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Elise Lockwood and Branwen Purdy

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

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Craig Swinyard and Sean Larsen

The purpose of this article is to elaborate Cottrill et al.'s (1996) conceptual framework of limit, an explanatory model of how students might come to understand the limit concept. Drawing on a retrospective analysis of 2 teaching experiments, we propose 2 theoretical constructs to account for the students' success in formulating and understanding a definition of limit. The 1st construct relates to the need for students to move away from their tendency to attend first to the input variable of the function. The 2nd construct relates to the need for students to overcome the practical impossibility of completing an infinite process. Together, these 2 theoretical constructs build on Cottrill et al.'s work, resulting in a revised conceptual framework of limit.

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Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison, Michelle Cirillo, Steven L. Kramer and James Hiebert

In our March editorial (Cai et al., 2019), we discussed the nature of significant research questions in mathematics education. We asserted that the choice of a suitable theoretical framework is critical to establishing the significance of a research question. In this editorial, we continue our series on high-quality research in mathematics education by elaborating on how a well-constructed theoretical framework strengthens a research study and the reporting of research for publication. In particular, we describe how the theoretical framework provides a connecting thread that ties together all of the parts of a research report into a coherent whole. Specifically, the theoretical framework should help (a) make the case for the purpose of a study and shape the literature review; (b) justify the study design and methods; and (c) focus and guide the reporting, interpretation, and discussion of results and their implications.

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Gwyneth Hughes, Jonathan Brendefur and Michele Carney

As the focus of mathematics education moves from memorization toward reasoning and problem solving, professional development for in-service teachers must model these activities while simultaneously increasing participants' mathematical knowledge. We examine a representative task from a mathematics professional development course that uses rational number operation as an opportunity for problem solving and modeling. Transcripts exemplify the growth teachers make in deeply understanding the content–division of fractions–while engaging in guided reinvention and classroom discourse. We propose 4 interconnected qualities of this task that allow participants to engage in and reflect on the process of guided reinvention: (1) authentic context with multiple solution methods, including visual; (2) cognitive dissonance; (3) deep engagement; and (4) impact on mathematical knowledge for teaching.

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Chris Rasmussen and Karen Marrongelle

Teaching in a manner consistent with reform recommendations is a challenging and often overwhelming task. Part of this challenge involves using students' thinking and understanding as a basis for the development of mathematical ideas (cf. NCTM, 2000). The purpose of this article is to address this challenge by developing the notion of pedagogical content tool. A pedagogical content tool is a device such as a graph, diagram, equation, or verbal statement that a teacher intentionally uses to connect to student thinking while moving the mathematical agenda forward. We tender two examples of pedagogical content tools: Transformational record and generative alternative. These two pedagogical content tools are put forth as instructional counterparts to the Realistic Mathematics Education (RME) design heuristics of emergent models and guided reinvention, respectively. We illustrate the pedagogical content tools of transformational record and generative alternative by drawing on examples from two classroom teaching experiments in undergraduate differential equations.

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Frederick A. Peck

activity that were historically not possible ( Cole, 1996 ; Vygotsky, 1987 ). Learning in Mathematics as Reinvention and Objectification of Mathematical Artifacts Many artifacts are present in schools, including instructional artifacts such as

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

( Freudenthal, 1973 ; Kuster et al., 2018 ). Specifically, the IOAA curricular materials include instructional units on groups and subgroups, isomorphism, and quotient groups. Each unit includes both a reinvention phase and a deductive phase. During the

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Lara Alcock, Paul Hernandez-Martinez, Arun Godwin Patel and David Sirl

undergraduate mathematics education, such studies have often been based in small specialist classes, and reports have focused on teacher and student activity ( Johnson, Caughman, Fredericks, & Gibson, 2013 ), on guided reinvention task sequences ( Larsen, 2013