Exploring student facility with "goes like'' reasoning in introductory physics
Charlotte Zimmerman, Alexis Olsho, Andrew Boudreaux, Trevor Smith, Philip Eaton, Suzanne White Brahmia
EExploring student facility with “goes like” reasoning in introductory physics
Charlotte Zimmerman, Alexis Olsho, and Suzanne White Brahmia
Department of Physics, University of Washington, Seattle, WA, 98103
Andrew Boudreaux
Deprtment of Physics, Western Washington University, Bellingham, WA, 98225
Trevor Smith
Department of Physics, Rowan University, Glassboro, NJ, 08028
Philip Eaton
School of Natural Sciences and Mathematics, Stockton University, Galloway, NJ 08205
Covariational reasoning—reasoning about how changes in one quantity relate to changes in anotherquantity—has been examined extensively in mathematics education research. Little research has been done,however, on covariational reasoning in introductory physics contexts. We explore one aspect of covariationalreasoning: “goes like” reasoning. “Goes like” reasoning refers to ways physicists relate two quantities througha simplified function. For example, physicists often say that “the electric field goes like one over r squared.”While this reasoning mode is used regularly by physicists and physics instructors, how students make senseof and use it remains unclear. We present evidence from reasoning inventory items which indicate that manystudents are sense making with tools from prior math instruction, that could be developed into expert “goes like”thinking with direct instruction. Recommendations for further work in characterizing student sense making as afoundation for future development of instruction are made. a r X i v : . [ phy s i c s . e d - ph ] J u l I. INTRODUCTION
A perhaps unexpected byproduct of the COVID-19 pan-demic is renewed clarity on how challenging it is for manyto conceptualize the exponential function. This is certainlynot novel; Albert Bartlett famously stated “The greatest short-coming of the human race is our inability to understand theexponential function”[1]. This has become a public issuein the face of the coronavirus epidemic. Headlines such as“What Does Exponential Growth Mean in the Context ofCOVID-19?,”[2] “The Exponential Power of Now,”[3] and“Is Poor Math Literacy Making It Harder For People To Un-derstand COVID-19 Coronavirus?”[4] have put conceptual-ization of function on the national stage.It is evident that quantitative literacy—the set of skills thatsupport the use of mathematics to describe and understandthe world—is important, and lacking, in the United States to-day. Quantitative literacy has many facets, including reason-ing about signed quantities, proportional reasoning and co-variational reasoning —conceptualizing change in one quan-tity with respect to change in another quantity [5–7]. Intro-ductory physics, a broadly-required college course with a fo-cus on quantifying and modeling nature, is an excellent placeto address this need.Proportional reasoning—reasoning about ratio as aquantity—has been identified as critical for success in physicsby physics educators and in Physics Education Research(PER). Early PER, confounded by student difficulties usingelementary mathematics in physics contexts, focused on iden-tifying specific reasoning difficulties such as the tendency touse additive, rather than multiplicative, strategies and the ten-dency of physics students to manipulate mathematical for-malism without understanding the physical meaning of theassociated quantities and operations [8–10]. By the early1980’s, studies in PER had begun to systematically documentand extend this body of work by using individual demonstra-tion interviews to explore student understanding of velocityas the ratio ∆ x/ ∆ t and acceleration as the ratio ∆ v/ ∆ t [11–13]. More recent work has examined the relationship betweenbasic reasoning ability, including proportional reasoning, andthe learning of physics content [14].Work on the role and challenge of proportional reasoningin physics contexts has included attention to scaling and func-tional reasoning. Arons points out, for example, that fewstudents “have formed any conception of the basic functionrelation between area and linear dimensions,” and that con-sequently, most students are “unaware that all areas vary asthe square of the length factor” [10]. We build on this bodyof work by integrating the language of covariational reason-ing established by Research in Undergraduate MathematicsEducation (RUME) community [6, 15–19]. Covariation en-compasses all functions that relate two or more quantities andconsiders multiple ways that one can think about those rela-tionships. For example, one can consider discrete covariation(if the radius is doubled, what happens to the electric field at apoint?), or continuous covariation (how does the field change smoothly as the radius is increased?) [6]. We suggest thatproportional reasoning is a subset of covariational reasoning,focused specifically on linear relationships and using ratiosthat have meaning as a single entity (such as velocity and ac-celeration).Physics educators regularly identify “thinking like a physi-cist” as a goal of introductory physics. In a 2019 studyof the ways in which experts use covariational reasoning tosolve introductory physics problems by Zimmerman, Olsho,Boudreaux, Loverude, and White Brahmia, it was noted thatphysics experts use functional reasoning by employing the“ ∝ ” symbol or phrases like “goes like” to illustrate relation-ships in statements like Area ∝ r , Force goes like /r ,etc. [20] This kind of “goes like” expert thinking is used torepresent a wide variety of simplified relationships betweenquantities, and is a desired outcome of introductory physicsinstruction. In his work on proportional reasoning, Arons as-serts that the capacity for scaling and functional reasoningwill not necessarily develop spontaneously [10]. Indeed, theneed for curricular intervention is evident from the currentliterature. What is less clear is what resources and emergentabilities students do have regarding quantitative literacy priorto physics instruction, and what educators can do to buildupon these skills to develop quantitative reasoning in theirstudents.This paper describes a study of students’ covariational rea-soning in physics contexts. It contributes to the work in math-ematics education, as well as to closing a gap in PER, whereit has been shown that reasoning in physics contexts is differ-ent from reasoning in purely mathematical ones [21, 22]. Wefocus on one expert-like facet of physics covariational reason-ing: “goes like” reasoning [20]. We will present some of theways an expert might use this kind of thinking, and some pre-liminary results that suggest while introductory students donot have strong facility with physics “goes like” reasoning,and their conceptualization of “goes like” is not improvingover a year long sequence in introductory physics, they dohave some productive resources and emergent abilities fromprior math courses that can be met and built upon with directinstruction to develop physics covariational reasoning skills.Recommendations for future work and curricular interven-tions are made. II. EXPERT REASONING ABOUT “GOES LIKE” “Goes like” reasoning refers to the simplified relationshipbetween two changing quantities that illustrates the behaviorof an evolving system. For example, consider a classic intro-ductory physics problem: a ball thrown from a cliff. An ex-pert might reason that if the ball’s initial height is increased,the final speed of the ball will also increase. They might rea-son further that the final speed of the ball “goes like” the thesquare root of the height. Here, “goes like” reasoning allowsthe expert to focus on the functional form of the relationshipbetween two changing quantities, and to ignore any constantsor pre-factors. This focus on co-varying quantities in turn al-lows for efficient problem solving, as the expected behaviorof the system can be quickly and clearly illustrated.We note that expert use of “goes like” reasoning relies onfacility with the mathematical functions involved, as well asthe experience and physics content knowledge that enable ex-perts to relate physical phenomena to those functions. Zim-merman et. al found that physics graduate students havestrong associations between certain routinely used physicsquantities that allow them to make inferences about relation-ships between quantities in a given problem. This simpli-fies problems to those they can solve more efficiently, or towhich they may already know the answer from experience[20]. Unlike novices, someone with substantial experiencewith physics is able to make claims such as “This probleminvolves a potential, which goes like /r ” or “This looks likescattering, so I expect it to be an exponential.”We don’t claim that novices do not have some ”compiledrelationships” between mathematical functions and physicalcontexts. To the contrary, in our experience many studentshave strong associations between functions that model realworld contexts, and we consider these to be resources forphysics learning. Many of these associations evolve fromprior math instruction and so are more suited to math con-texts than to a physics course—for example, where expertsmay associate circular motion with sinusoidal curves, intro-ductory physics students may more readily associate trigono-metric functions with right triangles.This led us to wonder what resources students in an in-troductory physics course are using to relate two quantities,and whether they include “goes like” reasoning. In partic-ular, we asked: do students enter introductory physics withthis skill already formed and ready to be applied from mathcourses? In addition, do their “goes like” reasoning skills im-prove after instruction in a physics class, where instructiontypically takes the form of experts modeling their reasoningand discussing it in lecture? To answer these questions, weprobed students’ covariational reasoning using items from aninventory currently in development: the Physics Inventory forQuantitative Reasoning (PIQL) [7, 23]. III. ASSESSING “GOES LIKE” REASONING
The PIQL measures fundamental aspects of mathemati-cal reasoning that are ubiquitous in physics modeling, i.e. physics quantitative literacy (PQL) . Development of this in-strument began with items targeting proportional reasoningand reasoning about sign and signed quantities, and has sincegrown to include additional items related to covariational rea-soning more broadly. During its development, the PIQL hasbeen administered over several years at a large research uni-versity in the Pacific Northwest. The test is given at the startof each of the three quarter-long courses that form a year-long introductory physics sequence. Here we focus on stu-dent responses to two PIQL items that we believe would illicit
FIG. 1. The Flag of Bhutan. The prompt associated with this imageasks students to select all of the following quantities that are largerby a factor of 1.5 when the length and width of the flag are bothincreased by a factor of 1.5: (a) The distance around the edge ofthe flag, (b) the amount of cloth needed to make the flag, (c) thelength of the curve forming the dragon’s backbone, (d) the diagonalof the flag, and (e) none of these. Students are prompted to chooseall answer choices that apply. We believe the correct answers to be(a), (c) and (d). “goes like” reasoning in experts: the Flag of Bhutan and Fer-ris Wheel [24, 25]. Other aspects of quantification and PQLare also involved in these responses, but will not be discussedin this paper.
A. Flag of Bhutan
In the Flag of Bhutan question, students are asked to de-termine what aspects of the flag would be larger by a factorof 1.5 if the length and the width were both increased by afactor of 1.5 (see Fig. 1). This item was originally designedas a scaling assessment to measure student facility with bothlinear and non-linear relationships, as some answer choicesdepend linearly on length and width (such as the length ofthe dragon’s backbone, or the distance around the edge of theflag) and the answer choice “the amount of cloth needed tomake the flag” depends on length times width [24]. Whilescaling was considered a facet of proportional reasoning atthe time, it was understood by the researchers that scalingwith non-linear functions is notably different than scalingwith linear relationships. We believe that this question can bere-examined in the context of discrete covariation and “goeslike” thinking.One of the challenges of the Flag of Bhutan item is thatit is a multiple-choice/multiple-response (MCMR) question.Thereby, its score is low compared to other items on the PIQLbecause these items are scored dichotomously for comparisonwith other multiple-choice/single-response items [26]. How-ever, the nature of the item does not fully account for the sig-nificantly low number of completely correct responses. Re-sults suggest a majority of introductory students struggle toreason without a linear equation as only 26% of students an-swer completely correctly, in contrast to instructors’ expecta-tions. Moreover, the percentage of students who answer thisquestion completely correctly does not change significantlythroughout the introductory sequence (25% in the first quar-ter, 25% in the second quarter, and 31% in the third quarter)suggesting that this kind of reasoning does not improve.One of the benefits of an MCMR item is that we can learnmore about student thinking by examining the partially cor-rect answers that students chose. The most common reasons astudent does not get the item completely correct are by not se-lecting either (c) or (d) in Figure 1. These were not chosen by55% and 43% of students respectively. Only 24% of studentsdo not choose (a). These results suggest that students do havefacility with directly linear relationships, such as perimeterto length and width, but have difficulty with more complexfunctional relationships, such as √ l + w , or those that donot have a known functional relationship, such as the dragon’sbackbone, even if the result is linear. Using tetrachoric corre-lation analysis, we found that students considered (c) and (d)together (either choosing both or declining to choose both)66% of the time. This suggests that even while the diagonalof the flag can be described by a geometric function and thebackbone cannot, the majority of students are able to real-ize that they have the same dependence on length and width.However, these results do not improve over the course of in-struction, suggesting that these early signs of “goes like” rea-soning might not be nurtured over the course of instruction todevelop students’ discrete covariational reasoning in the con-text of scaling. B. Ferris Wheel
Ferris Wheel asks students to choose an equation that rep-resents how the height of a Ferris wheel cart changes as afunction of the total distance it has traveled (see Fig. 2). Thisquestion was inspired by a Hobson and Moore study, and thedistractors were developed based on results from the Zimmer-man et al. study, and introductory student interviews [20, 25].Experts were given an animated version of the image in Fig-ure 2 in which the cart rotates with the Ferris wheel, andasked to produce a graph that relates the total distance trav-eled by the cart and the height of the cart [20, 25]. It wasobserved that the experts used time as a proxy for total dis-tance, noticing that both quantities described the evolutionof the system. They then demonstrated “goes like” reason-ing by making strong associations between the circular mo-tion presented in the animation and trigonometric functions:“the height goes like a trig function.” The authors refer tothese connections between quantities as compiled relation-ships [20]. In developing Ferris Wheel for the PIQL, we wereinterested to see if students also held compiled relationships,and if they were able to use “goes like” reasoning to solvethe problem. When reformating the item as multiple choice,we tried to choose distractors that represented other potentialcompiled relationships based on geometric shapes includingthe Pythagorean theorem, which students associate with trian-gles in interviews and in open-ended versions of other PIQLitems, and an expression containing the circumference, whichstudents associate with circles.
FIG. 2. Ferris Wheel. The prompt associated with this image asksstudents to identify which expression correctly identifies how h , theheight of the cart, directly changes with s , the distance traveled bythe rider, where the radius of the Ferris wheel is given by R : (a) h ( s ) = (cid:112) s + R , (b) h ( s ) = R exp( s/R ) , (c) h ( s ) = R − R cos( s/R ) , (d) h ( s ) = s / (2 πR ) Ferris Wheel was administered as part of the PIQL, andvalidation interviews were performed at another public uni-versity in the Pacific Northwest. We do not claim that thesetwo institutions represent identical populations; they oftenhave slightly different average scores on PIQL assessmentitems. Indeed, while a majority of students that took the PIQLat the large research university answered the question cor-rectly (58%), fewer than half of those interviewed chose thecorrect answer. However, the interviews do provide a broadlook into how some students are making sense of the problem.Based on the percentage of students who get the assessmentitems completely correct, this item appears to be consider-ably less challenging than Flag of Bhutan. However, it is notan MCMR question, so it cannot be compared directly [26].As before, we can explore what students may be think-ing by examining their incorrect answer choices. The mostcommon incorrect choices were (d) and (a) from Figure 2,with an answer rate 24.5% and 14.7% respectively that doesnot change significantly across the three courses. These re-sults suggest that the “circumference-like” distractor and the“Pythagorean-like” distractor are appealing to a significantfraction of the entire student population. We interpret theseanswer choices as unrefined “goes like” tendencies—theseare functions that are familiar to students from recent mathclasses, and have been fruitful in past experiences reasoningabout circles and triangles. Some students may not have read-ily accessible resources of “goes like” reasoning in physicscontexts, or a compiled relationship between circular-motionand trigonometry as demonstrated by physics experts, eventhough they are making sense with the tools they have.Validation interviews can provide some details into whatcompiled relationships students have formed, and how theymight be using them along with “goes like” reasoning to solvethe problem. It was found that nearly all students interviewedwere highly invested in answer choice (d), citing it as familiar.They often noted that it contains the expression for circumfer-ence of a circle, which most of those interviewed readily asso-ciated with the total distance traveled: “I’d say (d) because itsthe only one that has πR in there, which is the, essentially,the circumference formula.” Indeed, nearly all students inter-viewed began by defining the total distance traveled by thecircumference, and many returned to this definition through-out their problem solving process. While experts may realizethat focusing on circumference is not a productive method ofsolving this problem, we recognize this as a form of quan-titative reasoning—students demonstrate a strong compiledrelationship between distance and circumference. The keydifference is that experts are able to use distance as a quan-tity that describes the evolution of the system, while studentsare connecting total distance traveled (a quantity that changesin time) with the circumference of one revolution (a quan-tity that is fixed in this problem). Because the students in-terviewed didn’t spontaneously consider the total distance asit is changing , they didn’t demonstrate facility with expert-like “goes like” reasoning. They did not reach the point inthe problem where they could choose an expression for theheight as a function of total distance with confidence. Therewas only one student who articulated that the total distance isa changing quantity stating it represented “how much of thecircumference [the rider] has traveled,” but in this student’scase that line of reasoning was still used to support his selec-tion of answer choice (d).Interest in answer choice (a) was centered around reason-ing with triangles, and although none of the students inter-viewed chose (a) as their final answer, many grappled withits meaning. Every student interviewed verbally labelled op-tion (a) as “Pythagorean,” and many students drew an accom-panying triangle, demonstrating a strong compiled relation-ship with the expression itself and triangular geometry. Somerecognized right away that the Pythagorean approach wouldnot work, one stating, “(A) is the Pythagorean theorem, butthat doesn’t make sense because that’s linear distance.” Here,we interpret this as the student recognizing that Pythagoreantheorem uses linear distances, and the total distance traveledis not linear. Another student interviewed debated about thecorrectness of (a), stating, “This is like the Pythagorean theo-rem. . . if we do it like this, [the student draws a triangle withthe hypotenuse representing total distance] I guess you couldestimate [the total distance] as being a straight line.” Both ofthese students did not draw the triangle a physics expert mightexpect (with the radius as the hypotenuse), and most were un-certain about the expression presented because they had diffi-culty making sense of which quantities the sides of the trian-gle they drew were representing. However, their statementsdemonstrate sense making about the expression and its con-nection to right triangles, which we consider to be productive.When evaluating answer choice (c) that uses a trigonomet-ric expression, students continued to puzzle over how to drawthe appropriate triangle: “cosine gives me s over R . . . sothey’re saying the radius is the hypotenuse. How can that be?”Only one student interviewed made direct reference to theunit circle and was able to quickly recognize that “ θ is equalto arc length over the radius,” and that “the radius should bethe hypotenuse because the radius is the one thing that is mea-sured throughout the circle,” but then this student was drawn to the familiarity of (d) and eventually uses point by pointanalysis to choose her answer. These patterns suggest thatthe students interviewed have strong procedural facility with ageometric approach to Pythagorean theorem, but not concep-tual understanding about how it connects to circles. This gapin understanding between trigonometry learned and how it isapplied in physics was typical in the interviews. It is notablethat while students may comfortably reason about trigonome-try in the contexts of triangles and circles, many students maynot understand how that reasoning is used in physics contexts.Those that answered correctly in interviews often deter-mined their answer by plugging in points. Uniformly thisstrategy was approached as a last effort, suggesting that stu-dents don’t rely on other ways of making sense of the answerchoices and may consider plugging in numbers to be an ex-pert problem solving strategy. Typically, students using thismethod were choosing between option (c) and (d), howeverin one case the student tried all possible answer choices. Inparticular, students that did pick points to solve the problemchoose physically significant points, for example, the bottomand top of the Ferris Wheel where the height is at a minimumor maximum. This kind of problem solving—specificallychoosing physically relevant points to better understand thebehavior of the system—has been identified as an expert-likebehavior in previous studies [20]. IV. CONCLUSIONS
Ferris Wheel and Flag of Bhutan demonstrate that whilestudents have difficulty with physics “goes like” reasoning,they illustrate skills that could be used to develop physics co-variational reasoning with direct instruction. Responses toFlag of Bhutan show that students have strong “goes like”reasoning about linear relationships that could be developedinto “goes like” reasoning about non-linear relationships.Responses to Ferris Wheel demonstrate that students havestrong compiled relationships regarding right triangles andthe Pythagorean theorem, and circles and circumference, thatcould be developed into compiled relationships between cir-cular motion and trigonometric functions. As covariationalreasoning is integral to conceptualizing physics models, werecommend instructors consider including direct and explicitinstruction on relating quantities beyond demonstration intheir own teaching. Additional studies are needed to betterunderstand what kinds of covariational reasoning and com-piled relationships students have coming into introductoryphysics and are forming over the course of instruction. Cur-rently, appropriate curricular materials do not exist and needto be developed.
ACKNOWLEDGMENTS
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