Paul Christian Dawkins

The development and nature of problem-solving among first-semester calculus students

This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students’ problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students’ ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students’ performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students’ problem-solving performance in view of both their preparation for university calculus and the courses’ assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students’ problem-solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction.

Toward an Evolving Theory of Mathematical Practice Informing Pedagogy: What Standards for this Research Paradigm Should We Adopt?

In this chapter, we provide commentary on the four preceding chapters on proof in mathematics education. We contend that each of these chapters considers how Mathematical Practice can inform Pedagogy (MPP) research. We use these chapters to begin a discussion on what factors mathematics educators should consider when producing and evaluating MPP research. Each chapter seeks to inform mathematics education using the philosophy or history of mathematics. We argue that our field continues to borrow from these relevant fields without clear criteria for evaluating such research and without a framework for the transposition across disciplines. The chapters also all entail meta-mathematical learning goals for students and pre-service teachers. We raise questions about the exact intent of these learning goals and assessment of such learning whose answers would enhance the contributions of MPP research.

Student interpretations of axioms in planar geometry

ABSTRACT This study investigates students’ development of metamathematical understanding of axioms. Based on four semesters of experiments teaching neutral, axiomatic geometry, often through guided reinvention, I identify five categories of student interpretations of their axiomatizing activity. Similar to previously observed patterns in student interpretations of definitions and proofs, the most problematic student interpretations of axioms resulted from focusing exclusively on the referents of mathematical theory, which precluded generalization through abstraction. As a result of engaging in axiomatizing, study participants constructed several quite sophisticated views of axiomatizing compatible with various aspects of modern mathematical practice—what I call stipulated and formal interpretations.