Putting Differentials Back into Calculus

Dray, Tevian; Manogue, Corinne A.

doi:10.4169/074683410x480195

Cited by 23 publications

(11 citation statements)

References 14 publications

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“…(4)]-is automatically satisfied in the Differentials game. By taking the equations of state and ''zapping with d'' [21], one is working only with the derivatives in Eq. (4) (''nice sets'').…”

Section: Differentialsmentioning

confidence: 99%

“…(15)] and explicitly considering whether he had held entropy constant. Unlike most mathematicians, physicists are willing to work with differentials as intuitive objects standing for small changes [21]. Yet, as Elliott demonstrated, physicists are not always sure of the legitimacy of their use of differentials from a mathematics perspective.…”

Section: When Asked To Explain Further He Saidmentioning

confidence: 99%

“…began to explore the uses of differentials within the context of thermodynamics. This was a logical extension of the Vector Calculus Bridge Project [19], which found the vector differential to be a helpful bridge between vector calculus and upper-division electromagnetism courses and led to substantial reforms in how these courses are taught at Oregon State University [20,21].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Partial derivative games in thermodynamics: A cognitive task analysis

Kustusch

Roundy

Dray

et al. 2014

Phys. Rev. ST Phys. Educ. Res.

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Several studies in recent years have demonstrated that upper-division students struggle with the mathematics of thermodynamics. This paper presents a task analysis based on several expert attempts to solve a challenging mathematics problem in thermodynamics. The purpose of this paper is twofold. First, we highlight the importance of cognitive task analysis for understanding expert performance and show how the epistemic games framework can be used as a tool for this type of analysis, with thermodynamics as an example. Second, through this analysis, we identify several issues related to thermodynamics that are relevant to future research into student understanding and learning of the mathematics of thermodynamics.

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Section: Differentialsmentioning

confidence: 99%

Section: When Asked To Explain Further He Saidmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Partial derivative games in thermodynamics: A cognitive task analysis

Kustusch

Roundy

Dray

et al. 2014

Phys. Rev. ST Phys. Educ. Res.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In some interpretations of the differential, dx is not an independent object; rather, the d is a shorthand for a limit 19 -a notation that is not meaningful unless it is part of the derivative operator d=dx or the integral operator Ð dx. However, in differential geometry, the situation is different: the d refers to an exterior derivative, and it operates on objects called differential forms.…”

Section: Discussionmentioning

confidence: 99%

Distinguishing between “change” and “amount” infinitesimals in first-semester calculus-based physics

Korff¹,

Rebello²

2014

American Journal of Physics

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From the perspective of an introductory calculus course, an integral is simply a Riemann sum: a particular limit of a sum of small quantities. However, students connect those mathematical quantities to physical representations in different ways. For example, integrals that add up mass and integrals that add up displacement use infinitesimals differently. Students who are not cognizant of these differences may not understand what they are doing when they integrate. Further, they may not understand how to set up an integral. We propose a framework for scaffolding students' knowledge of integrals using a distinction between "change" and "amount" infinitesimals. In support of the framework, we present results from two qualitative studies about student understanding of integration.

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“…Essentially, the question is asked, given, say, y = x 2 , if I added a number named dy to all the ys, I would have to compensate by adding some other number dx to all the xs. This possibility is also explored, but not very fully, by Dray and Manogue (2010). Future work may entail discovering which approach is more beneficial to students-starting with slopes as a concrete tie-in to algebra and then pivoting to differentials, or starting more directly with differentials at the very beginning.…”

Section: Derivatives Vs Differentialsmentioning

confidence: 99%