Improving students' understanding of rotating frames of reference using videos from different perspectives
Stefan Küchemann, Pascal Klein, Henning Fouckhardt, Sebastian Gröber, Jochen Kuhn
aa r X i v : . [ phy s i c s . e d - ph ] F e b Improving students’ understanding of rotating frames of reference using videos fromdifferent perspectives
S. K¨uchemann, ∗ P. Klein, H. Fouckhardt, S. Gr¨ober, and J. Kuhn Department of Physics, Physics Education Research Group,Technische Universit¨at Kaiserslautern, Erwin-Schr¨odinger-Str. 46, 67663 Kaiserslautern, Germany Department of Physics, Integrated Optoelectronics and Microoptics Research Group,Technische Universit¨at Kaiserslautern, Erwin-Schr¨odinger-Str. 46, 67663 Kaiserslautern, Germany
The concepts of the Coriolis and the centrifugal force are essential in various scientific fields andthey are standard components of introductory physics lectures. In this paper we explore how stu-dents understand and apply concepts of rotating frames of reference in the context of an exemplarylecture demonstration experiment. We found in a
P redict − Observe − Explain -setting, that afterpredicting the outcome prior to the demonstration, only one out of five physics students correctlyreported the observation of the trajectory of a sphere rolling over a rotating disc. Despite this lowscore, a detailed analysis of distractors revealed significant conceptual learning during the obser-vation of the experiment. In this context, we identified three main misconceptions and learningdifficulties. First, the centrifugal force is only required to describe the trajectory if the object iscoupled to the rotating system. Second, inertial forces cause a reaction of an object on which theyact. And third, students systematically mix-up the trajectories in the stationary and the rotatingframe of reference. Furthermore, we captured students’ eye movements during the
P redict task andfound that physics students with low confidence ratings focused longer on relevant task areas thanconfident students despite having a comparable score. Consequently, this metric is a helpful tool forthe identification of misconceptions using eye tracking. Overall, the results help to understand thecomplexity of concept learning from demonstration experiments and provide important implicationsfor instructional design of introductions to rotating frames of reference.
I. INTRODUCTION
Rotating frames of reference play an important role ina variety of fields in physics. Accordingly, Coriolis andcentrifugal terms need to be considered for an accurateaccount of the theoretical description. While the Corio-lis force was originally introduced to describe the energytransfer in waterwheels, nowadays it is applied to prob-lems in meteorology [1, 2], oceanography [3], astrophysics[4], optics [5] and nuclear physics [6]. Given this widerange of applications, the Coriolis and the inertial cen-trifugal force are common topics in introductory physicscourses in college-level education and, accordingly, thereis a large number of experiments and online materials [7]which intend to demonstrate the Coriolis effect, i.e. theapparent deflection of an object by the Coriolis force.However, there are several shortcomings and false ac-counts, outlined below, potentially causing misconcep-tions and complications in students’ understanding.In this paper we explore how students understand and ap-ply concepts of rotating frames of reference in the directcontext of an exemplary lecture demonstration experi-ment. Therefore, we identify and study relevant miscon-ceptions, the non-obvious learning effect of experimentobservation and the relationship between response secu-rity and duration of focus on relevant areas (as measuredby eye tracking).The paper is structured in the following way. After this ∗ [email protected] introduction, an overview of the state of the art andthe preliminary work follows in the second chapter andthe third part explains the materials and methods usedin this work. The subsequent section contains the re-sults of the Predict-Observe-Explain test, including self-confidence ratings, student interviews and eye-trackingdata in the context of an exemplary demonstration ex-periment of rotating frames of reference. Then, theseresults are discussed in the context of previous litera-ture and, eventually in the final chapter, we conclude themanuscript with the main consequences of the results forphysics education research. II. STATE OF THE ART AND PRELIMINARYWORKA. Simplified conceptions of the Coriolis effect andthe centrifugal force
In simplified depictions of a curved trajectory of an ob-ject in a rotating frame of reference (RFR), the Coriolisforce is often presented as the cause for the deflection.However, according to Eq. (2) (see Appendix), the iner-tial centrifugal force (ICF) also acts on the object in avector sum with the Coriolis force. The fact that the in-ertial centrifugal force is a necessary quantity to describethe trajectory of an object in a rotating frame of refer-ence can be understood from two arguments of a thoughtexperiment.(A) Let us imagine a two-dimensional case where a planeflies in a uniform motion over a large rotating disc start-ing from the center of rotation. If an observer located onthe disc only used the Coriolis force for the descriptionof the curved trajectory, he/she would expect that theplane returns to the center of rotation at some point intime because the Coriolis force is always perpendicular tothe direction of motion thus leading to a circular trajec-tory. For an observer in a stationary frame of reference(SFR), however, it is obvious that this case would notoccur because the plane flies in a uniform motion due tothe absence of any real force. In reality, the plane wouldpursue a spiral trajectory for the observer in the RFRwhich is the consequence of the vector sum of the iner-tial centrifugal force and Coriolis force.(B) During the aforementioned motion of the plane, theabsolute value of the velocity | ~v ′ | in the RFR would in-crease according to Eq. (1) because the absolute value ofthe velocity in the SFR | ~v | and the angular velocity areconstant and | ~r | increases. Since the Coriolis force is al-ways perpendicular to the direction of motion, it cannotbe the reason for this apparent increase in | ~v ′ | . Only thecentrifugal force which points outwards from the centerof rotation can be responsible for this effect.The misconception that the centrifugal force only occurswhen the object is somehow coupled (e.g. by friction ora rope) to the rotating system [8] is potentially guidedby empirical experiences, such as the feeling of a forcepointing outwards. Here, the underlying problem is thata person when sitting in a carousel or in a car drivingthrough a turn actually feels a force acting on him orher, i.e. the body reacts to the force, because the personis actually partially coupled to the rotating system. Thisseems to be in conflict with the aforementioned character-istics of a virtual force. This cognitive dissonance can beresolved by discriminating between the centrifugal forcewhich occurs as a reaction to a Centripetal force (heretermed ”Reactive centrifugal force”, RCF) and the onewhich occurs as a virtual force in a RFR (”inertial cen-trifugal force”) [9]. Sometimes text books and scientificarticles lack this helpful linguistic distinction [8, 10–13].The reason for this could be that the mathematical equa-tions are the same, only the situations in which they oc-cur in and how they are perceived are different. For theoccurrence of the RCF, a coupling to the RFR is indeedrequired, for the occurrence of the ICF it is not. Accord-ingly, the RCF can be perceived when driving througha turn or sitting in a merry-go-round while the occur-rence of the ICF, for instance for a passenger in an airplane which flies in a uniform motion over a rotating disc,cannot be felt. B. Experimental lecture demonstrations andstudents’ understanding of the Coriolis force
Lecture demonstrations in the classical sense mean thedemonstrations of experiments by the lecturer during theclass while the students passively observe the presen-tation. The intention of the lecturer is often that the students process the information and understand theirobservations by integrating it into their concept knowl-edge [14]. Unfortunately, despite their regular use inintroductory physics lectures, it has been shown thatdemonstrations will have little effect on students’ con-cept learning if the students passively observe the exper-iments [15]. At the same time, the correct observationof a lecture demonstration is a necessary prerequisite forconcept learning [14].In this context, Predict-Observe-Explain (POE) is an in-teractive teaching scenario which can be implementedduring experimental lecture demonstrations [16–18].While it is sometimes proposed as an eight-step approach,here we reduce it to three central steps [19, 20]. First, inthe ”Predict” phase, the students are asked to make aneducated guess of the outcome of the experiment. Thisstep helps to initiate learning processes by reflecting onand relating to theoretical backgrounds and thus forminga mental model which links the theory to the experiment[20]. In the second, ”Observe” phase the experiment isdemonstrated and the students visually perceive its pro-cess and outcome. Here, students are expected to relatetheir observation to the previously anticipated result and,consequently, approve or reconsider their mental model[17]. In the final ”Explain” phase the outcome of theexperiment is revisited, typically by the teacher. In thispart the teacher explains the established link between thetheory and the outcome of the experiment.Previous research on the impact of POE scenarios haveshown that it is more effective for students’ learning andstudents pay more attention than during classical lecturedemonstrations in which the instructor only performs theexperiment and explains it in the framework of the es-tablished theory.To our knowledge there is no quantitative study whichexamines students’ misconceptions of Coriolis force andinertial centrifugal force. Stommel et al. report thatstudents consider the Coriolis effect as ”mysterious”and a result of ”formal mathematical manipulations” aspointed out by Persson [8, 13]. Previously observed mis-conceptions of students in mechanics imply, for instance,the ”Motion implies a force” misconception [21, 22]. Thismisconception potentially still persist in the students’understanding, thus complicating the students’ concep-tional learning of rotating frames of references and maytranslate to our study. We have accounted for these po-tential misconceptions in the posttest.
C. Analysis methods
1. Eye-tracking in educational research
During the POE tasks and the instruction (between
Observe and
Explain ) we have recorded the motion ofthe eyes of the students. In general, the eye-trackingtechnique has gained growing attention in educationalresearch in the past years, since several cognition-psychological and educational questions can be addressedwith this method. Theoretically, the eye-tracking tech-nique is founded on the eye-mind hypothesis, whichmeans that the visual focus is located on informationwhich are cognitively processed. This hypothesis wasoriginally formulated by Just and Carpenter [23] andwas later on experimentally confirmed by Kustov andRobinson [24]. Therefore, eye-tracking can be used, forinstance, to validate prevalent cognitive and multimediatheories [25], to reveal student strategies during problem-solving [26], and to discriminate between expert andnovice eye-gaze patterns [27], thus leading to improvedinstructional designs [28].In this context, Gegenfurtner et al concluded in a meta-analysis that experts, in comparison to non-experts, haveshorter fixation duration but more fixations on relevantareas and longer saccades [29], confirming a number oftheories, such as the theory of long-term working mem-ory [30] and the information-reduction hypothesis [31].In the context of physics education, eye-tracking has beenused, for instance, in the context of kinematic graphs andvector fields. Klein et al. found that high-performingstudents rather follow an equation-based approach thanlow-performers, thus executing more vertical and hori-zontal eye-movements during the interpretation of two-dimensional vector fields [28]. Apart from that, Mad-sen et al found that the response accuracy is correlatedto focus duration on relevant areas [32]. To our knowl-edge, eye-tracking has not been applied in the context ofdemonstration experiments in POE settings.
2. Self-confidence ratings
In this study, we use self-confidence ratings after thestudents have answered a question. These meta-cognitiveratings in a single choice format reflect the ability ofstudents to self-monitor their thought processes, whichcomprises a reflection of the understanding of the topicand the performance in the task [33]. In common inter-pretations of confidence ratings, the difference betweenthe confidence rating and the accuracy of the responseis termed bias . The bias is low for a student who hasa comparable confidence rating to his or her accuracyand, consequently, it would be high if the student tendsto over- or underestimate his or her performance. Thelevel of the bias is an indication for the calibration . Adeviation from a zero bias is an indication for a lack ofcalibration. The relatively robust effect of overconfidencecan be explained within the probabilistic mental model(PMM) theory proposed by Gigerenzer et al. [34, 35], inwhich confidence judgments are first a spontaneous con-sequence of a local mental model (LMM). In case, wherea LMM in the context of a specific task fails, a PMM iscreated in which the person retrieves probabilistic cuesfrom the environment. The mismatch between the cuevalidity and ecological validity, which is the true accountof a certain situation, might be one of two reasons for an overconfidence. The second potential reason within thePMM theory is that the set of information retrieved fromthe environment is not a representative selection for thereference set [34] and, for comparison, the reason is nota misled perception of the task difficulty [36, 37].In the field of physics education, Planinic et al. foundsignificantly higher confidence ratings for wrong answersin the area of Newtonian dynamics than in the area ofelectrical circuits, suggesting that concepts of Newtoniandynamics are more prone to misconceptions [38].In this work, we use the confidence ratings as an aid toidentify underlying misconceptions, which reveal them-selves when the student appears to be rather confidentwith an incorrect answer. Furthermore, this study ex-plores the influence of the level of calibration on the con-ceptual learning within a POE setting and relates theconfidence to eye-tracking metrics.Within this educational framework we formulate threeresearch questions: • What are the prevailing misconceptions of physicsstudents in the field of rotating frames of reference? • What is the influence of the demonstration experi-ment on learning about the outcome of the experi-ment? • Is there a specific eye-movement pattern which re-lates to the performance or confidence of physicsstudents within a POE setting?
III. MATERIALS AND METHODSA. Participants
The sample consists of 21 freshman students witha physics major of the Technische Universit¨at Kaiser-slautern, Germany. The students were participants of thelecture ”Experimental Physics 1” where they had seen ex-perimental demonstrations and the mathematical deriva-tion of the topic of rotating frames of reference in one lec-ture, one tutorial, one problem sheet, and one recitationsession prior to participation in this study. Participationin this study was voluntary and was compensated with10 Euro. The study took place several weeks before thefinal exam of the lecture and the students expected thatthe topic might be part of the exam.
B. Experimental setup
The setup consists of a rotating disc with a diameterof 55 cm which is connected to a motor that allows thedisc to rotate at a constant angular velocity (see Fig.1a and 1b). Initially, the sphere rests at the end of atilted rail which is attached to the rotating disc point-ing in the direction of the center of rotation (since therail is attached to the rotating disc the sphere receivesan initial tangential velocity component). As soon as therail passes a trigger, the sphere starts to roll down therail (from this acceleration the sphere receives an initialradial velocity component). The experiment is recordedfrom the top via two cameras. The first camera is con-nected to the stationary frame and does not move whilethe disc rotates. The second camera is attached to therotating disc, allowing the observation in the perspectiveof a rotating observer.In the stationary system, the sphere moves uniformly ina straight line on the left side of the disc in respect tothe center of rotation. Note here, that it does not runthrough the center because the resulting motion is a su-perposition of the tangential and the radial part. Thismeans that the answer (b) is correct in the stationaryframe of reference K (see Fig. 1d). In the rotating frameof reference, the trajectory d) describes the motion cor-rectly (see Fig. 1d). C. Study design
The study design is outlined in Fig. 1c. The pretestconsisted of three single choice items in a paper-penciltest assessing the understanding of essential representa-tions of vectors. Thus, we verified whether or not thestudents had visual understanding of typical depictionsof rotating frames of reference used in this study - a nec-essary prerequisite for learning from multiple visual rep-resentations as in the instruction part [39]. It was fol-lowed by an explanation of the experimental setup andthe procedure of the experiment (without demonstrationyet) by the instructor (see Fig. 1 b,c). In this phase, thestudents were allowed to ask questions.Afterwards, the students were asked two questions to an-ticipate the trajectory of the sphere in a stationary frameof reference (first) as well as in a rotating frame of ref-erence (second), each of them in a single choice format(see Fig. 1d for answer alternatives), which is termedthe
P redict phase. These two questions were computer-based and the eye movements were recorded. After eachprediction, the students were asked to rate their confi-dence on a four-point Liekert scale ranging from “veryconfident” to “very unconfident”.Then, the instructor demonstrated the experiment twice(part
Observe ). The students were allowed to observeit from every perspective. This part was supposed toclosely resemble an ideal situation of a lecture demon-stration. Then, the students were asked to answer thesame two questions as in the
P redict phase in order toreport their observation of the trajectory in the station-ary and in the rotating frame of reference. Again, weused eye-tracking and confidence ratings for these twocomputer-based items.Subsequently, the students received the computer-basedinstruction consisting of two text pages and six videos.The first page displayed a standard text book instructionof inertial forces including the equations of the Coriolis
FIG. 1. (a) The experimental setup for the demonstration ofrotating frames of references. (b) Top view of the rotating disc. (c) Study design where ∗ indicates the parts in which the eyemovements were recorded and (d) the alternative answers ofthe POE items in the stationary frame of reference K and therotating frame of reference K ′ . In both coordinate systemsthe distractors are identical. and centrifugal force. The second page explained the tra-jectory of the sphere rolling over a rotating disc in theparticular context of the previously demonstrated exper-iment. This page also contained two snapshots of thefinal location of the sphere during the experiment fromeach perspective (see Fig. 2) augmented with circles andarrows indicating the trajectory and velocity vectors inboth frames of reference.After this first instruction page, three videos from eachof the two perspectives (i.e. six videos in total) wereshown to the students. The first video showed the ex-periment recorded by the stationary camera in real time.It was augmented with the same information as in thesnapshots in Fig. 2. The two following videos were iden-tical to the first one but they were played in slow motion(4 × slower). The three videos recorded from the rotatingcamera were produced in the same format and played in FIG. 2. Snapshots during the final phase of the experiment inthe stationary frame of reference (a) and the rotating frameof reference (b). The black coordinate system K (axes x andy) is stationary and the green coordinate system K ′ (axes x ′ and y ′ ) rotates at the same angular velocity as the disc. the same order. The students had no option to pause orreplay the videos.After the instruction, the posttest in a paper-pencil for-mat and two computer-based questions followed. It con-sisted of seven true-false items, two items Explain (iden-tical to
P redict and
Observe ) and seven single choiceitems, two of which had a direct link to the experimentand they were posed in the Eye-tracking setting. Aftercompleting the posttest, the students were asked to com-ment on their responses of two single choice items fromthe posttest in an audio interview. The aim of the inter-view was to reveal potential misconceptions. Therefore,these two questions were directly motivated by the mis-leading depictions in literature (see above).
D. Eye-tracking equipment
The motions of the eyes were recorded using a To-bii X3-120 stationary eye-tracking system with a visual-angle resolution of 0 . ◦ and a sampling rate of 120 Hz.The questions were presented on a 22-inch computerscreen with a resolution of 1920 × IV. RESULTSA. Test scores of POE items
The test scores of the POE questions are shown in Fig.3a. The score in each POE part is the average score fromtwo questions about the trajectory of the sphere on therotating disc: The first question is about the trajectoryin a stationary coordinate system and the second one is about the trajectory in a rotating coordinate system.It is noticeable that the students have very low scores
FIG. 3. Test scores of POE items (a) and probability of dif-ferent error types (b). The inset in panel (a) shows the totalnumber of errors N of the POE items which refers to theanalysis in Sec. IV B (see also Tab. I) during the P redict phase. The demonstration of the ex-periment which is the only intervention between
P redict and
Observe has no significant effect on the score ( p =0 . Explain , 62 % of the students report the tra-jectories of the sphere correctly.The average confidence ratings do not change between
P redict (2 . ± .
7, where 1 = very confident and 4 =very unconfident) and
Observe (2 . ± . B. Analysis of distractors
For a deeper understanding of error sources and theinfluence of interventions, we divided the distractorsof the POE items into different types. The distractorsare displayed in Fig. 1d. The following example maydemonstrate the motivation for this process. A studentwho chooses a straight trajectory through the center ofrotation in the stationary frame of reference K has adifferent perception of the trajectory and a potentiallydifferent concept of the situation than a student whochooses a curved trajectory which is deflected to theright of the center in K , despite the fact that bothanswers are incorrect.In detail, we identified four different types of errorsamong the distractors of the POE task:I: Confusion of K and K ′ : When a student choosesa curved trajectory in K or a straight trajectory in K ′ . In K , this error type occurs when a studentchooses either one of the distractors (d), (e), (f) or(g). In K ′ this error type occurs when a studentchooses either one of distractors (a), (b), (c).II: Inversion: Here, the student chooses a distractorwhich depicts a trajectory to the right in respectto the center of rotation. Included distractors in K : (c), (e), (g). Included distractors in K ′ : (c),(e), (g).III: Initial condition: Here, the student does not con-sider that the sphere also has a tangential veloc-ity component in K and chooses the trajectorythrough center of rotation. Included distractors in K : (a). Included distractors in K ′ : (a).IV: Curvature: In this error, the student selects a dis-tractor with an incorrect curvature. Included dis-tractors in K : (f), (g). Included distractors in K ′ :(f), (g).Following this line of thought, the assignment the errortypes to the different distractors of the POE items im-plies that some distractors exhibit more than one error(see Tab. I). The number N of errors for one answer al-ternative ranges from 0 − K and 0 − K ′ . Fig. TABLE I. Error type (ET) and number of errors ( N ) for eachanswer alternative in K and K ′ .Distractor ET K N K ET K ′ N K ′ a III 1 I, III 2 b cor 0 I 1 c II 1 I,II 2 d I 1 cor 0 e I,II 2 II 1 f I,IV 2 IV 1 g I,II,IV 3 II,IV 2
3b shows the probability of each error during the POE tasks. The error probability displayed in this figure isthe average probability of the ones within the two coor-dinate systems. It is noticeable that, as a consequenceof the experiment demonstration, all errors are reducedbetween
P redict and
Observe except the inversion error.In fact, the average difference between the total num-ber of errors in
P redict ( N = 3 . ± .
40) and
Observe ( N = 2 . ± .
29) exhibits a significant medium effect(Cohen’s d = 0 . p < .
05, see inset of Fig. 3a). Incomparison, the average difference between the numberof errors in
Observe and
Explain ( N = 0 . ± .
03) ex-hibits a significant very large effect (Cohen’s d = 1 . p < . P redict and
Observe was found in the observation of the “initialcondition”, i.e. the students were able to correct theirprediction that the sphere does not go through the centerof rotation. In contrast, the observation of the demon-stration experiment did not affect the “Inversion”-error,i.e. the students did not recognize that the sphere hadbeen deflected to the left side in respect to the center ofrotation if they had previously predicted that the trajec-tory is located on the right side of the disc.The largest improvement between
Observe and
Explain ,as a consequence of the instruction, is the “Confusion”-error, i.e. after the instruction most students can relate alinear trajectory to the stationary frame of reference anda curved trajectory to the rotating frame of reference.And again, as previously observed between
P redict and
Observe , the instruction did not affect the “Inversion”-error, i.e. even after seeing the trajectory in an aug-mented photograph and in six videos those students whopreviously made the “Inversion”-error still fail to real-ize that the trajectory in both coordinate systems is lo-cated on the left side of the disc. Apart from this, the“Curvature”-error could be corrected entirely using theinstruction and also the error in the “initial condition”was made only by one student after the instruction.Tab. II shows the confidence ratings of each error
TABLE II. Average confidence ratings in respect to error typein the
P redict and the
Observe part. The numerical confi-dence values correspond to: 1 = very confident, 2 = confident,3 = unconfident and 4 = very unconfident.Error
P redict Observe type Confidence ConfidenceI 2.2 2.2II 2.6 2.4III 1.9 1.2IV 2.5 3.3 type during the
P redict and
Observe questions. It isnoticeable that the confidence ratings between
P redict and
Observe are very similar and no significant changecan be observed. The students which made an error oftype III had the highest confidence, particularly in the
Observe part. In contrast, the students which made anerror of type IV had the lowest confidence in the
Observe part.The confidence ratings neither correlate with the num-ber of errors ( r Pearson = 0 . p = 0 .
90) nor with theimprovement between the
P redict and the
Observe part( r Pearson = − . p = 0 . K and the rotating frame ofreference K ′ .In the context of confidence ratings in science educa-tion research, the Dunning-Kruger effect, which statesthat low-performing students rather tend to overestimatetheir performance, is often reported [40, 41]. Here, we arenot able to verify this effect due to the small number ofcorrect answers during the P redict and the
Observe part.
C. Student interviews
The confidence ratings suggested that there are smalldifferences between error types. To consolidate this find-ing and to identify misconceptions we performed studentinterviews after completing the posttest. In these inter-views we asked the students to comment on their answersof two particular questions Q1 and Q2 of the posttest. Inquestion Q1 the students were asked to name the forces,which are required to describe the trajectory of an air-plane flying from the center of an rotating disc outwardsin a uniform motion. In question Q2 the students wereasked to predict whether or not water would slosh overthe edge of a glass if the glass moves along a curved tra-jectory in K ′ but uniformly along a straight line in K .The two questions and the possible answers are outlinedin the Appendix. The interviews were conducted in Ger-man and, afterward, translated to English. Languageerrors were corrected to improve readability.Table III shows the probability of each possible answer. TABLE III. Distribution of answers of the two interview ques-tions Q1 and Q2 (see Appendix for the questions and possibleanswers). The correct answer is marked with a dagger.Distractor Q1 [%] Q2 [%]a) 28.6 † † In question Q1, the distractor (c) has the highest prob-ability. This corresponds to the answer that only theCoriolis force is required to describe the trajectory of theairplane flying over a rotating disc. In the interviews,all students who chose this answer either justified theirresponse by the thought that the airplane has no connec-tion to the rotating disc or argued that in the absence ofa centripetal force, no centrifugal one is required for thedescription of the trajectory. Here, we show two exam-ples of medium performing students M1 and M2 and one example of a high performing student H:Instructor: “Please comment on your answer of questionQ1.”
M1: “I ticked the third one, because actually only theCoriolis force would have to work. I originally assumedthat the Coriolis force is a counterforce of the centrifugalforce. But since this is wrong and actually the centrifu-gal force is the counterforce of the centripetal force andsince we have here, in my view, no centripetal force, thereshould be no centrifugal force here.”
In the comments, the student describes the role of thecentrifugal force as a counterforce to the centripetal force.This implies, that the student does not apply the conceptof the inertial centrifugal force and, instead, refers to theconcept of the reactive centrifugal force which occurs asa reaction to a centripetal force.And this is another example for the comments to a wronganswer in which the student assumes that only the Cori-olis force is necessary to describe the trajectory of theairplane.Instructor: “Please comment on your answer.”
M2: “The plane is deflected to the left from the point ofview of K ′ and as it flies and the air friction is neglected,it has no contact with the ground and therefore no cen-trifugal force has to act which somehow has to keep it ona circular path and therefore I think that you can neglectthat. But if now a person would rest in the center of K ′ ,he would see that the aircraft is apparently being deflectedto the left. The plane actually flies straight ahead, but thedisc on which the observer stands turns to the right. Andtherefore, seen in the rotating system, only the Coriolisforce acts which would deflect the aircraft.” Here, the student argues that the missing contact of theairplane to the rotating frame of reference is responsiblefor the description via the Coriolis force.Instructor: “Please justify your answer of question Q1.” H: “To describe the trajectory in K ′ , both the Coriolisand the centrifugal force are needed. This is the casebecause, first, we have a velocity of the airplane in therotating frame of reference. That’s why we need a Cori-olis force. And since there is a distance r ′ to the centerof rotation, which is the origin of the coordinate systemK’, there must also be a centrifugal force.” In the arguments, the student directly refers to the non-zero velocity v ′ of the object in K ′ in the equation of theCoriolis force F Cor (Eq. (3)) and to the non-zero distanceto the center of rotation r ′ which is a necessary compo-nent in the equation of of the centrifugal force F Cen (Eq.(4)).In question Q2, the distractor (e) has the highest prob-ability among the incorrect answers. This correspondsto the answer that the water is sloshed opposite to thedirection of the sum vector of Coriolis and centrifugalforce. This answer would be correct if there were realforces acting on the glass. This is the reason for the an-swer of the student M1 who selected this answer:Instructor: “Please tell us why you have selected this an-swer.”
M1: “The water spills out for sure, because of the iner-tia of the water, so it’s just a question of how it spillsout and I’ve decided to tick the answer (e), because ofthe idea that the water goes straight ahead and thus thedirection of motion is precisely directed opposite to theseforces. Because it does not matter to the water, whetherit is in the rotating system or not.”
In this reasoning to question Q2, the student seems con-vinced of the fact that Coriolis and centrifugal force causean effect in the stationary reference system K .And this is the comment to the answer of the high achiev-ing student H to question Q2.Instructor: “Please give a reason to your answer to ques-tion Q2.” H: “For the description of the trajectory, the Coriolis andcentrifugal force are introduced and in K the glass makesa straightforward uniform motion. But since both areonly apparent forces, they are only of relevance for thetrajectory description in K ′ and do not really affect theglass in the reference system K , the water does not spillover. So, in this straight uniform motion, no force actson the glass.” In this reasoning, the student refers to the fictitious char-acter of the Coriolis and centrifugal force and draws thecorrect solution by relating the uniform motion of theglass to the absence of forces in the stationary referencesystem K .The reasoning of student M2 is similar to the one of H,therefore it is not displayed here. D. Confidence levels affect visual focus
We were interested whether confidence ratings follow-ing the
P redict and the
Observe tasks have an influenceon the visual attention of the students. For this purpose,we divided the students in two groups: the first grouprated their confidence in this items with “confident” or“very confident”, the students in the second group ratedtheir confidence with “unconfident” or “very unconfi-dent”. Additionally, we designed a pattern of square-likeAreas of Interest (AOIs) with a size of 50 ×
50 pixels thatcovers all relevant areas (including the figure of the ro-tating disc, the coordinate system and the distractors)except the question text (see Fig. 4). Then, we com-pared the total visit duration and the size of the regionsof attention. Table IV shows the visit duration and thenumber of AOIs N AOIs which received a high focus (i.e.a focus which is longer than the average focus of eachstudent) of students who feel confident of their answerand those who are unconfident during the predict task.Here, N AOIs is a measure of the size of the area of focus,i.e. it indicates the spatial spread of attention.The analysis demonstrates that there is significant verylarge effect in the maximum visit duration and a signif-icant large effect in the average and total visit durationbetween confident and unconfident students during thepredict questions.
FIG. 4. Locations of square-shaped AOIs (50 ×
50 pixels)during the
P redict and
Observe task.TABLE IV. Maximum, average, and total visit duration onAOIs in seconds during the
P redict -task as well as the numberof AOIs ( N AOIs ) which exhibit a visit duration larger thanthe average one.Confident Unconfident d p
Max [s] 3.38 7.33 1.21 0.005Average [s] 0.47 0.76 1.11 0.005 N AOIs
Table V shows the visit duration and the number of
TABLE V. Maximum, average, and total visit duration onAOIs during the
Observe -task in seconds as well as the num-ber of AOIs ( N AOIs ) which exhibit a visit duration longerthan the average one.Confident Unconfident d p
Max [s] 2.74 5.68 0.68 0.16Average [s] 0.45 0.65 0.53 0.20 N AOIs
AOIs which received a high attention. It is noticeablethat the difference in the maximum, average, and totalvisit duration of confident and unconfident students isclearly reduced in comparison to the
P redict -questionsso that the effects are not significant during the
Observe -questions. Furthermore, the results indicate that there isno significant difference in N AOIs between confident andunconfident students, which means that the studied areafrom where information is processed is similar betweenthese two student groups.
V. DISCUSSION
In this work we demonstrated how students understandconcepts of rotating frames of reference and how theyapply their knowledge to understand a standard lectureexperiment in which they are supposed to report the tra-jectory of a sphere rolling over a rotating disc in a rotat-ing and in a stationary coordinate system.The presented study reveals a number of misconceptionsin the field of rotating frames of reference which leads toa number of promising suggestions for future instructionsof the topic.The “Confusion”-error, which is the error to think thatthere is a linear trajectory of the sphere in the rotatingframe of reference or a curved trajectory of the sphere inthe stationary coordinate system, was the most commonerror of students in the
P redict as well as in the
Observe items. This error could be successfully resolved via theinstruction using a fundamental theoretical review of ro-tating frames of references, augmented photographs andsix augmented videos. However, the results highlight thehigh difficulty of this topic for first-year physics students.This yields, for instance, the surprising observation thatthe “Inversion”-error is neither corrected during the ob-servation of the experiment nor during the instruction.This indicates that some errors require special attentionwhich potentially could be realized via the implementa-tion of cues [42] or via highlighting and discussing com-mon errors of students in advance. For this reason, it islikely that a briefer instruction could fail to transfer thelink between mathematical equations of the Coriolis andcentrifugal force and their application to the trajectoryof the sphere in a rotating and stationary frame of refer-ence.Furthermore, the item difficulty of the POE tasks provethe conceptual and perceptual complexity of the topicof rotating frames of references. Only one out of fivephysics students was able to report the observation ofthe trajectory of a sphere rolling over the disc correctlyin a single choice question. This is significantly less thanprevious reports of POE interventions [14, 15]. This im-plies that there was no obvious improvement which canunambiguously be attributed to the observation of the ex-perimental demonstration, which was previously pointedout by Crouch et al. [15]. However, the detailed analysisof distractors of the single choice questions during POEin combination with the identification of different errortypes reveals the hidden benefits of lecture demonstrationfor learning. The probabilities of all error types show de-creasing trends between
P redict and
Observe with theexception of the inversion error. We could identify that,particularly false accounts for initial conditions can becorrected. This type of analysis clearly reveals the bene-fits of lecture demonstrations on learning about the out-come of the experiment.
A. Misconceptions related to Coriolis andcentrifugal force
The interviews as well as the distractor analysis of thePOE items reveal prevailing misconceptions among first-semester physics students in the field of rotating frames ofreference. Nearly half of the participants (42.8 %) believe that the centrifugal force is only necessary to describe thetrajectory of an object in a rotating coordinate systemwhen there is a coupling of the object to the rotatingsystem. This misconception is likely to be attributed tocommon instructional connections of the inertial centrifu-gal force and the reactive centrifugal force which occursas a reaction to a Centripetal force, as for instance in acarousel. In the light of these results, we suggest to ver-bally discriminate between these two types of centrifugalforces. Apart from that, about one out of four physicsstudents (23.8 %) do not include the fictitious characterof inertial forces in their arguments and rather argue thatthey have the same effect on objects as real forces.
B. Eye-tracking reveals confidence
The Eye-Tracking analysis reveals a direct link betweenconfidence ratings and visit duration on AOIs during theitems of the
P redict phase. Students which are confidentof their answer spent significantly less time on the AOIsthan unconfident students. Despite this fact, unconfidentstudents distribute their attention on a similar-sized area.This observation is an interesting extension to previousresults and interpretations of long visit durations. For in-stance, Palinko et al. report that high visual attention onrelevant areas is related to a high mental effort [43]. Asa consequence, the visit duration has also been used as ameasure for (intrinsic or extrinsic) cognitive load withinthe framework of the Cognitive Load Theory [44]. Duringthe
Observe phase, there is no difference in the averageor total visit duration between confident and unconfidentstudents. The disappearance of the aforementioned re-lation between visit duration and confidence ratings inthe
Observe part might be attributed to the fact thatthe students have seen the exact same questions alreadyduring the
P redict phase and have naturally less time-on-task since the content of the page is already partiallyfamiliar to the students. This interpretation is supportedby an overall decrease of visit durations.Furthermore, we observe that students with low confi-dence levels and high visit durations in the
P redict itemsdistribute their focus on a similar-sized area as confidentstudents. This seems to indicate that unconfident stu-dents tried longer to extract the same amount of informa-tion as confident students. In the theoretical frameworkof Rau [39], the author points towards necessary pre-requisites for learning using multiple visual representa-tions. To identify and extract relevant information froma visual representation such as a graph, photograph, orschematic, students need visual representational under-standing, which refers to the conceptual knowledge ofhow a visual representation depicts information. In or-der to relate the information from two different visualrepresentations, as it is required in several parts of thisstudy, the students need connectional understanding oftwo or more representations. This knowledge refers to theability to identify relevant similarities between the repre-0sentation and to know about conventions for interpretingand combining the information from multiple represen-tations [39]. Embedding our results in this framework, itseems that unconfident students seem to try to developvisual and/or connectional understanding of the repre-sentations displayed in the
P redict -items.
VI. CONCLUSION
In this study we tested the conceptual learning ofphysics students during a POE task on rotating frames ofreference. The students had significant difficulties in pre-dicting and observing the correct trajectory of a sphere(total score of approx. 20 %) rolling over a rotating discin a stationary and a rotating coordinate system K and K ′ . Primarily, the low score can be attributed to the mis-conception of a confusion of the effects of inertial forcesin K and K ′ . Additionally, we found that some miscon-ceptions even withstood the instruction. Students whoinitially predicted that the sphere is deflected to the op-posite side on the disc (in respect to the actual trajectory)kept this conception during the Observe and
Explain part (“Inversion” error). This emphasizes the need foradditional instructional support in this topic for instancevia cues which highlight essential information.Furthermore, the results indicate that after the instruc-tion nearly half of the students answered that a centrifu-gal force will only be necessary if there is a coupling be-tween the object and the rotating system. In comparison,the misconception that an object shows a reaction to in-ertial forces in the same way as they do to real objectsonly persists in one quarter of the students.Within the POE task, the eye tracking analysis in combi-nation with confidence ratings showed that unconfidentstudents spent significantly more time extracting infor-mation than confident students. This finding demon-strates the cognitive activation particularly of unconfi-dent students during the
P redict phase. In contrast toprevious reports we found that passive observations ofexperiments, in fact, stimulate conceptual learning in adetailed distractor analysis which is not reflected in anincrease of total scores. At this point we cannot judge theimportance of this non-obvious learning behavior and ad-ditional research is necessary. In this way the results as-sist to understand conceptual learning during POE tasks.
VII. ACKNOWLEDGMENT
This work is funded by the Federal Ministry of Educa-tion and Research (BMBF; project: VorleXung; supportcode: 16DHL1001). The authors are responsible for thecontent of this contribution.
VIII. APPENDIXA. Theoretical background on rotating frames ofreference
When an observer examines motion of an object mov-ing uniformly in a stationary frame of reference (SFR)from a rotating frame of reference (RFR), the trajectoryappears to be curved in comparison to a trajectory whicha stationary observer (SO) would report. For instance, ifan object moved uniformly in a SFR, it would display acurved trajectory for a rotating observer (RO). The the-oretical description of the trajectory in a RFR requiresthe introduction of the centrifugal and the Coriolis force.They are “virtual forces” which means that Newton’sthird law of motion ( action = reaction ) does not hold forthem. In other words, both forces are not the result ofan interaction between two bodies but the consequenceof the motion within a RFR. If either one, the inertialcentrifugal or the Coriolis force, acts on a body, thereis no reaction from that body in the opposite direction.They are also called ”inertial forces” which emphasizesthe fact that the forces are caused by the inertia of themoving object. Typical examples include the motion ofclouds observed from the earth or a thrown ball observedfrom a person sitting in a rotating merry-go-round [11].The velocity ~v ′ of an object in a RFR which rotates witha constant angular velocity ~ω is given by the sum of thevelocity ~v of the object with position ~r in the SFR andthe negative tangential velocity − ~ω × ~r in the RFR : ~v ′ = ~v − ~ω × ~r. (1)The derivative d ~v ′ d t leads to the acceleration of the objectin the RFR [11]: ~a ′ = ~a + ~ω × ( ~r × ~ω ) + 2 ( ~v ′ × ~ω ) . (2)This equation shows the necessity of introducing addi-tional terms apart from the acceleration ~a in the SFRfor the mathematical description of the determination of ~a ′ . The second term in Eq. (2) corresponds to the iner-tial centrifugal acceleration and points radially outwardsfrom the axis of rotation. The third term is called theCoriolis acceleration which is perpendicular to the veloc-ity vector ~v ′ in the plane of motion.From Eq. (2) the terms for the Coriolis force follow: ~F Cor = 2 m ( ~v ′ × ~ω ) , (3)and the equation for the centrifugal force: ~F Cen = m~ω × ( ~r × ~ω ) . (4)In both equations m denotes the mass of the object. B. Questions for student interviews
The following questions were used during the studentinterviews:1Q1: Now imagine that, instead of the sphere, an air-plane flies straight and uniformly from the centerhorizontally outward over a rotating disk. The discrotates at constant angular velocity ~ω . The coor-dinate system K ′ has its origin in the center of thedisk and also rotates at the velocity ~ω . Please ne-glect air friction. Which forces are necessary to de-scribe the trajectory of the airplane in K ′ ? Pleasejustify your answer.Answers: a) With the help of Coriolis and centrifu-gal force, b) Only with the help of the centrifugalforce, c) Only with the help of the Coriolis force,d) No forces need to be introduced. FIG. 5. Figure of the moving water glass which correspondsto interview question Q2.
Q2: A glass is completely filled with water and movesin a straight line and uniformly in a stationary co-ordinate system K without friction over a rotatingdisk. The coordinate system K ′ rotates just likethe disc with the constant angular velocity ~ω . Thefigure below shows the trajectory of the glass in K ′ (see Fig. 5). To describe the motion in K ′ , a Cori-olis force and a centrifugal force are introduced. Isthe water sloshing over the edge? If so, in whichdirection? Please justify your answer.Answers: a) Yes, in the direction of the centrifugalforce, b) Yes, opposite to the direction of motion, c)Yes, in the direction of the Coriolis force, d) Yes,opposite to the direction of the Coriolis force, e)Yes, opposite to the sum vector of the Coriolis andcentrifugal force, f) No.2