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Comparing Covariational Reasoning of Experts in Physics and in Mathematics

Charlotte Zimmerman

Department of Physics

University of Washington

AAPT Summer Meeting, Provo UT

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Funding

University of Washington

This work was supported by:

Department of Physics

NSF

This work was supported by NSF DUE IUSE grants:

1832836, 1832880, and 1833050

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Collaborators

  • Alexis Olsho, University of Washington
  • Michael Loverude, California State University, Fullerton
  • Andrew Boudreaux, Western Washington University
  • Trevor Smith, Rowan University
  • Suzanne White Brahmia, University of Washington

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“Goes Like”

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Covariational Reasoning in Math Education Research

Covariational Reasoning

Holding in mind invariant relationships among quantities’ values as they vary in dynamic situations [1, 2].

  • Student use of covariational reasoning has been studied widely in Mathematics Education research
  • The framework developed by Carlson et al. identifies the mental actions used by mathematics students [3].
  • Do covariational reasoning mental actions look the same for mathematics students and physics students?

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Hobson and Moore Study [4]

  • 2017 study of covariational reasoning in Mathematics experts
  • Performed 10 think-out-loud interviews with graduate students (instructors for introductory sequence)
  • Examined the ways that experts expressed their thinking in 3 graphing tasks designed to elicit covariational reasoning.

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Expert

Instructor

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Research Question

Research Question

Do expert physicists demonstrate the same kinds of behaviors as expert mathematicians when solving covariational graphing tasks?

  • Replicated the Hobson and Moore study with physics graduate students
  • Looking for similarities and differences in their behaviors compared to the math students
  • To help us better understand what covariational reasoning students are coming into classes with and how it is different then what the instructor might do.

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Going Around Tacoma

A few students are going on a road trip, and decide to drive from Seattle to Portland. They want to avoid the traffic around Tacoma, and take the path shown in the animation. Draw a graph of the students’ distance from Tacoma vs. their total distance traveled.

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Going Around Tacoma

A few students are going on a road trip, and decide to drive from Seattle to Portland. They want to avoid the traffic around Tacoma, and take the path shown in the animation. Draw a graph of the students’ distance from Tacoma vs. their total distance traveled.

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Going Around Tacoma

Math Experts

Physics Experts

“10 miles total distance and it’s also 10 miles closer to [Tacoma]” [4]

“They’re going at constant speed… I’m just gonna draw it versus time, and then kinda parameterize it.”

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The Ferris Wheel Task

As shown in the animation, a cart moves around a Ferris wheel. Draw the graph of the height of the cart from the ground vs its total distance traveled in the space below.

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The Ferris Wheel Task

As shown in the animation, a cart moves around a Ferris wheel. Draw the graph of the height of the cart from the ground vs its total distance traveled in the space below.

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The Ferris Wheel Task

Math Experts

Physics Experts

“It’s not a constant rate of change… [it involves] increasing rates of change” [4]

“I’m just like, oh, trig function”

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Math Experts

Physics Experts

“Notice in the first x interval, there’s very little change in y… but then if you look at the next same amount of x, you get a much bigger change in y.” [4]

“So if I want it to go slow, fast, slow here, and then symmetrically on the other side…”

The Ferris Wheel Task

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Conclusions

Take Away

Expert physicists approach these problems differently than mathematics experts.

  • Reliance on familiar functions
  • “Neighborhood” Analysis
  • Students may not be prepared to think this way from math class.
  • As instructors, it is helpful to think about where our students are coming from in comparison to how we think and use mathematics.

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Further Work

  • Physics-Content Questions

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  • Implications on instruction and student reasoning

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Acknowledgements

  • Questions?
    • Come to my poster: Wednesday, 4:45

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  • Thank you!

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Faculty and Staff:

Peter S. Shaffer

Paula R. L. Heron

Lilian C. McDermott

Suzanne White Brahmia

Donna Messina

Alexis Olsho

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Graduate Students:

Anne Alesandrini

Jesse Ashworth

Dean Bretland

Jared Canright

Jeffery Commons

Sheh Lit Chang

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Yasmene Elhady

Lisa Goodhew

Adnan Kahn

Bert Xue

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References

[1] P. W. Thompson,“Quantitative reasoning and mathematical modeling,”in New

perspectives and directions for collaborative research in mathematical education (L. L. Hatfield, S.

Chamberlain, and S. Belbase, eds.), vol. 1, pp. 33–57, Laramie, WY: University of

Wyoming, 2011.

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[2] M. Carlson, M. Oehrtman, and N. Engelke, “The precalculus concept assessment:

A tool for assessing students’ reasoning abilities and understandings,” Cognition and

Instruction, vol. 28, no. 2, pp. 113–145, 2010.

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[3] M. Carlson, S. Jacobs, E. Coe, S. Larsen, and E. Hsu, “Applying Covariational

Reasoning While Modeling Dynamic Events: A Framework and a Study,” Journal for

Research in Mathematics Education, vol. 33, no. 5, pp. 352–378, 2002.

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[4] N. L. F. Hobson and K. C. Moore, “Exploring Experts’ Covariational Reasoning,”

in 20th Annual Conference on Research in Undergraduate Mathematics Education, pp. 664–

672, Moore & Thompson, 2017.