Students' understanding of gravity using the rubber sheet analogy: an Italian experience
SStudents’ understanding of gravityusing the rubber sheet analogy: anItalian experience
A Postiglione , , I De Angelis , Dipartimento di Matematica e Fisica, Universit`a degli Studi RomaTre, Rome (ITALY) INFN Sezione di Roma Tre, Rome, (ITALY)[email protected]
Abstract
General Relativity (GR) represents the most recent theory of gravity,on which all modern astrophysics is based, including some of the mostastonishing results of physics research. Nevertheless, its study is limited touniversity courses, while being ignored at high school level. To introduceGR in high school one of the approaches that can be used is the so-calledrubber sheet analogy, i.e. comparing the space-time to a rubber sheetwhich deforms under a weight. In this paper we analyze the efficacy of anactivity for high school students held at the Department of Mathematicsand Physics of Roma Tre University that adopts the rubber sheet analogyto address several topics related to gravity. We present the results of thequestionnaires we administered to investigate the understanding of thetopics treated to over 150 Italian high school students who participatedin this activity.
Keywords : gravity, Einstein, General Relativity, space-time, Secondary Educa-tion, hands-on activity, experimental activity
The most recent and successful theory describing gravity is the Theory of Gen-eral Relativity (GR), introduced by Albert Einstein in 1916 [1]. This theoryrelies on the concept of space-time, a four-dimensional entity that unifies space(which has three dimensions) and time (one dimensional). According to thistheory, the space-time change according to the objects placed in it: a massive1 a r X i v : . [ phy s i c s . e d - ph ] F e b bject (a galaxy, a star, a planet, a dog) deforms the space-time producinggravity, i.e. attracting nearby masses.All modern astrophysics, including some of the most important and recentdiscoveries of the field [2–5] is based on GR. But yet, its study is typicallyaddressed only in advanced University courses, while it is ignored at lower levelsof education, such as in high schools, where gravity is only described using theNewtonian theory.The reasons for this are many. The mathematical complexity of GR forcesto adopt a purely qualitative approach, which is not easy to realize withoutoversimplifying the concepts too much. Moreover, even if one decides to onlydeal with the basic concepts at a qualitative level, such as the space-time andits deformation, he/she will have to accept a new vision of the world, far fromthe everyday experience.In recent years, several efforts have been made in order to address theseissues and create activities suitable to introduce GR in school curricula [6–16].Several other works focus on quantitative analysis of students’ understandingof gravity and GR, ranging from very young kids to university students [17–30].This present paper aims at contributing to this discussion through the analysisof questionnaires administered to over 150 Italian high school students whoparticipated to an activity we built to treat gravity and GR.The model we used is the popular rubber sheet analogy (RSA), which com-pares the space-time to a rubber sheet that deforms under the weight of a mass,and allows to simulate the gravitational attraction through marbles and ballsthrown on the warped sheet. Although this model shows some critical pointsextensively addressed in the literature [13, 31–33], it represents a powerful ac-tivity [6, 11, 18, 34] that has already demonstrated to be very well welcomed byItalian high school teachers [35].For this reason, at the Department of Mathematics and Physics of Roma TreUniversity we built a structure that could support a lycra sheet, and we used itas RSA to realize an activity addressed to high school students that could dealwith different topics such as Kepler’s laws, gravity assist, gravitational lensingand black holes. We asked all participants to answer three questionnaires, onebefore the activity, one immediately after and one four months later. In this waywe investigated their understanding of the most important aspects addressedduring the activity, and the possible presence and persistence of misconceptionsor wrong beliefs.The remaining paper is organized as follows. In section 2 we briefly describethe structure we used to exploit the RSA, the activity and the correspondingquestionnaires. In section 3 we illustrate the results of our research, focusing onthe main aspects that come out. In section 4, we discuss our achievements andin section 5 we present our conclusion and suggest some future development ofour work. 2 Background and data collection
In order to carry out an activity that dealt with gravity using the RSA we builta circular structure of 1.8 meters in diameter in aluminum covered by a lycrasheet of about 2x1.5 meters (
Figure 1 ), in collaboration with the mechanicalshop of the INFN Roma Tre Section. We chose this size for the structure toensure that a group of about 25 people (a typical Italian high school class) couldcomfortably watch what showed on the sheet.
Figure 1: The 1.8 m diameter structure of space-time we built at the Department of Mathematicsand Physics of Roma Tre University in collaboration with the mechanical shop of the INFN RomaTre Section.
In the period January - February 2020 we used this structure to carry out ouractivity with 6 high school classes, engaging more than 150 students. Before, af-ter and four months after the activity, we asked the participants to answer threequestionnaires in order to explore their understanding of the topics addressed . The activity consists of a lesson lasting an hour and a half during which momentsof strong interaction with participants using the rubber sheet are alternated totheoretical insights using videos and photos. We start with a description of themodel of RSA: we first introduce the concept of space-time and how it is re-lated to gravity, using a central weight and some marbles; we then focus on thesimplifications that the usage of the rubber sheet implies, that can lead to mis-conceptions and wrong beliefs. We believe that bringing these misconceptionsto light helps participants to overcome them (an idea that has been confirmedby the results of the questionnaires, as we will see in the following). First ofall, we underline that space-time can be deformed by any mass (or energy) andnot only by big masses. Then, we clarify that the rubber sheet represents atwo-dimensional space-time. We thus point out that the space-time curvatureoriginated by a spherical object is symmetrical in all directions, and therefore The answers to our questionnaires were anonymous. This study was carried out in accor-dance with the principles outlined in IOP Science ethical policy. up and down , but only a near or far from the source ofgravity.After these first clarifications, we start to show the participants how thismodel, although simplified, can allow to visualize quite faithfully the way inwhich planets orbit the Sun, i.e. following Kepler’s laws. Once the basic rulesof the game are shown, we treat other examples of motion of celestial objects.We show the Earth-Moon system, the orbits of a planet around two stars andthe phenomenon of gravity assist, that can explain the typical voyage of a spaceprobe. It is worth reminding that throughout these activities the students areactively involved, and firsthand experience the behaviour of the marbles throw-ing them on the sheet: in this way their attention is kept high, and they becomemore willing to ask questions and join the discussion.At this point a fairly complete picture of how the space-time and its defor-mation describe the phenomena that the students have only studied in termsof the Newtonian gravity has been given. Then we focus on the topics fullyexplained only by GR. We start from the phenomenon of gravitational lensing,representing light with a marble that deflects its trajectory when approachingthe central weight. Through videos and photos, we then show the participantsthe consequences of this phenomenon, and how it is used by astrophysicists tocharacterize some celestial objects. We then introduce black holes, explain thatthey are compact objects, underline the fact that they strongly affect only thesurrounding region and restate that this attraction does not point downwards .Finally we talk about the gravitational waves, the way in which scientists havediscovered them and their usefulness in improving our knowledge of the Uni-verse. We use three questionnaires: one administered before the activity in order toanalyze the prior knowledge of the participants, one administered right afterthe activity so that we could investigate the knowledge acquired thanks to theactivity, and one questionnaire administered four months after the activity inorder to study the permanence of the knowledge obtained.All the questionnaires have the same structure: a first part that deals withthe age and the school attended by the participants, a second part that focuseson the Newtonian theory of gravity and a third part that considers more complextopics related to GR.When designing the answers to the questionnaires we paid attention inadding distractors so that we could investigate the presence of wrong beliefsand misconceptions.While the two questionnaires administered after the activity are identical,between the questionnaires administered before and after there is a slight dif-ference: the pre-questionnaire includes both multiple choice questions with 5alternatives (one of which was always “I don’t know”) and open questions, while4he post-questionnaires only used multiple choice questions with 5 alternatives,one of which could eventually be an open answer.
Overall we obtained 153 answers to the questionnaire administered just beforethe activity, 125 answers to the one administered just after the activity andonly 42 to the one administered four months later the activity. This relevantdecrease in the sample is mainly due to the fact that, unlike the other two, thelast questionnaire was administered remotely, and in the period when the Italianschools were closed due to the first Covid-19 lock-down; the teachers could thusnot follow their students’ compiling live.
Figure 2:
The age distribution of the participants before, after and four months after the ac-tivity. Since the Newtonian gravity is typically introduced during the third year, the majorityof the participants already treated it. The percentage of each answer is shown in brackets(left: pre-questionnaire, right: post-questionnaire). igure 3: The distribution of the answers to the question What is the space-time? . The percentageof each answer is shown in the legend in brackets (left: pre-questionnaire, right: post-questionnaire).Figure 4: The distribution of the answers to the questions related to the cause of the deformationof the space-time. The percentage of each answer is shown in the legend in brackets (left: pre-questionnaire, right: post-questionnaire). igure 5: The distribution of the answers to the question Which is the shape of the orbit that aplanet describes around the Sun? . The percentage of each answer is shown in the legend in brackets(left: pre-questionnaire, right: post-questionnaire).Figure 6: The distribution of the answers to the question
How do the planets move around theSun? . The percentage of each answer is shown in the legend in brackets (left: pre-questionnaire,right: post-questionnaire). igure 7: The distribution of the answers to the question Which planets orbit faster around theSun? . The percentage of each answer is shown in the legend in brackets (left: pre-questionnaire,right: post-questionnaire).Figure 8: The distribution of the answers to the question
If we want to reach Saturn with a probe,it is better to make it do . The percentage of each answer is shown in the legend in brackets (left:pre-questionnaire, right: post-questionnaire). igure 9: The distribution of the answers to the question What is a gravitational lens? . Thepercentage of each answer is shown in the legend in brackets (left: pre-questionnaire, right: post-questionnaire).Figure 10: The distribution of the answers to the question
What are gravitational waves? . Thepercentage of each answer is shown in the legend in brackets (left: pre-questionnaire, right: post-questionnaire). igure 11: The distribution of the answers to the question What is a black hole? . The percentage ofeach answer is shown in the legend in brackets (left: pre-questionnaire, right: post-questionnaire).Figure 12: The distribution of the answers to the questions related to Kepler’s second and thirdlaw by participants attending the third year of high school, before the activity. The percentage ofeach answer is shown in the legend in brackets. igure 13: The distribution of the answers to the questions related to Kepler’s second and third lawby participants attending the third year of high school, after the activity. The percentage of eachanswer is shown in the legend in brackets. Figure 2 , themajority of the participants already treated the Newtonian gravity in school.
To assess the participants’ understanding of the concept of space-time and itsdeformation, we asked for a definition of space-time (
Figure 3 ). Before theactivity, the 46% of the students’ answers is correct (
It’s the union of space andtime, who together form a 4-dimensional entity ), while the 34% of them statesthat it is 2-dimensional. The remaining 20% of the participants confuses thespace-time or with the fourth dimension, or with the speed (which in school isusually defined as the ratio between space and time) or says they do not know.After the activity, the percentage of correct answers increases: over 90% of theparticipants states that the space-time has 4 dimensions, while only the 6%still confuses the concept of space-time with the fourth dimension. After fourmonths this positive trend is confirmed, since the percentage of correct answersis 86% (see the Appendix).We then focused on the cause of the deformation (
Figure 4 ). Before theactivity with the rubber sheet, the majority of respondents (58%) says thatthey do not know if there is something that can deform the space-time, whilethe 1% states that nothing can deform it, while the 41% declares that it canbe deformed. In more detail, the 9% generically states that it can be deformed,the 28% gives a correct example of the source of the deformation, while the 4%gives a wrong example.Immediately after the activity, the 60% of the participants states that thecause of the deformation can be any mass, the 34% only refers to a big masswhile the 3% cites a force that points downward. The same trend can be foundfour months after the activity, when the answer “big mass”is cited by the 31%of the participants (see the Appendix).
Later, we investigate the participants’ knowledge of the three Kepler’s laws.Before the activity, the majority of the students already remember the firstKepler’s law, since the 90% states that the shape of the orbit that a planetdescribes around the Sun is elliptical (
Figure 5 ). The 4% says that planets havea random orbit, while only less than 1% admits they do not know. Immediatelyafter the activity the situation slightly improves, since the percentage of correctanswers stays high (95%), while the remaining 5% of the participants gives12rong answers. Even after four months, the 93% of the sample gives the correctanswer (see the Appendix).The Kepler’s second law shows a more considerable improvement (
Figure6 ). Before the activity, in fact, just over half (52%) of the participants correctlyanswer when asked about the way in which planets move around the Sun. Theothers (48%) think that the planets move slower when they are closer to the Sun(16%), that they always keep the same speed along the orbit (16%) and thatthey are faster both when they approach the Sun and when they move awayfrom it (15%). Right after the activity the percentage of correct answers risesto 83%, while all the other wrong answers reduce. After four months from theactivity about the 80% of the sample of respondents still remember the correctanswer (cfr. Appendix).Regarding Kepler’s third law, participants were asked which planets orbitfaster around the Sun (
Figure 7 ). If before the activity the 70% already knowhow to correctly respond, this percentage rises to 91% after the activity, and to83% four months after (see the Appendix).In order to investigate the comprehension of the gravity assist, we asked howspace probes travel in space (
Figure 8 ). In this case the 55% of the participantsdo not correctly answer the related question of the pre-questionnaire. After theactivity the majority of students (79%) have understood what is gravity assist,a trend that is confirmed after four months (76%).
Before the activity, only less than a half of the participants (44%) gives theright answer when asked to describe the phenomenon of gravitational lensing(
Figure 9 ), while the 41% declares they do not know at all. The remaining 16%confuses the phenomenon or with something related to a glass lens, or explainthe phenomenon using the Italian word “lens=lenti”meaning “slow”instead oflens. After the activity, the correct answers rise to 93% of the total, whilethe percentage of those who associate the phenomenon to a glass lens remainsunchanged (3%). Also after four months the percentage of correct answersconfirms the improvement of the result (98% of correct answers).Regarding the gravitational waves, before the activity, about 52% gives thecorrect definition of them (
Figure 10 ). The remaining half is made up of a21% who admits they do not know what they are, and a 27% who gives wronganswers confusing them with other topics they studied in school (waves formedon water) or with concepts they have heard of after the spread of the news aboutdiscovery (black holes and lasers). After the activity the 83% gives the correctdefinition. After four months we have a percentage of 81% of correct answers.Finally, we asked the participants to give us a definition of black holes(
Figure 11 ). Before the activity, the 57% does not feel confident enough togive an answer. Only less than the 2% gives completely wrong answers (the es-cape speed, nitrogen, a place where there is an emptiness). The remaining 41%gives an answer not far from the correct one: the 24% associates black holeswith a mass, a body or a dead star, the 9% with a deformation of space-time,13he 3% with something that absorbs everything, or nearby matter, or energy orlight, the 5% associates them with a very intense gravity or with something thatcannot be escaped. After the activity, the percentage of correct answers risesto 79%, giving that they associate black hole with something that attracts onlythe nearby matter. Only the 7% chooses the answer inserted as a distractor,i.e. that the black hole attracts matter downward . This is also confirmed by thetrend shown in the four months later answers: the 84% gives the right answer,while only the 7% recalls the idea of an object that attracts downward . Overall, the students who participated in our activity with the rubber sheethave significantly improved their knowledge about the topics addressed. Thistrend is also confirmed by the responses obtained four months after the activity.As regards the topics usually treated in school, the activity straightens theirknowledge and improves their understanding. The most significant case in thissense is Kepler’s second law, that the rubber sheet has helped to visualize andremember. In particular, while the first law was already known before theactivity by the great majority of students (90%), the second was rememberedonly by the 52% of the total. This trend could suggest that the second law isstudied in school in a more mnemonic way with respect to the first law, withoutsufficient understanding. In fact, if we consider only the students attending thethird year (who have just treated the Kepler’s laws in school, and thus whoshould remember them better), we note that only the 62% of them correctlyrecall the second law before the activity, while the first law reaches a percentageof the 95% (
Figure 12 ).Moreover, when verbally asked during the activity, the participants oftenremembered the statement of the second law “
A line segment joining a planetand the Sun sweeps out equal areas during equal intervals of time. ” but theyevidently did not understand its meaning. Only when they saw the motion ofthe marbles on the sheet they really understood it, as confirmed by the resultsof the post-questionnaires. In fact, after the activity, also the percentage ofcorrect answers for the second law becomes very high, as shown in
Figure 13 (86%).Regarding the topics that are not typically treated in school, like black holes,gravitational waves and gravitational lense, the data show a prior knowledgelower than the one related to Kepler’s laws, as might be expected.Despite this fact, also for these topics the improvement has been remarkable,since the percentage of correct answers increases from less then a half of the totalto almost the 100%. Our analysis also shows that the acquired knowledge seemsto be lasting, given that almost all of the respondents of our sub-sample continueto correctly answer even after four months from the activity. Moreover, it can beseen that, although these topics are not addressed in high school curricula, theywere not completely unknown by the students even before the activity. Thiscould be a sign of a widespread fascination for these topics, which encourage14he students to search for information even if they are not treated in school (asindicated by the fact that they give reasonable answers even before the activity:black holes are “dead stars”, “space-time deformations”, “something where youcan’t escape from”), or at least of a strong bond with current news (as suggestedby the answers we received for the gravitational waves, related to lasers or blackholes).Our analysis also suggests that the rubber sheet can also be effective in deal-ing with the misconceptions and wrong beliefs related to gravity. In particular,our data allow us to investigate two misconceptions: the fact that the deforma-tion of space-time is only due to big masses and the idea that gravity is alwaysa force that points downward. As regards the former, the data shows that ouractivity, even having faced this aspect, does not solve it completely, since thereis a substantial part of students (34%) that after the activity refers only tovery big masses when asked about the source of deformation. This means thata greater attention must be paid in fighting this idea throughout the activitywith the rubber sheet, emphasizing more than once that space-time is not onlyrelated to stars and galaxies, but also to objects with smaller mass, such thosewho populate our everyday life.Regarding the idea of a gravity that points downward, our data show that itcan indeed be fought using the RSA, since only the 3% of our sample choose thedistractor “a force that points downward” when asked to specify what deformsthe space-time. This means that the discussion that is made at the beginningof the activity, that deals with the absence of a privileged direction of gravityoutside the Earth, and with the limitations of the rubber sheet analogy, has paidoff. Our idea is also confirmed by the results obtained after four months fromthe activity, when the answer “downward” completely disappears, and also bythe results obtained for what concerns black holes.Overall, our analysis therefore leads us to think that the rubber sheet, albeitwith all its limitations, represents a formidable tool for introducing GR at highschool level, given that it helps students both to visualize how gravity worksand to remember it longer.
In this paper we presented the results of the questionnaires we have administeredto over 150 high school students who participated in the activity held at theDepartment of Mathematics and Physics of Roma Tre University that uses therubber sheet analogy to address several topics related to gravity. Our data showthat the rubber sheet can indeed be very useful not only in treating topics thatcan be explained by GR, but also to better understand and remember the topicsgenerally addressed in high school using Newtonian theory of gravity.In the next future we plan to continue to test our proposal involving anincreasing number of students. In particular, we hope to resume the activityin person, so that we could also follow more closely the compiling of the ques-tionnaire administered four months after the activity, in order to have statistics15omparable to the ones obtained with the questionnaires distributed right be-fore and right after the activity. This will help us to understand and quantifybetter the long-term effect of introducing the rubber sheet at high school level.
Acknowledgements
This work has been supported by the Italian Project ’Piano Lauree Scientifiche’and the Young Minds Section of Rome of the European Physical Society. Aspecial thanks goes to the mechanical shop of the INFN Roma Tre Sectionthat built our aluminum space-time structure. Also thanks to the teachers andstudents who participated in our activity at the Department of Mathematicsand Physics of Roma Tre University with their classes.16 ppendix
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