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Kathleen Melhuish

Many studies in mathematics education research occur with a nonrepresentative sample and are never replicated. To challenge this paradigm, I designed a large-scale study evaluating student conceptions in group theory that surveyed a national, representative sample of students. By replicating questions previously used to build theory around student understanding of subgroups, cyclic groups, and isomorphism with over 800 students, I establish the utility of replication studies to (a) validate previous results, (b) establish the prevalence of various student conceptions, and (c) reexamine theoretical propositions. Data analyzed include 1 round of open-ended surveys, 2 rounds of closed-form surveys, and 30 follow-up interviews. I illustrate the potential of replication studies to refine theory and theoretical propositions in 3 ways: by offering alternate interpretations of student responses, by challenging previous pedagogical implications, and by reevaluating connections between theories

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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,

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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.

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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.

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Estrella Johnson, Christine Andrews-Larson, Karen Keene, Kathleen Melhuish, Rachel Keller and Nicholas Fortune

Our field has generally reached a consensus that active-learning approaches improve student success; however, there is a need to explore the ways that particular instructional approaches affect various student groups. We examined the relationship between gender and student learning outcomes in one context: inquiry-oriented abstract algebra. Using hierarchical linear modeling, we analyzed content assessment data from 522 students. We detected a gender performance difference (with men outperforming women) in the inquiry-oriented classes that was not present in other classes. We take the differential result between men and women to be evidence of gender inequity in our context. In response to these findings, we present avenues for future research on the gendered experiences of students in such classes.