets into the field. They continue by discussing the application of such research to teaching practice. Finally, they consider new directions for such research. (Contains 21 footnotes.)

In two previous studies we investigated the non-routine problem-solving abilities of students just finishing their first year of a traditionally taught calculus sequence. This paper reports on a similar study, using the same non-routine first-year differential calculus problems, with students who had completed one and one-half years of traditional calculus and were in the midst of an ordinary differential equations course. More than half of these students were unable to solve even one problem and more than a third made no substantial progress toward any solution. A routine test of associated algebra and calculus skills indicated that many of the students were familiar with the key calculus concepts for solving these non-routine problems; nonetheless, students often used sophisticated algebraic methods rather than calculus in approaching the non-routine problems. We suggest a possible explanation for this phenomenon and discuss its importance for teaching. (Contains 5 footnotes, 3 figures, and 6 tables.) [Partial support for this research was provided by the Exxon Education Foundation.]

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., 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 apparent utility for reducing validation errors. We will then examine the relationship between the need for logic in validating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses do not 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 find it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reflection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge. (Contains 11 footnotes.)

Investigated (n=61) undergraduates' ability to unpack informally written mathematical statements into the language of predicate calculus in an introduction to proofs and mathematical reasoning. Found that students were unable to construct proofs or validate them. Appendices are "A Sample Validation" and "Building a Statement Image." (MKR)