Samuel Otten

Professional development (PD) for mathematics teachers often emphasizes transformative instructional change. However, a more modest, incremental approach may offer a higher likelihood of success in ways that complement transformational efforts. This Editorial discusses the potential advantages of incremental PD where teachers make small but meaningful improvements to their practice over time. We explore the differences between transformational PD and incremental PD as evidenced by the articles in this Special Issue. We also distinguish between varieties of incremental PD projects, from those that focus on a small piece of a larger instructional vision (piece-and-support incremental PD), those that set out to achieve a modest goal without appeal to a broader instructional transformation (modest-goal-with-support incremental PD), to a more radically modest approach that involves offering small, optional instructional suggestions without investments of time or training (nudge-based incremental PD). Ultimately, this Editorial, and this Special Issue as a whole, advocates for new conversations and investigations into how PD is structured and how the promoted instructional changes might spread among mathematics teachers.

For thousands of years, there was a gradual increase of carbon in Earth’s atmosphere. Students can model the historical data using linear functions. Then they can learn about climate change in the modern era by using a combination of exponential and periodic functions to explore carbon’s multi-faceted variation, distinguishing the recent trend (exponential) from the seasonal variation (trigonometric).

Broadly speaking, school mathematics has been placed in a double bind. From state policymakers, there is the culmination of decades of increasing requirements to teach more mathematics, which now often means a requirement of four full years of mathematics in high school or every student progressing through at least a second course in algebra (Remillard et al., 2017). From the mathematics education community, there is growing consensus around the call to ban tracking (e.g., National Council of Teachers of Mathematics, 2018; National Council of Supervisors of Mathematics & TODOS, 2016) because of the known harms of identifying and segregating students based on mathematical “ability.” But tracking is one of the primary strategies that schools (and communities) conceive of when trying to meet the demand for all students to complete more mathematics courses.

Reasoning-and-proving is a crucial part of students' mathematical experiences in secondary school. There is scholarly debate, however, on the extent to which proving at the secondary level needs to be formal and whether all students should be held to disciplinary standards of rigor. In this study, we investigated the notion of "proof for all" from the perspective of secondary mathematics teachers. We analyzed, using the framework of practical rationality, the justifications teachers gave for whether or not all students should learn proof. Based on interviews with twenty-one secondary teachers from a socioeconomically-diverse set of schools, we found that teachers differ in their opinions of who should learn proving but they were similar in their feelings of obligation toward individual student learning; some teachers cited obligations to individual students as a justification for teaching proving to all students and others cited those obligations as a justification for not teaching proving to some students. We also share teachers' perspective with regard to their obligations to the discipline, educational institutions, interpersonal dynamics among students, and the worldly relevance of mathematics education.

Teachers are implementing flipped instruction in an increasing number of mathematics classes but the research base is not yet well developed on this topic. Many studies of flipped instruction in mathematics have involved a small number of classes utilizing flipped instruction being compared to classes with non-flipped instruction, but this study design fails to account for variations in how teachers implement flipped instruction. There seems to be as much variation within flipped implementations and within non-flipped implementations as there are between the two categories. To aid systematic attention to variation, this article presents a framework for flipped mathematics lessons that identifies key features of the videos assigned as homework as well as features of the in-class time with students. The components of the framework allow for structurally similar implementations to be grouped together meaningfully whereas different implementations, though still under the banner of ‘flipped instruction’, can be distinguished from one another. The framework is illustrated with data from flipped mathematics classes at secondary and post-secondary levels.