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Accountability pressures and the Common Core State Standards for Mathematics have created complex demands for educators, especially early-career teachers (ECTs). Analyzing longitudinal data, including the social networks of 119 ECTs, we find that ECTs increase their ambitious mathematics instruction when their network members positively interpret accountability pressures and curricular standards as manifest in standardized tests and evaluation. This estimated effect is net of an ECT's rich covariates, including the levels of ambitious mathematics instruction at the beginning of the academic year. It is implied that not all ECTs experience accountability pressures and curricular standards in the same way--their experiences are affected by the immediate networks in which they are embedded. Corresponding guidance for teacher educators and administrators is given.

The COVID-19-induced closure of K-12 schools significantly impacted the field experiences of students enrolled in teacher preparation programs. We addressed this ongoing challenge by adapting an early field experience model for secondary teachers that shifted online mid-semester. The University Teaching Experience model deploys a cohort of preservice secondary mathematics teachers to support instruction in an introductory university mathematics course. When the designated mathematics course moved online, the preservice teachers were able to continue their field experience by facilitating small-group discussions in virtual breakout rooms. To understand the perspectives of the stakeholders participating in the online field experience, we conducted semi-structured one-on-one interviews with the preservice teachers, the mathematics course instructor, and the university mathematics students involved in this setting. Early results indicated that the preservice teachers were highly valued by both the course instructor and the undergraduate mathematics students. Additionally, the preservice teachers appreciated the opportunity to continue their field experience, albeit in the more limited format. We present themes which emerged from preservice teacher interviews and share guidance for teacher preparation program faculty interested in trying an online early field experience while access to K-12 classrooms is limited.

Preparing students placing into developmental mathematics for success in undergraduate mathematics and preparing future teachers for the increasing demands of K-12 school settings are both persistent, yet seemingly divergent, problems facing higher education. In this paper, we shed light on a model that attempts to address aspects of each problem and share preliminary evidence of success in a multi-year application of this model. The paper highlights the benefits of implementing the model, particularly to support collaboration between faculty in mathematics and teacher education.

This paper discusses the challenges and lessons learned from conducting observations to measure the quality of classroom practice for a large-scale study of elementary teachers' mathematics instruction. Specifically, this paper shares our process for obtaining valid data for quality of elementary mathematics instruction; what we learned can inform similar efforts by large, interdisciplinary teams, as well as by instructional coaches and evaluators. Considerations for selecting and testing the viability of a classroom observation instrument, as well as the importance of conducting inter-rater reliability throughout the data collection process, are discussed.

Early career teachers (ECTs) face unprecedented pressure to fulfill expectations of 'highly effective' teachers within their first few years of teaching. Lesson planning is an important precursor to effective instruction, yet little is known about how the social and institutional contexts where ECTs work influences their planning. In this paper, we report findings from a large-scale study of elementary ECTs' mathematics lesson planning practice to shed light upon the nature of ECTs' lesson planning and factors influencing their lesson planning practice. Informed by the conceptualization of the curriculum enactment process, we present our findings of 67 ECTs' planning practices across four school districts, with distinct profiles for each district as defined by districts' official curriculum and the instructional materials available for teachers and students to use. We found that differences in the designated curriculum in each district corresponded with differences in how groups of ECTs utilize curricular objectives and instructional materials, and how they engage with colleagues during the lesson planning process. Implications for the preparation and induction of ECTs are discussed.

Problem solving has long been a focus of research and curriculum reform (Kilpatrick 1985; Lester 1994; NCTM 1989, 2000; CCSSI 2010). The importance of problem solving is not new, but the Common Core introduced the idea of making sense of problems and persevering in solving them (CCSSI 2010, p. 6) as an aspect of problem solving. Perseverance is important for solving meaningful mathematical problems, and when student thinking drives the curriculum, a lack of perseverance is a real barrier to the development of interesting and significant mathematics. This article describes the efforts of a cross-grade level team of mathematics teachers from Holt Public Schools in Holt, Michigan, to understand what kinds of systemic supports, provided across grade levels, encourage students' perseverance when problem solving. The teachers used "lesson study" (Fernandez and Yoshida 2004) as the professional development structure to support their learning, and they conducted a series of lesson studies in three mathematics classrooms across grades 7, 9, and 10. The authors share what the team learned about supporting perseverance not just within particular lessons but also throughout students' mathematics experiences from middle to high school.

Against the backdrop of mentor teachers' reasoning about practice, we seek to understand shifts in intern teachers' reasoning about practice during a year in which they take their final methods course in the fall and then do intern teaching in the spring. The data we analyze consist of intern and mentor teacher study group discussions of repeated viewings of an animation representing mathematics classroom practice around the solving of linear equations. Our analysis utilizes Toulmin's framework of argumentation and the construct of professional obligations from the practical rationality of mathematics teaching. The analysis suggests that, in reasoning about practice, in the fall, intern teachers emphasized teachers' obligations to mathematics as a discipline and to students as individuals, but as they moved into teaching, their arguments about practice include greater attention to obligations to the institutional goals of schools and to their class as a whole.

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle-school students' (ages 12-14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that "tells why" the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle-school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.

Discussions about school mathematics often address the importance of reasoning and proving for building students' understanding of mathematics. However, there is little research examining how teachers enact tasks designed to engage students in justifying and proving in the classroom. This article presents results of a study investigating the processes and outcomes of implementing proof-related tasks in the classroom. The findings suggest that students' experiences with such tasks are insufficient for developing an understanding of what constitutes valid mathematical justification. (Contains 4 figures, 5 tables and 6 footnotes.)

A math classroom is complex. Teachers navigate through a lesson using a variety of activities and tools that are designed to address students' needs and help them build conceptual understanding. Manipulatives, drawings, and other such concrete representations can help students explore mathematical ideas in physically grounded manifestations. Manipulatives can also be used to help generate remarkable change over time in students' justifications, but little is known about how explorations with manipulatives can influence their justifications. The use of manipulatives in mathematics instruction does not guarantee meaningful learning outcomes. That said, aligning various components of a lesson, including explorations with manipulatives, to support students' justifying and proving is important. This article focuses on presenting and discussing actual classroom scenarios of students who are using manipulatives to explore and justify mathematical ideas. The importance of aligning all aspects of proof-related instruction, from pattern generalization to conjecture validation, is highlighted to show that students can be prepared to generate justifications that are based on general statements. (Contains 6 figures.)

The goal of this article is to help teachers both recognize and capitalize on classroom opportunities so that students can be meaningfully engaged in mathematical proof. To do so, the authors discuss students' thoughts about proof and their abilities to construct arguments. They also offer suggestions for teachers that are intended to support their efforts to provide students with meaningful experiences with proof throughout their middle school mathematics education. (Contains 4 figures.)

The article offers information on disciplinary boundaries between literary arts and mathematics and explores mathematical argumentation as a genre of literature to help teachers and students' engagement in argumentation.

The article focuses on the use of examples as powerful tools for developing reasoning skills in a mathematics classroom. It is said that students should be exposed to reasoning and sense making from the early elementary grades. It is said that students are required to work with mathematical proof in high school, and they should be prepared for this work at middle schools.