A framework for the natures of negativity in introductory physics
Suzanne White Brahmia, Alexis Olsho, Trevor I. Smith, Andrew Boudreaux
AA framework for the natures of negativity in introductory physics
Suzanne White Brahmia, Alexis Olsho, Trevor I. Smith, and Andrew Boudreaux Department of Physics, University of Washington, Box 351560, Seattle, WA 98195-1560, USA Department of Physics & Astronomy and Department of STEAM Education,Rowan University, 201 Mullica Hill Rd., Glassboro, NJ 08028, USA Department of Physics & Astronomy, Western Washington University, 516 High St., Bellingham, WA 98225, USA
Mathematical reasoning skills are a desired outcome of many introductory physics courses, particularlycalculus-based physics courses. Positive and negative quantities are ubiquitous in physics, and the sign car-ries important and varied meanings. Novices can struggle to understand the many roles signed numbers playin physics contexts, and recent evidence shows that unresolved struggle can carry over to subsequent physicscourses. The mathematics education research literature documents the cognitive challenge of conceptualizingnegative numbers as mathematical objects—both for experts, historically, and for novices as they learn. Wecontribute to the small but growing body of research in physics contexts that examines student reasoning aboutsigned quantities and reasoning about the use and interpretation of signs in mathematical models. In this paperwe present a framework for categorizing various meanings and interpretations of the negative sign in physicscontexts, inspired by established work in algebra contexts from the mathematics education research community.Such a framework can support innovation that can catalyze deeper mathematical conceptualizations of signedquantities in the introductory courses and beyond.
I. INTRODUCTIONA. Motivation
Experts in physics translate fluidly between differ-ent representations of phenomena. To an expert, aphysics equation “tells the story” of an interaction orprocess. For example, when reading the equation x ( t ) = +40 m + ( − m/s ) t + ( − . m/s ) t , an expertmay quickly construct a mental story of the co-variation ofposition and time of a projectile that starts m above theground and is launched with a speed of m/s vertically down-ward. Part of the challenge of learning physics is developingthe ability to decode symbolic representations in this manner.In these translation processes, experts can readily attributespecific meanings to positive and negative signs. In the ex-ample above, the positive sign in front of the term m indi-cates that the projectile starts at a position that is in the pos-itive direction from the origin, upward in this case becausethe gravitational acceleration is always downward and hap-pens to be negative in this expression. The positive sign afterthe m term indicates that the following term, ( − m/s ) t ,represents an additive change in the projectile’s position—a one-dimensional vector quantity that has the initial value40 m and is in the positive direction. The sign in front of m/s indicates that the projectile is launched downward .Other examples of the fluid interpretation of signs aboundin introductory physics: • In the equation (cid:126)F = − (cid:126)F , the negative sign signalsthat the force exerted by object 2 on object 1 is in theexact opposite direction as the the force exerted by 1on 2. • In the expression − ( − µ C), the first negative signindicates that a quantity of electric charge is being re-moved from an electrically neutral object, while the second negative sign indicates which of the two dif-ferent types of electric charge is being removed. • In Faraday’s law, E = − d Φ B d t , the negative sign re-minds the expert that the voltage induced by a chang-ing magnetic flux acts to oppose (rather than reinforce)the change that created it.In all of these cases, experts generally decode specificmeanings of the sign quickly and effortlessly, perhaps in mostcases without conscious awareness of the decoding processitself. Novices may need to spend considerable conscious ef-fort interpreting the sign, or may fail altogether to success-fully interpret it. Pitfalls likely to challenge the novice mightinclude a tendency to overgeneralize a particular interpreta-tion, or a lack of awareness that two nearby signs in an equa-tion may have completely different physical interpretations.The challenge for introductory physics teaching may be com-pounded if the already difficult cognitive task of interpretingsigns goes unaddressed when instructors themselves are notconsciously aware of the mismatch between their own abilityand the much lower skill level of novices.An additional layer of complexity is introduced by the im-plicit nature of the sign of many symbolically-representedquantities. For example, the expression ∆ U g is a stand-infor a generalized number of Joules, which could be positiveor negative. The sign in this case would indicate whetherthe potential energy of some system increased or decreasedduring a particular process. U g is also a stand in for a gener-alized number—but in contrast to ∆ U g , the sign of U g tellsan expert whether the energy of a system in a particular con-figuration is larger or smaller than the energy associated witha pre-established reference configuration.In a fast-paced introductory physics course, this nuancedinterpretation of sign may fall by the wayside, as instructorsattend to a host of other, perhaps more obvious challenges.Unfortunately, student difficulties with decoding sign, if un- a r X i v : . [ phy s i c s . e d - ph ] M a r addressed, may not spontaneously resolve. In such cases, dif-ficulties with signs could contribute to serious obstacles inthe development of overall quantitative reasoning and anal-ysis skills so highly prized by physicists. In this article, wedescribe an effort to systematically parse and document thevarious meanings of sign in introductory physics contexts.We hope that a taxonomy of this type can support instructorefforts to nurture their students’ ability to translate betweenrepresentations, and can support the efforts of physics educa-tion researchers to more fully understand the nature of studentreasoning about signs and signed quantities. B. Prior Research
Negative pure numbers represent a more cognitively diffi-cult mathematical object than positive pure numbers do forpre-college mathematics students [1]. Mathematics educa-tion researchers have isolated a variety of “natures of neg-ativity” fundamental to algebraic reasoning in the context ofhigh school algebra—the many meanings of the negative signthat must be distinguished and understood for students to de-velop understanding [2–4]. These various meanings of thenegative sign form the foundation for scientific quantifica-tion, where the mathematical properties of negative numbersare well suited to represent natural processes and quantities.Physics education researchers report that a majority of stu-dents enrolled in a calculus-based physics course struggled tomake meaning of positive and negative quantities in spite ofcompleting Calculus I and more advanced courses in math-ematics [5, 6]. Developing “flexibility” with negative num-bers (the recognition and correct interpretation of the multiplemeanings of the negative sign) is a known challenge in math-ematics education, and there is mounting evidence that rea-soning about negative quantity poses a significant hurdle forphysics students at the introductory level and beyond [7–9].Few published studies have focused on the use of the negativesign, i.e., negativity , specifically in the context of the mathe-matics used in physics courses. Studies conducted in the con-text of upper-division physics courses reveal robust studentdifficulties associated with interpretation of the vector natureof acceleration and its representation in Newton’s second law,and with contexts in E&M in which there are often multiplenegative signs, each with a separate meaning [7, 9].Brahmia and Boudreaux conducted studies to probe stu-dent difficulties with negative quantities at the introductorylevel. The authors constructed physics assessment itemsbased on the natures of negativity from mathematics ed-ucation research [10] and administered them to introduc-tory physics students in the introductory sequence of courses[5, 6, 11]. The authors report that students struggle to rea-son about signed quantity in the contexts of negativity typ-ically found in the introductory curriculum (e.g., negativework, negative direction of acceleration or electric field in onedimension), and they concluded that science contexts mayoverwhelm some students’ conceptual facility with negativ- ity. These studies reveal that signed quantities, and their vari-ous meanings in introductory physics, present cognitive diffi-culties for students that many don’t reconcile before complet-ing the introductory sequence. These difficulties then carryover into upper-division course work.
C. Contribution to the literature
The current study advances this body of research by intro-ducing a framework for categorizing the natures of negativityin introductory physics (NoNIP), analogous to the natures ofnegativity developed in the context of algebra [10]. While werecognize that students struggle with signed quantities moregenerally, we choose to focus on negatively signed quantitiesin this work because they are the only signed quantities forwhich the sign is always explicitly included (e.g., a velocityin one dimension of 3 m/s is typically assumed to be in thepositive direction). The intention is to provide a frameworkthat can help researchers and instructors characterize and ad-dress the mathematical conceptualization of signed quantityin introductory physics.In the next section, we describe the development of theNoNIP, including its basis in an analogous framework devel-oped by math education researchers. We present the NoNIP,along with examples of quantities and relationships that il-lustrate its use. We end Section II with a discussion of thevalidation of this framework in the context of introductoryphysics.In Section III, we analyze three recent studies in upper di-vision physics using the NoNIP framework as a lens throughwhich student cognition can be categorized and understood.Section IV describes our exploration of student understand-ing of “positivity” and the necessity of extending this work tosigned quantities more generally. We discuss our conclusionsand implications for instruction in Section V.
II. MODELING THE NATURES OF NEGATIVITY
In this section we discuss the need for and development ofa framework to understand the uses of the negative sign inintroductory-level physics. In section II A, we include workby researchers that has directly influenced and guided the de-velopment of the NoNIP framework. In section II B, we de-scribe the process by which this framework was developed,expert validated and, as a result of expert input, modified. Wepresent the current version of the framework in section II C.
A. Underpinnings
Our initial model for the natures of negativity in introduc-tory physics was based on the natures of negativity in elemen-tary algebra, described by Vlassis [10]. Vlassis summarizedthe work of mathematics education researchers, identifyingthree distinct algebraic natures of negativity. The first, re-ferred to as the “unary” nature, describes situations in whicha negative sign is used in close association with a single quan-tity, and includes the formal concept of a negative number(e.g., the number − ). Two additional natures signify mathe-matical operations: the “binary” nature describes various con-ceptualizations of the negative sign as it is used in subtraction(e.g., − ), while the “symmetrical” nature describesuse of the negative sign to invert (i.e., take the opposite of)a number or operation (e.g., the first negative sign in the ex-pression − ( −
5) = 5 ). Table I summarizes these algebraicnatures.We found that while most uses of the negative sign thatarise in introductory physics could be categorized using themap of Vlassis, the nuances of the physics described by themath were often lost. We also found ourselves tempted torepresent the physical meaning of a negative sign attachedto a single quantity ( “unary” category in Table I) using thecategories intended for operations, as negative quantities inphysics in some cases represent a process rather than just anamount (e.g., system). Moreover, it was difficult for physicsexperts to reach consensus when attempting to categorizesome quantities and relationships. The mathematical naturesof the negative sign described by the Vlassis framework gaveus key insights into the meaning of negative signs in the con-text of physics, but we ultimately concluded that the use ofmathematics as a language to describe physics quantities andrelationships (i.e., quantification ) requires a different catego-rization than pure algebra.Vlassis’s map is based in semiotics, the study of symbolsand their meanings. We believe this is appropriate for de-scribing meanings of the negative sign in mathematics. Tointerpret the meaning of the negative sign in the mathemat-ics used in physics, a blended processes framework is moreappropriate. Conceptual blending theory (CBT) provides aframework for understanding the integration of mathemati-cal and physical reasoning. In their theory, Fauconnier andTurner describe a cognitive process in which a unique men-tal space is formed by merging two (or more) separate mentalspaces [12]. The blended space can be thought of as a productof the input spaces, rather than a separable sum. According toCBT, development of expert quantification in physics wouldoccur not through a simple addition of new elements (physicsconcepts) to an existing cognitive structure (arithmetic), butrather through the creation of a new and independent cogni-tive space. This space, in which creative, quantitative analysisof physical phenomena can occur, involves a continuous in-terdependence of thinking about the mathematical and physi-cal worlds.The design of the NoNIP was further inspired by Sherin’swork on symbolic forms, which posits that. . . successful (physics) students learn to under-stand what equations say in a fundamental sense;they have a feel for expressions, and this under-standing guides their work. . . from the point ofview of improving instruction, it is absolutely critical to acknowledge that physics expertise in-volves this type of flexible and generative un-derstanding of equations. We do students a dis-service by treating conceptual understanding asseparate from the use of mathematical notations[13].We take the approach that a preliminary step in helpingstudents develop a feel for expressions in a way that can pro-ductively guide their work is to understand better how expertconceptualization is organized, and thus characterize an ex-pert’s feel for expressions in a way that is useful both to edu-cation researchers and instructors.We note that the physics contexts that are typically used asapplications in a mathematics course are limited. Nonethe-less, mathematics education research into student reasoningwith quantity helps to build a framework for thinking aboutmathematical objects in physics contexts. In their study in-volving middle school through graduate mathematics stu-dents’ conceptualization of negative quantity, Chiu identifiedthree categories of metaphorical reasoning used by middleschool students, undergraduate, and graduate mathematicsand engineering majors during problem-solving interviewsthat focused on arithmetic with signed numbers. The cat-egories isolated by the researchers are: motion (movementalong a number line), the manipulation of objects/opposingobjects (removing or acting in opposition), and social transac-tion (associated with the experiences of giving and exchang-ing) [14]. While these are metaphors in mathematics, they arein fact contexts in physics in which a conceptual mathemati-cal understanding is essential for learning the physics. In fact,the entire content of mechanics is focused on actual motionin space (not motion along an abstract number line). It is theinterplay between physical quantities and their representationthat motivates the creation of a negativity framework specificto physics.
B. Initial Development
To create a physics-specific map of the natures of nega-tivity, we began by generating a list of physics relationshipsinvolving an explicit negative sign (e.g., (cid:126)F spring = − k(cid:126)r ), andby considering base quantities (e.g., position, charge) and de-rived quantities (e.g., velocity, electric field) that can be neg-ative or associated with a negative sign. Our intent was todevelop a framework drawing on the practices and conven-tions of physics—which necessarily exist in a blended spaceof mathematics and physics—to make meaning of negativity.Learning scientists have used card-sorting tasks to investi-gate mental organization of disciplinary knowledge [15, 16].Experts are given cards showing various content with no pre-established groupings. They are then asked to sort the cardsinto groups that they feel make the most sense and describeeach group. Two of the authors employed a modified card-sorting task of the quantities and relationships that make up TABLE I. A map of the different uses of the negative sign in elementary algebra [10]Unary (Struct. signifier) Symmetrical (Oper. signifier) Binary (Oper. signifier)Subtrahend Taking opposite ofor inverting the operation CompletingRelative number Taking awayIsolated number Difference between numbersFormal concept of neg. number Movement on number line the introductory physics course. The sorting process wasmodified to focus specifically on the role of the negative signfor each quantity or process. The task resulted in broad cat-egorization based on physical similarities, gradually refinedthrough further discussion and comparison with other quan-tities. Creation of a “Change” category exemplifies this pro-cess. The importance in physics of change and of conservedquantities led us to categorize many different and seeminglydisparate uses of the negative sign in a category related tochange, because of an underlying connection to the calcula-tion of and reasoning about change. This includes the nega-tive sign as an operator (to calculate change or to signify thephysical removal of a quantity from a system); the negativesign as an indication of a decrease in a quantity; and the com-pound use of the negative for signifying a change in a quantityas well as calculating it.Along with
Change , the emergent categories were
Direc-tion and
Opposition . A fourth category,
Compound wasadded to account for cases that require interpretation of mul-tiple negative signs in a single context. We note that the
Di-rection and
Opposition categories are supported by the cate-gories isolated Chiu’s study [14]. Phenomena that arise dueto the parallel or antiparallel orientations of two quantitiesare ubiquitous throughout physics (e.g., speeding up/slowingdown, friction and air resistance, electromagnetic induction).Direction and Opposition are central natures of signed quan-tities in physics, and hallmarks of physics reasoning.To allow for further refinement, subcategories emergedwithin each of the main categories. Some subcategories alsospecify the mathematical function of the negative sign (as inthe
Change category’s subcategory
Difference (operator) ).Table II shows the resulting map of the natures of negativityin introductory physics. We do not attempt to further explainthe meanings of the categories and subcategories shown inthe table, as further research led us to reorganize this earlyversion of the NoNIP.The initial version of the NoNIP, Table II, was extensivelyvalidated. We assessed face validity by surveying introduc-tory physics textbooks, using NoNIP to categorize all in-stances of the use of negative signs. With one notable excep-tion (negative exponents), we found that all uses of the nega-tive sign could be categorized satisfactorily using the NoNIP.Expert validation of the NoNIP was conducted by perform-ing formal, semi-structured interviews with two experts eachin mathematics and physics. Our physics experts are expe-rienced introductory-level instructors; our mathematics ex- perts taught introductory- and intermediate-level undergradu-ate mathematics (for example, single- and multivariable cal-culus and differential equations), with sufficient backgroundin physics to understand physics-specific meanings (e.g., oneof our mathematics experts had an undergraduate degree inphysics). During interviews lasting 30–60 minutes, expertswere asked to comment on the appropriateness of the map inthe context of introductory physics, as well as any uses of thenegative sign that were not compatible with the NoNIP frame-work. Mathematics experts were also asked to comment onour interpretations of the negative sign from a more algebraicperspective. Expert comments were very supportive.
C. Steady state version of the NoNIP
While feedback from experts in interviews—and duringless formal interactions during conference presentations—was positive, comments led to a number of changes to theNoNIP. These changes were initially small, but ultimately ledto a re-organization. Each of the mathematics experts inter-viewed stressed the importance of recognizing the meaning of“zero” in any given context. This led us to consider the neg-ative sign in the “unary” sense (that is, attached to a singlequantity) separately from other uses, and allowed us to dis-tinguish more clearly between scalar and vector quantities.Comments by physics experts led us to consider when quan-tities are opposite one another (as − is opposite +5 , suchthat +5 + ( −
5) = 0 ) and when quantities oppose one another(as in Faraday’s law, E = − d Φ B d t ). Finally, we recognized theimportance of differentiating between the negative sign as anoperator and its other uses.Although we reorganized the NoNIP to have categoriesthat are more algebraically based, we believe that the majorstrengths of both the initial and revised versions is the physi-cal interpretation present in the main categories as well as thesubcategories.The revised NoNIP is shown below in Table III. Here, wefocus on the functions of the negative sign: specifying its usewith a quantity , defining the relationship between two quan-tities, or as an operation . The subcategories give meaningspecific to physics. We have removed the compound categorybecause it is not parallel to the others, which will be discussedmore fully below. It is worth noting that some quantities (suchas mechanical work and electric charge) appear in multiplecategories. This speaks to the challenges that students face TABLE II. Initial version of the natures of negativity in introductory physics, a framework for the different uses of the negative sign inintroductory physics (D) Direction (O) Opposition (Ch) Change (Co) Compound
1. Location 1. Opposite type 1. Removal (operator) 1. Scalar rates of change x Q (charge) − ( − µC ) d φ d t
2. Direction of motion 2. Opposes 2. Difference (operator) 2. Base + change v x , ∆ x (cid:126)F = − (cid:126)F E f − E i φ + d φ d t tp x (cid:126)F = − (cid:126) ∇ U (cid:126)p f − (cid:126)p i (cid:126)v + (cid:126)at
3. Other vec. quant. comp. E = − d Φ B d t
3. System scalar quantities 3. Products f ( x ) dxE x , B x (cid:126)F = − k(cid:126)r ∆ K, ∆ E E ( r ) drF x , L z
3. Scalar products ∆ S P ( V ) dVa x W = (cid:126)F · ∆ (cid:126)x
4. Scalar, vector change 4. Models ∆ p x , ∆ v x Φ = (cid:126)B · (cid:126)A ∆ E = E f − E i , ∆ V = V f − V i W net,ext = ∆ E
4. Above/below reference ∆ (cid:126)p = (cid:126)p f − (cid:126)p i (cid:126)F net = m(cid:126)a T (temperature) ∆ U = Q − W V (electric potential) when trying to decode and make sense of negative signs inintroductory physics.
1. Description of the natures of negativity in introductory physics
The
Quantity (Q) category is most similar to the mathe-matical
Unary category described by Vlassis (i.e., identifiesa number as negative), with the negative sign attached to asingle quantity. The Q category is subdivided for scalar (1)and vector (2) quantities; because the negative sign associ-ated with a quantity has different meanings for scalars andvectors, this subdivision is appropriate here.Four subcategories exist for scalar quantities. The first sub-category,
Type (1a) is reserved for electric charge. In thecase of electric charge, sign specifies the type of charge. Wecan identify no other contexts that use the negative sign thisway. The subcategory
Change (1b) is for scalar quantitiessuch as ∆ E , as well as scalar time rates of change, such as d φ d t , where a negative sign typically indicates that the quan-tity decreases with time. We also include differentials in thiscategory, as a negative differential indicates an infinitesimaldecrease in a quantity. Next, scalar quantities such as temper-ature, electric potential, and energy can be negative relativeto an arbitrary reference point “0”; these quantities are in thesubcategory Comparison to reference (1c). Finally, we con-sider scalar quantities for which the sign carries importantphysical meaning that is an artifact (sometimes arbitrary) ofgiven
Models/convention (1d). We consider quantities suchas heat Q (which is negative for a system when heat is trans-ferred out of that system), the net external work done on asystem (which, when negative, signifies that the mechanicalenergy of the system decreases), and current i (which is neg-ative when opposite to the sense arbitrarily decided to be pos-itive) to be described by this subcategory.For vector quantities, we consider two subcategories forvector components: Direction from origin (2a), which de-scribes the direction relative to a coordinate system or origin, and describes quantities such as components of position, elec-tric field, and velocity; and
Direction of change (2b) for com-ponents of differences of vector quantities, such as changesin momentum and in velocity. We consider these two vectorsubcategories to be distinct from each other. For isolated vec-tor components such as E x , we consider a single vector andhow it compares to a coordinate system. For vector compo-nents such as ∆ (cid:126)v x , the direction tells us something about thedifference between two vectors.Although the category Relationship (R) is most similarto the algebraic category
Symmetrical (operational signifier) (i.e., when a negative sign is used to take the opposite of orinvert a number or operation), we do not consider the neg-ative sign to be indicative of an operation for the quantitiesdescribed in this category; rather, the negative signs in thiscategory signify how quantities relate to each other. Whilein mathematics a negative sign may be used as an operatorthat defines one quantity as another’s opposite, in physics thenegative sign may describe that two quantities are inverselyrelated. In this category (unlike in the Quantity category) wedid not immediately sub-divide relationships by their scalar orvector nature. As stated above, the meaning of the negativesign is distinctly different for scalar and vector quantities. Foruse in relationships with explicit negative signs, however, themeaning of the negative sign is very similar for many scalarand vector relationships.Our first subdivision is for relationships in which quantities
Oppose each other. The relationships in this subdivision havean explicit negative sign, as the negative sign signifies the op-posing nature of the relationship between quantities. Thereare two subcategories in this division. In the
Scalar (1a) cat-egory, we have relationships such as that described by Fara-day’s Law, where the EMF opposes the time rate of changeof the magnetic flux. Similarly, in the
Vector (1b) category,we have vector relationships such as Hooke’s Law, where theforce exerted by a spring opposes the spring’s displacement.In the next subdivision, we consider quantities that are
Op-posite to each other. These are not quantities that are in-
TABLE III. Current version of the natures of negativity in introductory physics(Q) Quantity (R) Relationship (O) Operation
1. Scalar 1. Opposes 1. Removal (physical) a. Type (charge only) a. Scalar − ( − µ C ) , m total − m a b. Change/rate of change E = − d Φ B d t
2. Difference (temporal) ∆ E, dV, d φ d t b. Vector E f − E i c. Comparison to reference (cid:126)F = − (cid:126) ∇ U (cid:126)p f − (cid:126)p i T, V, E, t (cid:126)F spring = − k(cid:126)r
3. Difference (other) d. Models/Convention
2. Opposite Q − W ) W net,ext , Heat ( Q ) , Current ( i ) +5 µ C + ( − µ C ) = 0 Pathlength difference ( ∆ D )
2. Vector (cid:126)F = − (cid:126)F Distance from equilibrium ( ∆ x )a. Direction from origin
3. Relative Orientation
Electric potential difference ( ∆ V ) x, E x , v x (cid:126)F · ∆ (cid:126)x , (cid:126)E · (cid:126)A
4. Removal (modeling) b. Direction of change
4. Negative exponents I net = I disk − I hole ∆ p x , ∆ v x e − tτ , r − versely related by a physical relationship such as Faraday’sor Hooke’s Law. Despite this, we put such quantities in theRelationship nature, as the negative sign still indicates thatthese quantities are opposite to another quantity . Examplesinclude positive and negative charge, and the members of aNewton’s Third-Law Force Pair [17].We include Relative Orientation for scalar products suchas those used to calculate mechanical work ( (cid:126)F · ∆ (cid:126)x ) and elec-tric flux ( (cid:126)B · (cid:126)A ). A negative scalar product indicates that thefactor vectors have components that are oppositely-oriented.We include scalar products in the relationship category be-cause, as with other expressions and quantities in this cate-gory, a negative sign says something about how two quanti-ties (in this case, the factor vectors) relate to each other.Finally, we added a subcategory for Negative Exponents .This is included in the Relationship category because nega-tive exponents typically describe how one quantity relates toanother: how a quantity decreases in time may be describedby exponential decay, such as in circuits or damped harmonicoscillators; or how a quantity decreases in space is often de-scribed by r . Recognizing the equivalence between a quan-tity with a negative exponent and the inverse of that quantitywith a positive exponent is vital and (in our experience) non-trivial. This addresses a major shortcoming that existed inthe original version of the NoNIP, which did not allow forstraightfoward categorization of negative exponents.Our third base category is Operation (O) , for instanceswhen the negative sign is used to perform the mathemati-cal function of subtraction. This category is similar to thealgebraically-based category “Binary” described by Vlassis,which also describes uses of the negative sign to indicate sub-traction. There are four subcategories: 1)
Removal ; 2)
Dif-ference (change) ; 3)
Difference (other) ; and 4)
Removal(modeling) . The
Removal category is for uses of the neg-ative that signify a physical removal of some quantity (suchas electric charge or mass) from a system, whereas the
Dif-ference (change) subcategory is used for calculating the tem-poral change in a quantity, such as E f − E i or (cid:126)p f − (cid:126)p i . The third subcategory is for the operation used to calculate dif-ferences that are not necessarily temporal in nature. An ex-ample is for the calculation of displacement for Hooke’s law,with ∆ (cid:126)x = (cid:126)x displacement − (cid:126)x equilibrium . Here, we aren’t consid-ering ∆ (cid:126)x = (cid:126)x f − (cid:126)x i , only the displacement from equilibrium.Similarly, pathlength difference ∆ D can be determined bysubtracting one pathlength from another. Non- ∆ differences,such as the subtraction of work done by a system from theheat supplied to the system, as in the First Law of Thermo-dynamics, are also included in this subcategory. The fourthsubcategory ( Modeling (removal) ) is reserved for the non-physical removal of one quantity from another for the pur-poses of modeling a more complex situation. This type ofsubtraction is typified by the calculation of the moment ofinertia of a solid such as a disk with a hole.
2. Compounding multiple natures of the negative sign
Expert feedback led us to recognize that creating a
Com-pound category in the original NoNIP somewhat hid thecognitive difficulty students encounter unpacking the multi-ple natures of negativity from a single expression or equa-tion. Because of the many possible combinations of multiplesigned quantities appearing in an expression, each compoundcontext poses a unique challenge. It is in these compoundcontexts that even very strong students struggle most. A sig-nificant difference between the revised version of the NoNIPand the original version shown in Table II is the new version’slack of a Compound category. We believe that Quantity, Re-lationship, and Operation represent the individual natures ofnegativity in introductory physics thoroughly. The task ofcombining them in compound cases is an example of the so-phisticated reasoning with familiar mathematics that is char-acteristic of well-developed quantitative literacy.Many models in physics involve more than one unique na-ture of negative quantities; we consider the compound expres-sions ubiquitous in physics to be incomprehensible without
Compound
Relation- shipQuantity ⃗ F = kq q r ( ⃗ r − ⃗ r ) Operation E f − E i = ∫ ⃗ F ⋅ Δ ⃗ x R3Q2bQ2aO2Q2aQ1aO3
FIG. 1. Compound use of negative signs typical of introductoryphysics: Coulomb’s law, the force of object 1 on object 2 (left),and the work-energy theorem (right). flexibility between the three main natures of negativity. Fig-ure 1 maps two examples to the current NoNIP, which revealvarious natures of negativity from all three categories thatmust be understood in order to make physical sense of cen-tral ideas in introductory physics. One must note that somequantities (such as electric charge) appear under multiple na-tures, as the sign carries multiple meanings. Moreover, insome cases, such as Coulomb’s Law, multiple negative signsmay “cancel out.” Such “hidden” negative signs make keep-ing track of the signs of individual components more chal-lenging. When Coulomb’s Law is used with two charges with opposite signs, students must also make sense of what the re-sulting negative sign implies. For physics experts, a negativesign implies that the force between the two charges is attrac-tive, but this meaning is also described by the mathematics:that (cid:126)F on 1 by 2 is oppositely directed to (cid:126)r = (cid:126)r − (cid:126)r . We contendthat this physical interpretation of a quantified relationship isa hallmark of expert-like reasoning.The identification of multiple meanings of negative signsin a single context is challenging and the ability to do so isassociated with expert-like thinking. It should not be consid-ered to be a skill at the same level as identifying the meaningin less complex cases; rather, it should be viewed as a cul-mination of reasoning with and about sign. We argue thatflexibility between the various natures that appear in NoNIPmay better prepare students for the challenge of combiningthem into a single equation. Indeed, much of the research ofmore advanced student reasoning about sign has been in thecontext of compounded use of the negative sign. In the fol-lowing section, we describe such research, using the NoNIPas a framework by which student difficulties can be under-stood. In doing so, we provide examples of how to use the3-nature NoNIP to categorize compound quantities and rela-tionships.Like the original version of the NoNIP, this version hasbeen validated using reviews of introductory-level physicstextbooks as well as formal interviews with mathematics andphysics experts. The validation interviews were especiallyuseful in refining the characterization of the negative sign in introductory physics, leading to the creation of new subcate-gories as well as reclassification of some quantities and rela-tionships. We consider the current version to be in a steadystate. III. APPLYING THE FRAMEWORK
In this section, we use the current, steady-state version ofNoNIP as an analytical lens through which to view four re-cently published studies in physics and calculus that mostlyinvolve advanced physics or math students.Bajracharya, Wemyss, and Thompson investigated upper-division student understanding of integration in the contextof definite integrals commonly found in introductory physics,but with all physics context stripped from the representation.Specifically, the variables typically used in physics contextswere replaced with x and f ( x ) [8]. Their results suggest dif-ficulties with the criteria that determine the sign of a defi-nite integral. Students struggle with the concept of a negativearea-under-the-curve, and in particular negative directions ofsingle-variable integration. In research related to student un-derstanding of integration and negativity, Sealey and Thomp-son interviewed undergraduate and graduate students to un-cover how they made sense of a negative definite integral. Un-dergraduate (beyond introductory) and graduate mathematicsstudents had difficulty making meaning of a negative differ-ential in the context of integration [18]. The struggles theseresearchers described can be seen through the lens of NoNIPas struggle with the product of the integrand, f ( x ) (generic Qin NoNIP), and the differential, d x (Q1.b in NoNIP), each ofwhich can independently be negative. Making meaning of thenegativity of the integrand (generic Q in NoNIP) was less ofa struggle for the students in these studies than was the notionof a negative differential (Q.1b in NoNIP), which has appli-cation throughout physics. The researchers report that “noneof the students thought about d x as a signed quantity on theirown accord, but with prompting from the interviewers, somewere able to do so.” Encountering the differential as a smallchange in quantity, and being provided opportunity to thinkabout it in this way, provides a context that has been shown tohelp [19]. NoNIP can support this kind of explicit reasoningabout sign in this important context.A study conducted by Hayes and Wittmann situated in thecontext of sophomore-level mechanics investigates the nega-tive signs and quantities associated with the equation of mo-tion of an object thrown downward, with non-negligible airresistance [7]. The interviewed student struggles with treat-ing one-dimensional acceleration as a signed quantity, andfeels there should be an additional negative sign included toindicate that the acceleration is “negative,” or opposing themotion. The authors explain student difficulties with negativ-ity creating a notion of implicit and explicit “minus” signs.They conclude that the multiple natures of the negative signare a source of cognitive conflict that manifests as sensemak-ing about “outer and inner minus.” An “outer minus” is anegative sign that is assigned, for example, by choice of coor-dinate system; an “inner minus” is one that is associated withvariables that may be negative—buried in the physics mean-ing. Through the lens of NoNIP, “minus” is an operator, andnegative signs are used to represent many mathematical ob-jects and relationships in physics. In this particular contextof one-dimensional motion, both the (one-dimensional) ac-celeration and initial velocity have “explcit” signs that carryphysical meaning. In the framework of NoNIP, the studentstruggles with Q.2a in the contexts of one-dimensional accel-eration and velocity. The negative sign that modifies the cv term is used as R.1b, to indicate that the force is in the op-posite direction to the velocity. In the process of combiningthese terms, the student struggles to make sense of the equa-tion of motion. The cognitive load of negativity associatedwith the individual terms contribute to a higher-level strug-gle of making physical sense. The burden of these compoundnatures of the negative sign resulted in roughly half of thisgroup of physics majors feeling it was necessary to includean extra negative sign in the equation of motion so that themathematics would match the physics.In their study of negativity in junior-level Electricity andMagnetism, Huynh and Sayre describe the in-the-momentthinking of a student solving for the electric field due to twoequal and opposite charges along the axis that passes throughthem [9]. The authors focus on student reasoning about thesign of the electric field vector component along the axis ofsymmetry in three regions of space—to the left of one charge,between the two charges and to the right of the other charge.They report that four students solve this problem in an oralexam, and none get the directions correct on their first at-tempt.The solution involves an algebraic superposition of thefield due to each charge individually. The authors de-tail one representative student’s development of an increas-ingly blended approach that is situated in a mental space in-formed by both mathematical and physical concepts. Thestudent starts reasoning about the direction of the field us-ing Coulomb’s law by (unintentionally) combining multiplenatures of negativity into one. In Coulomb’s law, the signscombine multiplicatively from two sources: the direction ofthe displacement vector, (cid:126)r − (cid:126)r , and the sign of the sourcecharge (see Fig. 1). The student first uses the canceling fea-ture that multiplying two negative numbers always results ina positive number, without explicitly considering the sourceof each negative sign, and then reflects based on physics con-siderations why that approach doesn’t make sense.The student considers ˆ x to be a proxy for ˆ r , without con-sidering that ˆ r is connected to the physics (the difference ofthe two position vectors with respect to an origin) while ˆ x is afeature of the coordinate system. By making this substitution,the student glosses over an important source of negativity inthe final solution and it keeps him from being able to calculatethe direction of the electric field that he predicts using physicsprinciples. His conceptual understanding of the physics isstrong, but he can’t make his calculation match because he isn’t considering that the source of the negative signs havedeep physical meaning beyond the charges involved. Seenthrough the lens of NoNIP we can see evidence of the studentfirst conflating the natures Q.2b with Q.2a, not recognizingthat they are the same thing. The authors summarize the stu-dent’s confusion regarding the relative signs of the contribu-tions to the electric field due to each charge individually:This result clearly conflicts with their relative di-rection because he has double associated theiropposite direction with inappropriate applicationof destructiveness. [The student] tries hard to de-termine where another negative sign could comefrom, such as the denominator, to cancel onenegative sign for the whole term. Finally, he de-cides to absorb the destructive meaning of thesign into the opposite-direction meaning of theelectric field vector and changes the second neg-ative sign of the whole term back into a plus sign,which supports the fact that they are in oppositedirections. However, [the student] has not con-sidered the sign commensurate with the relativedirection of ˆ E and ˆ x , leading to his solution hav-ing the opposite sign to the correct answer [9].Collapsing the signs using arithmetic rules is a common ap-proach first tried by the students in this study, which focuseson the multiplicative rules of signed numbers rather than thephysics of the meaning of the signs. Next, the student rar-efies his approach as he considers more carefully the naturesof negativity in the context of the problem. The student re-flects “. . . I should have figured it out . . . which direction itis. This is exactly what is changing signs, not necessarily thesign of the charge.” After reconciling the basic level, thenhe struggles with R.1b and its equivalence to Q.2b. The au-thors report that when the student moves on to the other tworegions, which is essentially repeating the same reasoning se-quence, the student encounters the same struggles but he isfaster at obtaining a solution that matches his physical under-standing of the system. The fact that he doesn’t automaticallyand quickly solve the remaining two regions is evidence thatthis kind of reasoning is difficult.The authors conclude, and we agree, that the most so-phisticated challenge occurs when the natures negativity arecombined—the compound nature presents its own challengein addition to the challenge associated with each nature indi-vidually. This case study reveals the cognitive difficulty whenthree natures of the negative sign must be made sense of in thecontext of a single equation, and illustrates the challenges as-sociated with reasoning about the natures of negativity, evenfor strong majors. We believe it also reveals a hierarchy thatlends plausibility to the NoNIP model being representative ofemergent expert-like reasoning.This section, in which previously published work is in-terpreted through the NoNIP framework, demonstrates thebenefits of NoNIP: it allows comparisons across studies thatinvolve different physics contexts and even different math-ematical approaches. Both the Huynh & Sayre and Hayes& Wittmann studies involve upper-division topics and situa-tions involving negative signs [7, 9]; otherwise there are fewsimilarities. The Hayes & Wittmann paper is situated in clas-sical mechanics and difficulties with “inner and outer” neg-ative signs, while the Huynh & Sayre paper investigates stu-dent ability setting up integrals in E&M. By using the NoNIPframework, we can see that the two papers discuss similaraspects of the use of the negative sign. These similaritiesare much more specific than just using a negative sign inupper-division contexts. Further, the study by Bajracharya,Wemyss, Thompson [8] is also about using negative sign inan upper-division context, but applying the NoNIP frame-work reveals a use of a different nature of negativity thanthe other two physics papers. We believe this illustrates thepower of the NoNIP for researchers: identifying similaritiesacross (and differences between) studies that are more thanjust superficial. IV. EXTENSION TO SIGNED QUANTITIES
Although we focus in this paper on our categorization ofthe negative sign in introductory physics, we recognize thatstudents must make sense of the meaning of positive quanti-ties and relationships as well. Our focus on the negative signin this work is due to the assumption of “positivity” when aquantity has no explicit sign. While the practice of assum-ing an unsigned number has an implicit positive sign in frontof it perhaps poses few problems understanding pure num-bers, quantities in physics that aren’t signed may be eitherunsigned scalars (e.g., speed, mass, time), or quantities inwhich the positive sign holds meaning (e.g., component ofvelocity, change in energy).To investigate student understanding of signed quantitiesmore generally, we administered three questions about posi-tive or negative quantities in a multiple-choice format, shownin Table 2, to students enrolled in a calculus-based introduc-tory physics course at a large, diverse, public R1 university.Each student received either all three negative or all three pos-itive versions. Figure 3 shows the results from the negativeand positive versions of the mechanics items ( N pos = 242 , N neg = 309 ).To determine the effect size, we use the odds ratio. It is auseful effect size measure that describes the likelihood of anoutcome occurring in the treatment group compared with thelikelihood of the outcome occurring in the control group byforming a ratio of the two. An odds ratio of would indicatethat the odds are exactly the same. If we consider the hypoth-esis that positive quantities pose fewer challenges for studentsthan negative ones, we can consider the positive questions asthe control and the negative questions as the treatment.For items ME2 and ME3, effect sizes determined fromodds ratios are . and . respectively, which imply that stu-dents struggle with both positive and negative versions with ME1: An object moves along the x-axis, and the acceleration ismeasured to be a x = +8 m/s .Consider the following statements about the “+” sign in“ a x = +8 m/s ”. Pick the statement that best describes theinformation this positive sign conveys about the situation.a. The object moves in the positive directionb. The object is speeding up.c. The object accelerates in the + x -direction d. Both a and be. Both b and cME2: A hand exerts a horizontal force on a block as the blockmoves on a frictionless, horizontal surface. For a particularinterval of the motion, the work W done by the hand is W = − . J . Consider the following statements about the“ − ” sign in the mathematical statement “ W = − . J.” Thenegative sign means:I. the work done by the hand is in the negative directionII. the force exerted by the hand is in the negative directionIII. the work done by the hand decreases the mechanicalenergy associated with the blockWhich statements are true?a. I onlyb. II onlyc.
III only d. I and II onlye. II and III onlyME3: A cart is moving along the x-axis. At a specific instant,the cart is at position x = − m. Consider the followingstatements about the “ − ” sign in “ x = − m.” Pick thestatement that best describes the information this negative signconveys about the situation.a. The cart moves in the negative directionb. The cart is to the negative direction from the origin c. The cart is slowing downd. Both a and be. Both a and cFIG. 2. Examples of a multiple-choice questions probing studentunderstanding of signed quantities. ME1 is an example of a positive-quantity question, while ME2 and ME3 are negative-quantity ques-tions. roughly equal likelihood on these questions. The effect sizefor item ME1 is . , which is a statistically small effect sizeindicating that students find the negative version of this ques-tion slightly more difficult than the positive version [20]. Ourexperience as instructors has led us to recognize that studentstend to inappropriately associate negative (positive) accelera-tion with decreasing (increasing) speed regardless of the co-ordinate system. We suspect that the small difference in stu-dent performance on the two versions of item ME1 are re-lated to this difficulty, and thus we do not interpret the findingas evidence that students experience inherent difficulty with0 majoring in physics, chemistry, biology, and other STEM fields. In each case, students were concurrently enrolled in some level of calculus course. To investigate the first research question, we administered MC versions of the ME items at the R1 institution in Spring 2016 to 551 students completing the second course in the physics sequence (which includes mechanics applications and thermodynamics). Half of these students (at random) received modified versions of the items, in which the negative quantities used in the original versions were replaced with positive quantities. To investigate the second research question, we extended the study described earlier (Brahmia & Boudreaux, 2016), in which the ME items were administered at the R1 institution in Fall 2015 at the end of both the first course in the sequence (which covers mechanics) and the third course (which covers EM). We examined responses on free-response versions (n=84, ME; n=138, EM), and made changes to the wording and the MC distractors of some items. These changes are described in the footnotes of the appendix. The modified versions of all items were then administered in Winter 2016 at the regional university, at both the start and end of the second course of the three-quarter physics sequence (which covers EM). Table I summarizes the administration of assessment items. Table I:
Administration of items in introductory, calc-based physics courses to assess student understanding of signed quantities. (FR=free-response, MC=multiple-choice)
Institution Administered at Math Pre/Co Req. Item context Item format
R1 University (2015/16) End of 1 st sem. course PreCalc, Calc I ME (negative quantities only) MC and FR End of 2 nd sem. course Calc I ME (neg. and pos. quantities) MC only End of 3 rd sem. course Calc II, III EM (neg. only) MC and FR Regional Univ. (2016) Beginning and end of 2 nd qtr. course Calc I, II ME and EM (mod. wording and choices; neg. only) MC only Findings
Figure 2 shows results from the positive and negative versions of items ME1-ME3. For all three item pairs, a chi-square test of significance yields p-values > 0.6. For items ME2 and ME3, effect sizes determined from log odds ratios are < 0.8 (Borenstein et al., 2009). The effect size for item ME1 is 0.13, a statistically small effect size. Prior research has found that physics students tend to inappropriately associate negative (positive) acceleration with decreasing (increasing) speed. We suspect that results on item ME1 are related to this difficulty, and thus do not interpret the finding as evidence that students experience inherent difficulty with negatively signed quantities (i.e., relative to positive quantities). Results overall suggest that students struggle to make sense of positive values just as they do with negative values, when the sign is an explicit part of the quantity. Figure 3 shows results from the original and modified versions of ME1 (see Appendix). We use these results to address the first part of the 2 nd research question. Note that on the original version, choice “e” would be correct if the motion were assumed to be in the +x-direction. We note that on the modified version, given at the regional university, 17% fewer of the students P e r c e n t c o rr e c t Negative Positive
Figure 2:
Positive vs. Negative
FIG. 3. Percentage of students who answered correctly for positiveand negative versions of mechanics items, N pos = 242 , N neg = 309 ;the error bars represent the binary standard error. negatively-signed quantities (i.e., relative to positive quanti-ties).These results indicate that students have difficulty inter-preting the meaning of the sign of a quantity, regardless ofthe sign ; students may not recognize that the sign specifiesthe direction of a vector component relative to a coordinatesystem, or that the sign of a scalar quantity such as work in-dicates how the energy of a system changes.Informal conversations with students also indicate that stu-dents sometimes fail to see the significance of the relationship between two quantities that are positively correlated (e.g., thatNewton’s Second Law tells us that the acceleration of a sys-tem is always in the same direction as the net force exertedon that system). Such an understanding is crucial not onlyfor nominally causal relationships, but also for understandingfeedback loops and accumulated change. While interpretingnegative signs is a pressing issue in physics learning becausenegative signs are explicitly used, these difficulties fall underthe umbrella of student difficulties with the interpretation ofsigned quantities in general—which has not been studied indepth. V. CONCLUSION
Negative signs in physics have nuanced and varied inter-pretations that can pose a challenge, even to majors. In thispaper, we present the NoNIP framework for categorizing ex-pert uses of the negative sign which has undergone expert val-idation and revision; it is presented in its steady state. We an-ticipate that the NoNIP table can be useful across the physicseducation research community. For researchers, the NoNIPtable can serve as a map of the natures of negativity, and astarting point for thinking about sign, as part of the broadercontext of mathematical reasoning development in physicscontext. We have demonstrated how viewing published re-search through the NoNIP framework can help bring out pat-terns between the findings that weren’t clear before. We present evidence that students struggle to make sense ofpositive signs as much as they do negative signs. Student dif-ficulties with positivity aren’t as noticeable because in prac-tice experts assume the absence of sign means the quantityis positive, so, unlike a negative quantity, there is no symbolthere to decode. We intend for the research presented in thispaper to extend to signed quantities more generally, and in-crease awareness that students will benefit from making senseof the meaning of positive quantities as well as negative. Thesign of a quantity, along with the magnitude of the quantityand its units, are part of what defines a quantity and how weunderstand it in most physics contexts [21].As a tool for instructional development, NoNIP can benefitboth curriculum developers—who can use the NoNIP table tohelp guide their efforts to situate signed quantity reasoning inthe broader context of the materials that they develop—andinstructors. We close by presenting some recommendationsfor instruction that can inform instructors about the organiza-tion of their expertise, which can thus influence how they talkabout and present material to the novices in their classroom.Developing this awareness can help students become morecognizant of the natures of both positivity and negativity inphysics. Acknowledging the nuances, rather than assumingthe mathematics to be trivial, can create access for studentsthat otherwise might not exist. We offer three suggestions asa start, fully anticipating that expert instructors will devisetheir own ways also:1. Quantities that are inherently signed quantities shouldbe prefaced with a negative sign when the quantity isnegative, and a positive sign when the quantity is posi-tive, e.g., x o = +40 m. Priming students to expect thatreal-world quantities often have signs associated withthem that carry meaning, and that “no sign” is a differ-ent kind of quantity than a positively signed quantity,can help establish a physics habit of mind that the signcarries scientific meaning, and eventually that vectorquantities have different mathematical properties thanscalar ones.2. Orientation (along a particular axis) and sense (posi-tive or negative) are not always explicit in coordinatesystems. In problems associated with motion, align-ing the positive coordinate axis with the direction ofmotion eliminates the need for signed quantities whendiscussing velocity. This choice, however, could be amissed opportunity to distinguish between orientationand sense. The opposite coordinate choice can primestudents to consider the signed nature of position, ve-locity, and subsequent vector quantities they encounter.3. Sign and operation are often conflated using an equalssign (e.g., −
3) = 5 − ), and unsigned numbers areassumed positive. Adding a negative quantity and sub-tracting a positive one often have different meaningsin physics contexts (e.g., adding electrons). Althoughthese operations yield the same arithmetic results, con-flating them may lead students to struggle with the dis-1tinctions between sign and operation. We suggest usingthe term “minus” for the operation of subtraction, andthe term “negative sign” to describe the symbol.In addition to enriching subsequent physics learning, a fo-cus on natures of sign in physics contexts can also enrich thecorequisite mathematics learning. Sealey and Thompson re-port on a context in which physics helps math students makesense of negativity in calculus. The researchers observed thatinvoking a physics example of a stretched spring helped cat-alyze sensemaking—the physics helped them to make senseof an abstract binary nature of the negative sign [18]. We sug-gest that there is symbiotic cognition possible in which bothmathematics and physics learning can be enriched by con-ceptualization of the other, and that reasoning about sign pro-vides a rich context. We present NoNIP as a representation of the natures of negativity providing a step in that direction. ACKNOWLEDGMENTS
We thank Peter Shaffer for his support with data collec-tion that informed this work, and the entire Physics EducationGroup at the University of Washington for vibrant discussionand feedback. Brian Stephanik’s feedback and physics con-tent expertise was also instrumental in the construction of theNoNIP. We thank Roy Montalvo at Rutgers University forsharing his creative software innovation that helped to makethe data collection run smoothly. We also thank Laurie Smithfor fruitful conversations regarding negative exponents. [1] Jessica Pierson Bishop, Lisa L Lamb, Randolph A Philipp, IanWhitacre, Bonnie P Schappelle, and Melinda L Lewis, “Ob-stacles and affordances for integer reasoning: An analysis ofchildren’s thinking and the history of mathematics,” Journal forResearch in Mathematics Education , 19–61 (2014).[2] Aurora Gallardo and Teresa Rojano, “School algebra. syntac-tic difficulties in the operativity,” Proceedings of the XVI Inter-national Group for the Psychology of Mathematics Education,North American Chapter , 265–272 (1994).[3] Patrick W Thompson and Tommy Dreyfus, “Integers as trans-formations,” Journal for Research in Mathematics Education ,115–133 (1988).[4] Terezinha Nunes, “Learning mathematics: Perspectives fromeveryday life,” Schools, mathematics, and the world of reality, 61–78 (1993).[5] Suzanne Brahmia and Andrew Boudreaux, “Exploring studentunderstanding of negative quantity in introductory physics con-texts,” in Proceedings of the 19th Annual Conference of RUME (2016) p. 79.[6] Suzanne Brahmia and Andrew Boudreaux, “Signed quantities:Mathematics based majors struggle to make meaning,” in
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