In this article, I argue for a renewed focus in mathematics education research on procedural knowledge. I make three main points: (1) The development of students' procedural knowledge has not received a great deal of attention in recent research; (2) one possible explanation for this deficiency is that current characterizations of conceptual and procedural knowledge reflect limiting assumptions about how procedures are known; and (3) reconceptualizing procedural knowledge to remedy these assumptions would have important implications for both research and practice.
Jon R. Star
Jon R. Star
The present issue of JRME features three articles—Melhuish (2018); Jamil, Larsen, and Hamre (2018); and Thanheiser (2018)—that involve, at least to some degree, replication of prior published studies. In each of these articles, the authors provide a clear rationale for the importance of the work, and in all three cases, the importance of replication of prior work is discussed. The authors of these three articles point out that the scientific community writ large values replication, and they also note that replication is quite rare in educational research generally and in mathematics education research more specifically.
Jon R. Star
In this rejoinder to Baroody and colleagues (2007), I point out that there are several areas of agreement between my position and that of Baroody et al.—most notably that both procedural knowledge and conceptual knowledge are of critical importance in students' learning of mathematics. However, there are issues on which Baroody et al. and I do not agree. In particular, I elaborate on the idea that procedures can be known deeply, flexibly, and with critical judgment—positive learning outcomes that are exclusively about students' knowledge of procedures and not necessarily a result of connections to conceptual knowledge.
Kathleen Lynch and Jon R. Star
Although policy documents promote teaching students multiple strategies for solving mathematics problems, some practitioners and researchers argue that struggling learners will be confused and overwhelmed by this instructional practice. In the current exploratory study, we explore how 6 struggling students viewed the practice of learning multiple strategies at the end of a yearlong algebra course that emphasized this practice. Interviews with these students indicated that they preferred instruction with multiple strategies to their regular instruction, often noting that it reduced their confusion. We discuss directions for future research that emerged from this work.
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.
Kristie J. Newton and Jon R. Star
This study involved a promising practice-based professional development activity called model teaching, where teachers collaboratively wrote and then enacted a lesson plan to a “class” of fellow teachers. Analysis of videos during the activity suggested that playing the role of “students” was especially effective as a way to highlight student thinking and to help teachers consider pedagogical strategies for addressing student difficulties. The activity also provided evidence of teacher learning from the professional development experience. Five teachers were followed throughout the school year, and findings suggested that although implementation varied, much of what was learned transferred to the classroom. We report on this variation and the extent of transfer, and we discuss affordances and limitations of the model teaching activity.
Jon R. Star and Kuo-Liang Chang
How Chinese Learn Mathematics: Perspectives from Insiders. Lianghuo Fan, Ngai-Ying Wong, Jinfa Cai, and Shiqi Li (Eds.). (2004). Singapore: World Scientific Publishing, 592 pp. ISBN 981-256-014-9 $88 (hb). ISBN 981-270-414-0 $52 (pb).
Jon R. Star, Beth A. Herbel-Eisenmann, and John P. Smith III
New mathematics curricula serve middle grades students well when they provide students with richer and more accessible introductions to a wide range of mathematical content. New curricula also serve teachers well when they lead us to examine and reflect on what and how we teach. When these curricula enter our working lives and conversations, we are often forced to question exactly what is “new” about them and how this “newness” may affect our students' learning. To address this issue and, we hope, to support further reflection and discussion, we take a closer and more careful look at what is new in one middle school curriculum's approach to algebra. The curriculum we examine is the Connected Mathematics Project (CMP) (Lappan et al. 1998), particularly the eighth-grade units, but the issue of what is new in algebra is relevant to many other innovative middle school curricula, as well.
Jon R. Star, Martina Kenyon, Rebecca M. Joiner, and Bethany Rittle-Johnson
Consider the following, perhaps familiar, scenario. A mathematics teacher is circulating around the classroom, looking over the shoulders of students who are busy solving linear equations such as 3x + 2 = 5x + 8. The teacher notices that one student, Paul, persists in using his own somewhat idiosyncratic and quite inefficient strategy (see fig. 1). Although Paul's strategy is not fundamentally incorrect, the extra steps required can lead to more calculation errors and wasted time.
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.