The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

### Elise Lockwood and Branwen Purdy

### Kristen Lew and Juan Pablo Mejía-Ramos

This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing. Three main themes emerged: First, mathematicians believed that mathematical language should obey the conventions of academic language, whereas students were either unaware of these conventions or unaware that these conventions applied to proof writing. Second, students did not fully understand the nuances involved in how mathematicians introduce objects in proofs. Third, mathematicians focused on the context of the proof to decide how formal a proof should be, whereas students did not seem to be aware of the importance of this factor.

### Donald W. Hight, Robert J. Kansky and Don K. Richards

The Profrssional-Year Program. a component of the Training Teacher Trainers Project (or TTT), was initiated during the 1969-70 school year al Indiana University in cooperation with the Monroe County Community School Corporation. Presently the project involves 92 undergraduate students who are completing methods courses and student teaching simultaneously in three schools in Monroe County.

### Kevin C. Moore

A growing body of literature has identified quantitative and covariational reasoning as critical for secondary and undergraduate student learning, particularly for topics that require students to make sense of relationships between quantities. The present study extends this body of literature by characterizing an undergraduate precalculus student's progress during a teaching experiment exploring angle measure and trigonometric functions.

### Matthew Inglis and Lara Alcock

This article presents a comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians. Participants' eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short purported proofs; (b) compared with mathematicians, undergraduate students spend proportionately more time focusing on “surface features” of arguments, suggesting that they attend less to logical structure; and (c) compared with undergraduates, mathematicians are more inclined to shift their attention back and forth between consecutive lines of purported proofs, suggesting that they devote more effort to inferring implicit warrants. Pedagogical implications of these results are discussed, taking into account students' apparent difficulties with proof validation and the importance of this activity in both schooland university-level mathematics education.

### Katye O. Sowell and Katharine W. Hodgin

Undergraduate students who are preparing to teach mathematics need as many opportunities as possible to engage in simulated or actual teaching. Planning lessons, constructing situations to facilitate learning, and communicating with learners are skills that are best learned through practice. At East Carolina University, the students enrolled in a course entitled “The Teaching of Mathematics” are given two quite different opportunities to teach under supervision during the quarter preceding their student teaching in a secondary school.

### Edited by E.W. Hamilton

Who reads research articles? All classes of readers of such a journal as this should—college professors, because trends that inspire other, more exhaustive research are sometimes detected in short reports and pilot studies; graduate tudents, because it is a way of keeping up with the t hinking of the profession; undergraduate students, because it is a start toward knowledge acquired on a basis other than an authoritarian one and a start toward acquaintance with professional literature and scholarship.

### Grace M. Burton

When Rebelsky (1964) asked graduate and undergraduate students to complete drawings of glasses of water by illustrating the position the liquid would assume when the glass was tilted, she was surprised to discover not only that many students were unable to do that seemingly simple task but also that the performance of women was significantly poorer than that of men. Although Piaget and Inhelder (1963) made no comments regarding sex differences in their prototypic study in this area, differences in favor of males have been found in almost all subsquent studies (Ray, Georgiou, & Ravizza, 1979).

### Jean M. Shaw

When we attended the Smithsonian Museum's traveling exhibition on kaleido-scopes at our University Museum, several undergraduate students made an interesting discovery. Pan of the exhibit featured a device with hinged mirrors and colored objects that could be arranged on a table; participants arranged the objects and explored the different effect that they could see reflected in the mirrors. When the students placed a colored strip between the ends of the mirrors and looked directly into the mirrors, which were positioned to create an angle with measure 90 degrees, they saw a square (fig. 1a). When they changed the angle of the mirror, they saw a regular pentagon or hexagon.

### Matthew Inglis and Lara Alcock

We recently reported a study in which undergraduate students and research mathematicians were asked to read and validate purported proofs (Inglis & Alcock, 2012). In our eye-movement data, we found no evidence of the initial skimming strategy hypothesized by Weber (2008). Weber and Mejía-Ramos (2013) argued that this was due to a flawed analysis of eye-movement data and that a more fine-grained analysis led to the opposite conclusion. Here we demonstrate that this is not the case, and show that their analysis is based on an invalid assumption.