When negative is not "less than zero": Electric charge as a signed quantity
Alexis Olsho, Suzanne White Brahmia, Andrew Boudreaux, Trevor Smith
WWhen negative is not “less than zero”: Electric charge as a signedquantity
Alexis Olsho, Suzanne White Brahmia, Andrew Boudreaux, and Trevor Smith
Electromagnetism (E&M) is often challenging for students enrolled in introductory college-level physicscourses. Compared to mechanics, the mathematics of E&M is more sophisticated and the representationsare more abstract. Furthermore, students may lack productive intuitions they had with force and motion.In this article, we explore the mathematization of electric charge. Specifically, we explore how difficultieswith positive and negative signs can arise for learners who approach integers primarily as positions on anumber line. In sections 1 and 2, we discuss the nuances of electric charge as a physical quantity andsituate it in a body of work by mathematics and physics education researchers to characterize the usesand meanings of signs. In section 3, we describe preliminary research that illustrates the effect of wordingdifferences on student reasoning about electric charge as a signed quantity. Finally, in Section 4, we discussimplications for instruction.
Physicists mathematize the world—that is, physicists create mathematical quantities and relationships toanalyze and explain real-world phenomena [1].
Quantification of electric charge (that is, the representationof electric charge as a quantity) is an example of a specific—perhaps idiosyncratic—use of sign in physics,with “positive” and “negative” acting as labels for different charge states of matter. Indeed, as Arons pointsout, “the names are perfectly arbitrary and could just as well have been chosen to be ‘red’ and ‘blue’ . . . or‘charming’ and ‘revolting’” [2]. The choice of which type of charge to associate with the electron is arbitraryin a further sense—the quantity − µ C is not inherently negative; instead, the “ − ” simply indicates whichtype of charge is present, while the number (with unit) indicates the amount of that type that is present.Consider the two statements “ − ◦ F < +5 ◦ F” and “ − µ C < +5 µ C.” The former is unambiguous:temperatures are conceptualized using a number line and its implied ordering of values. The negativesign signifies a value lower than an established reference, while the positive sign signifies a higher value.The values − ◦ F and +5 ◦ F are not symmetric in a physically meaningful way; using the Celsius scale,the numerical values would no longer have the same absolute value, and would have the same sign. Thestatement “ − ◦ F < +5 ◦ F” simply indicates one temperature is higher than the other.The statement “ − µ C < +5 µ C,” however, introduces ambiguity. The negative and positive signs heresignify which of two types of electric charge is in surplus. − µ C and +5 µ C denote symmetric states, withthe same amount of surplus of the different types of electric charge, while the value 0 would correspond tothe physical state of balance of equal amounts of the two types. The two symmetric charge states wouldstill be represented by numbers with the same absolute value if electric charge were operationalized with adifferent unit of measure. To a physicist, the best answer to the question “Which of these two objects hasmore net charge?” might be “Neither!” We thus argue that − µ C and +5 µ C are better conceptualizedas equal amounts of departure from balance, rather than values ordered on a number line. We furtherargue that the statement “ − µ C < +5 µ C,” due to its close association with number line ordering, canobscure the physically meaningful insight that − µ C and +5 µ C represent states with the same amountsof unbalanced charge.We offer an analogy to color charge. In quantum chromodynamics, each type of color charge (e.g., red)has a corresponding negative (antired). A particle with a red color charge of +5 units thus sums to zeronet color charge with its anti-particle, of red color charge − a r X i v : . [ phy s i c s . e d - ph ] J u l han a positive charge, but not that it “has less charge” than a positive charge. This suggests that fortemperature, the sign does not signify any fundamental opposition of complementary types, but rather anordering along a number line, with 0 denoting a reference.Positive and negative signs are well-suited as labels for electric charge. Because charge is conserved, andthe two types of charge are complementary, positive and negative charges behave as real numbers underaddition and subtraction. For example, the equation 0 − ( − µ C) = +5 µ C concisely describes the removalof − µ C from an electrically neutral object, leaving the object with a net charge of +5 µ C. Moreover, thequantitative statement of Coulomb’s Law ( (cid:126)F , = kq q r ˆ r ) exploits the multiplicative properties of positiveand negative numbers to to express the empirical rule “like charges repel, and unlike charges attract.”Despite the correspondence between electric charge and real numbers, care must be taken when dis-cussing net charge . An object with a net charge of − µ C typically contains a relatively large amountof balanced positive and negative charge, and a small amount of unbalanced negative charge. (Note thathere we use “amount” to refer to an inherently positive quantity; i.e., it is possible to have “an amount ofnegative charge” but not “a negative amount of charge.”) On this basis, we offer a working definition of“net charge”: an object with net charge + Q has an amount Q of unbalanced charge of the positive type,while an object with net charge − Q has an amount Q of unbalanced charge of the negative type. (Q nowrepresents a positive number, rather than a signed number—this definition is making explicit the use ofa sign to represent one of two types, and the use of a positive number to represent an amount.) In mostphysics contexts, the amount of balanced positive and negative charge is unknown and unimportant. Achange from electrically neutral to a non-zero net charge of either sign corresponds to an increase in theamount of unbalanced charge. Our investigation of student reasoning about signed quantities in physics (such as electric charge) has beeninformed by the work of mathematics education researchers. Vlassis studied the development of “flexibility”with the negative sign, finding that understanding and applying different meanings of the negative sign wascorrelated with ability to solve linear equations with one unknown [3]. Vlassis synthesized these meaningsin a map of the “natures of negativity” in algebra. Inspired by Vlassis’s map, and motivated by the relativelack of research on negative physics quantities, we undertook development of a framework for the naturesof negativity in physics [4]. In creating the framework, we identified the use of sign as an identifier of type as unique to physics.Bishop et al. identified productive strategies in students’ reasoning about negative numbers prior toformal instruction [5]. Two of these strategies interested us in particular: number-line reasoning, which“leverages the sequential and ordered nature of numbers,” and magnitude reasoning, which relates numbers(including negative numbers) “to a countable amount or quantity” [5]. As discussed above, “net charge”can be understood as the amount of unbalanced charge present, where the sign specifies the type ofcharge in surplus. This way of thinking is well-served by the magnitude-based reasoning strategy fornegative numbers, which is associated with “the view of a number having magnitude or substance.” In thisapproach, negative numbers may “evoke the idea of opposite (directed) magnitudes” [5]. This contrastswith the number-line-based reasoning strategy, in which students consider quantities as ordered positionson a number line.When using a number line to support addition and subtraction, “one typically treats the start andresult as locations on the number line and the change as a distance” [5]. We do not assert that using anumber line to aid in adding or subtracting would result in incorrect calculations about electric charge.However, ordering reasoning may lead students to treat a net charge of -5 units as intrinsically less than anet charge of 0 units. The student might then plausibly fail to consider the implicit meaning of the sign asa signal for which of the two types of charge is in surplus. While further research could reveal whether ornot such confusion is indeed prevalent, we here simply identify the possibility that the ubiquitous orderingreasoning associated with positive and negative values arrayed on a number line could “crowd out” thedesired physical reasoning involving two distinct types of electric charge.2 student has two electrically neutral spheres, A and B. Initially, sphere A has exactly the same number of protons andelectrons as sphere B. The student touches the spheres to each other. After the spheres touch, the charge on sphere A ismeasured to be q A = µ C, and the charge on sphere B is q B = +5 µ C.Which of the following statements best describe the charges on the spheres after the spheres touch each other? Selectthe statement(s) that must be true.
Choose all that apply.
A. The net charge on sphere A is greater than the net charge on sphere B.B. The net charge on sphere A is less than the net charge on sphere B.C. † The net charge on sphere A is neither greater than nor less than the net charge on sphere B.D. † The number of charged particles in sphere A is greater than the number of charged particles in sphere B.E. The number of charged particles in sphere A is less than the number of charged particles in sphere B.F. None of these.
A student has two electrically neutral spheres, A and B. Initially, sphere A has exactly the same number of protons andelectrons as sphere B. The student touches the spheres to each other. After the spheres touch, the charge on sphere A ismeasured to be q A = µ C, and the charge on sphere B is q B = +5 µ C.Which of the following statements best describe the charges on the spheres after the spheres touch each other? Selectthe statement(s) that must be true.
Choose all that apply.
A. The amount of unbalanced charge on sphere A is greater than the amount of unbalanced charge on sphere B.B. The amount of unbalanced charge on sphere A is less than the amount of unbalanced charge on sphere B.C. † The amount of unbalanced charge on A is equal to the amount of unbalanced charge on sphere B.D. † The number of charged particles in sphere A is greater than the number of charged particles in sphere B.E. The number of charged particles in sphere A is less than the number of charged particles in sphere B.F. None of these.
Figure 1: Versions 1 (top) and 2 (bottom) of the
Charged Spheres questions, intended to probe studentreasoning about net charge and “negativity” in charge. Responses C and D are both correct for bothversions (shown with daggers).
Our investigation of student reasoning about electric charge as a signed quantity began with the develop-ment of a multiple-response test item, the Charged Spheres question, as part of a suite of items designed toinvestigate student interpretation of negative signs in physics contexts [6]. The Charged Spheres questionproved challenging for introductory students, and also, during expert validation of the item, for physicsgraduate students.To investigate further, we modified the item, creating the two different versions shown in Figure 1.These two versions involve the same physical context and wording of the question stem, but use differentwording for the answer choices. In the first version, the answer choices refer to “net charge,” while in thesecond version, the answer choices refer to “the amount of unbalanced charge.” For both versions, answerchoice B is consistent with number-line reasoning—which treats a negative number as less than zero—whilechoice C is consistent with magnitude-based reasoning—which relates a negative number to a countableamount of something. On both versions, selecting both C and D is the correct response.We used the modified Charged Spheres question to explore which wording, if either, might be associatedmore strongly with the interpretation of sign as an indicator of type in the context of electric charge.We wondered whether the more descriptive language of the second version would more strongly cue acomparison of two positive quantities (i.e., the amount of surplus charge on Sphere A and the amount onB), and, correspondingly, whether this more descriptive language might suppress reasoning associated withorder on a number line. We emphasize that answer choice C in the first version of the question (“the netcharge on Sphere A is neither greater than nor less than the net charge on Sphere B”) does not imply thatthe net charge of the spheres are equal. Validation interviews suggested that students were interpreting3 .000.250.500.751.00 A B C D E
Response (a) F r a c t i on o f P opu l a t i on Charged Spheres: Net Charge
Response (b) F r a c t i on o f P opu l a t i on Charged Spheres: Unbalanced Charge
Figure 2: Response frequencies for two versions of the
Charged Spheres question shown in Fig. 1. Note:values do not add to unity because students may choose as many or as few answers as they think arecorrect. Responses C and D are both correct for both versions. Error bars represent the 95% confidenceintervals calculated using the Wilson method for binomial distributions [7, 8, 9].the question as intended. Students choosing C typically explained that “ − ” and “+” are used as labels forcharge type. Moreover, students choosing B justified this answer using reasoning suggesting an overarchingbelief that “negative is less than positive” for net charge.The Charged Spheres question was administered as part of an online, ungraded pretest in the secondquarter of the calculus-based introductory physics sequence at a large public research university. Thestudents were enrolled in three different sections of the same course, with question version being randomlyassigned by section. There were no significant differences in the sections’ average midterm or final examscores. Though the sections had different instructors, instruction was standardized, and all lecturers usedthe term “net charge” or simply “charge” during lecture, consistent with the course textbook. The taskwas administered after all relevant lecture instruction on charge. Each student saw only one version of thequestion.Results from the Charged Spheres question are summarized in Figure 2. On version 1, 67% of students(105 of N = 158) included answer choice C, that the net charge on sphere A is neither greater than norless than the net charge on sphere B. On version 2, however, 93% of students (171 of N = 183) includedchoice C, that the two spheres have equal amounts of unbalanced charge. (A binomial test indicates thatthis difference is statistically significant, with p < . p < . Instructional implications
Reasoning about the concept of electric charge, and of “net charge” in particular, presents a greater learningchallenge than students and instructors might initially recognize, in part due to subtleties in the use ofpositive and negative signs to characterize complementary charges. We offer three suggestions for promotingstudent learning of charge, anticipating that expert instructors will also devise their own approaches.1.
Include explicit language involving “unbalanced charge”
Many instructors and textbooks usethe phrase “magnitude of the net charge” when asking students to consider the amount of chargein surplus. We view this as a missed opportunity to emphasize the subtle and usually implicitinterpretation of sign as the signifier of type in the context of electric charge. Our findings suggestthat students may not spontaneously recognize that − Discourage assumption of “positivity” : We caution against an assumption of positivity whendiscussing charge. We suggest that instructors specify “+5 µ C” (rather than simply “5 µ C”) whendiscussing a positive net charge. The sign of any quantity carries meaning. For electric charge, the signspecifies the type, which in turn determines how the object will interact with other charged objects.Priming students to expect that real-world quantities have associated signs that carry meaning, andthat an unsigned quantity is different than a positively-signed quantity, can help establish a physicshabit-of-mind of actively seeking meaning in the sign. We also believe clearly labeling positive andnegative charge with sign will aid in student recognition that variable or unknown amounts of charge(“ Q ”) could be either positive or negative [4].3. Explicit instruction on zero-sum pairs : We recognize that many students may not have con-sidered ways of conceptualizing integers outside of positions on a number line. Because we believethe number-line strategy can be an obstacle when it is applied to electric charge, we suggest explicitinstruction on the quantification of opposites as zero-sum pairs (i.e., pairs of numbers which sumto zero). There are a number of curricula that present strategies for understanding integers with afocus on zero-sum pairs [10]. Explicit instruction in positive and negative numbers as complementsrather than positions on either side of zero on a number line may help students understand betterthe quantification of electric charge.
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