Although preservice elementary school teachers (PSTs) lack the understanding of multidigit whole numbers necessary to teach in ways that empower students mathematically, little is known about their conceptions of multidigit whole numbers. The extensive research on children's understanding of multidigit whole numbers is used to explicate PSTs' conceptions of these numbers. A grounded theoretical approach leads to the development of a framework for PSTs' conceptions of multidigit whole numbers, and that framework facilitates description of their conceptions and their difficulties in the context of the standard algorithms. The framework also enables discussion of the PSTs' performance in other contexts.
Elementary prospective school teachers (PSTs) often struggle to understand why they need to relearn the mathematics that they think they already know. In this set of replication studies, I address this struggle in 3 ways: First, by increasing and varying the participant pool, I replicate Thanheiser's (2009) study, which shows that PSTs do not yet understand number in a way that enables them to teach it. Second, I introduce, validate, and examine the effect of using a survey instead of interviews, thus changing the methods of the original study. Third, I move beyond Thanheiser's original study and show that an interview designed to help PSTs assess their own knowledge accurately correlates with more sophisticated conceptions at the end of the course. Based on these findings, I posit that such an interview could be used to help PSTs learn the mathematics that they need to teach.
Kathleen Melhuish and Eva Thanheiser
As mathematics education researchers, our goal in publishing papers is to advance the field. To contribute in this manner, we must value not just novelty but also rigorous science that tests the generalizability of work in our field. This is especially important in education research, where it is impossible to have the clear, delineated, randomized studies that may exist in the hard sciences. Each study is situated in any number of contextual variables, from the particular group of students and teachers to the nature of any particular school setting. In this issue, we present two sets of replication studies (Melhuish, 2018, and Thanheiser, 2018) aiming to confirm, refute, and expand prior work. In the same issue, Schoenfeld (2018) and Star (2018) comment on these studies by raising greater questions about when replication studies are warranted in mathematics education, which studies should be published, and what exactly is meant by replication studies. We respond to the challenges posed by Schoenfeld and Star by making two points. To meet generalization goals,
Eva Thanheiser and Amanda Jansen
Engaging prospective elementary teachers (PTs) in participating productively by making their exploratory (rough draft) thinking public during class discussions remains a constant challenge for instructors of mathematics content courses for teachers, in part because of perspectives incoming PTs may hold about interacting in academic settings. In this article, we share the effects of an intervention designed to confront PTs' incoming perspectives. PTs were provided with opportunities to label the level of completeness and correctness of their thinking before they displayed and discussed their written work publicly during a mathematics content course for teachers. Results indicated that labeling their work increased PTs' level of comfort with sharing their thinking and awareness of the value of doing so. PTs also reported that the label served as a reflection tool. The label increased the PTs' productive disposition in terms of comfort level with taking intellectual risks when doing mathematics and reflecting on their work.
Jodi Fasteen, Kathleen Melhuish and Eva Thanheiser
Prior research has shown that preservice teachers (PSTs) are able to demonstrate procedural fluency with whole number rules and operations, but struggle to explain why these procedures work. Alternate bases provide a context for building conceptual understanding for overly routine rules. In this study, we analyze how PSTs are able to make sense of multiplication by 10five in base five. PSTs' mathematical activity shifted from a procedurally based concatenated digits approach to an explanation based on the structure of the place value number system.
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.
Kathleen Melhuish, Eva Thanheiser and Joshua Fagan
In classrooms, students engage in argumentation through justifying and generalizing. However, these activities can be difficult for teachers to conceptualize and therefore promote in their classrooms. In this article, we present the Student Discourse Observation Tool (SDOT) developed to support teachers in noticing and promoting student justifying and generalizing. The SDOT serves the purpose of (a) focusing teacher noticing on student argumentation during classroom observations, and (b) promoting focused discussion of student discourse in teacher professional learning communities. We provide survey data illustrating that elementary-level teachers who participated in professional development leveraging the SDOT had richer conceptions of justifying and generalizing and greater ability to characterize students' justifying and generalizing when compared with a set of control teachers. We argue that the SDOT provides both an important focusing lens for teachers and a means to concretize the abstract mathematical activities of justifying and generalizing.
Eva Thanheiser, Randolph A. Philipp, Jodi Fasteen, Krista Strand and Briana Mills
Helping prospective elementary school teachers (PSTs) recognize that they have something useful to learn from university mathematics courses remains a constant challenge. We found that an initial content interview with PSTs often led to the PSTs' changing their beliefs about mathematics and about their understanding of mathematics, leading to the recognition that (a) there is something to learn beyond procedures, (b) their own knowledge is limited and they need to know more to be able to teach, and (c) engaging in the mathematical activities in their content courses will lead them to learning important content. Thus, such an interview can set PSTs on a trajectory characterized by greater motivation to learn in their content courses.
Randolph A. Philipp, Rebecca Ambrose, Lisa L.C. Lamb, Judith T. Sowder, Bonnie P. Schappelle, Larry Sowder, Eva Thanheiser and Jennifer Chauvot
In this experimental study, prospective elementary school teachers enrolled in a mathematics course were randomly assigned to (a) concurrently learn about children's mathematical thinking by watching children on video or working directly with chil-dren, (b) concurrently visit elementary school classrooms of conveniently located or specially selected teachers, or (c) a control group. Those who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not. Furthermore, beliefs of those who observed in conveniently located classrooms underwent less change than the beliefs of those in the other groups, including those in the control group. Implications for assessing teachers' beliefs and for providing appropriate experiences for prospective teachers are discussed.