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Obstacles to Mathematization in Physics: The Case of the Differential

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Abstract

The process of the mathematization of physical situations through differential calculus requires an understanding of the justification for and the meaning of the differential in the context of physics. In this work, four different conceptions about the differential in physics are identified and assessed according to their utility for the mathematization process. We also present an empirical study to probe the conceptions about the differential that are used by students in physics, as well students’ perceptions of how they are expected to use differential calculus in physics. The results support the claim that students have a quasi-exclusive conception of the differential as an infinitesimal increment and that they perceive that their teachers only expect them to use differential calculus in an algorithmic way, without a sound understanding of what are they doing and why. These results are related to the lack of attention paid by conventional physics teaching to the mathematization process. Finally, some proposals for action are put forward.

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Notes

  1. Artigue et al. (1989), Berry and Nyman (2003), Ferrini-Mundy and Gaudard (1992), Ferrini-Mundy and Graham (1991), Labraña (2001), Mahir (2009), Nagle et al. (2013), Orton (1983a, b), Porter and Masingila (2000), Schneider (1991, 1992), Tall (1985, 1992), Thompson (1994), among many others.

  2. See, for example, Meredith and Marrongelle (2008), Hu and Rebello (2013), Sealey (2014), Von Korff and Rebello (2014), and Wilcox et al. (2013).

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Acknowledgments

We would like to thank the reviewers for careful reading and insightful suggestions that greatly improved this manuscript.

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Authors and Affiliations

  1. Didáctica de las Ciencias Experimentales, Facultad de Educación, U. de Almería, Ctra. Sacramento, s/n. La Cañada de San Urbano, 04120, Almería, Spain

    R. López-Gay

  2. Département de Mathématiques, Lycée Français d’Alicante “Pierre Deschamps”, Camino de Marco, 21, 03560, El Campello (Alicante), Spain

    J. Martínez Sáez

  3. Didáctica de las Ciencias Experimentales, Facultad de Educación, U. de Alicante, Apdo. correos, 99, 03080, Alicante, Spain

    J. Martínez Torregrosa

Authors
  1. R. López-Gay
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  2. J. Martínez Sáez
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Correspondence to R. López-Gay.

Appendix: Document and Guidelines for the Semi-Structured Interview

Appendix: Document and Guidelines for the Semi-Structured Interview

PROBLEM STATEMENT: We know that the density of air (ρ) decreases with height (h) according to the following equation: ρ = 1.29·(1–0.000125·h). That equation is written for the International System, that is, if h is written in metres, density is obtained in kg/m3. The value h = 0 represents sea level. What would be the mass of a cylindrical column of air measuring 1 m 2 at the base that rises 2000 m above sea level?

figure a
  1. 1.

    Why do you take that step? (Be it the step from increment to differential or from incremental quotient to differential quotient)

  2. 2.

    What is the meaning of that expression? (We insist on searching for an explanation that goes beyond the use of key words or literal reading)

  3. 3.

    What could be the value of dm, dV…? (If they answer with a numerical value, we check on its meaning and functional nature)

  4. 4.

    How must that expression be read? Is it correct to isolate the differential? (We are referring to the expression of the derivative, and want to know if they consider it as a true differentials quotient)

  5. 5.

    What is the meaning of those integrals? (They may adhere to the idea of the anti-derivative, or go further and identify Riemann sums)

  6. 6.

    Why is the result of that integral precisely that? (We will inquire to see if they are capable of justifying why the integral of a differential is a macroscopic increment, or why infinite sums are necessarily calculated using anti-derivatives)

  7. 7.

    Do you understand properly when your teacher or the textbook use differential calculus in physics lessons?

  8. 8.

    Where have you best learnt the use and meaning of differential calculus, in physics or maths lessons?

  9. 9.

    In general, do you think that the use of differential calculus makes students like physics more or less? Why do you think so? And do you think that is the case for you too?

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López-Gay, R., Martínez Sáez, J. & Martínez Torregrosa, J. Obstacles to Mathematization in Physics: The Case of the Differential. Sci & Educ 24, 591–613 (2015). https://doi.org/10.1007/s11191-015-9757-7

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