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 15 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 Dubinsky, Schoenfeld, & Kaput (Eds.), Research in Collegiate Mathematics Education III (pp. 114–162). AMS.
- 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 Lesh & Doerr (Eds.), Beyond Constructivism in Mathematics Teaching and Learning (pp. 465–478). 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. Cognition and Instruction, 113–145.
- Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In Carlson & Rasmussen (Eds.), Making the Connection (pp. 27–42). MAA.
- Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In Berenson & Coulombe (Eds.), PME-NA Proceedings. NCSU.
- Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In Kaput, Carraher & Blanton (Eds.), Algebra in the Early Grades (pp. 95–132). Erlbaum.
- Thompson, P. W. (1985). Experience, problem solving, and learning mathematics. In Silver (Ed.), Teaching and Learning Mathematical Problem Solving (pp. 189–243). Erlbaum.
- Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In Harel & Confrey (Eds.), The Development of Multiplicative Reasoning in Mathematics Learning (pp. 179–234). SUNY Press.
- Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In Gravemeijer et al. (Eds.), Symbolizing and Modeling. Kluwer.
- Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In Hatfield, Chamberlain & Belbase (Eds.), New Perspectives for Collaborative Research (pp. 33–57). U. Wyoming Press.
- Thompson, P. W. (2013). In the absence of meaning. In Leatham (Ed.), Vital Directions for Research in Mathematics Education (pp. 57–93). Springer.
- Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In Cai (Ed.), Compendium for Research in Mathematics Education (pp. 421–456). NCTM.
- Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In Coxford (Ed.), 1994 NCTM Yearbook (pp. 79–92). 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.
- Baş-Ader, S., & Carlson, M. P. (2022). Decentering framework: A characterization of graduate student instructors’ actions to understand and act on student thinking. Mathematical Thinking and Learning, 24(2), 99-122.
- 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, Hackenberg & Norton, Eds.). Springer.
- Carlson, M. P., & Moore, K. (2015). The Role of Covariational Reasoning in Understanding and Using the Function Concept. In E. Silver & P. Keeney (Eds.), Lessons Learned From Research (pp. 279-291). NCTM.
- 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, Özgür Zembat, Arslan & Thompson, Eds.). 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. Journal of Mathematical Behavior, 73, 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. Journal of Mathematical Behavior, 27(4), 297-310.
- Hower, J., Merkin, R., & Wells, L. (2023). A Faculty learning community implements research-based curriculum and pedagogy to redesign precalculus. PRIMUS, 33(5), 463-486.
- Kuper, E., & Carlson, M. (2020). Foundational ways of thinking for understanding the idea of logarithm. Journal of Mathematical Behavior, 57, 100740.
- 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.
- McNicholl, T. H., Frank, K., Hogenson, K., Roat, J., & Carlson, M. P. (2021). Improving student success and supporting student meaning-making in large-lecture precalculus classes. PRIMUS, 31(7), 792-810.
- 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. Journal of Mathematical Behavior, 45, 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, 66-74.
- O’Meara, J., Carlson, M. P., O’Bryan, A. E., & Vaidya, A. (2025). Network-Based Trajectory Analysis of a Precalculus Course. PRIMUS, 1-26.
- 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 Mundry & Stiles (Eds.), Professional Learning Communities for Science Teaching (pp. 89-106). NSTA Press.
- 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. 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 Teacher 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, Arizona State University.
- Bowling, S. (2014). Conceptions of function composition in college precalculus students. Unpublished Ph.D. dissertation, Arizona State University.
- Cox III, F. E. (2005). Secondary precalculus professional learning communities: A structure for teacher development. Unpublished Ph.D. dissertation, 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, Arizona State University.
- Frank, K. M. (2017). Examining the development of students’ covariational reasoning in the context of graphing. Unpublished Ph. D. dissertation, Arizona State University.
- Infante, N. M. E. (2007). Students’ understanding of related rates problems in calculus. Unpublished Ph.D. dissertation, Arizona State University.
- Jacobs, S. (2002). Advanced Placement BC calculus students’ ways of thinking about variable. Unpublished Ph.D. dissertation, 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, Arizona State University.
- Lock, K. (2023). Investigating the role of relative size reasoning in students’ understanding of precalculus ideas. Unpublished Ph.D. dissertation, Arizona State University.
- Marfai, F. S. (2017). Characterizing teacher change through the perturbation of pedagogical goals. Unpublished Ph.D. dissertation, Arizona State University.
- Moore, K. (2010). The role of quantitative reasoning in precalculus students’ learning central concepts of trigonometry. Unpublished Ph.D. dissertation, Arizona State University.
- O’Bryan, A. E. (2018). Exponential growth and online learning environments: Designing for and studying the development of student meanings in online courses. Unpublished Ph.D. dissertation, 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, Arizona State University.
- Smith, N. N. (2008). Students’ emergent conceptions of the Fundamental Theorem of Calculus. Unpublished Ph.D. dissertation, Arizona State University.
- Strom, A. (2008). A case study of a secondary mathematics teacher’s understanding of exponential function: An emerging theoretical framework. Unpublished Ph.D. dissertation, Arizona State University.
- Tallman, M. (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, 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 a list of some of the relevant conference proceedings sharing work from the Pathways research team.
- Carlson, M., Bowling, S., Moore, K., & Ortiz, A. (2007). The role of the facilitator in promoting meaningful discourse among professional learning communities of secondary mathematics and science teachers. In Lamberg & Wiest (Eds.), PME-NA 29 Proceedings (pp. 841-848).^
- 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. MSP LNC 2012, NSF web publication.
- Carlson, M. P., Slemmer, G., Moore, K., Teuscher, D., & Joyner, K. (2011). Key Variables for Establishing and Sustaining Highly Effective Professional Learning Communities. MSP LNC 2011, NSF web publication.
- Clark, P. G., Carlson, M., & Moore, K. (2007). Documenting the emergence of “speaking with meaning” as a sociomathematical norm. In Lamberg & Wiest (Eds.), PME-NA 29 Proceedings (pp. 872-874).
- Engelke, N. (2008). Developing the solution process for related rates problems using computer simulations. Research in Undergraduate Mathematics Education Conference Proceedings.
- 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. In ASEE/IEEE FIE 2008. IEEE.
- Marfai, F., Moore, K., & Teuscher, D. (2011). The influence of a teacher’s decentering moves on students engaging in reflective thinking. In Wiest & Lamberg (Eds.), PME-NA 33 Proceedings (pp. 138-146).
- McClain, K., Carlson, M., Coe, E., & Saldanha, L. (2009). The emergence of norms for mathematical argumentation. In Swars, Stinson & Lemons-Smith (Eds.), PME-NA 31 Proceedings (pp. 288-295).
- Meylani, R., & Teuscher, D. (2011). Calculus readiness: Comparing student outcomes from traditional precalculus and AP Calculus AB with a novel precalculus program. In Wiest & Lamberg (Eds.), PME-NA 33 Proceedings (pp. 778-786).
- Moore, K. C., Carlson, M. P., & Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. In SIGMAA RUME 2009 Proceedings. NC State.
- Moore, K. C., Carlson, M. P., & Teuscher, D. (2011). Using research-based curriculum to support shifts in teachers’ pedagogical understandings. MSP LNC 2011, NSF web publication.
- Moore, K. C., Teuscher, D., & Carlson, M. P. (2011). Exploring shifts in a teacher’s key developmental understandings and pedagogical actions. In Wiest & Lamberg (Eds.), PME-NA 33 Proceedings (pp. 1673-1681).
- Musgrave, S., & Carlson, M. P. (2016). Transforming graduate students’ meanings for average rate of change. In Fukawa-Connolly et al. (Eds.), RUME 2016 Proceedings (pp. 809-814). WVU.
- Musgrave, S., & Carlson, M. P. (2016). Understanding and advancing graduate teaching assistants’ knowledge for teaching. In Fukawa-Connolly et al. (Eds.), RUME 2016 Proceedings (pp. 340-354). WVU.
- O’Bryan, A. E. (2020). A research-based approach to developing, refining, and assessing student learning in an online precalculus course. In INTED 2020 Proceedings, Valencia, Spain.
- O’Bryan, A. E. (2020). Quantitative reasoning and symbolization activity: Do individuals expect calculations and expressions to have quantitative significance? In RUME 2020 Proceedings, Boston, MA.
- 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 Fukawa-Connolly et al. (Eds.), RUME 2016 Proceedings (pp. 1192-1200). WVU.
- Oehrtman, M., Carlson, M., Martin, J., & Sutor, J. (2010). Coherence and change in teacher professional learning communities. MSP LNC 2010, NSF web publication.
- Oehrtman, M., Carlson, M., Sutor, J., Agoune, L., & Stroud, C. (2009). Meaningful Collaboration in Secondary Mathematics and Science Teacher Professional Learning Communities. RUME 2009 Proceedings (28 pages).
- Rasmussen, C., Bressoud, D., & Carlson, M. (2015). Who are the students that switch out of calculus and why do they switch? AERA 2015 Proceedings (pp. 149-156).
- Sander, G., & Carlson, M. P. (2016). On the use of dynamic animations to support students in reasoning quantitatively. In Fukawa-Connolly et al. (Eds.), RUME 2016 Proceedings (pp. 1262-1270). WVU.
- Teuscher, D., Moore, K. C., & Carlson, M. P. (2011). Interaction between teacher’s questions and student discourse. MSP LNC 2011, NSF web publication.
- Underwood, K., & Carlson, M. P. (2012). Understanding how precalculus teachers develop knowledge for teaching rate of change. RUME 2012 Proceedings (149-157).