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One of the challenges of university mathematics courses in secondary teacher preparation is incorporating pedagogical discussions. The focus in a mathematics course is--and should be--on mathematics. But research also suggests that without addressing pedagogical implications these content courses are not meaningful to secondary teachers' future classroom practice. The thrust of this paper is exploring ideas for how to leverage mathematical practice in university mathematics courses--and, in particular, what have been described as Pedagogical Mathematical Practices (PMPs). The paper reports on a study of (n = 10) pre- and in-service mathematics teachers that explored the viability of the PMP construct, with the intent of specifying particular PMPs. Drawing on interviews with teacher participants who had recent experiences in an inquiry-oriented discrete mathematics course, the study reports on the ways in which they identified a set of mathematical practices as being productive pedagogically. The study contributes a teacher-perspective on the construct of PMPs, including the identification of four PMPs from the study data: explicit visualization; multiple approaches; concrete exemplification; and informal justification. Implications for their potential use in university mathematics courses with regard to teacher education are discussed.

Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers--by which we mean teachers of students with ages, approximately, of 12-18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.

This paper provides an empirical exploration of mathematics teachers' planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman's (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed.

Shulman (1986) conceptualized pedagogical content knowledge (PCK) at the intersection of content knowledge and pedagogical knowledge. In this paper, I re-explore the intersection of content--in our case, mathematics--and pedagogy, not in terms of domains of knowledge, but in terms of domains of practice. I conceptualize, describe, and exemplify what ideas at the intersection of mathematical practices and pedagogical practices might mean--which I refer to as pedagogical mathematical practices (PMPs)--and then explore what the potential implications for such a construct might be in teacher education, and how it might be useful for, and used by, the field.

This article explores secondary teachers' opportunities to learn from an innovative real analysis course, as reflected in their actual classroom teaching. The course used cases of teaching as a site for applying mathematics and developing pedagogical mathematical practices. This article explores particular teaching moments in (N = 6) secondary teachers' classrooms, and the attributions they gave for why they engaged in those teaching practices. Teachers engaged in instructional practices that exemplified course objectives, and their attributions for their actions contribute a teacher perspective on opportunities to learn in teacher education from (advanced) mathematical coursework. Results highlight cases of teaching and modeled instruction as catalysts of change and as opportunities to develop pedagogy from mathematical activity, and vice versa.

Empirical work that connects content knowledge to teaching practice has become increasingly important in discussions around a professional knowledge base for teaching. This paper aims to continue that work, reporting a single exploratory case study of one elementary mathematics teacher. The purpose of the study was to investigate how introducing teachers to advanced mathematics, particularly ideas in abstract algebra, might inform and influence their instructional practices for teaching early algebra topics about arithmetic structure. The findings indicate that instructional practices did change, especially in ways that more explicitly incorporated structure in early algebra with students, and fostered increased reasoning and sense-making about these ideas. Implications for mathematics teacher education are discussed.

Most prospective secondary mathematics teachers in the United States complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically-motivated instructional model to design modules for a real analysis course that could inform secondary teachers' actionable content knowledge and pedagogy. The theoretical model and designed curriculum launches the study of advanced mathematics content, in this case, real analysis, via authentic 7-12 classroom situations, abstracts the secondary mathematics, and uses that content to motivate the presentation of the advanced mathematics content. Subsequently, the curriculum then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. This study evaluated the success of the curriculum in terms of the students' proficiency with real analysis, challenging secondary mathematics content, and the use of that content in teaching situations via both written post-tests and interviews and showed the viability of the model and experimental curriculum.

Most prospective secondary mathematics teachers complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically motivated instructional model to design modules for a real analysis course that could inform secondary teachers' pedagogy, focusing on how this model was implemented in a single module about "attending to scope." The central aim is to document how teachers' experience in this real analysis course influenced their subsequent teaching. In the majority of lessons we observed from six participating teachers, the following occurred: (i) the teacher explicitly attended to the scope of statements discussed in class; and (ii) in post-class interviews, the teacher highlighted their deliberate attention to scope and attributed this behavior to their experience in the experimental real analysis course. We discuss the design approach, its relation to teachers' actions in the classroom, and potential implications for teacher education in advanced content courses.

Combination problems are a cornerstone of combinatorics courses, but little research has been done examining the ways that students perceive and differentiate among different combination problems. In this article, we investigate how mathematics education students (n = 18) in a discrete mathematics course view two categorically different combination problems (Category I and II combination problems). In particular, we look at how participants conceptualized each problem's sets of outcomes, counting processes, and formulas, while also exploring the means by which students justified their relationships. Review of the data collected suggests that students tend to be less consistent and have more trouble utilizing and justifying combinations with a collection of ordered indistinguishable objects (Category II) than they do with a collection of unordered distinguishable objects (Category I). Based on these findings, we provide recommendations for the teaching and learning of combinations in combinatorics education.

The aim of this collaborative study was to understand what factors beginning secondary mathematics teachers attribute their success to in the classroom, regardless of their preparation program. Further description of how and when beginning teachers reported acquiring important teaching attributes provides a perspective on how they make the transition to teaching. A large-grain analysis of critical developmental moments, pre-, during, or post-program, contributes to the conversation about teacher education, highlighting valuable aspects of the preparation process for beginning teachers. The results have implications for informing the types of students mathematics education programs should try to attract or recruit, and defining areas on which teacher education programs should focus and where practicum or internship components might be incorporated into the preparation process.

In this paper we report on a survey designed to test whether or not students differentiated between two different types of problems involving combinations--problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count ordered sequences of two (or more) indistinguishable objects (a less obvious application of combinations). We hypothesized that novice students may recognize combinations as appropriate for the first type but not for the second type, and our results support this hypothesis. We briefly discuss the mathematics, share the results, and offer implications and directions for future research.

Research-based guidelines for learning variation exist (e.g., Franklin et al., 2007; Garfield, delMas, & Chance, 2007), but little is known about how teachers plan to teach standard deviation, or how these plans align with recent recommendations. In this article, we survey lesson plans designed by inservice and preservice secondary mathematical teachers. We report on the accuracy, technology usage, and visual representations in the lesson plans. We consider how many elements are used, the level of conceptual development, and the mathematical nature. Findings support differences between preservice and master's level students in education, as well as a tendency by in-service teachers to teach in alignment with prior learning experiences, despite professional development. Implications for teacher education and curricular development are offered.