Doing math with mathematicians to support pedagogical reasoning about inquiry-oriented instruction

  • Christine Andrews-Larson
  • Estrella JohnsonEmail author
  • Valerie Peterson
  • Rachel Keller


Given the prevalence of research in undergraduate mathematics education focused on student reasoning and the development of instructional innovations that leverage student reasoning, it is important to understand the ways undergraduate mathematics instructors make sense of these innovations. We characterize pedagogical reasoning about inquiry-oriented instruction relative to vertices of the instructional triangle (content, students, and instructor). Through this lens, we analyze conversations of twenty-five mathematicians who elected to attend a workshop on inquiry-oriented instruction at a large national mathematics conference. Identifying differences in talk between two breakout groups, we argue that deeper mathematical engagement in task sequences designed for students supported deeper engagement in students’ mathematical reasoning and engendered reasoning about instruction that was more frequently accompanied by rationale based in mathematics or students’ reasoning about mathematics. Importantly, deeper mathematical engagement was observed when the discussion facilitator prompted participants to engage through a mathematical lens rather than an instructional lens.


Inquiry-oriented instruction Undergraduate mathematics Instructional change Pedagogical reasoning 



This research was supported by National Science Foundation award numbers #1431595, #1431641, and #1431393.

Supplementary material

10857_2019_9450_MOESM1_ESM.pdf (172 kb)
Supplementary material 1 (PDF 171 kb)


  1. Andrews-Larson, C., Wawro, M., & Zandieh, M. (2017). A hypothetical learning trajectory for conceptualizing matrices as linear transformations. International Journal of Mathematical Education in Science and Technology,48, 1–21.CrossRefGoogle Scholar
  2. Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education,59(5), 389–407.CrossRefGoogle Scholar
  3. Bouhjar, K., Andrews-Larson, C., Haider, M., & Zandieh, M. (2018). Examining students’ procedural and conceptual understanding of eigenvectors and eigenvalues in the context of inquiry- oriented instruction. In S. Stewart, C. Andrews-Larson, M. Zandieh, & A. Berman (Eds.), Challenges in teaching linear algebra. Berlin: Springer.Google Scholar
  4. Braun, B., Bressoud, D., Briars, D., Coe, T., Crowley, J., Dewar, J., Ward, M. (2016). Active learning in post-secondary mathematics education. CBMS News, Retrieved from
  5. Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal,97(1), 3–20.CrossRefGoogle Scholar
  6. DeCuir-Gunby, J. T., Marshall, P. L., & McCulloch, A. W. (2011). Developing and using a codebook for the analysis of interview data: An example from a professional development research project. Field Methods,23(2), 136–155.CrossRefGoogle Scholar
  7. Elo, S., & Kyngäs, H. (2008). The qualitative content analysis process. Journal of Advanced Nursing,62(1), 107–115.CrossRefGoogle Scholar
  8. Fairweather, J. (2008). Linking evidence and promising practices in science, technology, engineering, and mathematics (STEM) undergraduate education. Washington, DC: Board of Science Education, National Research Council, The National Academies.Google Scholar
  9. Florensa, I., Bosch, M., Gascón, J., Ruiz-Munzon, N. (2017). Teaching didactics to lecturers: A challenging field. In Proceedings of the 11th Congress of European Research in Mathematics Education, Dublin, Ireland. HAL Id: hal-01941653.Google Scholar
  10. Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., et al. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences,111(23), 8410–8415.CrossRefGoogle Scholar
  11. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  12. Henderson, C., Beach, A., & Finkelstein, N. (2011). Facilitating change in undergraduate STEM instructional practices: An analytic review of the literature. Journal of Research in Science Teaching,48(8), 952–984.CrossRefGoogle Scholar
  13. Horn, I. S., & Little, J. W. (2010). Attending to problems of practice: Routines and resources for professional learning in teachers’ workplace interactions. American Educational Research Journal,47(1), 181–217.CrossRefGoogle Scholar
  14. Hurtado, S., Eagan, K., Pryor, J. H., Whang, H., & Tran, S. (2012). Undergraduate teaching faculty: The 2010–2011 HERI faculty survey. Retrieved from
  15. Iannone, P., & Nardi, E. (2005). On the pedagogical insight of mathematicians: ‘Interaction’and ‘transition from the concrete to the abstract’. The Journal of Mathematical Behavior,24(2), 191–215.CrossRefGoogle Scholar
  16. Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.Google Scholar
  17. Johnson, E. (2013). Teachers’ mathematical activity in inquiry-oriented instruction. The Journal of Mathematical Behavior,32(4), 761–775.CrossRefGoogle Scholar
  18. Johnson, E., Caughman, J., Fredericks, J., & Gibson, L. (2013). Implementing inquiry-oriented curriculum: From the mathematicians’ perspective. The Journal of Mathematical Behavior,32(4), 743–760.CrossRefGoogle Scholar
  19. Johnson, E., Keller, R., & Fukawa-Connelly, T. (2018). Results from a national survey of abstract algebra instructors: Understanding the choice to (not) lecture. International Journal for Research in Undergraduate Mathematics Education,4(2), 254–285.CrossRefGoogle Scholar
  20. Johnson, E., & Larsen, S. (2012). Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior,31, 117–129.CrossRefGoogle Scholar
  21. Kogan, M., & Laursen, S. L. (2013). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. Innovative Higher Education,39, 1–17.Google Scholar
  22. Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2018). Inquiry-oriented instruction: A conceptualization of the instructional principles. PRIMUS,28(1), 13–30.CrossRefGoogle Scholar
  23. Kuster, G., Johnson, E., Rupnow, R., & Wilhelm, A. (2019). The inquiry-oriented instructional measure. International Journal for Research in Undergraduate Mathematics Education,5(2), 181–204.CrossRefGoogle Scholar
  24. Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics,105(5), 227–239.CrossRefGoogle Scholar
  25. Lai, Y., & Weber, K. (2014). Factors mathematicians profess to consider when presenting pedagogical proofs. Educational Studies in Mathematics,85(1), 93–108.CrossRefGoogle Scholar
  26. Larsen, S., Johnson, E., & Bartlo, J. (2013). Designing and scaling up an innovation in abstract algebra. The Journal of Mathematical Behavior.,32(4), 776–790.CrossRefGoogle Scholar
  27. Larsen, S., Johnson, E., & Scholl, T. (2016). The inquiry oriented group theory curriculum and instructional support materials. Retrieved November 25, 2019, from
  28. Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics,67(3), 205–216.CrossRefGoogle Scholar
  29. Laursen, S. L., Hassi, M. L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study. Journal for Research in Mathematics Education,45(4), 406–418.CrossRefGoogle Scholar
  30. Mason, J. (2002). Researching your own practice: The discipline of noticing. Abingdon: Routledge.CrossRefGoogle Scholar
  31. Nardi, E., Jaworski, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: From” tricks” to” techniques”. Journal for Research in Mathematics Education 284–316.Google Scholar
  32. President’s Council of Advisors on Science and Technology (PCAST). (2012). Engage to excel: Producing one million additional college graduates with Degrees in Science, Technology, Engineering, and Mathematics. Washington, DC: The White House.Google Scholar
  33. Rasmussen, C., Keene, K. A., Dunmyre, J., & Fortune, N. (2018). Inquiry oriented differential equations: Course materials. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
  34. Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics. The Journal of Mathematical Behavior,26(3), 189–194.CrossRefGoogle Scholar
  35. Rasmussen, C., & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education,37(5), 388–420.Google Scholar
  36. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education,8(3), 255–281.CrossRefGoogle Scholar
  37. Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.Google Scholar
  38. Speer, N., & Hald, O. (2008). How do mathematicians learn to teach? Implications from research on teachers and teaching for graduate student professional development. In M. Carlson & C. Rasmussen (Eds.), Making the Connection: Research and practice in undergraduate mathematics education (pp. 305–218). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  39. Speer, N. M., Smith, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior,29, 99–114.CrossRefGoogle Scholar
  40. Speer, N. M., & Wagner, J. F. (2009). Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education,40(5), 530–562.Google Scholar
  41. van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education,24(2), 244–276.CrossRefGoogle Scholar
  42. Viirman, O. (2015). Explanation, motivation and question posing routines in university mathematics teachers’ pedagogical discourse: A commognitive analysis. International Journal of Mathematical Education in Science and Technology,46(8), 1165–1181.CrossRefGoogle Scholar
  43. Wagner, J., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. Journal of Mathematical Behavior,26, 247–266.CrossRefGoogle Scholar
  44. Wawro, M., Rasmussen, C., Zandieh, M., Sweeney, G., & Larson, C. (2012). An inquiry-oriented approach to span and linear independence: The case of the Magic Carpet Ride sequence. PRIMUS,22(7), 1–23.Google Scholar
  45. Wawro, M., Zandieh, M., Rasmussen, C., & Andrews-Larson, C. (2013). Inquiry oriented linear algebra: Course materials. Available at This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
  46. Winsløw, C., Barquero, B., De Vleeschouwer, M., & Hardy, N. (2014). An institutional approach to university mathematics education: From dual vector spaces to questioning the world. Research in Mathematics Education,16(2), 95–111.CrossRefGoogle Scholar
  47. Zandieh, M., Wawro, M., & Rasmussen, C. (2017). An example of inquiry in linear algebra: The roles of symbolizing and brokering. PRIMUS,27(1), 96–124.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Christine Andrews-Larson
    • 1
  • Estrella Johnson
    • 2
    Email author
  • Valerie Peterson
    • 3
  • Rachel Keller
    • 4
  1. 1.School of Teacher EducationFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsVirginia TechBlacksburgUSA
  3. 3.Department of MathematicsUniversity of PortlandPortlandUSA
  4. 4.Department of MathematicsVirginia TechBlacksburgUSA

Personalised recommendations