Mathematical Reasoning in Physics
Photo Credit: Matt Shirley
Quantitative Literacy: the nuanced ways that we use mathematics in every day contexts to make sense of the world.
My research focuses on developing quantitative literacy for everyone —- regardless of prior educational experiences. We believe that physics is an excellent place to teach quantitative literacy as it is often required for a wide variety of STEM majors, cited as a course that teaches people “problem solving". Thinking quantitatively is an essential skill in physics, STEM majors, and across all fields in the wider world. Quantitative literacy includes developing mathematical models for observed phenomenon, creating and interpreting graphical representations, and making sense of what affect changes to a part of the system have on the larger model.
“Let it Flow”: Introductory Physics for Life Science
Self-Efficacy: the belief that you are capable of doing something, even if you don’t have the skills to do it yet.
Life science and chemistry students are often required to complete physics, and for many it understandably feels like an unsurmountable hurdle or a box to check to get to medical school. My research at Cornell involves re-designing the Introductory Physics for Life Science course along three goals: (1) helping students see physics as authentically relevant for life science students; (2) improving students sense of self-efficacy by integrating calculus ideas without assuming procedural fluency; and (3) improving student critical thinking with experimentation and data analysis for use across the science disciplines.
Current Areas of Research
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Covariational Reasoning
Covariational reasoning — how one quantity changes with respect to changes in another, related quantity — is at the heart of modeling in physics. Mathematics education researchers have long studied covariational reasoning and shown that it is essential for student conceptual reasoning about rates of change. Our research group is currently investigating the differences between covariational reasoning in physics and in math for both students and experts in order to determine what kinds of curricular interventions may be most effective towards moving the needle of quantitative literacy.
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The PIQL and The GERQN
The Physics Inventory for Quantitative Literacy, or PIQL, is a mathematical reasoning inventory designed to measure aggregate student reasoning about mathematical concepts that are foundational to introductory physics. It is a validated instrument designed for calculus-based physics courses, and available for widespread use. The Generalized Equation-based Reasoning inventory for Quantity and Negativity is the algebra-based version of the PIQL that I developed and validated as part of my graduate work.
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Physics for Life Science
A key element of my research is developing curricular materials that help students see themselves as capable of studying, understanding, or using physics in whatever career they pursue. Of particular interest to me is the large and increasing population of students interested in the life sciences, many of whom do not yet see physics as accessible or useful to them. I believe strongly in the interdisciplinary nature of science; this population offers an opportunity to include interdisciplinary skills and attitudes as direct learning outcomes of a physics course.
Grants
S. Brahmia, T. Smith, Collaborative Research: Measuring and Improving Physics Quantitative Literacy throughout the Undergraduate Curriculum, NSF Award #2214765. (Awarded 2022, posting in progress).
I contributed significantly to the writing of this grant, including the research design and methodology. The project includes funding for my thesis work on the Generalized Equation-based Reasoning inventory for Quantity and Negativity (GERQN).
Academic Awards
Graduate Research Fellowship Award, National Science Foundation (2020-2023)
Karrer Prize, University of Washington (2023)
Graduate Student Excellence in Teaching Award, University of Washington (2021)
Excellence in First Year Teaching Award with Distinction, University of Washington (2020)
Mellam Scholar Award, University of Washington (2018)
Clare Boothe Luce Research Scholar, Carleton College (2012)
Publications
Journal Articles
C. Zimmerman, T. Smith, A. Olsho, P. Eaton, A. Boudreaux, S. White Brahmia, Assessing physics quantitative literacy development in algebra-based physics. Phys. Rev. Phys. Educ. Res., 21 (2), 10.1103/lnd6-pxyt (2025).
A. Olsho, C. Zimmerman, S. White Brahmia, A framework for covariational reasoning in physics. Phys. Rev. Phys. Educ. Res., (revise and resubmit)
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia, Empirical evidence of the inseparability of mathematics and physics in expert reasoning about novel graphing tasks. Phys. Rev. Phys. Educ. Res., (revise and resubmit)
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia, Expert covariational reasoning resources in physics graphing tasks. Phys. Rev. Phys. Educ. Res., (revise and resubmit)
A. Olsho, T. Smith, P. Eaton, C. Zimmerman, A. Boudreaux, S. White Brahmia, Online test administration results in students selecting more responses to multiple-choice-multiple-response items Phys. Rev. Phys. Educ. Res., 19, 013101 (2023).
S. Brahmia, A. Olsho, T. Smith, A. Boudreaux, P. Eaton, and C. Zimmerman, Physics Inventory of Quantitative Literacy: A tool for assessing mathematical reasoning in introductory physics Phys. Rev. Phys. Educ. Res., 17 (2), 020129 (2021).
C. Zimmerman, M. White, and M.-E. Baylor, Effects of varying interfacial surface tension on macroscopic polymer lenses. Optical Engineering, 54 (9), (2015).
Doctoral Thesis
C. Zimmerman, (2023) Characterizing and Assessing Covariational Reasoning in Introductory Physics Contexts. University of Washington, Department of Physics. Defense presentation recording can be accessed here.
Peer-Reviewed Conference Proceedings
C. Zimmerman and S. White Brahmia, Accumulation as a tool towards blending reasoning about quantity and rate of change in physics contexts The Learning and Teaching of Calculus Across Disciplines 2: Proceedings 2025, (Awaiting print).
B. Boyle, T. Smith, C. Zimmerman and S. White Brahmia, Validating Shorter Versions of the Physics Inventory of Quantitative Literacy Physics Education Research Conference 2024 Proceedings, 2024, (p-).
E. Henry, C. Zimmerman, J. Goldak and S. White Brahmia, Exploring Mathematical Intuition and its Role in Physics Problem-Solving Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education, 2024 (p.1472-1774).
T. Smith, Z. Bischoff. B. Boyle, J. Sayers, C. Zimmerman, P. Eaton, A. Olsho, and S. White Brahmia, Creating Statistically Equivalent Versions of a test of Quantitative Literacy in Physics Contexts Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education, 2024, (p.1447-1450).
Q. Guo, C. Zimmerman, and S. White Brahmia, The Journey of Quantitative Literacy Development: Insights from Physics Majors Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education, 2024, (p.1437).
C. Zimmerman, A. McCarty, S. White Brahmia, A. Olsho, M. De Cock, A. Boudreaux, T. Smith, and P. Eaton, Assessing physics quantitative literacy in algebra-based physics: lessons learned Physics Education Research Conference 2022 Proceedings, 2022, (p. 519-524).
A. Olsho, C. Zimmerman, A. Boudreaux, T. Smith, P. Eaton, and S. White Brahmia, Characterizing covariational reasoning in physics modeling Physics Education Research Conference 2022 Proceedings, 2022, (p. 335-340).
T. Smith, P. Eaton, S. White Brahmia, A. Olsho, C. Zimmerman, and A. Boudreaux, Analyzing Multiple-Choice-Multiple-Response Items Using Item Response Theory Physics Education Research Conference 2022 Proceedings, 2022, (p. 432-437).
Y. Elhady, C. Zimmerman, and S. White Brahmia, Effects of facilitating collaboration in large-enrollment introductory physics courses Physics Education Research Conference 2021 Proceedings, 2021, (p. 117-122).
C. Zimmerman, A. Olsho, S. Brahmia, A. Boudreaux, T. Smith, and P. Eaton, Exploring student facility with "goes like'' reasoning in introductory physics Physics Education Research Conference 2020 Proceedings, 2020 (p. 605-610).
C. Zimmerman, A. Olsho, M. Loverude, and S. Brahmia, Identifying Covariational Reasoning Behaviors in Expert Physicists in Graphing Tasks Research in Undergraduate Mathematics Education Conference Proceedings, 2020 (p. 976-981).
S. Brahmia, A. Olsho, A. Boudreaux, T. Smith, C. Zimmerman A Conceptual Blend Analysis of Physics Quantitative Literacy Reasoning Inventory Items Research in Undergraduate Mathematics Education Conference Proceedings, 2020 (p. 853-858).
A. Olsho, C. Zimmerman, S. Brahmia, A Framework of Covariational Reasoning in Introductory Physics Research in Undergraduate Mathematics Education Conference Proceedings, 2020 (p. 1263-1264).
T. Smith, P. Eaton, S. Brahmia, A. Olsho, C. Zimmerman, and A. Boudreaux, Toward a valid instrument for measuring physics quantitative literacy Physics Education Research Conference 2020 Proceedings, 2020, (p. 490-496).
A. Olsho, S. Brahmia, C. Zimmerman, T. Smith, P. Eaton, and A. Boudreaux, Online administration of a reasoning inventory in development Physics Education Research Conference 2020 Proceedings, 2020, (p. 376-381).
A. Boudreaux, S. Kanim, A. Olsho, S. Brahmia, C. Zimmerman, and T. Smith, Toward a framework for the natures of proportional reasoning in introductory physics Physics Education Research Conference 2020 Proceedings, 2020 (p. 45-50).
C. Zimmerman, A. Olsho, S. Brahmia, M. Loverude, A. Boudreaux, and T. Smith, Toward understanding and characterizing expert physics covariational reasoning Physics Education Research Conference 2019 Proceedings, 2019 (p. 693-698).
Presentations
Invited Talks
C. Zimmerman, “A framework for developing generative, graphical reasoning activities,” Michigan State University Physics Education Research Lab Seminar, planned Oct. 29th, 2024
C. Zimmerman, “‘Thinking like a physicist:’ research into how physics experts reason mathematically in introductory physics contexts,” Stockton University Physics Colloquium, Apr. 12, 2024; Virtual.
C. Zimmerman, “‘Thinking like a physicist:’ research into how physics experts reason mathematically in introductory physics contexts,” Providence College Physics Seminar, Nov. 15, 2023; Providence, RI.
C. Zimmerman, “‘Thinking like a physicist:’ research into how physics experts reason mathematically in introductory physics contexts,” Mount Holyoke College Physics Seminar, Nov. 14, 2023; South Hadley, MA
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia, “The inseparability of mathematics and physics reasoning: a study of physics experts generating graphs,” Western Pennsylvania Section, American Association of Physics Teachers Virtual Conference, Nov. 4, 2023; Virtual.
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia, “The inseparability of mathematics and physics in expert reasoning about graphs,” University of Washington, Bothel Physics Department Seminar, Oct. 31, 2023; Bothel, WA.
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia, Characterizing expert covariational reasoning in physics. Western Washington University Science, Math, and Technology Education (SMATE) Program Colloquium. May 24, 2023; Bellingham, WA.
C. Zimmerman, A. Olsho, A. Boudreaux, S. White Brahmia Data linearization and graphical interpretation as methods of developing covariational reasoning. Research in Undergraduate Mathematics Education Conference 2022. Feb. 24, 2022; Boston, MA.
C. Zimmerman, J. Canright, S. White Brahmia, A conceptual blend analysis of covariational reasoning in physics labs. Physics Education Research Conference 2021. Aug. 4, 2021; Virtual.
Conference Workshops / Sessions Organized
Education Research at the Interface of Mathematics and Science: Examining Courses for Non-Majors, Organizers: C. Zimmerman, Ben Geller, Steve Bennoun. Research in Undergraduate Mathematics Education Conference, Feb. 27, 2025; Alexandria, VA.
Mathematical Reasoning in Algebra-Based Physics, Organizers: C. Zimmerman, Suzanne White Brahmia, Trevor Smith, Alexis Olsho. Community Connections Workshop at the Physics Education Research Conference, July 11, 2024; Boston, MA.
Considering covariational reasoning in math and physics. Organizers: C. Zimmerman. Speakers: Paul Emigh, Cameron Byerley, Michael Loverude, and Darío González. Talk Symposium at the Physics Education Research Conference, Aug. 5, 2021; Virtual.
Education Research at the Interface of Mathematics and Science: Perspectives on Quantitative Modeling Workshop. Organizers: B. Schermerhorn, C. Simmons, C. Zimmerman. Research in Undergraduate Mathematics Education Conference, Feb 27, 2020; Boston, MA.
Contributed Talks
C. Zimmerman and S. White Brahmia, “Accumulation as a tool towards blending reasoning about quantity and rate of change in physics contexts” The Learning and Teaching of Calculus Across Disciplines 2, June 17, 2025; Milan, Italy.
C. Zimmerman, N. G. Holmes, S. White Brahima, “Using assessment to drive the development of algebraic reasoning activities,'' American Physics Society Global Meeting, Mar. 18, 2025; Los Angeles, CA.
C. Zimmerman, M. de Cock, P. van Kampen, S. White Brahmia, “Assessing students’ physics quantitative literacy in Ireland, Belgium, & the U.S.: Towards insights for teacher preparation,” World Conference in Physics Education, Aug. 27, 2024; Krakow, Poland.
C. Zimmerman, A. Olsho, S. White Brahmia, “CORP: Covariational Reasoning in Physics Framework,” Research in Undergraduate Mathematics Education Conference, Math-Science Working Group, Feb. 22, 2024; Omaha, NB (presented virtually).
C. Zimmerman, A. K. Snow, P. Heron, P Shaffer, “Investigating the intersection be- tween the Tutorials in Introductory Physics and mathematical reasoning”, American Association of Physics Teachers Conference Winter Meeting, Jan. 9th, 2024; New Orleans, LA.
C. Zimmerman, I. Carter, S. White Brahmia Examining student reasoning about graphs with salient, distracting features. American Association of Physics Teachers Conference Winter Meeting 2023, Contributed Talk
C. Zimmerman, M. De Cock, S. White Brahmia Assessing physics quantitative literacy focused on conceptualizing algebraic ideas American Association of Physics Teachers Conference Summer Meeting 2022, Contributed Talk
C. Zimmerman, A. Olsho, P. Shaffer, S. White Brahmia Student emergent sense-making about quantitative modeling in introductory physics labs. American Association of Physics Teachers Conference Virtual Winter Meeting 2021, Contributed Talk [recording available]
C. Zimmerman, A. Olsho, A. Boudreaux, S. White Brahmia. Exploring student use of "goes like" thinking when linearizing data American Association of Physics Teachers Conference Virtual Summer Meeting 2020, Contributed Talk
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia Identifying covariational reasoning behaviors in expert physicists in graphing tasks Research in Undergraduate Mathematics Education Conference 2020, Contributed Talk
C. Zimmerman, A. Olsho, M. Loverude, S. White Brahmia Comparing covariational reasoning of experts in physics and in mathematics American Association of Physics Teachers Conference Summer Meeting 2019, Contributed Talk