The article discusses the importance of the ordering of symbols in physics formulas and identify conventions that rule the standard form for how formulas are written and interpreted. The elements of this form consisting of constants, parameters, and variables have been identified. Instructors are encouraged to make this convention explicit and encourage its use to assist students

Jackson, Martin

Moelter, Matthew J.

English

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University of Puget SoundSound IdeasAll Faculty Scholarship Faculty Scholarship11-1-2012Formulas in Physics Have a “Standard” FormMatthew J. MoelterMartin JacksonUniversity of Puget Sound, martinj@pugetsound.eduFollow this and additional works at: http://soundideas.pugetsound.edu/faculty_pubsThis Article is brought to you for free and open access by the Faculty Scholarship at Sound Ideas. It has been accepted for inclusion in All FacultyScholarship by an authorized administrator of Sound Ideas. For more information, please contact soundideas@pugetsound.edu.CitationMoelter, Matthew J, and Martin Jackson. "Formulas in Physics Have a "standard" Form." Physics Teacher. 50.8 (2012): 472-474. Print.Formulas in Physics Have a “Standard” FormMatthew J. Moelter and Martin Jackson  Citation: The Physics Teacher 50, 472 (2012); doi: 10.1119/1.4758146 View online: http://dx.doi.org/10.1119/1.4758146 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/50/8?ver=pdfcov Published by the American Association of Physics Teachers  Articles you may be interested in “Developing Assessments for the Next Generation Science Standards” by the NRC Phys. Teach. 52, 126 (2014); 10.1119/1.4862133  All graduate medical physics programs should have an original research component Med. Phys. 38, 2315 (2011); 10.1118/1.3533902  Physics Representations in State Standards Phys. Teach. 45, 262 (2007); 10.1119/1.2731265  Bringing extra-solar planets into the introductory physics classroom Phys. Teach. 39, 120 (2001); 10.1119/1.1355173  Response to teaching modern physics Phys. Teach. 39, 4 (2001); 10.1119/1.1343418    This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:207.207.127.233 On: Mon, 13 Oct 2014 23:11:08472	 The Physics Teacher ◆ Vol. 50, NoVember 2012        DOI: 10.1119/1.4758146A student solved the problem correctly using the ap-propriate reasoning (concentric spherical Gaussian surface, constant electric field magnitude on surface, area of sphere, charge enclosed, etc.). Here is the student’s answer for part (a),3                                .              (1) To the practiced eye this appears a bit odd. Why? The for-mula is not written in the “standard” form (see below). But does this really matter? For this student, yes.Part (b) of the problem asked the student to make a sketch based on the result from the first part. The student’s sketch was a horizontal line on properly labeled Er-versus-r axes. During a face-to-face discussion after the exam, it became clear that the student did not “see” the factor of r2 buried in the denominator when attempting to recognize and sketch the functional form of the spatial dependence of the field.The student’s answer, Eq. (1), contains all of the correct symbols in a mathematically acceptable position (products, numerator, denominator, etc.). From a mathematical stand-point the answer is exactly “right,” by which we mean that if you substituted numbers for each of the symbols you would obtain the correct numerical result. However, colleagues with whom we shared examples such as this described the formu-las as “awkward,” or “difficult to interpret,” or “confusing.” What is it that makes them seem odd? The “standard” form of physics formulasExamining the displayed formulas in almost any physics text reveals a standard convention used when physicists write formulas. Let’s look at two typical formulas:              (2)and r The placement of each of the symbols in these formulas appears in an arrangement that follows canonical (but un-written) rules. Almost all expressions used in physics texts conform to a “standard” order in which each of the terms is written in the form (constants)(parameters)(variables).           (3)If the term is a fraction, the ordering can be applied to the numerator and denominator separately or factored into a product. The ordering is also applied within the arguments of transcendental functions.A first step in helping students learn to read and write Formulas in Physics Have a “Standard” FormMatthew J. Moelter, California Polytechnic State University, San Luis Obispo, CA Martin Jackson, University of Puget Sound, Tacoma, WA We discuss the importance of the ordering of sym-bols in physics formulas and identify implicit conventions that govern the “standard” form for how formulas are written and interpreted. An important part of writing and reading this form is understanding distinctions among constants, parameters, and variables. We delineate these conventions and encourage instructors to make them explicit for students.In most cases the natural phenomena described by physi-cal theories are represented in the language of mathematics. To succeed, students must understand this language both in terms of the underlying mathematics and how it is used in physics. Physics curricula typically require students to have mathematical (pre)co-requisites along with their physics courses. Mathematics, as used by physicists, is a language complete with a grammar and notational conventions. In particular, formulas used by physicists have particular con-ventions that help to expose different aspects of the physical content. Romer1 has an excellent discussion of the process of “reading the equations” of physics. More recently, Hewitt2 encourages the conceptual understanding of the equations as a guide to problem solving. However, this type of under-standing is only possible if one can read the language of the formulas. We suggest that there are implicit grammatical rules concerning the writing of formulas in physics, and that these conventions should be explicitly identified and taught to enhance student understanding.Example of a student’s formulaA student’s improper use of mathematical notation can readily lead to confusion for both the student and instructor. Many of us are familiar with the following difficulties: mix-ing upper and lower case (M versus m); failing to properly indicate the vector nature of a quantity (F = ma ); or careless use of sub/superscripts, m2 versus m2 versus m2. These are examples of incorrect symbol usage and students are typically instructed on their proper use. There is, however, a more sub-tle issue regarding the way symbols are arranged in a formula, even when all the symbols are present and correct.Consider the following exam problem posed to students in an introductory calculus-based course:(a) Use Gauss’s law to find the electric field outside of an ob-ject with a spherically symmetric charge distribution and total charge 2Q. (b) Sketch the magnitude of the electric field outside the sphere as a function of the distance from the center of the sphere, Er versus r.  This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:207.207.127.233 On: Mon, 13 Oct 2014 23:11:08The Physics Teacher ◆ Vol. 50, NoVember 2012                                      473 [everything]            (4) [parameter/variable]          (5) [variable]           (6)From the second version the experienced reader can readily see that the field doubles upon doubling the charge density or halving the distance. This “standard” form is important when physicists interpret and discuss the physical meaning of a formula.While some students will write formulas properly by imi-tation or through repetition, many will not. Because there is such a strong adherence to the formula conventions, we can help students communicate more effectively by being explicit about these conventions. To help students better use and ap-preciate the power of the mathematical description of the world, we suggest that instructors (and textbook authors) ex-plicitly discuss the conventions regarding the “standard” form and the ordering of quantities in formulas. This might be done by following a treatment similar to that presented here. This should include a discussion surrounding the distinctions among constants, parameters, and variables. As with most things, achieving facility requires repetition and practice with timely feedback. Therefore, instructors should provide explicit practice in working with the standard form and should enforce the standard form when evaluating student work. For example, instructors could ask students to rewrite Eq. (1) in “standard” form and then expect one of the following:                                  (7) In evaluating student work, we might invent a new copyedit-ing symbol such as “NSF” for “not in standard form.”ConclusionPhysics formulas follow implicit conventions and are writ-ten in a “standard” form. We have identified the elements of this form consisting of constants, parameters, and variables, in that order. To assist students, instructors are encouraged to make this convention explicit and encourage its use.AcknowledgmentsWe thank T. Bensky, E. Demsetz, and J. Keller for helpful discussions. MJM thanks the Physics Education Research Group in the School of Physics at the Dublin Institute of Technology for their hospitality during a sabbatical leave. formulas using this standard order is to help them classify the various quantities at hand into the categories of constants, parameters, and variables. Constants:  These are mathematical or physical quantities that never change such as • numbers in fractional or decimal form:  , 4, 2.43, … • named numbers: e, π, γ, … • physical constants:4    G, e0, c, kB,  … If more than one “constant” appears, then they are usually ordered in the sequence given above, e.g., in 4pe0, the order is number/named number/physical constant.Parameters:  These “quantities are constant for a particu-lar experimental run but can change from run to run.”5 Parameters are quite important and many times crucial to our physical understanding. Let’s take the example of a ball dropped from rest near the surface of a planet, where the position is given by  To appreciate the significance of the parameters h0 and g, think in terms of making measurements. For each experi-mental run, the quantities h0 and g have a fixed value, while t and y change during the experiment. In another experi-mental run, we might change the initial height, so h0 would be different, or we could go to a different planet and g would change. Thus, h0 and g are not “constants” in the same way that π or c represent particular mathematical or physical constants. So, a parameter is a quantity such as h0 or g that is constant for a particular experimental run, but might change from run to run. Variables:   These are usually the quantities of most physical interest and correspond to time, position, electric field, or ….  They are most like the inputs and outputs of the math-ematical expression of functions.Of course one person’s parameter might be another person’s variable. The distinction generally depends on the context and which quantity is of mathematical or physical interest.Be explicit with students: The “standard” ordering mattersWhen physicists discuss phenomena using mathematics, the “standard” ordering can be a crucial part of the discus-sion. Typically we want to understand the essential behavior of one physical quantity as a function of some other quantity.For example, how does the electric field depend on dis-tance or on charge? Specifically, consider the magnitude of the electric field for a long, thin rod with uniform linear charge density l0. We show three versions of the formula that exhibit different emphases [as indicated in square brackets]:  This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:207.207.127.233 On: Mon, 13 Oct 2014 23:11:08474	 The Physics Teacher ◆ Vol. 50, NoVember 2012References1. Robert H. Romer, “Reading the equations and confronting the phenomena–The delights and dilemmas of physics teacing,” Am. J. Phys. 61, 128–142 (Feb. 1993).2. Paul G. Hewitt, “Equations as guides to thinking and problem solving,” Phys. Teach. 49, 264 (May 2011).3. Discussion and follow-up interview with student in calculus-based introductory course for science and engineering stu-dents.4. In advanced courses, especially those in high energy or particle physics, common practice is to use a system of units such that ħ = 1 or c = 1 and the like. However, this is not common at the introductory or intermediate level, so we ignore it here.5. Regarding parameters we closely follow the discussion in An-drew F. Rex and Martin Jackson, Integrated Physics and Calcu-lus (Addison-Wesley, New York, 2000), p. 39.Martin Jackson has been a professor of mathematics at the University ofPuget Sound since 1990.  In collaboration with colleagues from physics, he has team-taught a course that integrates the introductory physics and calculus sequences. The text Integrated Physics and Calculus, co-authored with Andrew Rex, is based on that course.Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, WA 98416; martinj@pugetsound.eduMatthew Moelter is a professor of physics at Cal Poly and teaches all manner of introductory physics. He has particular interests in physics education and condensed matter.Department of Physics, California Polytechnic State University, San Luis Obispo, CA 93407; mmoelter@calpoly.eduPhysics teachers...get your students registered for the preliminary exam in the U.S. Physics Team selection process.All physics students are encouraged to participate in the American Association of Physics Teachers’ Fnet=ma Contest!The Fnet=ma Contest is the United States Physics Team selection process that leads to participation in the 44th Annual International Physics Olympiad (IPhO) in Copenhagen, Denmark, July 7-15, 2013. The U.S. Physics Team Program provides a once-in-a-lifetime opportunity for students to enhance their physics knowledge as well as their creativity, leadership, and commitment to a goal.For program information and registration visit: http://www.aapt.org/physicsteam(Registration is open now through December 14, 2012)American Astronomical Society This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:207.207.127.233 On: Mon, 13 Oct 2014 23:11:08

Formulas in Physics Have a “Standard” Form

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Formulas in Physics Have a “Standard” Form

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