Recent Papers

Fonger, N. L., Stephens, A., Blanton, M., Isler, I., Knuth, E., Gardiner, A. (2018). Developing a learning progression for curriculum, instruction, and student learning: An example from mathematics education. Cognition and Instruction, 36(1), 30-55.

Abstract. Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as deter- mined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curricu- lum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodol- ogy for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and devel- opment, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypoth- esized curricular framework and progression and to measure change in stu- dents. thinking over time as a means to validate both the e ectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning pro- gressions in mathematics education with a focus on the concept of mathemat- ical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning pro- gressions work in mathematics education research and practice.

Fonger, N. L. (2018). A design-based research partnership to support students’ coordination of computer algebra systems and paper-and-pencil. International Journal for Technology in Mathematics EducationPDF 2018_Fonger_IJTME_CAS Activity Structure

Abstract. In this study I examine the issue of how to support students’ coordination of computer algebra systems (CAS) and paper- and-pencil in solving mathematics problems. Together with a classroom teacher I designed and conducted a collaborative teaching experiment in a ninth-grade algebra classroom to understand how to support students’ coordination of tools. I introduce a predict-act-reflect-reconcile activity structure for coordinating CAS and paper-and-pencil in solving problems involving linear expressions and equations and adopt this as an analytic lens to examine the role of curriculum materials in impacting students’ learning. Findings suggest the activity structure of predict-act-reflect-reconcile is a potentially fruitful starting point for coordinating CAS and paper-and- pencil in classroom activity, but not sufficient on its own as a task design principle without focused classroom discussion and support during the enacted curriculum. This research points to the importance of teaching students how to communicate about their process of using multiple tools while solving problems, especially the mathematical thinking that students engage in while reconciling differences across tool- based representations.

Fonger, N. L., Davis, J., Rohwer, M. L. (2018). Instructional supports for representational fluency in solving equations with computer algebra systems and paper-and-pencil. School Science and Mathematics, 118(30), 30-42. doi:10.1111/ssm.12256 PDF available here.

Abstract. This research addresses the issue of how to support students’ representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students’ representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students’ change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students’ persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.


Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M., Knuth, E., Gardiner, A. M. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143-166. PDF from:

Abstract. In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detail- ing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive pattern- ing to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.

Fonger, N. L., Reiten, L., Strachota, S., Ozgur, Z. (2017). Engaging in research: Why? How? Now! Mathematics Teacher, 110(6), 462-465. JSTOR ResearchGate

Introduction. Why should teachers engage in research studies? As a community, teachers and researchers are concerned with addressing critical issues in math education. NCTM’s web resources and conferences, as well as the pages of this journal, give evidence of a growing community and an expand- ing body of work supporting NCTM’s (2012) position of linking research and practice—a “border crossing” between the world of research and the world of teaching (Silver 2003). Despite these initiatives, an emerging issue remains: How do we work together to cultivate a two-way exchange of professional knowledge (Heid et al. 2006)? Based on our experiences as math educators, we address the following questions: Why should teachers engage in research? What might teachers’ roles be in a research project? How do teachers get involved?

Davis, J. D., & Fonger, N. L. (2015). An analytical framework for categorizing the use of CAS symbolic manipulation in textbooks. Educational Studies in Mathematics. ResearchGate

Abstract The symbolic manipulation capabilities of computer algebra systems, which we refer to as CAS-S, are now becoming instantiated within secondary mathematics textbooks in the United States for the first time. While a number of research studies have examined how teachers use this technology in their classrooms, one of the most important factors in how this technology is used in the classroom is how it is embedded within curricular resources such as textbooks. This study introduces readers to an analytical framework for examining CAS-S within textbooks and presents the results of its application to three secondary U.S. mathematics textbook units involving polynomial functions. The framework consists of two components: application of CAS-S and reflection on CAS-S uses. The analyses identified differences among the three textbook units in pedagogical intent and task connectedness involving CAS-S. The majority of CAS-S tasks were coded as involving low procedural complexity and there were few instances in which the technology was used in the construction of proofs. Textbook developers asked students to reflect on the visible CAS-S result as opposed to the invisible process leading to those results. The implications of these results as well as the potential transformative role of CAS-S are discussed.

Fonger, N. L. (2014). Equivalent expressions using CAS and paper-and-pencil techniques. Mathematics Teacher, 107(9), 688-693. JSTOR ResearchGate


Fonger, N. L. (2012). Shed new light on student thinking with a representational lens. Consortium: The newsletter of the consortium for mathematics and its applications, 102, 1-6. ResearchGate


Fonger, N. L. (2011). Lessons learned as a novice researcher: The case of a pilot study in mathematics education. The Hilltop Review, 4(2), 55-62. Retrieved October 25, 2011, from ResearchGate


Hedican, E. B., Kemper, J. T., & Lanie, N. M. (2007). Modeling biomarker dynamics with implications for the treatment of prostate cancer. Computational and Mathematical Methods in Medicine, 8(2), 77-92.