A Constructivist Approach to Teaching Special Relativity

8/22/2019 A Constructivist Approach to Teaching Special Relativity

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A Constructivist Approach to Teaching Special RelativitySubmitted by Jason JenningsFor Prof. Robert Sargent, Curriculum in Practice II

‘Speed of light’, ‘space-time continuum’ and ‘E = mc2’ are all terms that mystify students when theystudy Special Relativity. It is one of the most fundamental theories in modern physics for describingthe universe yet it is not fully comprehended by most people. Contemporary high school physicscurriculum requires students to gain some understanding of the theory. The Atlantic Canada Science

Curriculum: Physics 11/12 document states the following specific course outcome:

“Students will be expected to apply quantitatively the law of conservation of mass and energyusing Einstein’s mass-energy equivalence.” (p. 140)

To put it more simply, students must understand E = mc2. This outcome is achieved specificallywithin the scope of the quantum theory of the atom. It would be easy for a teacher to quickly state theequation, briefly explain the according to Einstein mass can be converted to energy and vice versa,and move quickly into applications of mass-energy equivalence in situations like nuclear reactions.This approach may be used to save time and ‘cover the curriculum’ in order to have studentssuccessfully write the provincial exam.

E = mc2 is one of many aspects of the greater Theory of Special Relativity, first published by AlbertEinstein in 1905. To completely understand the nuances of the atomic world, students must firstunderstand the mechanics of the greater universe. Special Relativity would logically lend itself to astudy of quantum theory and modern physics.

In the interests of time and curriculum compaction, a teacher might be tempted to explicitly state thepostulates of the theory and its ramifications to students. Since understanding Special Relativityrequires a completely new paradigm for explaining observed phenomena, students may be leftinappropriately rejecting classical physics studies in lieu of this radical theory simply because thegreater scientific community (the teacher and Einstein included) dictate so. Conceptions from

classical and relativistic theories that contradict each other may exist simultaneously in the minds of learners leading to poor understanding of experiences and experimental data. Students may alsofeel disappointed after years of studying a physics that now fails to explain what is really going onwith the universe. By using a constructivist approach to teaching Special Relativity, students canengage in collaborative, formative discussions that allow them to feel connected to this paradigm fromtheir classical experiences.

Before analyzing how students can constructively learn Special Relativity, it is necessary to providesome background information of the theory. It consists of two postulates:

1. (principle of relativity): The laws of physics (including electrodynamics) are the same in all

inertial frames of reference.2. (invariance of c ): Light is always propagated in empty space with a definite velocity c that is

independent of the state of motion of the emitting body or on the state of motion of observer measuring it. (http://en.wikipedia.org/wiki/Special_relativity#Postulates)

In other words, all motion is relative, except for light (whose speed in a vacuum is roughly 300 000000 m/s). Four main consequences result from these postulates:

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1. Time Dilation: the time lapse between two events is varied from one observer to another, butis dependent on the relative speeds of the observers' reference frames. The equation for timedilation is:

−

=

2

2

1c

v

t t o

(t o is the time in the observer’s frame of reference; t is the time in the ‘moving’ frame of reference

by the observer)

2. Lack of Simultaneity of Events: two events that occur simultaneously to one observer mayoccur at different times to another observer.

3. Length Contraction: the length of an object as measured by one observer may be smaller from the results of measurements of the same object made by another observer. The equation

−=

2

2

1c

v LL o

(Lo is the length of an object in the observer’s frame of reference; L is the length of an object in the‘moving’ frame of reference by the observer)

4. Apparent Increase of Mass with Increased Velocity and Conservation of Momentum:

E = mc 2

−

=

2

2

1c

v

mv p

Constructivism is a pedagogical theory that purports that students construct knowledge from

preconceptions. When faced with good analogies and probing questions from teachers, studentsraise new questions that either accept a previously-held notion about a concept or refute it andreplace it with a better working theory or schema. Constructivist educators believe that learning isstudent-centered, reflective and socially mediated. (Llewellyn, p. 28) Instead of using prescribedtextbooks and lab experiments, students work in groups to pose and answer relevant questions ondata from different sources and manipulatives. Teachers facilitate group conversations usingSocratic Dialogue that includes “What if…” and “I wonder…” kinds of questions. Groups must makesense of available evidence, identify patterns and relationships and come to consensus on a commonframework of explanations. Teachers assess student development through roving observations andevaluations can be made through student exhibitions, portfolios and journals. Research intoconstructive practices show a high success rate in moving students from the ‘concrete’ to the

‘abstract’. Students eventually become formal, operational learners, equipped with not only theknowledge of a concept but the tools to test the concept against new information that may arise.

Traditional science classes begin as lectures and develop into hands-on activities. In theconstructivist classroom, student would engage in hands-on activities and then proceed to limitedlecture presentations. Overall, constructivist teaching is encapsulated in the 5E Learning Cycle:






In order for students to make the necessary conceptual change, they must initially becomedissatisfied with their existing conceptions; secondly, their scientific conception must be intelligible;thirdly, it must also be plausible; finally, fourthly, the new concept must be useful in explaining new,various situations. Teaching Special Relativity poses a great challenge. Its notions are counter-intuitive to even those students who have taken physics courses. In Relativity Visualized, LewisCarroll Epstein compares Einstein’s reasoning in defense of Special Relativity to be radical andfoolish. In order to explain the constancy of the speed of light, Einstein assumed that time, likespace, was relative, not absolute as is normally observed. Epstein comments on Einstein’s theory:“If a door in a house won’t close two things can be done. The door can be changed by planning or re-hanging. Or the house can be changed by going down to the foundation with house jacks and jackingup the building until the door will close…Jacking around with the foundation is usually a stupidapproach.” (p. 25) At first glance, it appears that challenging previously-held physics notions of spaceand time that are ‘absolute’ may leave students with no schema upon which to develop anunderstanding of Special Relativity. Experimental data is a good way to present the ‘reality’ of space-time relationships that force students to challenge their notions.

The following is a sample guide to teaching Special Relativity through a constructivist approach,utilizing the 5E Learning Cycle:

Engagement:

In the Engagement stage, students are placed in groups of three or four. One student is asked to bethe speaker. In this role, a speaker would moderate between all conversations and expressedopinions in the group. The speaker also keeps the pace of the activity, advancing through differentsteps. A second student is asked to be a recorder of information. The recorder is expected tocontribute to the conversation. Lastly, the remaining students act as skeptics. The skeptics’ role is toquestion all opinions and points of view for clarification and understanding and raise possibleinconsistencies with thinking. In later activities, students will be encouraged to take on a different rolethan previously held. Students may also be reorganized into different groups.

Once in groups, students are told that they will be playing “The Relativity Game”. In the RelativityGame, each group will be asked to answer a series of questions. All groups are posed the samequestions. Each group is asked to discuss each question thoroughly and write down one answer tharepresents a consensus from the group. The recorder is asked to record any difficulties or further questions that arise from the discussion. After 15-20 minutes, groups assemble in a common areaand play the action round of The Relativity Game. A game host (i.e. the teacher) poses the questionsto all groups and each group provides its answers. The goal is for each group to match its answerswith those answers predicted by Special Relativity. The teacher actives surveys all groups during thediscussion round of The Relativity Game, noting important conceptions, misconceptions andquestions that arise. Further noting takes place during the action round when groups compare their answers to each other and to Special Relativity. This game is meant to create disequilibrium within

Engage

Evaluate Explore

Elaborate/Extend Explain






the minds of students between previously held, classical conceptions about time, length, speed andmass and the consequences of Special Relativity. Sample questions are outlined below:

1. You and two of your friends are traveling in three different space ships. These ships aretraveling in a line with equal separation and speed. You and your friends wish to know your common speed. You, in the middle ship, send out a flash of light. If the ships are not moving,the signal will reach one friend in the lead ship and the other friend in the rear shipsimultaneously. If the ships are in formation, can their constant speed by determined be

examining how much more time it takes the signal to reach the lead ship than to reach the rearship? Why?

2. You are comfortably sitting on a rocket traveling straight in the air at 250 m/s. You throw arock in the direction of the moving rocket at 5 m/s. What is the speed of the rock from your viewpoint? What is the speed of the rock from the viewpoint of a friend on the ground?

3. You are comfortably sitting on a rocket traveling straight in the air at 250 m/s. You turn on aflashlight. What is the speed of the photons produced by the flashlight from your viewpoint?What is the speed of the photons from the viewpoint of a friend on the ground?

4. You are comfortably sitting on a rocket traveling straight in the air at the speed of light. Youturn on a flashlight. What is the speed of the photons produced by the flashlight from your viewpoint? What is the speed of the photons from the viewpoint of a friend on the ground?

5. Jill takes a ride in a spaceship moving at 80% of the speed of light. Her destination is a star 4light-years away. From Jill’s frame of reference, what distance in light-years does she travel tothe distant star?

6. TRUE or FALSE: Can a massive particle increase its speed continuously without bound?

As a culminating exercise for the Relativity Game, students are asked to organize their “confusions”in a journal. This journal will be kept throughout the unit on Special Relativity. The teacher willregularly read journal entries to assess proper development of concepts and identify anymisconceptions in order to plan for possible remediation lessons.

Exploration:

As an introductory activity to the Exploration stage, the teacher leads students in a class discussionsurrounding the counterintuitive nature of Special Relativity. Students are encouraged to share theirthoughts from their journals.

Following the discussion, students are divided into groups of two or three. Groups are instructed togo online to several physics websites that offer simulations in Special Relativity. Through thesesimulations, students are asked to collect data that confirm the new knowledge developed in theRelativity Game. Groups investigate (in order) time dilation, length contraction and energy-massequivalence. Several websites offer simulations, like ActivPhysics Online

(http://wps.aw.com/aw_knight_physics_1/0,8722,1123708-nav_and_content,00.html )

This website contains simulations entitled Relativity of Time, Relativity of Length and The ComptonEffect (see Tables 1-3). These particular simulations pose several questions to the user and provide

java applets in order to collect data to answer questions.


http://wps.aw.com/aw_knight_physics_1/0,8722,1123708-nav_and_content,00.html






Table 1 – ActivPhy sics webp age displaying Relat iv i ty of Time simu lat ion






Table 2 – ActivPhy sics webp age displayin g Relat iv i ty of Length s imulat ion






Table 3 – ActivPhy sics webp age displaying Compto n Effect simulat ion

Other useful online simulations that demonstrate the effects of relativity are available on the Fermilabwebsite. The Relativity Challenge (http://www-ed.fnal.gov/data/phy_sci/relativity/student/index.shtml) takesusers through a virtual tour of the Fermilab experiment E687, which produce subatomic particlescalled mesons that travel at speeds approaching the speed of light. The virtual tour allows studentsto examine real-world data and with some algebra and statistical regression derive the conversionequation for time dilation.


http://www-ed.fnal.gov/data/phy_sci/relativity/student/index.shtml



http://www-ed.fnal.gov/data/phy_sci/relativity/student/index.shtml



Table 4 – Fermilab’s Virtu al Tour of th e Relativ i ty Ch allenge ( ht tp: / /www-

ed.fnal.gov/data/phy_sci/relat iv i ty/student/ index.shtml )






Table 5 – Fermilab’s Virtual To ur fo r examinin g m ass-energy equ ivalence ( ht tp: / /www-

ed.fnal.gov/samplers/hsphys/act iv i t ies/student/ )

Another online simulation from Fermilab involves confirm of energy-mass equivalence or E = mc2 using real-world data obtained from the D-Zero Experiment, a proton-antiproton collision in Fermilab’sparticle accelerator. Students predict the mass increase in the particles, which travel a near-lightspeed and confirm their prediction with experimental data. Both these online activities from Fermilabcontain not only student web pages but also teacher web pages that help facilitate learning in the siteImportant assessment rubrics are also included. Rubrics, particularly those that are based onprocess, allow constructivist teachers to gain evidence for learning among students.






What Happens When Things Go Near the Speed of Light?Assessment - Relativity Rubric

Student Researcher: _____________________

Partners: _____________________ _____________________ _____________________

ExceedsExpectations

MeetsExpectations

Does Not MeetExpectations Score

Points Earned 3 2 1 or 0

Experience the discrepancy between frame-dependent measurements.

Predict how far an energetic particle will travel during its lifetime.

Compare these predictions todata.

Derive the velocity dependent correction factor that is used inrelativity.

Either draw their own plots or study those provided to determinethe nature of a plot and a best fit tothose plots.

Perform a wee bit of algebra onthe best-fit curve to derive acorrection factor for timemeasurements.

TOTAL SCORE

OVERALL POINT SCORE: _____________________

COMMENTS:

Table 6 – Student Assessment Rubric for the Relativity Challenge (http://www-ed.fnal.gov/data/phy_sci/relativity/student/assess.html)

The Exploration stage may takes several classes to complete, given the multitude of onlinesimulations and activities in which students could participate. To conclude this stage, students wouldbe expected to complete a journal entry. They would be expected to adequately describe timedilation, length contraction and energy-mass equivalence. As well, they would be expected to providebrief examples of experimental data that confirms these relativistic concepts. Finally, students wouldbe asked to highlight any inconsistencies between classical and relativistic physics. This could beaccomplished by having students from a Venn diagram to visually separate the common anddiscrepant characteristics of old and new theories. An example of such an exercise would be: “In theVenn diagram below, describe facts about time, length, mass and reference frame.”






Explanation:

For the Explanation stage of the unit on Special Relativity, a cooperative learning process called the‘jigsaw’ is employed. The teacher has students compare Venn diagrams from their journals in smallgroups of three or four. Students may discuss similarities and differences in their diagrams, and arefree to modify their diagrams. These groups are then instructed to combine with another group and

perform the same analysis of diagrams. Finally, the teacher leads a class discussion in order toconstruct an overall Venn diagram on the whiteboard. The teacher pays particular attention tovocabulary usage among students and any analogies that students use to explain phenomena. Theteacher can also use this discussion to answer questions for clarification and highlight new questionsthat arise.

The students are then exposed to the relativistic equations for time dilation, length contraction andenergy-mass equivalence. Students are required to perform several calculations and verifydeveloped conceptions. It may also be necessary to revisit some of the online simulations to confirmexperimental data with calculations. The derivation of the equation for time dilation is appropriate forthe algebraic skill level of most high school students. The derivation requires little more than Grade

10 algebra and trigonometry (i.e. the Pythagorean Theorem). The gamma factor

−2

2

1c

v (or

Lorentz transform, as it is called in more advanced physics courses) is evident through this derivationDerivations of the equations for length contraction and revelation of the gamma factor unfortunatelyare not so evident. Advanced calculus is required to construct this equation. Students may betroubled by this reality but can take solace knowing that calculations are confirmed by experimentaldata. Conventional derivations of E = mc2 requires similar higher-level mathematics.

An excellent resource that can be used by any physics student or teacher is Lewis Carroll Epstein’sRelativity Visualized. Epstein takes readers through Gedanken or thought experiments that challenge

conventional thinking, but contain a sense of logic. He provides an acceptable framework for explaining Special Relativity and providing context and extension to classical physics models. Clear illustrations and humor are used to lure readers into constructing new models for explaining time,space, mass and energy. Surprisingly, Epstein uses little mathematics to explain models. For introductory physics students, this departure from equations and calculations may greatly reduceanxiety about learning Special Relativity and establish a high comfort level for acceptance of newideas. In Relativity Visualized, E = mc2 is actually derived using simple algebra and beginner physicsconcepts. Epstein begins by discussing how energy packets or photons can ‘act’ like massiveparticles by colliding with actual massive particles (or the Compton Effect). He goes on to explainhow energy has mass and mass has energy. Paul Hewitt calls mass “congealed energy” in

Classical Relativistic






Conceptual Physics. Epstein discusses the characteristics of near-light speed particles inaccelerators (like Fermilab). Here, particles like electrons gain mass as they approach “c”.

Now the amount of energy put into something is the amount of force applied to it multiplied bythe distance the force pushes it…So energy = force x distance . The amount of distance athing moves is simply the time it spends traveling multiplied by its speed. So energy = force x

t ime x sp eed , or E = fts . In the case of something traveling at nearly the speed of light, addedenergy can only add mass. If something traveling at a constant speed gains mass, a force is

required to keep the increasing mass traveling at the constant speed. For example, if aconveyor belt is moving at constant speed while the mass of sand riding it grows, then a motoris required to exert a force on the belt to keep it moving…If the rate at which sand falls on thebelt is constant, the force necessary to keep the belt moving is proportional to the belt’s speedDouble the speed and you double the required force. If the speed of the belt is maintained asa constant, the necessary force is proportional to how fast he mass is riding the belt isincreasing. Double the rate at which sand falls on the belt and you double the driving forcerequired to keep the belts speed constant. If the sand stops falling on the belt, the requiredforce becomes zero, and the belt would coast by itself at constant speed were if not for pulley(motor) friction. So the force required to keep the belt in motion is force = speed x rate or f = s r . The rate at which sand falls is expressed as rate = mass/ t ime or r = m/t …Take the force

equation, erase the r and replace it with m/t, and you get f = sm/t . Now take the energyequation E = fts , and erase the f and replace it with sm/t , E = (sm/t)(ts). Simplify this mess bycanceling the two times and you get E = ms s …If a thing like the electron, is moving at nearlythe speed of light, its speed can hardly change and that speed is called c . Only its mass canincrease with added energy. So you have it in the bag, E = mcc or E = mc 2 . (pp. 122-5)

Epstein’s analogies and paradigms are thought provoking yet leave the reader with a greater understanding of Relativity. Several chapters in Relativity Visualized could be used as discussionmaterial for small groups or entire classes to add greater depth to explanation and increaseunderstanding. Chapters 1 (classical theory), 2 (the constancy of the speed of light), 3 (time dilationand length contraction), 4 (methodology to calculating relativistic effects), 5 (constructing a model for

why objects cannot exceed ‘c’), 7 (how speed affects mass and energy) and 8 (mass-energyequivalence) deals specifically with Special Relativity. Chapter 1 reviews ‘classical’ relativity from theviewpoints of Galileo and Newton and sets the stage for an acceptance of Special Relativity. Epsteinconstructs a transition between the traditional model and a new one. He alludes that classicalmechanics for objects traveling at speeds much lower than ‘c’ is simply a confined case of a broader theory that includes objects that approach ‘c’ and light itself. Paul Hewitt states the CorrespondencePrinciple: “if a new theory is valid, it must account for the verified results of the old theory in theregion where both theories apply.” (p. 665) Students may question how the new theory was receivedby the scientific community and wonder if their frustrations also felt by others. It may then becomforting to realize that many prominent scientists were initially perplexed by Einstein’s hypothesesand worked to validate the Correspondence Principle.

For the purposes of chapter analysis, students can be divided into groups of three or four. Again,speaker, recorder and skeptic(s) are chosen. Students are assigned a chapter to read previously forhomework. Once in groups, students discuss the main points of the chapter. Groups are assigned asection of the chapter and instructed to create a sentence for each paragraph that clearly summarizesthe main points of each paragraph. Then the class reassembles and each group presents their summary sentences. Students are encouraged to keep notes in their journals for further reference.






Elaborate/Extension:

For this stage, students may be lead through a variety of activities. One such activity is to participatein several Gedanken experiments and provide reasoning in the form of journal entries. ConceptualPhysics and Relativity Visualized provide several questions of this nature. Students may be assignedapplication problems that require calculations for the relativistic effects on time, length and mass.Given the availability of several online simulations via the Internet, students may search for differentwebsites that offer simulations on Special Relativity. They can rate these websites for presentation,

clarity and content (i.e. verification of derived results from simulations with Special Relativity). For students that are able to create website and java applets or Flash simulations, projects can bedevised that allow these student to create their own simulations and online learning environments forothers.

Evaluation:

The Evaluation stage of the constructivist approach to teaching Special Relativity can take manyforms. Traditionally paper-and-pencil tests can be given. These tests can contain objectivequestions, calculations and free-response items. Since students generally perform better onevaluation activities that are similar to those activities used to teach them throughout the unit, group

assessments would prove helpful. Groups could be created and provided a series of Gedankenexperiments to perform. Through performance rubrics that evaluate group and individual work, theinstructor can give valuable insight into the progression from the Relativity Game to this point.Questions that are similar to those posed in the Relativity Game would establish referents for improvement. Interviews with groups and individuals can also provide strong evidence for learningand compliment the collaborative nature of the learning process in general.

Einstein once said that “if at first the idea is not absurd, then there is no hope for it.”(http://www.thinkarete.com/quotes/by_teacher/albert_einstein/) This comment speaks to the heart of

constructivist teaching. The constructivist teacher aims to raise, not only proper conceptions from theminds and experiences of new learners, but also misconceptions. Special Relativity should then be‘absurd’ in the beginning. Only then may students confront absurdity and demand a better frameworkto eliminate it. When teachers accept where a student stands initially with relativity’s postulates andactively builds from that point using social structures within the classroom and available experimentaevidence, students can begin to explain the universe in a more reasonable way.


http://www.thinkarete.com/quotes/by_teacher/albert_einstein/






Works Cited

Epstein, Lewis Carroll. Relativity Visualized. Insight Press, San Francisco, CA. 1991.

Hewitt, Paul G. Conceptual Physics. Addison Wesley Longman, Inc. Menlo Park, CA. 1999.

Llewellyn, Douglas. Teaching High School Science Through Inquiry. Corwin Press. ThousandOaks, CA. 2005.

Province of Nova Scotia. Student Services. Department of Education. Atlantic Canada ScienceCurriculum: Physics 11/12. 2002.

“ActivPhysics Online”. Addison Wesley Longman. April 1, 2006.<http://wps.aw.com/aw_knight_physics_1/0,8722,1123708-nav_and_content,00.html >

“Fermilab Physical Science Data”. Fermilab. April 1, 2006.<http://www-ed.fnal.gov/data/physical_sci.html

“Special Relativity Postulates”. Wikipedia. April 1, 2006.

<http://en.wikipedia.org/wiki/Special_relativity#Postulates>

<http://www.thinkarete.com/quotes/by_teacher/albert_einstein/>


http://www-ed.fnal.gov/data/physical_sci.html





http://www-ed.fnal.gov/data/physical_sci.html


A Constructivist Approach to Teaching Special Relativity

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