Like the previous volumes, RCME VI has an unusually informative preface. The editors begin with a brief history of the RCME volumes and their niche in the mathematics community. They continue with an overview, including summaries of the articles and comments on the place of each article in the larger context. We recommend reading the preface before proceeding to any individual article.

As pointed out in the preface, the eight articles represent diversity in several dimensions: country, methodology, and topic (calculus/real analysis, complex analysis, group theory proofs, and learning styles in graduate education).

Jon R. Star and John P. Smith III, “An Image of Calculus Reform: Students’ Experiences of Harvard Calculus.”

Kelly K. Chappell, “Effects of Concept-Based Instruction on Calculus Students’ Acquisition of Conceptual Understanding and Procedural Skill.”

Maria Ángeles Navarro and Pedro Pérez Carreras, “Constructing a Concept Image of Convergence of Sequences in the van Hiele Framework.”

Niels Grønbæk and Carl Winsløw, “Developing and Assessing Specific Competencies in a First Course on Real Analysis.”

Peter Danenhower, “Introductory Complex Analysis at Two British Columbia Universities: The First Week — Complex Numbers.”

Ghislaine Gueudet-Chartier, “Using Geometry to Teach and Learn Linear Algebra.”

Keith Weber, “Investigating and Teaching the Processes Used to Construct Proofs.”

Janet Duffin and Adrian Simpson, “The Transition to Independent Graduate Studies in Mathematics.”

Research in Undergraduate Mathematics Education (RUME) is a young field and young fields go through growing pains. As we read, we felt that this set of papers was stronger than previous sets, which we take as evidence that RUME is maturing nicely as a field.

Toward that goal, despite so many good things about this book/journal, our academic training compels us to find room for improvement. In this case, we would like to see the journal enforce more careful use of language, especially language that has specific meaning in statistics. We say this because RUME makes use of statistical procedures so careful use of language seems to be an appropriate expectation. For example, at least one article used the word “significant” to describe some comparative results, but the analyses did not seem to involve any inferential statistical procedures. We don’t expect all researchers in RUME to be statistics experts, but we should be sufficiently literate in the fields whose methods we borrow that we do not misuse the language from those fields.

We will briefly discuss each of the eight articles.

In “An Image of Calculus Reform: Students’ Experiences of Harvard Calculus”, Star and Smith offer a perspective on the University of Michigan’s implementation of the Harvard Consortium calculus curriculum. The authors offer convincing arguments for including student voices in the research about reform courses: “…although authors of reform curricula clearly articulate the ways that their textbooks and recommended pedagogies differ from more traditional approaches (e.g., Harvard Consortium’s ‘rule of four’), there is no evidence that students see the new curricula in the same light as they were intended. … [Furthermore], students’ perceptions of reform curricula can provide explanations for why an implementation was, or was not, successful. Although grades are often used to determine success, they provide little or no information to account for or explain outcomes” (p. 3).

This article is part of a larger project and reports on the in-depth study of 19 students. As the authors point out, although the sample size may seem small, the number of interviews (80), including repeat interviews with the same students over time, was considerably higher, resulting in a large volume of rich data. The conclusion that “students’ reactions to and judgments about the U-M program were relatively independent of their grades” (p. 21) is one to add the arsenal for combatting the claim that students’ opinions are mere reflections of their grades. The researchers provide a creative way of considering achievement via GPAs, but the heart of the report is what they learned from the students. For example, “the increased emphasis on conceptual understanding and more balanced reliance on multiple representations… are both central features of the Harvard curriculum, yet the U-M students either had difficulty naming those differences (the focus on conceptual understanding…) … or never did (multiple representations).” (p.16).

What was clear from the article is that, as at the K-12 level, teachers and students bring beliefs about the teaching and learning of mathematics into these courses. Changing those beliefs is quite a challenge, especially when teachers and students have a history of being immersed in traditional practices, and creating that change takes time and persistent attention.

Chappell’s article on “Effects of Concept-Based Instruction on Calculus Students’ Aquisition of Conceptual Understanding and Procedural Skill” was also about teaching and learning in calculus. Her focus, though, was specifically on teaching with a conceptual emphasis vs. teaching with a procedural emphasis. The study, which augmented an analysis of test performance with student focus groups, involved 144 students and 4 instructors. The author concludes that

Concept-based instructional programs can effectively facilitate the development of student understanding without sacrificing skill proficiency.

Conceptual understanding supports the acquisition of procedural skills.

- Knowledge acquired with understanding can be extended to solve unfamiliar problems more readily than strictly procedural knowledge. (p. 54)

and

Major themes that emerged from whole-class interviews included the importance of conceptual foundations for learning and understanding procedures, the value of concept-based instruction for extending understanding to new problem solving situations, the need for a balance between effective concept-based and procedural teaching, and the challenge students felt they faced in adjusting to a concept-based learning environment. (p. 55)

Although some standard threats to validity persist (e.g., lack of random sampling or random assignment), the researchers produced a solid research design that addressed many more of the usual threats to validity than most studies manage. The article is written in a very detailed, concise way; provides useful, informative tables and figures; and addresses quite well many of the questions that tend to occur to readers of research papers.

“Constructing a Concept Image of Convergence of Sequences in the van Hiele Framework”, by Navarro and Pérez Carreras, reads somewhat like we would expect a paper in a psychology research journal to read. The work combines qualitative clinical interviews (n = 20) and a quantitative written questionnaire (n = 301) to classify students into categories of understanding, building on frameworks by van Hiele, Roberts, and Sierpinska. It wasn’t entirely clear to us what the focus of the paper was. At times, it seemed to be about understanding study thinking; at other times, it seemed to be about providing evidence for the success of a computer-based teaching strategy.

We were impressed by the insight to student understanding that the analysis offered, but we do not believe that the research design allowed for cause-effect conclusions to be drawn about the teaching strategy. In fact, in the quantitative analysis, the authors found that the “percentages [of students at each level of understanding] are similar to those obtained in the clinical interviews”. We interpret this result to imply that students who experienced the authors’ teaching strategy reached levels of understanding in the same proportions as students who experienced unspecified strategies.

In our experience, articles such as this one are difficult for novice researchers to understand — a result of the language, sentence structure, and section layout of the paper. Experienced researchers with a background in frameworks used to study learning processes will likely get more out of this article than novices. The article assumes enculturation in the field; for example, “The RUMEC group analyzed student conceptions of the sequence concept (Mathews, 2001). This work was performed with the usual methodology of this group” (p. 87). The authors do explain the “methodology” but do not explain what the RUMEC group is. Those who have been around for two decades or more know that this group was the Research in Undergraduate Mathematics Education Community that spawned the first Special Interest Group of the MAA: the SIGMAA on RUME. Yet a novice researcher would not necessarily know of this group or the impact of the group on RUME as a field. Thus, we would recommend that someone new to this field find an experienced person to read this article with them to help explain the framework and methodology used and assumed to be known to the reader.

Although the study by Grønbæk and Winsløw, “Developing and Assessing Specific Competencies in a First Course on Real Analysis”, takes place in the context of a real analysis course, the methodology (didactic engineering) should be useful in other contexts as well — which the authors point out several times. By “specific competency”, the authors mean “a person’s potential to perform a general type of mathematical action in the context of a specific area of mathematical content”. Toward the goal of helping students to navigate course content, the authors re-designed a first course in real analysis via a list of Specific Competency Goals (SCGs). The list of SCGs is in an appendix to the article (one of the features we really like about the RCME series is the space allowed for appendices). We especially like the authors’ first conclusion: “SCG descriptions are necessary but not sufficient” (p. 130).

We also find in this article considerable evidence for the need for ongoing specific professional development for teaching assistants — new teaching strategies are challenging even for experience, motivated instructors, even more so for people new to teaching. Like the Navarro and Pérez Carreras article discussed above, Grønbæk and Winsløw’s article, as written, will be most accessible to experienced researchers. Novices (researchers and instructors) will likely need some help in understanding the ideas and methodology. On the other hand, the authors move toward building this bridge by providing concrete examples and detailed appendices.

The title of “Introductory Complex Analysis at Two British Columbia Universities: The First Week — Complex Numbers”, accurately describes the study reported. Author Danenhower observed three courses at two universities and interviewed 21 students (a total of 54 task-based interviews) from these courses. The very detailed analysis was set in reification and APOS models, as well as in the LBP and Dreyfus schemes. The conclusions are succinct and do not over-reach. A sample conclusion is that “The students studied had a good grasp of the Cartesian vector forms and polar forms, but slightly less skill translating from on form to the other. At least half the students did not have good judgment about when to shift from one form to another. All shifting was between algebraic extension and polar vector forms, as students had almost no skill with other forms, such as the symbolic form” (p. 164). The article ends with thoughts about pedagogy and curriculum, suggested by the research, and with suggestions for future research directions.

This is one of the most readable articles we have read in all six volumes of RCME. The study builds on related and prior work, which is presented in a way that not only make clear its relevance but also makes that work accessible to readers. We would highly recommend this article as a model to doctoral students and other novices who would benefit from an example of strong work presented in a useful, candid, and unpretentious form. The author even offers the incisive point that “the term ‘realizing’ the denominator is not in common use in complex analysis, but it should be… Multiplying a complex number by its complex conjugate always produces a real number, but does not necessarily produce a rational number, so the accurate term is ‘realizing’ the denominator” (p.145). Good point!

We were impressed, and humbled, by Gueudet-Chartier’s understanding that many Americans who would be reading the article, “Using Geometry to Teach and Learn Linear Algebra” would likely have little or no conception of the mathematics curriculum in France. The article includes a quick, informative introduction to the history of discussions on teaching linear algebra in France. The author does a fabulous job of coming back around to this history in the analysis and conclusions: “Most of the mathematicians in France advocate one of two opposite approaches. The first group advocates a structural approach to linear algebra, without geometrical or figural models. The second group recommends a geometry course before linear algebra so that geometry can then provide models and the associated drawings. In both cases, however, the mathematicians do not develop a figural model specifically for linear algebra; their drawings are only used in a geometrical context (p. 190).

Evidence from the study reported in the article seemed to support a third option, which the author calls, “…a geometric model *inside* linear algebra” (p. 181). Data were gathered by way of a questionnaire responded to by 31 mathematicians who had recently taught linear algebra; an in-depth study of the textbook *Linear Algebra Through Geometry*, by Banchoff and Wermer (1991); class observations and interviews with the teacher and eight students. The author draws conclusions such as, “it appears that linear algebra cannot be taught nor learned as a mere generalization of a geometry… Yet, a geometric model can be helpful, especially because the associated figural model confers on the geometric model the appearance of concreteness” (pp. 189-190).

The article involves some theory that takes some work to understand, but the author does a good job of providing an introduction. We doubt that novice readers could become experts based on this introduction, but they should get a good sense from this article of what the theories have to offer in order to determine if they should pursue further study of those theories.

In another well-written article, Weber reports on “Investigating and Teaching the Processes Used to Construct Proofs”. The context is undergraduate abstract algebra, specifically, group homomorphisms. The premise is that even when students understand facts and theorems, and even when they understand the concept of proof (two of the reasons identified in the literature that students struggle with constructing proofs), they may still struggle because they lack effective decision-making strategies. Building on prior work about proofs and proving (some of which was the author’s), the author “designed a recursive procedure in which students first would determine what structure they needed information about, then choose a method to find that information, then determine what type of information would be needed to apply that method…” (p. 204). He tested this theory in a clinical setting with five undergraduate participants. The results of the study are promising.

What we find most compelling about this article is the author’s candor about the limitations of the study and his ability to frame the results both in the context of the limitations and in the context of the potential the strategy has for improving students’ ability to complete proofs. We do hope that the author will continue his work and that others will join him on this path. We also hope that instructors of abstract algebra will consider experimenting with this suggestions in their own classes. We could easily envision giving this article to colleagues who teach those classes, both to hear their perspectives and, hopefully, to inspire them to consider the ideas presented.

In “The Transition to Independent Graduate Studies in Mathematics”, Duffin and Simpson analyze interviews with 13 current PhD students at a medium-sized UK university to exploring the transition from undergraduate to graduate work in mathematics. They classified most of these students’ pre-graduate learning styles as *natural* (where new knowledge is connected to prior knowledge, thus enhancing depth of understanding and meaning of new material) or *alien* (where new knowledge is memorized, procedures are emphasized, and new concepts are independent cognitive structures, with no immediate link to prior knowledge). In general, students with natural learning styles transitioned smoothly to independent graduate study, while those with alien learning styles had to adapt to the new academic structure they encountered in graduate school. Upon entering graduate school, students with alien learning styles tended to adopt a more natural learning style or to adopt a third learning style, *coherence*, in which students focus on the “logical and structural coherence of new material” (p. 240). Three students failed to fit any of these cognitive styles; one of these students was described as having a *flexible* cognitive style because he could “adopt a number of different approaches depending on the value he attributed to the material he was trying to learn” (p. 242).

The authors conclude that most students’ cognitive styles were challenged on some level by the transition to graduate school. Most of these students adapted easily; others had to dramatically shift their cognitive tendencies to cope with the new academic and intellectual demands. Duffin and Simpson suggest that both undergraduate and graduate education could be modified to smooth the undergraduate-graduate transition for different types of learners.

While this study made interesting contributions to previous work in the theory of cognitive styles, the authors acknowledge that they did not include the perspectives of students who could not adapt their learning styles to cope with graduate work and who subsequently dropped out of PhD programs. Obtaining data from these students could be a beneficial direction for future research in order to understand the full range of possibilities of learning style adjustments between undergraduate and graduate work.

This article was thorough, well-written, and engaging. Furthermore, the use of case studies and quotations helped to clarify and illustrate the distinctions between the types of cognitive styles. By engaging in conversation with students about undergraduate-graduate cognitive style adjustments, Duffin and Simpson manage to highlight the key issues facing students during this important academic and intellectual transition. This article could offer much-needed perspective on the transition to graduate school for current and prospective graduate students as well as for graduate and undergraduate mathematics professors and program coordinators.

Teri J. Murphy is a Professor of Mathematics at the University of Oklahoma. Her research area is undergraduate science, engineering, and mathematics education. Sarah L. Marsh is a Ph.D. student in the Department of Mathematics at the University of Oklahoma. Her research area is undergraduate mathematics education.