A Constructivist Approach to Teaching Special Relativity

Submitted by Jason Jennings

For Prof. Robert Sargent, Curriculum in Practice II

‘Speed of light’, ‘space-time continuum’ and ‘E = mc²’ are all terms that mystify students when they study Special Relativity.  It is one of the most fundamental theories in modern physics for describing the universe yet it is not fully comprehended by most people.  Contemporary high school physics curriculum 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 energy using Einstein’s mass-energy equivalence.” (p. 140) 

To put it more simply, students must understand E = mc².  This outcome is achieved specifically within the scope of the quantum theory of the atom.  It would be easy for a teacher to quickly state the equation, 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 students successfully write the provincial exam. 

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

In the interests of time and curriculum compaction, a teacher might be tempted to explicitly state the postulates of the theory and its ramifications to students.  Since understanding Special Relativity requires a completely new paradigm for explaining observed phenomena, students may be left inappropriately rejecting classical physics studies in lieu of this radical theory simply because the greater 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 also feel disappointed after years of studying a physics that now fails to explain what is really going on with the universe.  By using a constructivist approach to teaching Special Relativity, students can engage in collaborative, formative discussions that allow them to feel connected to this paradigm from their classical experiences.

Before analyzing how students can constructively learn Special Relativity, it is necessary to provide some 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)

In other words, all motion is relative, except for light (whose speed in a vacuum is roughly 300 000 000 m/s).  Four main consequences result from these postulates:

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

(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 may occur 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

(L_o 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 = mc²

Constructivism is a pedagogical theory that purports that students construct knowledge from preconceptions.  When faced with good analogies and probing questions from teachers, students raise new questions that either accept a previously-held notion about a concept or refute it and replace it with a better working theory or schema.  Constructivist educators believe that learning is student-centered, reflective and socially mediated.  (Llewellyn, p. 28)  Instead of using prescribed textbooks and lab experiments, students work in groups to pose and answer relevant questions on data from different sources and manipulatives.  Teachers facilitate group conversations using Socratic Dialogue that includes “What if…” and “I wonder…” kinds of questions.  Groups must make sense of available evidence, identify patterns and relationships and come to consensus on a common framework of explanations.  Teachers assess student development through roving observations and evaluations can be made through student exhibitions, portfolios and journals.  Research into constructive 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 the knowledge 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 the constructivist classroom, student would engage in hands-on activities and then proceed to limited lecture presentations.    Overall, constructivist teaching is encapsulated in the 5E Learning Cycle:

In order for students to make the necessary conceptual change, they must initially become dissatisfied 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, Lewis Carroll Epstein compares Einstein’s reasoning in defense of Special Relativity to be radical and foolish.  In order to explain the constancy of the speed of light, Einstein assumed that time, like space, 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 jacking up the building until the door will close…Jacking around with the foundation is usually a stupid approach.” (p. 25)  At first glance, it appears that challenging previously-held physics notions of space and time that are ‘absolute’ may leave students with no schema upon which to develop an understanding 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 be the speaker.  In this role, a speaker would moderate between all conversations and expressed opinions in the group.  The speaker also keeps the pace of the activity, advancing through different steps.  A second student is asked to be a recorder of information.  The recorder is expected to contribute to the conversation.  Lastly, the remaining students act as skeptics.  The skeptics’ role is to question all opinions and points of view for clarification and understanding and raise possible inconsistencies with thinking.  In later activities, students will be encouraged to take on a different role than 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 Relativity Game, each group will be asked to answer a series of questions.  All groups are posed the same questions.  Each group is asked to discuss each question thoroughly and write down one answer that represents 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 area and play the action round of The Relativity Game.  A game host (i.e. the teacher) poses the questions to all groups and each group provides its answers.  The goal is for each group to match its answers with those answers predicted by Special Relativity.  The teacher actives surveys all groups during the discussion round of The Relativity Game, noting important conceptions, misconceptions and questions 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 the minds of students between previously held, classical conceptions about time, length, speed and mass 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 are traveling 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 ship simultaneously.  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 rear ship?  Why?
2. You are comfortably sitting on a rocket traveling straight in the air at 250 m/s.  You throw a rock 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 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?
4. You are comfortably sitting on a rocket traveling straight in the air at the speed of light.  You turn 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 4 light-years away.  From Jill’s frame of reference, what distance in light-years does she travel to the 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 will regularly read journal entries to assess proper development of concepts and identify any misconceptions in order to plan for possible remediation lessons.

Exploration:

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

Following the discussion, students are divided into groups of two or three.  Groups are instructed to go online to several physics websites that offer simulations in Special Relativity.  Through these simulations, students are asked to collect data that confirm the new knowledge developed in the Relativity Game.  Groups investigate (in order) time dilation, length contraction and energy-mass equivalence.  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 Compton Effect (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. 

Table 1 – ActivPhysics webpage displaying Relativity of Time simulation

Table 2 – ActivPhysics webpage displaying Relativity of Length simulation

Table 3 – ActivPhysics webpage displaying Compton Effect simulation

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

Table 4 – Fermilab’s Virtual Tour of the Relativity Challenge (http://www-ed.fnal.gov/data/phy_sci/relativity/student/index.shtml)

Table 5 – Fermilab’s Virtual Tour for examining mass-energy equivalence (http://www-ed.fnal.gov/samplers/hsphys/activities/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’s particle accelerator.  Students predict the mass increase in the particles, which travel a near-light speed and confirm their prediction with experimental data.  Both these online activities from Fermilab contain not only student web pages but also teacher web pages that help facilitate learning in the site.  Important assessment rubrics are also included.  Rubrics, particularly those that are based on process, allow constructivist teachers to gain evidence for learning among students.

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 online simulations and activities in which students could participate.  To conclude this stage, students would be expected to complete a journal entry.  They would be expected to adequately describe time dilation, length contraction and energy-mass equivalence.  As well, they would be expected to provide brief examples of experimental data that confirms these relativistic concepts.  Finally, students would be asked to highlight any inconsistencies between classical and relativistic physics.  This could be accomplished by having students from a Venn diagram to visually separate the common and discrepant characteristics of old and new theories.  An example of such an exercise would be:  “In the Venn 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 small groups of three or four.  Students may discuss similarities and differences in their diagrams, and are free 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 to construct an overall Venn diagram on the whiteboard.  The teacher pays particular attention to vocabulary usage among students and any analogies that students use to explain phenomena.  The teacher can also use this discussion to answer questions for clarification and highlight new questions that arise.

The students are then exposed to the relativistic equations for time dilation, length contraction and energy-mass equivalence.  Students are required to perform several calculations and verify developed conceptions.  It may also be necessary to revisit some of the online simulations to confirm experimental data with calculations.  The derivation of the equation for time dilation is appropriate for the 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  (or Lorentz transform, as it is called in more advanced physics courses) is evident through this derivation.  Derivations of the equations for length contraction and revelation of the gamma factor unfortunately are not so evident.  Advanced calculus is required to construct this equation.  Students may be troubled by this reality but can take solace knowing that calculations are confirmed by experimental data.  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’s Relativity 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 reduce anxiety about learning Special Relativity and establish a high comfort level for acceptance of new ideas.  In Relativity Visualized, E = mc² is actually derived using simple algebra and beginner physics concepts.  Epstein begins by discussing how energy packets or photons can ‘act’ like massive particles by colliding with actual massive particles (or the Compton Effect).  He goes on to explain how energy has mass and mass has energy.  Paul Hewitt calls mass “congealed energy” in Conceptual Physics.  Epstein discusses the characteristics of near-light speed particles in accelerators (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 by the distance the force pushes it…So energy = force x distance.  The amount of distance a thing moves is simply the time it spends traveling multiplied by its speed.  So energy = force x time x speed, or E = fts.  In the case of something traveling at nearly the speed of light, added energy 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 a conveyor belt is moving at constant speed while the mass of sand riding it grows, then a motor is required to exert a force on the belt to keep it moving…If the rate at which sand falls on the belt is constant, the force necessary to keep the belt moving is proportional to the belt’s speed.  Double the speed and you double the required force.  If the speed of the belt is maintained as a constant, the necessary force is proportional to how fast he mass is riding the belt is increasing.  Double the rate at which sand falls on the belt and you double the driving force required to keep the belts speed constant.  If the sand stops falling on the belt, the required force 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 = sr.  The rate at which sand falls is expressed as rate = mass/time 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 energy equation E = fts, and erase the f and replace it with sm/t, E = (sm/t)(ts). Simplify this mess by canceling the two times and you get E = mss…If a thing like the electron, is moving at nearly the speed of light, its speed can hardly change and that speed is called c.  Only its mass can increase with added energy.  So you have it in the bag, E = mcc or E = mc².  (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 discussion material for small groups or entire classes to add greater depth to explanation and increase understanding.  Chapters 1 (classical theory), 2 (the constancy of the speed of light), 3 (time dilation and 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-energy equivalence) deals specifically with Special Relativity.  Chapter 1 reviews ‘classical’ relativity from the viewpoints of Galileo and Newton and sets the stage for an acceptance of Special Relativity.  Epstein constructs a transition between the traditional model and a new one.  He alludes that classical mechanics 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 Correspondence Principle:  “if a new theory is valid, it must account for the verified results of the old theory in the region where both theories apply.”  (p. 665)  Students may question how the new theory was received by the scientific community and wonder if their frustrations also felt by others.  It may then be comforting to realize that many prominent scientists were initially perplexed by Einstein’s hypotheses and 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 for homework.  Once in groups, students discuss the main points of the chapter.  Groups are assigned a section of the chapter and instructed to create a sentence for each paragraph that clearly summarizes the 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 participate in several Gedanken experiments and provide reasoning in the form of journal entries.  Conceptual Physics and Relativity Visualized provide several questions of this nature.  Students may be assigned application 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 different websites 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 be devised that allow these student to create their own simulations and online learning environments for others.

Evaluation:

The Evaluation stage of the constructivist approach to teaching Special Relativity can take many forms.  Traditionally paper-and-pencil tests can be given.  These tests can contain objective questions, calculations and free-response items.  Since students generally perform better on evaluation 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 Gedanken experiments to perform.  Through performance rubrics that evaluate group and individual work, the instructor 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 learning and 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 the minds 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 framework to eliminate it.  When teachers accept where a student stands initially with relativity’s postulates and actively builds from that point using social structures within the classroom and available experimental evidence, students can begin to explain the universe in a more reasonable way.

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.  Thousand

Oaks, CA.  2005.

Province of Nova Scotia. Student Services. Department of Education. Atlantic Canada Science

Curriculum: 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>