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Eva Thanheiser, Amy Ellis and Beth Herbel-Eisenmann

In this Research Commentary, 3 JRME authors describe the process of publishing their research in JRME. All 3 authors published parts of their dissertation in JRME and are sharing their stories to help (new) researchers in mathematics education better understand the process and to offer (experienced) researchers in mathematics education a tool that can be used to mentor their less experienced colleagues and students. The authors address preparing, conceptualizing, and writing a manuscript as well as responding to reviewers.

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Robert Ely

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

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Dirk De Bock, Johan Deprez, Wim Van Dooren, Michel Roelens and Lieven Verschaffel

Kaminski, Sloutsky, and Heckler (2008a) published in Science a study on “The advantage of abstract examples in learning math,” in which they claim that students may benefit more from learning mathematics through a single abstract, symbolic representation than from multiple concrete examples. This publication elicited both enthusiastic and critical comments by mathematicians, mathematics educators, and policymakers worldwide. The current empirical study involves a partial replication–but also an important validation and extension–of this widely noticed study. The study's results confirm Kaminski et al.'s findings, but the accompanying qualitative data raise serious questions about their interpretation of what students actually learned from the abstract concept exemplification. Moreover, whereas Kaminski et al. showed that abstract learners transferred what they had learned to a similar abstract context, this study shows also that students who learned from concrete examples transferred their knowledge into a similar concrete context.

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Leone Burton and Candia Morgan

In this article we report on part of a study of the epistemological perspectives of practicing research mathematicians. We explore the identities that mathematicians present to the world in their writing and the ways in which they represent the nature of mathematical activity. Analysis of 53 published research papers reveals substantial variations in these aspects of mathematicians' writing. The interpretation of these variations is supported by extracts from interviews with the mathematicians. We discuss the implications for students and for novice researchers beginning to write about their mathematical activity.

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Sheri A. Hanson and Thomas P. Hogan

We examined computational estimation skill of 77 college students who estimated answers to problems presented in brief intervals. We categorized 23 “think-aloud” estimation strategies used by 45 participants in individual follow-up sessions. Some categories were based on strategies found in previous studies; others were based on responses in this study. Although students correctly estimated answers to most problems on addition and subtraction of whole numbers, they performed poorly on multiplication and division of decimals and subtraction of fractions. Students were more successful in solving computational problems than in estimating answers. Scores on the estimation tests showed substantial correlation with SAT Mathematics scores and with a direct measure of computational skill, but they did not significantly correlate with SAT Verbal scores.

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Annie Selden and John Selden

This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited—perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.

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John P. Smith III and Jon R. Star

Research on the impact of Standards-based, K–12 mathematics programs (i.e., written curricula and associated teaching practices) and of reform calculus programs has focused primarily on student achievement and secondarily, and rather ineffectively, on student attitudes. This research has shown that reform programs have competed well with traditional programs in terms of student achievement. Results for attitude change have been much less conclusive because of conceptual and methodological problems. We critically review this literature to argue for broader conceptions of impact that target new dimensions of program effect and examine interactions between dimensions. We also briefly present the conceptualization, design, and broad results of one study, the Mathematical Transitions Project (MTP), which expanded the range of impact along those lines. The MTP results reveal substantial diversity in students' experience within and between research sites, different patterns of experience between high school and university students, and surprising relationships between achievement and attitude for some students.

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Jon R. Star, John P. Smith III and Amanda Jansen

Research on the impact of Standards-based mathematics and reform calculus curricula has largely focused on changes in achievement and attitudes, generally ignoring how students experience these new programs. This study was designed to address that deficit. As part of a larger effort to characterize students' transitions into and out of reform programs, we analyzed how 93 high school and college students perceived Standards-based and reform calculus programs as different from traditional ones. Results show considerable diversity across and even within sites. Nearly all students reported differences, but high-impact differences, like Content, were not always related to curriculum type (reform or traditional). Students' perceptions aligned moderately well with those of reform curriculum authors, e.g., concerning Typical Problems. These results show that students' responses to reform programs can be quite diverse and only partially aligned with adults' views.

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Gabriel J. Stylianides and Andreas J. Stylianides

Although students of all levels of education face serious difficulties with proof, there is limited research knowledge about how instruction can help students overcome these difficulties. In this article, we discuss the theoretical foundation and implementation of an instructional sequence that aimed to help students begin to realize the limitations of empirical arguments as methods for validating mathematical generalizations and see an intellectual need to learn about secure methods for validation (i.e., proofs). The development of the instructional sequence was part of a 4-year design experiment that we conducted in an undergraduate mathematics course, prerequisite for admission to an elementary (Grades K–6) teaching certification program. We focus on the implementation of the instructional sequence in the last of 5 research cycles of our design experiment to exemplify our theoretical framework (in which cognitive conflict played a major role) and to discuss the promise of the sequence to support the intended learning goals.

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Izaak Wirszup

Mathematic's Staff of the College, University of Chicago, Chicago, Illinois. Enrichment of our school mathematics curriculum can have many forms and many purposes. This paper suggests a more central role for enrichment, and reports the first conclusions of a study of the uses of enrichment in Russian schools.