Experts’ Understanding of Partial DerivativesUsing the Partial Derivative Machine

David Roundy∗, Eric Weber†, Grant Sherer∗ and Corinne A. Manogue∗

∗Department of Physics, Oregon State University, Corvallis, Oregon 97331†College of Education, Oregon State University, Corvallis, Oregon 97331

Abstract. We developed the Partial Derivative Machine (PDM) in response to difficulties we encountered in teaching studentsabout mathematical concepts involving partial derivatives and total differentials that are needed in thermodynamics. ThePartial Derivative Machine is a system that has four observable and controllable properties: two forces and two positions.However, of these four properties only two may be controlled independently. This context-dependence of independent anddependent variables enables the same sort of mathematical flexibility (and confusion) that is present in thermodynamics.Because the PDM is easy to use and understand, we hypothesized that it would allow us to explore the nature of experts’thinking about derivatives, even those unfamiliar with ideas in thermodynamics. In this paper, we present results frominterviews with experts from several disciplines, as we explore how they understand partial derivatives when given anambiguous prompt. The research question guiding this work is "How do experts think about partial derivatives?"

Keywords: partial derivative, thermodynamics, experimentPACS: 01.40.Fk, 01.40.G, 05.70.-a

INTRODUCTION

Thermo is hard [1–4]. In a recent national workshopon the upper-division curriculum, approximately 1/3 ofthe faculty indicated, in an informal show of hands,that they are uncomfortable enough with the content ofthermodynamics that they would be reluctant to teach it.

There are various reasons why thermodynamics ishard. One reason is that thermodynamics involves anumber of different quantities, e.g. entropy, temperature,pressure, volume. Of these four quantities, two are in-dependent, but which two are controlled independentlydepends on the context. Partial derivatives represent im-portant physical quantities, and this ambiguity in the in-dependent variables makes it crucial to pay careful atten-tion to which quantities are held fixed.

Thermodynamics is the first time that our students en-counter scenarios in which the quantities held fixed whentaking a partial derivative are ambiguous. In mathemat-ics courses, students are taught that when taking partialderivatives, all the independent variables are held fixed,or at least all variables are able to be systematically var-ied and held constant. Nevertheless, through experiencein the classroom we have found that most students comeinto our course with a firm belief that when taking a par-tial derivative everything else is held fixed.

We do not think that the issues we have observed withpartial derivatives are limited to students. Indeed, we hy-pothesize that many of the issues we have observed aredue to the ways in which different disciplines use andthink about derivatives and partial derivatives. Moreover,by studying experts’ thinking about partial derivatives,

we will obtain a better benchmark for comparison in thestudy of students’ thinking about those same ideas. Inthis study, we conducted small group interviews with ex-perts in several STEM disciplines. These interviews aremost similar to a clinical interview except that the groupsetting provides a means for participants to listen andrespond to each others’ ideas, rather than just the inter-viewer’s. In the remainder of this paper, we will intro-duce the Partial Derivative Machine (PDM), and give theresults of our analyses from the expert interviews.

What is a derivative? Students need to be fluent inlooking at derivatives using multiple perspectives. Weconsider four different ways to understand and thinkabout a derivative, each of which is useful in differentscenarios.

1. A ratio of small changes. This is the limit definitionof a derivative, and seems to be the first seen bystudents and the first forgotten.

2. A physical measurement to determine its value [5].This involves measuring the small change in onequantity resulting from imposing a small change inanother, and finding their ratio.

3. The slope of the tangent to a curve.4. The result of algebraic manipulation of a symbolic

expression. For many students this is the primaryunderstanding of “finding a derivative.”

In the first two cases, it is most natural to think of aderivative as a number. One picks a point to take thederivative and finds a ratio as a number. While that num-ber will be different at other points—making the deriva-

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tive actually a function—this aspect of the derivativemay often be ignored. When considering the slope ofthe tangent to a curve, it is clearer that the derivative isa function, but it is also natural to think of the deriva-tive as a number. When using the symbolic approach, thederivative is inherently a function, and while that func-tion could be evaluated at a point, its value cannot bedetermined until its functional form is known.

What is a partial derivative? Partial derivatives differfrom ordinary derivatives in important ways. How weunderstand this difference can vary with how we under-stand derivatives.

1,2. When considering a ratio of small changes or ameasurement, a partial derivative requires that weask not only which quantities are changing, but alsowhich quantities to hold fixed.

3. A tangent line turns to a tangent plane in two dimen-sions, and a partial derivative becomes the slope ofthe plane in a given direction at a given point.

4. The procedure to find a partial derivative of a sym-bolic expression is identical to that for an ordinaryderivative, provided there are not interdependenciesamong the variables in the expression. It is unsur-prising that many of our students believe that a par-tial derivative means “everything else is held fixed.”

In thermal physics, the quantities that are being heldfixed are context-dependent. How we respond to thisambiguity depends deeply on our concept of a derivative.

THE PARTIAL DERIVATIVE MACHINE

We have developed and used two versions of the PartialDerivative Machine (PDM). The first version of this de-vice is documented in [6], and features a central systemthat is attached to four strings. The simplified version ofthis device—which will be discussed in this paper—isshown in Fig. 1, and consists of a fixed elastic system,which is constructed of springs and strings. In both ver-sions, the elastic system may be manipulated using twostrings independently. Each of these two strings has ascalar position that can be measured with a measuringtape and a tension that can be adjusted by adding to orremoving weights from a hanger. Detailed instructionson constructing a Partial Derivative Machine, includinga parts list and photographs of additional central systems,are available on our Paradigms website [7].

The usefulness of the PDM emerges because it is anexact mechanical analogue for a thermodynamic sys-tem. The system contains a potential energy U (analo-gous to the internal energy) that cannot be directly mea-sured. The system has four directly measurable—andcontrollable—state properties: two positions x and y and

the system

thumb nuts

weights

flags

vertical pulleys

spring

horizontal pulleys

FIGURE 1. The Partial Derivative Machine.

two tensions Fx and Fy. These four state properties playroles analogous to volume, entropy, pressure and temper-ature in a thermodynamic system.

As in thermodynamics, the choice of independent vari-ables is context-dependent. While it is experimentallyeasiest to control the two weights as independent coor-dinates while measuring positions, it is sometimes theo-retically more convenient to view the positions as the in-dependent coordinates. Most notably, when using workto determine the potential energy, the positions are the“natural” variables, as seen in the total differential that isanalogous to the thermodynamic identity:

dU = Fxdx+Fydy (1)

We can relate this total differential to the mathematicalexpression

dU =

(∂F∂x

)ydx+

(∂F∂y

)xdy. (2)

By equating coefficients of dx and dy, we can find ex-pressions for the two forces as partial derivatives of thepotential energy. This enables us to clarify the interde-pendence of the four directly observable quantities.

We use the PDM to teach a mathematical introduc-tion to thermodynamics prior to our junior-level coursein thermal physics, Energy and Entropy. This introduc-tion uses seven contact hours, and covers the range ofmathematical topics generally taught in undergraduatethermodynamics. We begin with total differentials andintegration along a path, discuss partial derivatives andchain rules, mixed partial derivatives and Maxwell rela-tions, and end with Legendre transformations. Through-out both this mathematical introduction and Energy and

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Entropy, a focus is placed on connecting the mathemati-cal expressions with tangible reality [8].

EXPERT INTERVIEWS

To gain insight into our research question, we performedthree expert interviews, each of which lasted around anhour. We began by introducing our experts to the PDM,and showing them how to manipulate the machine. Thenwe gave them the following prompt:

“Find∂x∂Fx

.”

The purpose of the prompt was to understand their think-ing about notation, their thinking about derivatives andpartial derivatives, and how they related those ways ofthinking to the PDM. During the interview the systemwas visible as in Fig. 1, which is in contrast to our class-room practice of hiding the system under a black box.

When we provided this prompt, we had not defined ei-ther x or Fx, but rather let the interviewees discuss whatthese quantities might mean. After they had discussed themeanings of these terms, and we agreed that their mean-ing was sufficiently clear to us, we clarified if necessarythat x was the position of one flag (i.e. one string) and thatFx was the tension in that string, which was determinedby the weight.

An explicit goal of this task—in fact, a pedagogicaldesign feature of the PDM—was to explore our expertsunderstanding of partial derivatives outside the worldof symbolic manipulation. We asked experts to “find” apartial derivative for which they are given no functionalform, thus forcing them to move outside the wide arrayof partial derivative games involving manipulation ofsymbolic expressions [4].

During the course of the three interviews, a numberof themes emerged. These themes were a combinationof issues we noticed as we did the interviews and issuesthat emerged as we conducted the initial phases of dataanalysis. In the sections below we describe these themesand articulate how we saw each group of interviewees inthe context of that theme.

We interviewed three groups of experts: physicists, en-gineers and mathematicians. For the physicists we inter-viewed one associate professor and one full professor.Both use computational methods in their research, whichis in astrophysics and optics. We interviewed three engi-neers: one chemical engineer who is a full professor withconsiderable research and teaching expertise in thermo-dynamics, and two engineers who study student think-ing and epistemology in engineering. Finally, we inter-viewed two mathematicians who are both assistant pro-fessors and whose research is in mathematics educationat the collegiate level.

Identifying x and Fx

Physicists The physicists’ comments from the outsetof the interview led us to believe that they had a similarunderstanding to ours. They noted that x and y were po-sitions, and soon after concluded that Fx was a tension onone string. However, they did spend some time thinkingthat the position x might be an internal property of thesystem, noting that the position of one end of the springwas fixed in the system.

Engineers The engineers began the interview by al-most immediately moving to measure the derivative us-ing calculations and data collection. With some press-ing, they revealed that they were measuring Fx as tension.However, it was not until the middle of the interview thatthey chose to identify x as the position of one particularstring which led to them reevaluating their initial calcu-lations for the derivative. Prior to this they expressed thetwo positions as xL and xR.

Mathematicians From the outset of the interview,the two mathematicians were extremely puzzled by thesubscript on Fx, noting that “we have not seen this typeof notation before” and “it looks like a derivative but weare unsure what the symbols are.” While they attended tothe position of the strings, they did not interpret Fx as theforce related to x until they were told this.

What is a derivative?

Physicists The physicists immediately (in under 2minutes) jumped to measuring ratios of small changes,and also quickly checked that their changes were smallenough by verifying that the relationship between thetwo quantities was linear. They never sought to determinea symbolic expression. that could describe the system, anissue that was important for other groups.

The physicists did articulate that the derivative itselfis a function, but did not focus on trying to determinea functional form in order to differentiate it. Instead,they understood they were at best going to determine anapproximation of the derivative and proceeded to focuson a ratio of small changes.

Engineers The engineers also immediately found aratio of small changes and verified that the relationshipbetween the quantities was linear. After some time, theydetermined they could create a single-variable plot ofthe points they collected and look at the tangent line toapproximate the derivative. However, this was their onlymention of using a graph in the entire interview.

Like the physicists, the engineers were also aware thatthe derivative was not a constant number, and their mea-surements which entailed varying different properties ofthe system in a systematic way support that the were fo-cused on finding an approximation of the derivative.

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Mathematicians The mathematicians required 30minutes and four prompts from the interviewer before fi-nally recognizing that they could develop an experimen-tal measurement for the derivative. They took data, andsaw that larger changes in x indicated a greater deriva-tive, but did not construct a ratio from their measure-ments. They repeatedly returned to speculation regard-ing the functional form of x, even after being promptedto find a numeric value. In the process, one of the mathe-maticians asked the question, “What is the nature of whatwe are trying to find? Is it a number? a function?”

Eventually they recognized that since the derivativeis the slope, a graph would allow them to measure aslope, and they decided to arrange their data in “orderedpairs” of (Fx,x). This approach led them to determinean approach using slopes equivalent to a ratio of smallchanges, although they never did actually perform a di-vision or write down a ratio.

What about y and Fy?Physicists One physicist decided that Fy must be held

fixed when measuring ∂x∂Fx

, because we were taking aderivative with respect to Fx. Thus if we took the inversederivative, y would have been held fixed. He stated that“this is what he was taught.” When he considered fixingthe thumb nut, he stated that this (physical act?) wouldchange x to be a different function, a function of Fx and y.

When asked how many independent variables therewere, the physicists very quickly recognized that onlytwo variables were independent, and soon acknowledgedthat you could do a change of variables so differentpairs could be considered independent. Based on theirgestures, we interpret that they used “physical” reasoningto decide they could fix either quantity. They expressedthis in terms of x = x(Fx,Fy) and Fy = Fy(Fx,y).

Engineers Like the physicists, the engineers discussedholding either y or Fy fixed, and assumed at first the forceneeded to be held fixed. Late in the interview, one of theengineers asked the others how many independent vari-ables they thought there were, and it was not immediatelyobvious that the others saw that there were only two in-dependent variables. They wrote down a function x withthree arguments x(Fx,Fy,y).

Mathematicians The mathematicians were veryquick (once they knew what the variables meant) torecognize that x depends on Fx and Fy, and made ef-fective use of the PDM to qualitatively investigate thedependence of the derivative ∂x

∂Fxon other quantities.

They observed that holding y fixed would give a differ-ent result than holding Fy fixed well before making anyquantitative measurements, but did not investigate thisdependence any further. They simply assumed (like onephysicist) that the other force was the “right” quantity tohold fixed.

CONCLUSIONS

Although we have not observed in physics studentsthe mathematicians’ specific confusion regarding Fx,the use of x and y as the two independent positionvariables—particularly in variables that are not spatiallyorthogonal—has consistently created confusion both inexperts and novices. In future, we intend to name ourfour variables x1, x2, F1, and F2.

There is a spectrum of the ability of the experts to rec-ognize and exploit the definition of a derivative as a ratioof small changes to answer the interview prompt. Thisdefinition was the most obvious to the physicists, but wenote that the two physicists that we interviewed both hadextensive computational experience, and this definitionis a standard numerical method. We intend to interviewadditional physicists in order to obtain a more represen-tative sample. Interestingly, in spite of the prominenceof the limit definition of a derivative in mathematicscourses and texts, our mathematicians took the longest toleverage this definition. This suggests to us that our stu-dents who will have received most of their training aboutderivatives from mathematicians may require scaffoldingto learn to use this definition explicitly and thoughtfullyin our courses.

All three groups of experts reached the conclusion thatwhen differentiating with respect to Fx, Fy was more nat-ural to hold fixed than y. The prompt was deliberatelyambiguous and there is no physical need for this assump-tion. We will think further about the pedagogical impli-cation for this surprising result.

ACKNOWLEDGMENTS

The funding for this project was provided, in part,by the National Science Foundation under Grant Nos.DUE 0618877, DUE 0837829, DUE 1023120, and DUE1323800.

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