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Pathways Mathematics

Research Related to Pathways Courses

Pathways is a research-based and research-proven mathematics curriculum and professional development project. Our goal is to support instructors in facilitating engaging, coherent, student-centered lessons that promote deep learning of critical mathematical ideas.

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Pathways Research

Pathways research has been funded by two NSF grants as well as ongoing support from curriculum adoptions. The Pathways program is built on decades of research on students’ mathematical learning, both in general and related to specific ideas proven critical for success in calculus and STEM fields. The Pathways research team has contributed novel research in areas such as quantitative and covariational reasoning, teacher change, professional development, curriculum development, scaling of curricular innovations, and many more.

Projects Pathways is based on two branches of mathematics education research. In recent decades, cognitive learning researchers have helped mathematics educators understand the ways in which students learn mathematical ideas as well as the implications of understanding key mathematical ideas in certain ways on future learning. Furthermore, researchers have made great strides in understanding the kinds of mathematical content knowledge for teaching that best supports effective instruction and how teachers interacting with a conceptually-oriented curriculum come to develop this special kind of content knowledge. The Pathways course materials, instructor supports, and professional development training models are all designed to leverage key findings in these research areas as well as to contribute to the growing body of literature by generating new insights about teaching and learning mathematics.

Pathways course materials are continuously revised and strengthened based on qualitative and quantitative research on student performance and have been in continual use for over 10 years in high schools and universities across the United States. Evidence suggests that students in Pathways courses are more successful than students using other curricula and are better prepared for Calculus.

Background Research

The Pathways program is built on decades of research related to general issues of students’ mathematical learning, understanding the ideas most critical for students’ success in calculus and STEM fields, and related to specific mathematical ideas such as rate of change, graphing, and algebraic reasoning. What follows is a short list of SOME of the relevant literature that inspired the Pathways project.

  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education III (pp. 114-162). Providence, RI: American Mathematical Society.
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33, 352–378.
  • Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating a Models and Modeling Perspective with Existing Research and Practice. In R. Lesh & H. Doerr (Eds.), Beyond constructivism in mathematics teaching and learning: A models & modeling perspective (pp. 465-478). Hillsdale, NJ: Lawrence Erlbaum.
  • Carlson, M. & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem solving framework, Educational Studies in Mathematics, 58, 45-75.
  • Carlson, M., Oehrtman, M., Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: A tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. Cognition and Instruction, 113-145.
  • Oehrtman, M., Carlson, M., & Thompson, P.W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 27-42). Washington, DC: Mathematical Association of America.
  • Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.), Proceedings of the Annual Meeting of the Psychology of Mathematics Education – North America. Raleigh, NC: North Carolina State University.
  • Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.
  • Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 189-243). Hillsdale, NJ: Erlbaum.
  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179-234). Albany, NY: SUNY Press.
  • Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing and Modeling In Mathematics Education. Dordrecth, The Netherlands: Kluwer.
  • Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education WISDOMe Monographs (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming Press.
  • Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education, pp. 57-93. New York: Springer.
  • Thompson, P.W., & Carlson, M. P. (2017). Variation, covariaton, and functions: Foundational ways of thinking mathematically. In J. Cai (ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics.
  • Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In A. Coxford (Ed.), 1994 Yearbook of the NCTM (pp. 79-92). Reston, VA: NCTM.
  • Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually, Part I: A teacher’s struggle. Journal for Research in Mathematics Education, 25(3), 279-303.

Pathways Publications

Pathways researchers have contributed to the field of mathematics education in many areas, including scaling curricular innovations, professional development, teacher change, and quantitative and covariational reasoning. The following papers and book chapters emerged from research conducted in the context of the Pathways project.

  • Carlson, M. P., Bas-Ader, S., O’Bryan, A. E., & Rocha, A. (2024). The Construct of Decentering in Research on Student Learning and Teaching. In Piaget’s Genetic Epistemology in and for Ongoing Mathematics Education Research. Dawkins, P. C., Hackenberg, A. J., & Norton, A. (Eds.), Berlin: Springer.
  • Carlson, Marilyn P., Moore, Kevin (2015). The Role of Covariational Reasoning in Understanding and Using the Function Concept. In E. Silver, & P. Keeney (Eds.) Lessons Learned From Research. The National Council of Teachers of Mathematics, pp. 279-291.
  • Carlson, M. P., O’Bryan, A. E., & Rocha, A. (2022). Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships. In Quantitative Reasoning in Mathematics and Science Education. Karagöz Akar, G., Özgür Zembat, I., Arslan, S., & Thompson, P. W. (Eds.), Berlin: Springer.
  • Carlson, M. P., O’Bryan, A. E., Strayer, J. F., McNicholl, T. H., & Hagman, J. E. (2024). Considering, piloting, scaling and sustaining a research-based precalculus curriculum and professional development innovation. The Journal of Mathematical Behavior, 73, pp. 101-126.
  • Clark, P., Moore, K. & Carlson, M. (2008). Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. The Journal of Mathematical Behavior. 27(4), 297-310.
  • Madison, B. L., Carlson, M. P., Oehrtman, M., & Tallman, M. (2015). Conceptual precalculus: Strengthening students’ quantitative and covariational reasoning. Mathematics Teacher, 109(1), 54-59.
  • Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. Journal of Mathematical Behavior, 31(1), 48-59.
  • Musgrave, S., & Carlson, M. P. (2016). Understanding and advancing graduate teaching assistants’ mathematical knowledge for teaching. The Journal of Mathematical Behavior, Volume 45, March 2017, pp. 137-149.
  • O’Bryan, A. E. (2020, Spring). You Can’t Use What You Don’t See: Quantitative Reasoning in Applied Contexts. OnCore: Journal of the Arizona Association of Teachers of Mathematics, pp. 66-74.
  • Oehrtman, M., Carlson, M., & Vasquez, J. A. (2009). Attributes of content-focused professional learning communities that lead to meaningful reflection and collaboration among math and science teachers. In S. Mundry & K. E. Stiles (Eds.), Professional Learning Communities for Science Teaching: Lessons from Research and Practice National Science Teachers Association Publication Series. Arlington, VA: NSTA Press, pp. 89-106.
  • Tallman, M., Carlson, M. P., Bressoud, D., & Pearson, M. (2016). A characterization of calculus I final exams in U.S. colleges and universities. International Journal of Research in Undergraduate Mathematics Education, 2(1), 105-133.
  • Teuscher, D., Moore, K., & Carlson, M. (2015). Decentering: A construct to analyze and explain teacher actions as they relate to student thinking. Journal of Mathematics Teacher Education, 1-24. doi: 10.1007/s10857-015-9304-0
  • Thompson, P.W., Carlson, M.P., Silverman, J. (2007). The design of tasks in support of teachers’ development of coherent mathematical meanings. Journal of Mathematics Teachers Education, 10, 415-432.

Dissertation Studies

Pathways has supported the work of many graduate students since 2006, and their research has directly contributed to the ongoing cycle of research, reflection, and modification that drives improvements in the curricula.

  • Bloom, I. (2008). Promoting and characterizing the problem solving behaviors of prospective high school mathematics teachers. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Bowling, S. (2014). Conceptions of function composition in college precalculus students. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Cox III, F. E. (2005). Secondary precalculus professional learning communities: A structure for teacher development. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Flores, E. G. K. (2018). Sparky the Saguaro: Teaching experiments examining students’ development of the idea of logarithm. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Frank, Kristin M. (2017). Examining the development of students’ covariational reasoning in the context of graphing. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Infante, N. M. E. (2007). Students’ understanding of related rates problems in calculus. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Jacobs, S. (2002). Advanced Placement BC calculus students’ ways of thinking about variable. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Larsen, S. (2004). Supporting the guided reinvention of the concepts of group and isomorphism: A developmental research project. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Lock, K. (2023). Investigating the Role of Relative Size Reasoning in Students’ Understanding of Precalculus Ideas Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Marfai, Frank S.. (2017). Characterizing teacher change through the perturbation of pedagogical goals. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Moore, Kevin (2010). The Role of quantitative reasoning in precalculus students’ learning central Concepts of Trigonometry. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • O’Bryan, Alan E. (2018). Exponential growth and online learning environments: Designing for and studying the development of student meanings in online courses. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Rocha, A. (2023). An Investigation into the Relationships Among Teachers’ Mathematical Meanings for Teaching, Commitment to Quantitative Reasoning, and Actions. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Smith, N. N. (2008). Student’s emergent conceptions of the Fundamental Theorem of Calculus. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Strom, April (2008). A case study of a secondary mathematics teacher’s understanding of exponential function: An emerging theoretical framework.. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.
  • Tallman, Michael (2015). An examination of the effect of a secondary teacher’s image of instructional constraints on his enacted subject matter knowledge. Unpublished Ph.D. dissertation, School of Mathematical and Statistical Sciences, Arizona State University.

Conference Proceedings

Pathways researchers have been very active over the years in sharing our work and findings with the mathematics education community. The following is the list of all relevant conference proceedings sharing work from the Pathways research team.

  • Carlson, M., Bowling, S., Moore, K., & Ortiz, A. (2007). The Emergence of Decentering in Facilitators of Professional Learning Communities of Secondary Mathematics and Science Teachers. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 841-848). Reno: University of Nevada.
  • Carlson, M. P., Moore, K. C., Teuscher, D., Slemmer, G., Underwood, K., & Tallman, M. (2012). Affecting and Documenting Shifts in Secondary Precalculus Teachers’ Instructional Effectiveness and Students’ Learning. Proceedings of the 2012 Math and Science (MSP) Learning Network Conference (LNC). Washington, D.C.: National Science Foundation.
  • Carlson, M., Oehrtman, M., & Teuscher, D. (2010, January). Transforming the professional development culture and quality of mathematics and science instruction within a secondary school. Proceedings of the Eighth Annual Math and Science Partnership (MSP) Learning Network Conference (LNC). Washington, D.C, web publication.
  • Carlson, M. P., Slemmer, G., Moore, K., Teuscher, D., and Joyner, K. (2011). Key Variables for Establishing and Sustaining Highly Effective Professional Learning Communities. Proceedings of the 2011 Math and Science (MSP) Learning Network Conference (LNC). Washington, D.C.: National Science Foundation, web publication.
  • Clark, P. G., Carlson, M., & Moore, K. (2007). Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 872-874). Reno: University of Nevada.
  • Engelke, N. (2008). Developing the solution process for related rates problems using computer simulations. In Conference on Research in Undergraduate Mathematics Education.
  • Krause, S., Culbertson, R., Carlson, M. & Oehrtman, M. (2008). High school teacher change, strategies, and actions in a professional development project connecting mathematics, science, and engineering. Proceedings of the 38th ASEE/IEEE Frontiers in Education Conference. Saratoga Springs, NY.
  • McClain, K., Carlson, M., Coe, E., & Saldanha, L. (2009). The emergence of norms for mathematical argumentation: Contributing to a framework for action. Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 288-295). Atlanta, GA: Georgia State University.
  • Moore, K. C., Carlson, M. P., Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. Proceedings for the Twelfth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education Conference. Raleigh, NC: North Carolina State University. http://rume.org/crume2009/Moore1_LONG.pdf
  • Moore, K. C., Carlson, M. P., and Teuscher, D. (2011). Using Research-based Curriculum to Support Shifts in Teachers’ Key Pedagogical Understandings. Proceedings of the 2011 Math and Science Partnership (MSP) Learning Network Conference (LNC). Washington, D.C.: National Science Foundation, web publication.
  • Moore, K. C., Teuscher, D., and Carlson, M. P. (2011). Exploring shifts in a teacher’s key developmental understandings and pedagogical actions. In Wiest, L. R., & Lamberg, T. (Eds.), Proceedings of the 33rd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1673-1681). Reno, NV: University of Nevada, Reno.
  • Musgrave, S., & Carlson, M. P. (2016). Transforming graduate students’ meanings for average rate of change. In T. Fukawa-Connolly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the Nineteenth Annual Conference on Research in Undergraduate Mathematics Education. Pittsburgh, PA: West Virginia University, pp. 809-814.
  • O’Bryan, A. E. (2020). A research-based approach to developing, refining, and assessing student learning in an online precalculus course. Proceedings of the 14th Annual International Technology, Education and Development Conference, Valencia, Spain.
  • O’Bryan, A. E. & Carlson, M. P. (2016). Fostering teacher change through increased noticing: Creating authentic opportunities for teachers to reflect on student thinking. In T. Fukawa-Connolly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the Nineteenth Annual Conference on Research in Undergraduate Mathematics Education. Pittsburgh, PA: West Virginia University, pp. 1192-1200.
  • Oehrtman, M., Carlson, M., Martin, J., & Sutor, J. (2010, January). Coherence and change in teacher professional learning communities. Proceedings of the Eighth Annual Math and Science Partnership (MSP) Learning Network Conference (LNC). Washington, D.C, web publication.
  • Oehrtman, M., Carlson, M., Sutor, J., Agoune, L. & Stroud, C. (2009). Meaningful Collaboration in Secondary Mathematics and Science Teacher Professional Learning Communities, Proceedings of the Twelfth Conference on Research in Undergraduate Mathematics Education, 28 pages, Web publication at http://rume.org/crume2009/Oehrtman_LONG.pdf.
  • Rasmussen, C., Bressoud, D., Carlson, M. (2015). Who Are the Students that Switch out of Calculus and Why Do They Switch? American Education Research Association (AERA) Conference Proceedings, pp. 149-156.
  • Sander, G. & Carlson, M. P. (2016). On the use of dynamic animations to support students in reasoning quantitatively. In T. Fukawa-Connolly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the Nineteenth Annual Conference on Research in Undergraduate Mathematics Education. Pittsburgh, PA: West Virginia University, pp. 1262-1270
  • Teuscher, D., Moore, K. C., and Carlson, M. P. (2011). Interaction Between Teacher’s Questions and Student Discourse. Proceedings of the 2011 Math and Science Partnership (MSP) Learning Network Conference (LNC). Washington, D.C.: National Science Foundation, web publication.
  • Underwood, K. & Carlson, M. P. (2012). Understanding how precalculus teachers develop mathematic knowledge for teaching the idea of rate of change. In Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education. Portland, OR. (149-157).