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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.

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Creating Opportunities for Visualization of Data: Applying STEM Education Research (COViD-TASER)

Our National Science Foundation RAPID grant (DUE- 2032688) incorporates a diverse project team to investigate how people interpret media used quantitative data representations (QDRs) of COVID-19 data. Drawing on our respective areas of expertise, we also produce novel QDRs to support individuals in making data-informed decisions regarding their behavior, personal health risk, and the health risk of others. We accomplish project goals in three phases.​ In Phase I, the project team investigates a diverse population to produce differentiated models of participants’ QDR interpretations and juxtapositions of these models that reveal key conceptual categories across participants. In Phase II, the project team applies findings from Phase I and STEM education research to create research-based, project-designed QDRs while simultaneously investigating the extent these QDRs better support individuals in understanding the pandemic. In Phase III, the project team enacts an active dissemination plan in order to draw attention to project generated knowledge and products.

Quantitative and Covariational Reasoning Directory

Gathering your favorite researchers in one place

Quantitative and Covariational Reasoning (QR and CR) represent two intertwined ways of thinking. The former involves constructing and reasoning about measurable attributes, while the latter specifically refers to reasoning about how measurable attributes change in tandem. A growing body of researchers have conducted research into students', teachers', and citizens' QR and CR across numerous contexts and settings. Whether key K-16 mathematics concepts, pandemic data, STEM work, or everyday life, QR and CR play a crucial role in supporting individuals' generativity, flexibility, and learning.

In response to the proliferation of QR and CR researchers and important work in the area, the following project was launched to provide a QR and CR directory. The directory lists researchers and their contact information, and it highlights a few of their key scholarly works. 

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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? 

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GAMMA-CAT: Generalization Across Multiple Mathematical Areas-Classrooms and Teaching

National Science Foundation Award #1920538

GAMMA-CAT explores how productive mathematical generalization can be supported in whole-classroom settings. Drawing on their research expertise, the project team investigates students' classroom generalizations and the instructional, task, and pedagogical supports for fostering generalizing in the mathematical domains of algebra, advanced algebra, trigonometry, calculus, and combinatorics in Grades 6 - 16. Project results are reported through various research- and practice-based resources including papers, presentations, instructional materials, and pedagogical practices.


Generalization is a critical component of mathematical reasoning, with researchers and policymakers recommending that it be central to the education of all students at all grade levels. This recommendation poses serious challenges, however, given the research base identifying students' difficulties in creating and expressing appropriate generalizations and the challenges teachers face in supporting generalization in the classroom.

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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