We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first videorecorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed 6 students to see what they understood from this lecture. The students did not comprehend the ideas that the professor cited as central to his lecture. Based on our analyses, we propose 2 factors to account for why students did not understand these ideas.

### Kristen Lew, Timothy Patrick Fukawa-Connelly, Juan Pablo Mejía-Ramos and Keith Weber

### Timothy Fukawa-Connelly, Keith Weber and Juan Pablo Mejía-Ramos

This study investigates 3 hypotheses about proof-based mathematics instruction: (a) that lectures include informal content (ways of thinking and reasoning about advanced mathematics that are not captured by formal symbolic statements), (b) that informal content is usually presented orally but not written on the board, and (c) that students do not record the informal content that is only stated orally but do if it is written on the board. The authors found that (a) informal content was common (with, on average, 32 instances per lecture), (b) most informal content was presented orally, and (c) typically students recorded written content while not recording oral content in their notes.

### Michael Petricig

In recent years many colleges have noticed an increase in the number of student applicants who are underprepared in the basic skills of reading, writing, and mathematics. At Chapman College, a large percentage of freshmen have been required to take remedial courses in these subjects. Furthermore, many students were not performing well in the remedial course in Intermediate Algebra. To improve the success rate for students in this course, a method for teaching it was developed that combined individualized instruction with the traditional lecture approach. The method was easy and inexpensive to implement, and it required a minimum of reorganization.

### Roberta J. Flexer

A 3-year study was conducted to compare the effectiveness of laboratory and lecture strategies in a course for prospective elementary teachers on the structure of the number system. Experimental and control groups were compared in terms of achievement in the course; retention on a follow-up examination given 6 months after the end of the course; attitude toward mathematics; performance in student teaching of mathematics; and responses to a questionnaire concerning the course. Within the experimental limitations of this study, there was no reason to conclude that either strategy was superior to the other. Implications concerning the design of preservice courses in mathematics are discussed.

### Harold L. Schoen

Teachers will be interested in reading about an attempt to strive toward individualization without sacrificing the benefits of group instruction. Have you tried to modify your teaching? Share your experience with the readers of the Mathematics Teacher.

### Fred Pigge and Irvin H. Brune

Teachers of methods in elementary mathematical education usually seek somehow to create, maintain, and extend students' mathematical understandings. Often their students come to them with three to six semester hours of credit in mathematics for futu re elementary teachers. Not similarly countable, however, are the students' stores of usable mathematical concepts. Nearly always their instructors recognize that these students need a new look, if not a thorough review, into some key ideas. So the question arises, How best can we handle such a sharpening of mathematical understandings?

### Thomas Jones

### Edited by Eugene D. Nichols

MOST educators will agree that more should be done to attenuate student failures at all grade levels and in all subject areas. The research project to be discussed here was designed to explore classroom techniques that would be conducive to more effective learning. The present research focuses upon ninth-grade general mathematics students who had underperformed in the regular session, necessitating their participation in a summer session.

### Samuel Otten, Wenmin Zhao, Zandra de Araujo and Milan Sherman

on the videos for their flipped lessons ( de Araujo, Otten, and Birisci 2017a ). Whether discovering ready-to-use videos or creating their own, teachers can find the process daunting. And creating a video lecture is not the same as lecturing in front

### C. C. Camp

1 Lecture delivered before tho Freshmen of the Arts and Science Oollego, University of Nebraska, January 15 and 16, 1928.

### Edited by Irving Allen Dodes, Philip Peak, Jack Price and Norman Schaumberger

Sophisticated but lucid lectures on systems, games, continuity, filters, approximation theory, and programming systems.—Dodes.