Research in undergraduate mathematics education (RUME) is a relatively young field that built momentum in the 1980's and 1990's and made a very public debut as the first Special Interest Group of the MAA (SIGMAA) at the New Orleans Joint Mathematics Meetings in 2001 (

http://www.rume.org). The first volume in this series,

*Research in Collegiate Mathematics Education*, was published in 1998, in part as a response to a need for a publication outlet dedicated to RUME. As the preface to this fifth volume says,

"This and the four previous volumes serve purposes similar to those of a journal. Each presents readers with peer-reviewed research on questions regarding the teaching and learning of collegiate mathematics" (p. vii)

There are other journals in which RUME is published (e.g., the *Journal for Research in Mathematics Education* and *Educational Studies in Mathematics*), but those journals have a wider focus and a history of publishing research in the K-12 arena. The editorial policy available in the final pages of the volume states:

"The papers published in these volumes will serve both pure and applied purposes, contributing to the field of research in undergraduate mathematics education and informing the direct improvement of undergraduate mathematics instruction. The dual purposes imply dual but overlapping audiences and articles will vary in their relationship to these purposes. The best papers, however, will interest both audiences and serve both purposes" (p. 205).

The fifth volume in this series includes the following seven articles:

- Maria Trigueros and Sonia Ursini, "First-year Undergraduates' Difficulties in Working with Different Uses of Variable."
- Abbe Herzig and David T. Kung, "Cooperative Learning in Calculus Reform: What Have We Learned?"
- Cheryl Roddick, "Calculus Reform and Traditional Students' Use of Calculus in an Engineering Mechanics Course"
- Pessia Tsamir, "Primary Intuitions and Instruction: The Case of Actual Infinity"
- Kirk Weller, Jule M. Clark, Ed Dubinsky, Sergio Loch, Michael A. McDonald, and Robert R. Merkovsky, "Student Performance and Attitudes in Courses Based on APOS Theory and the ACE Teaching Cycle"
- David E. Meel, "Models and Theories of Mathematical Understanding: Comparing Pirie and Kieren's Model of the Growth of Mathematical Understanding and APOS Theory"
- Jack Bookman and David Malone, "The Nature of Learning in Interactive Technological Environments: A Proposal for a Research Agenda Based on Grounded Theory"

As with the previous four volumes in the series, the editors made a point of including an unusually detailed preface, the bulk of which is an overview of the articles in the volume. More than just abstracts, however, these overviews place each article in a larger context and pull together ideas from related articles. To add further context, the opening paragraphs of the preface in this fifth volume include a bit of history and a look at present trends. In particular, these paragraphs offer a clarification of the internal organization of the articles — a clarification that also adds insight into the state of the field. I think this sort of progress report is especially important for young fields that are growing quickly in terms of both membership and visibility. In all of the volumes, I find the preface to be one of the most valuable resources available. At the very least, I would recommend that readers who want to filter their to-be-read stack spend some time reading not only the abstracts that appear with the actual articles but also the overviews in the preface.

One of the most powerful advantages of this publication outlet is sufficient space for authors to publish thorough descriptions of their research. For example, Trigueros and Ursini were able to include not only the 65-item questionnaire they use, in both English and Spanish, but also the relationship of each item on the questionnaire to the corresponding category of their framework. Most of the pieces also include instruments in the appendices. Detail of this kind allows the reader to inspect and reflect on the research at a deeper level and should facilitate related research. Where most journals do not allow space for these kinds of additional information, this particular set of volumes emphasizes the importance of this information both for purposes of reading the specific articles, but also for facilitating the advancement of the field as a whole.

In the remainder of this review, I offer my personal reaction to each of the seven chapters in this volume. This was a difficult exercise. Graduate school trained me to be critical as I read the literature and, thus, my inclination is to seek out weaknesses in the articles. Yet I wanted this review to be more about the kinds of articles that appear in this series to illustrate the niche that this publication outlet fills. To this end, for each chapter, I try to offer a few sentences about the content and a few sentences about how I envision making use of the article in my own professional life.

- Trigueros and Ursini investigated "First-year Undergraduates' Difficulties in Working with Different Uses of Variable," using a written questionnaire (65 items) completed by 164 students. They defined three categories for variable: "as specific unknown," "as general number," and "in functional relationships," with specific details for what they consider as evidence that a student understands variable in a particular way. The sample consisted of students that I would classify as weak in mathematics, as opposed to, perhaps, calculus-ready students. The authors indicate that they interviewed four students as well, but I think this is not the heart of the study. As I said above, the instrument they used is provided in its entirety in the appendix, along with a delineation of how each item fits into their classification framework. Unfortunately, it is not clear how the items were scored, although it appears that they were scored as correct/incorrect. The teaching and learning of algebra in college is certainly an important topic and I would urge anyone interested in this topic to take a very close look at this instrument and its development.
- The article, "Cooperative Learning in Calculus Reform: What Have We Learned," by Abbe Herzig and David T. Kung, offers a good overview of the history of cooperative learning and of its use in various reform calculus models. Herzig and Kung then completed a quantitative study, specifically attempting to control for factors such as length of the class period — quite a challenging undertaking.
As with many educational studies, the investigators ultimately ended up with additional confounding variables such as prior experience with calculus. In order to decrease confounding variables, the researchers reduced the pool used for analysis. They started with an n of 313, conducted pieces of the analysis on only 189 who met the restricted criteria. While they justify these decisions, the results naturally leave me wondering about the students who were not represented in the analysis.

In my opinion, this article makes the case, explicitly and implicitly, that isolating cooperative learning as a factor in student learning of calculus is a challenging task. While I might disagree with some of the statements and claims made by the authors, I agree wholeheartedly with the authors' conclusions that "having focused for a decade on the question 'Does this work?' researchers now need to move on to the next generation of studies, attempting to answer the question 'Why and how does this work, and for whom?'" (p. 48). Certainly, I have added this article to my list of references on cooperative learning.

- Roddick undertook another challenging task in the article, "Calculus Reform and Traditional Students' Use of Calculus in an Engineering Mechanics Course". The primary data were interviews with six students, three of whom had completed a "traditional" calculus sequence and three of whom had completed the "Calculus &
*Mathematica*" sequence. In my mind, this article touches on two topics: students' use of calculus in a subsequent course and the comparison of students who learned calculus in different instructional settings. The participant pool lends itself very well to an in-depth exploratory investigation of students' use of calculus. However, this sample size is much too small to draw the kinds of comparative conclusions that the researcher wishes to make. I say that as a person who conducts qualitative research myself. There are simply too many alternative explanations for the noted differences. However, when combined with other work in this area, this article contributes evidence to the discussion. In this sense, it belongs in any complete bibliography about the impact of computer algebra systems on student learning of calculus. Yet I feel more confident that the article belongs in general bibliographies about student understanding (and "transfer") of calculus ideas.
Not to nitpick, but this article also suffers from distracting editing mistakes. In two instances, several paragraphs are repeated in their entirety. And several references are missing. I do recognize that editing problems happen regularly (including in my own work!) and may have occurred elsewhere in the volume as well. I just happened to be very aware of them in this article.

- In the article, "Primary Intuitions and Instruction: The Case of Actual Infinity," Tsamir investigated student understandings of infinity. The sample was 181 students from teachers colleges in Israel, some of whom had studied Cantorian set theory and others who had not. Questionnaire items investigated student inclinations to use or accept one-to-one correspondence, inclusion, and single infinity as criteria for comparing infinite sets. The first part of the questionnaire was administered twice ("Stage I" and "Stage III"), with the second part acting as a mediation device in between ("Stage II"). The article reports results from the initial completion of the first part, then goes on to report the change in responses after completion of the second part and re-completion of the first part. Despite the relatively large sample size, the only statistics offered in the article are percentages. The author also conducted interviews with 20 students, but does not offer (in this article) a systematic analysis of those data. Summatively,
"after having been presented in Stage II with illustrations of the use of one-to-one correspondence as a criterion for comparing infinite sets, at Stage III there was an increase among all participants in the acceptance of one-to-one correspondence. However, the rate of [students who had not studied Cantorian set theory] and even [those who had] who accepted inclusion was unchanged, while the rate of acceptance of single infinity rose" (p.90).

The article points out early on that even some prominent mathematicians initially held similar beliefs about infinity. However, the article does not emphasize sufficiently, in my opinion, that it then makes sense that these ideas are difficult for students. The author does offer a list of possible instructional interventions and, certainly, if I were teaching a Cantorian set theory course, I would turn to this article to help me think through likely student misconceptions and as a starter set of activities for students to work through.

- The action-process-object-schema (APOS) theory of student learning, and the related activities-class discussion-exercises (ACE) teaching cycle, have been evolving since the mid-1980's. APOS/ACE experts — Weller, Clark, Dubinsky, Loch, McDonald, and Merkovsky — teamed together in "Student Performance and Attitudes in Courses Based on APOS Theory and the ACE Teaching Cycle" to produce a much-needed synthesis of 14 prior studies related to APOS. The authors provide detailed evidence from these papers to support their claims
"that instruction based on upon APOS Theory yields better results than what one would expect within a traditional setting. Moreover, these papers provide strong evidence supporting the contention that the research and curriculum development framework based on APOS Theory and the ACE Teaching Cycle is a reasonable approach to describe and to enhance student learning of mathematics" (p.128).

I wish that the authors had included in an appendix the instruments used by the studies; it would have been convenient to have those instruments all in one place like this. Nevertheless, I look forward to giving this article to mathematics education doctoral students as an introduction to this theory.

- Another article that is a must-read for doctoral students in mathematics education is Meel's "Models and Theories of Mathematical Understanding: Comparing Pierie and Kieren's Model of the Growth of Mathematical Understanding and APOS Theory". The article includes "a brief history of 'understanding'", which gives a fantastic summary of models, ideas, and perspectives over the decades. This historical overview sets the stage for an in-depth examination of APOS theory and Pirie & Kieran's model of understanding. After giving an overview of each of these frameworks, Meel methodically compares and contrasts the two theories along several dimenstions. As part of this process, he first verifies that each model meets Shoenfeld's criteria for a theory. This article is a must-read because it offers glimpses into many ideas while leading the reader through an in-depth study of two current, much-talked-about theories.
- I have been looking for an article that I would want to recommend to people who are interested in research related to the use of computer algebra systems for teaching mathematics. The chapter by Bookman and Malone, "The Nature of Learning in Interactive Technological Environments: A Proposal for a Research Agenda Based on Grounded Theory," is such an article. In the end, the authors offer the following list of questions as a research agenda (p. 199):
(1) What is the role of the instructor in this envoronment?

(2) What types of behavior and thinking processes are students engaged in as they work together in front of the computer, and how can the modules be written to facilitate students' self-monitoring and effective collaborative interaction?

and

(3) What opportunities and obstacles are raised by the technology itself?

Rather than starting with a theory, as other chapters in this volume did, these authors allowed the data to generate a model (this process is known as "grounded theory"). Seven vignettes of student pairs working on mathematics problems using a CAS and HTML documents illustrate the development of the list of questions. Not only did I find the content of the chapter compelling, I also found the writing to be unusually engaging. My only real complaint is more sadness than annoyance: out of 10 students depicted in the vignettes that the authors present, only one is female.

Teri J. Murphy (tjmurphy@math.ou.edu) is associate professor of mathematics at the University of Oklahoma. She has a Ph.D. in mathematics education, an M.S. in mathematics, and an M.S. in applied mathematics from the University of Illinois at Urbana-Champaign. Her research specialty is undergraduate mathematics education.