After experiencing years of procedural teaching in K-12 mathematics classrooms, many students arrive at college with ideas about, and approaches towards, mathematics that are not helpful to their learning. Students’ prior experiences and misconceptions can then negatively impact their experiences in university STEM courses. This paper describes a short course in the “big ideas” of calculus, that offered students an approach of problem-based learning, combined with mindset messages, otherwise known as a “mathematical mindset approach”. The mixed-method study considered how a ‘mathematical mindset’ teaching intervention impacted the learning, achievement, and beliefs of incoming college students, finding that the intervention significantly changed students’ ideas about mathematics, their own potential, and the value of collaboration. At the end of the course students also significantly improved their achievement on assessments of problem solving and collaboration. Importantly the course allowed students to believe in their own potential and to approach mathematics with a growth mindset, suggesting a role for such courses in students’ mathematics pathways.

In this paper we compare and contrast our experiences in using standards-based grading in different courses and across two separate institutions to explore the related tradeoffs and subtleties in designing and implementing such grading systems, guided by innovation diffusion theory. We summarize our individual courses, use Linda Nilson's criteria for evaluating grading systems as a framework for our shared reflections, and present questions for instructors to consider when designing a standards-based grading course. Despite the differences in courses, students, and institutions we had considerable overlap in the challenges we faced and the benefits we experienced.

While the nautilus shell is often represented in popular culture as an example of a golden spiral, according to many mathematicians it is not. In this paper we examine multiple arguments for and against considering the nautilus as a “golden” spiral and offer a semi-structured task that is accessible to middle school students and beyond to begin their own investigation. Our hope is that asking, what would the nautilus say, can serve as a starting point for children and adults alike to push against the walls we so often draw around mathematical questions and begin to see where their creativity takes them as they venture outside of the maths box.

Over the past ten years, the authors of this paper have implemented a series of project-based and inquiry-based learning initiatives in the calculus sequence at our institution. These initiatives took place in a department with an existing strong culture supporting teaching innovation and improvement. In this paper we describe three initiatives in detail. We describe how in one, a syllabus redesign allowed for more instructor autonomy. In another, we were able to significantly improve student buy-in. We describe key support we received from our department and institutional leadership at each stage, as well as from working in community with each other. Finally, by looking at the long arc of change in our department, we hope to encourage others who are encountering roadblocks. By working in community, we have created an environment where active learning can flourish.

Many education systems aim to support teachers in moving toward conceptually oriented mathematics instruction. The 2023 California Mathematics Framework provides one example of this effort by organizing standards around big ideas and emphasizing mathematical connections. In this study, we report on an early investigation of teachers’ learning about, and efforts to enact, the framework’s big ideas approach. We followed a cohort of elementary teachers who participated in a year-long professional learning fellowship. Drawing on multiple qualitative data sources, we examine the affordances and constraints teachers experienced as they learned to teach mathematics conceptually through big ideas, as well as the ways that the framework's conceptual structure supported their instructional decision-making. Teachers reported meaningful shifts in their teaching, which they attributed to increased student engagement, a sense of instructional freedom, and growing confidence in themselves as mathematical thinkers. Classroom observations corroborated these reported shifts by documenting teachers' enactment of key features of conceptual mathematics instruction. At the same time, teachers identified important constraints, particularly tensions between conceptual teaching and procedural district assessment systems. Across teachers, the big ideas approach provided a shared conceptual structure for organizing instruction around connected mathematical ideas while supporting teachers’ professional judgment in how those ideas were enacted in classrooms. To illustrate variation in teachers’ experiences, we present four cases representing different trajectories of learning and enactment. We conclude with implications for the design of professional learning and the use of coherent conceptual frameworks to support teachers' learning and instructional decision-making.

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