To determine what was taught requires investigating what was learned. 

"What radical constructivism may suggest to educators is this: the art of teaching has little to do with the traffic of knowledge, its fundamental purpose must be to foster the art of learning."

- Ernst von Glasersfeld

Picture a traditional textbook, drawn up by a mathematician in her or his office as a way to sort out and present her or his understandings to another. Such a textbook can provide beautiful expositions of mathematical ideas, but it is typically absent of something critical: an image of the learner. Teaching and curricula often emphasize an adult's mathematics, a mathematics as already learned.


Drawing from the radical constructivist movement in mathematics education, I take a scientific-inquiry approach to developing models of students' mathematical learning. My specific interest is understanding how students' quantitative and covariational reasoning is a generative foundation for their learning precalculus and calculus concepts. Through better understanding students' mathematical learning, I work to construct epistemic learners and ultimately design instructional experiences that foreground their mathematical experiences.

What does it mean for a student to understand a concept? 

What do we take as evidence of a student understanding a concept?

What instructional features support students in constructing concepts?




To understand something requires an attempt to change it.


A great project inspires those involved to learn more. 

The quality of our work is defined by what students learn.

CAREER: Advancing Secondary Mathematics Teachers' Quantitative Reasoning

Advancing Reasoning addresses the lack of materials for teacher education by investigating pre-service secondary mathematics teachers' quantitative reasoning in the context of secondary mathematics concepts including function and algebra. The project extends prior research in quantitative reasoning to develop differentiated instructional experiences and curriculum that support prospective teachers' quantitative reasoning and produce shifts in their knowledge. Three interrelated research questions guide the project: (i) What aspects of quantitative reasoning provide support for prospective teachers' understanding of major secondary mathematics concepts such as function and algebra? (ii) How can instruction support prospective teachers' quantitative reasoning in the context of the teaching and learning of major secondary mathematics concepts such as function and algebra? (iii) How do the understandings prospective teachers hold upon entering a pre-service program support or inhibit their quantitative reasoning? 

GAMMA: Generalization Across Multiple Mathematical Areas

National Science Foundation Award #1419973

The recommendation to make generalization a central component of mathematics instruction from elementary school through undergraduate mathematics poses serious challenges in light of the research base that identifies students' difficulties in creating and expressing correct mathematical generalizations and the challenges teachers face in supporting students' abilities to generalize. Furthermore, although student difficulties are well documented, the instructional conditions necessary for fostering generalization are not well understood, particularly at the secondary and undergraduate levels. This project addresses these challenges by developing a comprehensive framework characterizing productive mathematical generalization in Grades 8-16 and identifying instructional interventions that can support correct generalizing. The project occurs within multiple mathematical domains extending from middle school to undergraduate mathematics, including algebra, geometry, calculus, and combinatorics. The project investigators will leverage student interviews, teaching experiments, and design experiment methodologies in order to characterize the processes of generalizing and to identify the instructional conditions that support productive generalization. The results of the project will identify specific tasks and activities fostering student generalizing in a diversity of mathematical settings, which will be of practical use to teachers, school districts, teacher educators, and university instructors.

Rational Reasoning: Research-Based Education

Curricula including Precalculus: Pathways to Calculus

Our Materials are research developed and refined to impel student success & learning. Our support staff and professional development team will assure your successful implementation of Pathways curriculum.

Pathways materials are fully developed for beginning algebra through precalculus mathematics. Course content can be customized to your syllabus. Aligned calculus materials are currently under development.

Our materials and professional development are based in 25 years of sustained inquiry into student learning of precalculus and beginning calculus. We continually adapt our materials based on data and input from our users.


The Pathways Precalculus curriculum consists of 12 modules. Each module contains 6 to 12 in-class investigations that include problems and prompts that engage students in making meaning, building connections and understanding ideas. Students learn general approaches for solving problems that enable them to emerge as competent and confident problem solvers.

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