The Limits of Prediction: Students’ Conceptions ofChaotic Behavior in Nonlinear Systems

Edward J. StamasMaster’s ThesisUniversity of California, BerkeleySchool of Education

Abstract:

A design experiment utilizing multiple representations of linear and nonlinear phenomenato facilitate students’ understandings of classical and modern physics concepts wasconducted. Over the course of nine 50-minute instructional periods in January 2006, anenthusiastic, ethnically diverse group of 33 high school students in the San Francisco Bayarea participated in a curriculum designed by this author. All students were finishingtheir first semester of physics with the author as their teacher. Prior to the educationalintervention, students were surveyed about their epistemological views on predictabilityin physics, the limitations of measurement, sensitivity to initial conditions, and holismversus reductionism. Students engaged in multiple inquiry-based investigations in thephysics of systems, which included hypothesis generation, experimental design, graphicalrepresentations, and using calculators and Boxer-based computer simulations asexperimental tools (diSessa, 1995, 2000). Calculators were utilized to employ iterationalgorithms demonstrating sensitivity to initial conditions (Burger and Starbird, 2000).After experimenting with simple pendulums, students constructed nonlinear, magneticchaotic pendulums and used them in experiments investigating the dynamics of chaoticsystems. In the final week of the intervention, small groups of students engaged inscaffolded inquiry with Boxer providing a computer representation of the chaoticpendulum. Throughout the curriculum, qualitative data was obtained through studentinterviews and written responses on worksheets. Post-assessments surveys of students’knowledge of nonlinear dynamics were statistically contrasted with pre-assessments,providing quantitative data to supplement qualitative findings. Overall, there is evidencethat the educational intervention helped students understand modern physics concepts inchaos theory and changed students’ epistemological beliefs regarding how much ispossible to know about a system.

Acknowledgements

Jeanne Bamberger, Professor Emeritus, Massachusetts Institute of Technology;Dr. Andrea A. diSessa, Dr. Kathleen Metz, Dr. Orit Parnafes, Janet Casperson,Karen Chang, Lauren Barth-Cohen, Michael Leitch, Katie Lewis, PatternsResearch Group, University of California, Berkeley, School of Education; Dr.Michael Ranney, Dr. Edward Munnich, Janek Nelson, Luke Rinne, AndrewGalpern, Myles Crain, Kelvin, Nelson, Reasoning Group, University ofCalifornia, Berkeley; Dr. Christine Diehl, University of California, Berkeley,

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School of Education; Dr. Jerry Gollub, Haverford College; Dr. Ralph Abraham,University of California, Santa Cruz; Dr. Bryan Brown, Stanford UniversitySchool of Education; Mark Kelsey, Harvard University School of Education; Dr.Michelle Spitulnik, David Futterman, Jose Zavaleta, UC Berkeley School ofEducation; Daniel Wolf-Root, Todd Sheehy, Josh Spira, Glen Botha, JenniferTune, Jully Yi, Jill Bergan, Alton Lee, Abbey Novia, Pauline Huang, KristinaDuncan, Effie Hsu, MACSME program, UC Berkeley School of Education; EdLay, Boxer programmer; High School Technology Support Staff; and forinspiration, people who have passed away: Dr. Ilya Prigogine, Dr. Carl Sagan, Dr.David Bohm, Dr. Richard Feynman

Introduction

Traditionally, the first semester of high school physics is devoted to elementaryclassical mechanics. Common physical phenomena such as projectile motion andcollisions are modeled algebraically. Approximate, or ideal models of physical systemsare made in the process of modeling them mathematically; e.g., small forces areneglected and numbers are rounded. Underlying these approaches is the assumption thatsmall differences in the initial values of a problem’s critical variables will not result indrastically different results later in time. Some historians of science have noted that thisassumption—that arbitrarily small influences cannot have arbitrarily large effects—wasonce the “basic idea of Western science” (Gleick, 1987). In the second half of the 20th

century, numerous physicists, mathematicians, chemists, and meteorologists working inthe realm of nonlinear dynamics proved that this assumption is not true for all physicalsystems. Systems for which the assumption does not hold were dubbed “chaotic,” andcan behave unpredictably.

Can high school students understand modern physics concepts like “sensitivity toinitial conditions” by engaging in classroom experimentation? Duit and Komorek (1997,2001) have had success doing so. This research project attempts to duplicate theirfindings. It also attempts to see if students’ views on the degree to which physicalsystems are predictable can be influenced by an educational intervention. Anoverarching interest motivating this project was the effect to which traditional physicscurricula give students the impression that all systems are predictable. An alternativeepistemological position suggested by chaos theory is that some systems are inherentlyunpredictable because they are infinitely sensitive to their fundamental parameters. Doestraditional physics education give students the wrong impression of the epistemology ofsystems? Answering such a question might involve surveying students before they beginstudying physics, and again when they have finished. In this research project, studentswere not surveyed prior to instruction in physics, but after having completed the firstsemester of an intermediate level high school physics course. Then, students participatedin an educational intervention in which they investigated a chaotic algorithm inmathematics, a chaotic system in the laboratory, and a computer simulation of this systemwritten in the Boxer programming language (diSessa, 2000). Afterward, data fromstudents’ post-assessment surveys indicated the extent to which students’ views on thenature of the physical world changed by conducting investigations in the physics andmathematics of nonlinear systems.

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What is a Chaotic System?

A chaotic system is defined as one that shows sensitivity to initialconditions. That is, any uncertainty in the initial state of the given system,no matter how small, will lead to rapidly growing errors in any effort topredict the future behavior…In other words, the system is chaotic. Itsbehavior can be predicted only if the initial conditions are known to aninfinite degree of accuracy, which is impossible (Gollub and Solomon,1996).1

“Chaos” derives from the Greek word “caoz,” meaning vast chasm or void.Chaotic systems are dynamical2 systems that, under specific parameter values, can beunpredictable. Though used extensively in science, the term “system” is difficult todefine. In general, a “system” is an ensemble of related elements comprising a whole.Elmer (2002) defines “system” as “a self-contained entity, or an abstract mathematicalmodel for such an entity,” and lists the pendulum as an archetypical example.Mathematically, the dynamics of chaotic systems cannot be described using lineardifferential equations. Prior to computers, nonlinear differential equations wereextremely difficult to solve, so physicists preferred to base their theories on lineardifferential equations. In the late 19th and early 20th centuries, electrodynamics andquantum mechanics were successfully developed using linear differential equations. Bythe 1950s the physics of microscopic atomic scales was being applied to engineering, butphysical phenomenon closer to everyday experience, such as turbulence and fluiddynamics, were still poorly understood. Like the weather and an organism’s success inan ecosystem, turbulence and fluid dynamics are nonlinear processes.

In the early 1960s, Edward Lorenz, an MIT meteorologist, used a computer and asimple system of nonlinear equations to model convection in the atmosphere. To hissurprise, Lorenz found that systems with only a few variables can display highlycomplicated, unpredictable behavior. Meteorological outcomes in Lorenz’s model wereextremely dependent on slight differences in the initial value of one variable, and themodel accurately described real-world phenomena. Linear meteorological modelsquickly became obsolete. In his seminal 1963 paper, Lorenz introduced the phrase“sensitivity to initial conditions” and wrote about how a butterfly flapping its wings inBeijing could theoretically affect the weather thousands of miles away several days later.Thus, sensitivity to initial conditions came to be known as “the Butterfly Effect.”

The scientific community came to a consensus that most systems in the real worldare nonlinear to some extent, and can exhibit chaotic behavior under certaincircumstances. Examples include the weather, generational fluctuations in biologicalpopulations, fluid flow, turbulence, mechanical and electrical oscillatory phenomena,heart and brain activity, planetary orbits, economies, and plate tectonics. Sensitivity to

1 Dr. Jerry Gollub was one of my physics professors as a Haverford Collegeundergraduate, and was consulted as an advisor to this project. His work is cited inGleick (1987) and Briggs and Peat (1990).2 changing in time

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initial conditions can occur without chaos as well: simply multiplying two slightlydifferent small numbers by an extremely large number will result in two divergentproducts. However, if a system is mathematically “bounded,” meaning that its variablesstay within a finite range, sensitivity to initial conditions will result in chaotic behavior.Chaotic behavior is also transitive; “transitivity” means that, given enough time, anytrajectory in phase space will pass through all points within its bounds with equalprobabilities (Nemirovsky, 1993). In thermodynamics, this is known as the “ergodichypothesis.”

Previous Educational Research

Jonassen et al. (1997) states that chaos theory is a scientific perspective that callsinto question “many traditional assumptions about learning systems,” but his researchconcerns applying chaos theory to instructional design itself. Ironically, academicjournal searches for “chaos” return more articles attempting to apply modern physicsconcepts to educational research itself than ones that discuss how to teach chaos theory inphysics classrooms. The few curricular treatments of chaos theory that have beendiscussed in educational research literature are documented here.

Ricardo Nemirovsky (1993) had students investigate a Lorenzian Water Wheel, arather complicated nonlinear system. He found that students expressed several intuitions.One was that periodic regimes are the basic modes of behavior: "If a few irregularitiesappeared, they dismissed them as exceptions, imperfections, or little mistakes. Thestudents felt that the irregularities had to be explained. It was as if periodic motion wasnatural and unproblematic, whereas irregularities were puzzling…the students explainedthat the water wheel was predictable only to the extent that its motion was periodic.When no periodic pattern was discernible, students experienced ‘tensions.’”(Nemirovsky, 1993). This result was used to help formulate hypotheses H1, H2, and H3below.

Another intuition Nemirovsky found was that a big number of affecting variablescauses erratic change: "George expressed that something is unpredictable when it isaffected by too many variables,” like the weather; however, a student named Oscardisagreed: “but the wheel displays both periodic and non-periodic behavior withoutchanging the number of variables” (Nemirovsky, 1993). George’s intuition helped formthe basis of sub-hypotheses H2f below.

Another intuition was that hidden periodicities underlie all irregularities. Ana andPaul stated that, “a system with a small number of degrees of freedom (perceived ascontrollable) eventually has to become periodic or predictable” (Nemirovsky, 1993).This intuition helped form the basis of sub-hypotheses H2a below. Other students statedthat irregular trajectories reflect discrete randomness (see H2g below). Nemirovsky alsofound that some students learned that trajectories are determined by the initial conditions(see hypothesis H4 below) and that unstable behavior may be due to being in the borderbetween two regimes in a region of unstable equilibrium.

Adams and Russ (1992) conducted a unit of study for gifted fourth and fifthgraders about mathematical periodicity and chaos and the underlying physical processesthat produce these phenomena. Hands-on activities, data analysis tools and computeraids were used for instruction in simple periodic motion (as in the pendulum), complex

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superposition of motions (in vibrations), and chaotic sequences (in stock prices). Theirresults indicate that young students were able to understand these concepts to a certainextent.

A reason for teaching about modern physics has been identified by Italianresearcher Olivia Levrini (2006). She found that high school physics students preferredlearning about abstract concepts in quantum mechanics more compared to learning aboutless abstract, traditional physics concepts. This may be because students are moreinterested in the philosophical implications of modern physics. Chaos theory certainlyhas many philosophical implications.

Several articles in The Physics Teacher describe attempts to introduce chaoticpendulums into the classroom for teaching about chaos. Cassoro et al. (2004) attached aspark generator to a magnetic pendulum to record evidence of its chaotic trajectory onthermally activated paper. Oliver (1999) recorded students’ qualitative explorations ofthe interaction between gravitational potential energy, magnetic potential energy, andkinetic energy by studying a magnetic pendulum’s chaotic behavior.

In 1993, Cornilsen, a student teacher in Germany, embarked on a research projectsimilar to the one described in this thesis. In “The Magnetic Pendulum as a Way toUnderstand the Basic Idea of Chaos Theory” (a Master’s project), Cornilsen indicatedthat he had students read two pages about chaos theory from a new physics text.Students had to answer questions about it. One week later, they were interviewed aboutthe magnetic pendulum, which was not explained in the text. Results showed that Grade10 students are able to understand the basics of chaos theory, and opened the door tofurther research about teaching chaos theory in high school.

German educational researchers Reinders Duit, Michael Komorek, and JensWilbers at the Institute for Science Education (IPN), University of Kiel, Germany, haveconducted the majority of published research in this field. They state that, “...so far, thereappear to be no studies available that address the learners' preinstructional point of view.[Chaos theory] challenges the idea of the deterministic predictability of natural eventswhich is paradigmatic in traditional physics” (Duit & Komorek, 1997). Their findingsshow how students changed their minds about predictability, and that curriculum teachingchaos theory at the high school level can be successful. 3

Komorek, Duit, Bucker and Naujack collaborated in a 2001 article about students’“Learning Process Studies in the Field of Fractals,” focusing on the question of whetherthe core ideas of chaos theory and fractals “can be understood by students at the age of15-17.” These researchers note that studies on how students learn about nonlinearsystems through new teaching materials (like experiments) are “almost non-existent.”The German researchers view their work as “preliminary,” but their results encouragethem in their attempts to make the core ideas accessible to 15-16 year old students.”Students experimented with fractal patterns created through electrolysis. When theexperiment was repeated, the same fractal pattern formed. Students were interviewed,and described the pattern as “random.” Regarding students’ differing conceptions of“random,” Komorek, et al. (2001) found one group of students who believed that due to alarge number of variables, a system’s behavior only appears to be random, but is actuallydeterministic. They were not surprised when the same pattern repeated. A second group

3 See http://www.ipn.uni-kiel.de/abt_physik/nlphys/index_eng.html

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believed that random behavior is irregularity, and not determined by principle. However,these students did not question why a similar pattern arose twice.

Researchers at IPN in Kiel use a three-part framework that includes analysis ofcontent structure, empirical investigations, and the continuous reevaluation of theconstruction of educational models. In accordance with the constructivistepistemological position, they assume there is no 'true' content structure of a particularcontent area. "What is commonly called the content structure is the consensus of theparticular scientific community. Every presentation of the consensus [even in textbooks]is an idiosyncratic reconstruction by the referring author informed by the specific aimsthe author explicitly or implicitly holds.” Texts are analyzed “to reconstruct contentstructures in such a way that 'elementary' features [key ideas] are emphasized” (Duit &Komorek, 1997). A similar process was followed when the curricular intervention wasdeveloped for this research project.

The Patterns Project

“A pattern is an identifiable structure with a particular set ofrelationships that is quite general and surprisingly powerful forexplaining and analyzing phenomena in the world.”

This research project has been inspired and guided by the Dr. Andrea A.diSessa’s Patterns Research Group at the University of California, Berkeley School ofEducation. Coming from the background of conceptual change research, the “Patterns ofChange and Control” project seeks to identify the basic patterns in physics and use themto teach physics concepts. From the perspective of the Patterns Group, chaos is a pattern,as are oscillation, stability, balancing, equilibration, randomness, threshold, andresonance. These patterns share the qualities of context independence (generality),explanatory power, the ability to be modeled mathematically, and inherent simplicity.Students are given hands-on examples and interactive Boxer-based simulations and theiractions and comments are recorded. Analysis of videotaped evidence seeks to identifystudents’ thinking about a pattern and the ways in which they might come to betterunderstand it. Understanding a pattern means seeing its defining features (it essence),identifying what it includes and what it does not (its extension), modeling the patterns’relations, and the pointing out important differences between different embeddings of thesame pattern.

Since patterns are context independent, they can be represented in multiple ways,and have “ multiple embodiments.” DiSessa believes that a beneficial educational task isto give students the opportunity to look at a number of different situations that embodythe same concept or pattern. Concrete, specific knowledge of a wide range of situationsis necessary for understanding a pattern (diSessa, 2005). By presenting students withseveral different examples of the same pattern, similarities and differences betweenmultiple embodiments are illuminated, allowing conceptual change to occur.

DiSessa’s view contrasts with the dominant view in the field of cognitivepsychology: that students presented with several examples of the same general idea areable to overlook the differences and focus only on the similarities between these multipleapproaches. In contrast, diSessa argues that, the concrete, situation specific details in

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each embodiment are extremely important in students' constructions of conceptual ideas.While it is certainly valuable to compare representations, the details of any singlepresentation of a pattern cannot be overlooked. In practical terms, this means thatstudents must be presented with multiple embeddings in order to truly understand thepattern; one example is not sufficient.

Taking diSessa's view into account means that technology can best be utilized inconjunction with other representational forms, such as a real-world, laboratory-basedapproach and/or a theoretical, mathematics-based approach. The details specific to eachrepresentation should not be swept under the rug; rather, students' understandings areenhanced and conceptual change occurs best when each approach is presented (andconstructed in students' minds) as one valid perspective among multiple perspectives on alarger, overarching metaconcept. Since it can be argued that the perspective that studentsgain from a technologically enhanced educational support will always differ from thealternative perspectives more traditional supports provide, technology has the potential towiden learners' ways of looking at scientific and mathematical ideas, in turn providing abasis for deeper understanding.

In this project, and the patterns project in general, the use of computer simulationsis not gratuitous. Simulations are necessary to highlight a key concept in nonlineardynamics: deterministic unpredictability. By analyzing a real-world chaotic system suchas a magnetic pendulum, double pendulum, or chaos bowl, it is fairly easy for students tosee that within certain ranges of starting points (initial conditions), that future behavior isunpredictable. However, the fact that these systems are also deterministic is impossible tosee without a computer simulation. The reason for this is that chaotic systems can bedescribed by their extreme sensitivity to initial conditions. The degree of sensitivity isinfinite. The initial conditions must be exactly the same -- to an infinite number ofdecimal places -- in order for the system's trajectory (the path of the ball or the pendulumbob) to be the same. In practice, it is impossible to pick two starting points that lie inexactly the same position. Thus, in traditional labs, students never see the same trajectorytwice. Without a computer simulation, students may grasp the concept that some systemsare unpredictable, yet not see that the system's trajectories are completely determined byinitial conditions. Thus, some students may inaccurately conclude that chaotictrajectories are random or based on probability functions, which is not true.

One of the goals of this project was to facilitate in students an understanding thatchaotic behavior is unpredictable, yet deterministic. In order for a meaningful predictionto be made, conditions must be known to an infinite degree of precision, which, inpractice, is impossible. This concept is both philosophically interesting and fundamentalto understanding the nature of chaos.

Participants

This project was devised, organized and conducted by the author when he was astudent teacher in a San Francisco Bay Area physics classroom. A cooperating teacherwas present, but chose not to participate in this research. The author taught for oneperiod each day in the cooperating teacher’s classroom. During this period, 33 studentswere instructed, and 33 participated in this research project. 22 students were in 11th

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grade, and eleven were in 12th grade. 26 students were male and seven were female. Nostudents had identified special needs or disabilities.

Students’ backgrounds and abilities in mathematics and science were in theintermediate range for their school. There were two other physics classes offered at thehigh school: Conceptual Physics, which is for students weaker in math, and AP physics,which is for students stronger in math. The prerequisites for the intermediate level classof students participating in this research are: (1) having passed chemistry, (2) a C orbetter in algebra II, or (3) to be currently taking those classes. Although the 33 studentparticipants had experience in algebra, some were still experiencing difficulties with it.

Socially, the students did a satisfactory job working with others. For the groupwork portions of the educational intervention, groups were assigned based onobservations of students’ social dynamics. As was the case in class, students had noobserved difficulties getting along with each other that were severe enough to interferewith their learning processes. Throughout the project, participants communicated witheach other extensively, although conversations were not always on topic.

This research was conducted in one of the most ethnically diverse high schools inthe United States. In the project, students groups were all ethnically heterogeneous.From surveys given on the first day of class, four students identified Vietnamese as alanguage spoken in the home. One listed Cambodian, five listed Tagalog, three listedChinese, two listed Spanish, and one listed Arabic as languages spoken in the home.Students’ ethnicities were never officially documented, but three students appeared to beof mostly European descent, one appeared to be East Indian, one appeared to be Latino,and one appeared to be African American. The most common ethnicity was Filipino.Several students were of mixed ethnicities, and several were first generation Americans.

All students were designated to be proficient English speakers, but 17 students(52%) indicated on a first day of class survey that in the home they speak a languageother than English, or in addition to English. Some students did not answer the question,so the actual number may be higher. Although none were officially recognized as ELL orESL students, some students had major difficulties writing and speaking English. Nolinguistic resources or information about students’ backgrounds were provided to theteacher.

Setting

The community in which the research was conducted is one of the most ethnicallydiverse in the United States. Overall, the community’s support for education is high.There are a large number of immigrants in the community, but the community’s diversityis not completely due to recent immigration; many families of many different ethnicitieshaving been living in the area for several generations. The community is not as poor asothers in the San Francisco Bay area, but pockets of poverty, linguistic isolation, and loweducational attainment rates mark it. In its largest census tract, 54% of the adults over theage of 25 do not have a high school degree, and 30% of these did not reach the ninthgrade.

The high school in which this research was conducted is one of the largest in theUnited States. It has over 4000 students, and its “population status” is the “Urban Fringeof a Large City,” according to the California Department of Education. Almost 80% of

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its students are of color, with over 20% from immigrant families. There are 1091students in 9th grade, 1053 in 10th grade, 1132 in 11th grade, and 973 in 12th grade. 769classes are held at the school, and the average class size is 31.0. There are 1383computers at the school and an average of 3.2 students per computer. 159 classroomshave Internet access. In the 2004/2005 school year, the school’s API base was 692, andits statewide rank was 6, slightly above average. Similar schools ranked 3. In2004/2005, 12% of the school’s students were African American, 23% were Asian, 19%were Filipino, 26% were Latino, 2% were Pacific Islander, and 17% were white (not ofHispanic origin). 25% of students participated in the free or reduced price lunchprogram, NSLP. In the physics classroom in which this research was conducted, therewere six functioning computers.

The author began teaching non-calculus based physics to the 33 studentparticipants on the first day of the 2005-06 school year in early September, 2005. Theresearch was conducted in January, 2006. Prior to beginning the research, the author hadcovered chapters 1 through 11, or pages 1 through 240, in Merrill Physics: Principles andProblems by Paul W. Zitzewitz, et. al. From September 2005 to January 2006, thefollowing topics were taught: “What is Physics?,” “A Mathematical Toolkit,”“Describing Motion: Velocity,” “Acceleration,” “Forces,” “Vectors,” “Motion in TwoDimensions,” “Universal Gravitation,” “Momentum and Its Conservation,” “Work,” and“Energy.” No topics outside of elementary mechanics were covered in detail.

Goals

In chronological order, the goals of this research project were to:1. Develop and implement a curriculum to teach modern physics concepts in chaostheory, complexity theory, and nonlinear systems theory to urban high school students.2. Assess students’ epistemological and conceptual views regarding the nature of physicalsystems before, during, and after the curricular intervention.3. Analyze data for evidence of students understanding the modern physics concepts todiscover how the curricular intervention helped students learn them.4. Analyze data for evidence indicating that students’ epistemological and conceptualviews regarding the nature of physical systems changed.

Goal 1 was completed prior to the teaching event described in the procedurebelow. In goals 2 and 4, “epistemological” refers to students’ views on what it is possibleto know about a physical system; in other words, acquire information about thespecificities of what cannot be known about the system.

Goal 3 was accomplished through reviews of both videotapes of students’ wordsand actions during the intervention and copies of student-completed instructionalhandouts. It was assumed that the structure of the intervention and the instructor’scomments were together responsible for students’ gains in understanding the modernphysics concepts. The degree to which the actions of the primary investigator / teacher /author of this project influenced students’ understanding was deemed too difficult todistinguish, especially given the bias inherent in the instructor also being the writer of thecurriculum. Verbatim fragments of dialog and students’ written comments werereviewed and quoted in the curriculum description section of this thesis. Goal 3 providedevidence for all hypotheses and sub-hypotheses.

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Goal 4, in conjunction with Goal 2, provided evidence for hypotheses H1 and H2,including sub-hypotheses H2a, H2b, H2c, H2d, H2d, H2e, H2f, and H2g (see below).

Procedure

Prior to the intervention, each student filled out a two-page pre-assessment survey(see Appendix C). Two different versions of each page were produced, with questionsworded and ordered differently. This was done in order to diagnose the effects ofquestion wording on student responses. For example, statements worded using the word“not” in one version were worded without the “not” in the other version. This way, theextent to which students answered questions affirmatively could be documented.

The two versions of the first page of pre-assessment questions are hereafterreferred to as A and B, and are followed by the question number. The two versions of thesecond page are C and D. Approximately one quarter of the class received pages A andC, another quarter A and D, a third, B and C, and a fourth received versions B and D.

The educational intervention took place over a span of eight 50-minuteinstructional periods from January 11-23, 2006. On the first day, the class as a wholeperformed a dueling calculators activity that was modified from an activity described inthe innovative instructional text The Heart of Mathematics (Burger and Starbird, 2000).The class was divided in two. Half entered in 0.510 as an initial seed value into theircalculators. The other half entered a number less than but as close to 0.510 as theresolutions of their calculators would allow. Students then performed an iterativealgorithm, multiplying by 180 and hitting the sine function key. Students saw that as thenumber of iterations increased, values began to diverge. However, students with thesame make and model calculator had the exactly the same results if they started with thesame seed value.

On the second day, a quarter of the class investigated stable and unstableequilibrium states using bowls and marbles. Another quarter observed chaotic turbulencein a water faucet (Gleick, 1987; Briggs and Peat, 1989). These activities were scaffoldedas inquiry investigations, but students did not appear to learn very much from them. Theremaining half of the class worked on problems unrelated to this intervention, and neverperformed the “faucet” and “bowl” activities.

On the third day, half of the class began investigating the energy dynamics in asimple pendulum, a tie-in with prior class content. They drew plots of the pendulum’sswing in phase space. Then, they performed inquiry investigations of the magneticpendulum, writing hypotheses before conducting experiments. One experiment involvedseeing if the pendulum bob followed the same trajectory if released twice fromapproximately the same point in space. At the end of the period, students wereencouraged to design and perform their own experiments, but few did so. On day four,the other half of the class followed the same curriculum.

On the fifth and sixth days, groups of eight students (a quarter of the class)observed computer generated plots of a chaotic pendulum in phase space. They thenexperimented with a Boxer simulation of the magnetic pendulum, first drawinghypotheses of what a map matching the starting and final positions of the pendulum bobwould look like. The Boxer program was used to obtain simulated data.

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On the seventh and eighth days, a real magnetic pendulum was displayed togroups of eight, and students devised explanations of its behavior. They then used theBoxer program as in days five and six.

On January 24, 2006, students filled out post-assessment surveys. Two versionsof this instrument were distributed: E and F. Just as in the pre-assessments, questionswere worded and ordered differently in E and F in order to diagnose the effects of slightvariations, such as including a negative modifier.

Date Students Description of ActivitiesJanuary 10 n=27 pre-assessments completedJanuary 11 n=28 dueling calculators activityJanuary 12 n=16

Group 1: QP, AP, MR, JMGroup 2: KC, DS, PA, FRGroup 3: WL, DN, BT, ACGroup 4: MS, MV, MP, EZ

half of each group observed turbulence in awater faucet, the other half investigatedstable and unstable equilibrium states usingbowls and marbles

January 13 n=16Group 1: QP, AP, MR, JMGroup 2: KC, DS, PA, FRGroup 3: WL, DN, BT, ACGroup 4: MS, MV, MP, EZ

-mechanical energy in a simple pendulum-phase space plots of simple pendulum-experimentation with magnetic pendulum:hypothesis1, observation1, hypothesis2,observation2, hypothesis3, observation3

January 17 n=16Group 5: CC, TL, IC, SSGroup 6: CP, JCh ,MT, IPGroup 7: JG, CS, BS, MStGroup 8: AB, JQ, ED, JC

-mechanical energy in a simple pendulum-phase space plots of simple pendulum-experimentation with magnetic pendulum:hypothesis1, observation1, hypothesis2,observation2, hypothesis3, observation3

January 18 n=8Group 2: KC, PA, FR, EZGroup 6: CP, JCh ,MT, IP

-observation of computer generated phasespace plots of a chaotic pendulum-experimentation with Boxer simulation ofmagnetic pendulum

January 19 n=8Group 3: WL, DN, BT, ACGroup 7: JG, CS, BS, MSt

-observation of computer generated phasespace plots of a chaotic pendulum-experimentation with Boxer simulation ofmagnetic pendulum

January 20 n=8Group 4: MS, MV, MP, DSGroup 8: AB, JQ, ED, JC

-observation of real world magneticpendulum-experimentation with Boxer simulation ofmagnetic pendulum

January 23 n=8Group 1: QP, AP, MR, JMGroup 5: CC, TL, IC, SS

-observation of real world magneticpendulum-experimentation with Boxer simulation ofmagnetic pendulum

January 24 n=30 post-assessments completedTable 1. This chart outlines the curricular intervention.

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Hypotheses

Hypothesis H1, “prior predictability”: After learning traditional high school levelclassical mechanics and prior to the modern physics educational intervention, moststudents believe that all physical systems are predictable.

It is possible that the view of the nature of physics that students get fromtraditional curricula implicitly implies or explicitly indicates that all phenomena can benumerically analyzed and all future events can be predicted, given the right measuringdevices.

Hypothesis H2: During the educational intervention, students will learn modern physicsconcepts about the nature of chaotic physical systems and change their beliefs about theepistemology of physical systems:

Sub-hypothesis H2a, “limited of prediction”: Students will learn that in some systems(chaotic systems), there are limits to what it is possible to know about the system’s futurebehavior, since no measurement of the system’s parameters can be infinitely precise. Inother words, some systems are not predictable.

Sub-hypothesis H2b, “modeling uncertainty”: After the intervention, students will have agreater tendency to disagree with the view that given the right measuring devices, allsystems can be modeled in a way that allows their futures to be predicted.

Traditional high school physics instruction is based around the measurements oflinear systems, or linear models of real world systems. Such instruction may givestudents the impression that all systems’ futures can be predicted using models. Nomodel can predict the results of chaotic behavior in a nonlinear system, because nodevice can store an infinite amount of information. After students have experimentedwith nonlinear systems, students may see that there exist systems that cannot be perfectlymodeled, even with computers.

Sub-hypothesis H2c, “sensitivity to initial conditions”: During the intervention, studentswill learn the concept of “sensitivity to initial conditions,” and afterward they will have agreater tendency to believe that small influences in a system can sometimes produce largechanges in the future behavior of the system.

Gleick (1987) states that the assumption that “very small influences can beneglected” once lay “at the philosophical heart of science.” If traditional physicsinstruction leads students toward this assumption, can an educational intervention allowstudents to see that it is not always true?

Sub-hypothesis H2d, “examples of chaos”: Before the intervention, students will havedifficulties providing examples of systems in which sensitivity to initial conditionsoccurs. After, students will be able to provide physics definitions of “initial conditions,”“chaos,” and examples of chaotic systems. Some students will be able to demonstratetheir comprehension of these concepts in writing.

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Sub-hypothesis H2e, “holistic view”: The educational intervention will cause students tomove away from reductionist epistemologies and adopt a more holistic view of physicalsystems.

Holism is the idea that all the properties of a given system cannot be determinedor explained by the sum of its component parts alone. Instead, the system as a wholedetermines how the parts behave. Reductionism is the view that the nature of complexthings can always be explained by more fundamental things, such as a system’scomponents. In the philosophy of science, advocates of holism often cite chaotic systemsas examples of phenomena that cannot be adequately explained by reducing them to theircomponents. Advocates of reductionism argue that chaotic systems can still be reducedto their parts, though the way those parts interact with each other spawns emergentproperties.

Sub-hypothesis H2f, “few variables”: Before the intervention, students believe chaoticsystems must have many variables. After interacting with physical systems with fewvariables, students realize that chaotic systems with few variables can exist.

This sub-hypothesis was inspired by the work of Nemirovsky (1993), who founddiffering views among students during their explorations of a chaotic water wheel.

Sub-hypothesis H2g, “limited to probability”: Before the intervention, students maybelieve that a probability is never all that can be known about a system. The interventionwill show students that sometimes in physics, it is only possible to know the probabilitythat something will happen.

Introductory instruction in quantum mechanics commonly states that by theHeisenberg Uncertainty Principle, it is only possible to know the probability that anelectron is some distance away from the atomic nucleus.4 Even if students have heardabout uncertainty in the atom, they may not necessarily believe that, epistemologically, aprobability is all that can be known about a macroscopic system’s future behavior. Theintervention may change students’ views because, in the magnetic pendulum, theprobability of the pendulum bob ending up above a given magnet is 1 divided by thenumber of magnets (as long as the magnets are an equal distance apart and an equaldistance from where the bob would rest without any magnets). However, the path of thebob—and the magnet below the bob when it comes to rest—is not predictable.

Hypothesis H3, “complexity”: During the intervention, students will see how nonlinearsystems exhibit both order and chaos, with windows of key variable ranges that result inperiodicity mapped within variable ranges resulting in chaos.

This pattern is sometimes called “complexity.”5 Order and periodicity arefrequently observed in “windows” of variable ranges above and below variable ranges

4 Probability in quantum mechanics may be the result of non-deterministic or randomprocesses. Chaos, on the other hand, is not randomness; it is deterministic.5 Prigogine (2003) writes, “Complexity is a property of systems that for given boundaryconditions have more than one possible solutions. Also in complex systems long rangecorrelations appear between components for very short-range local interactions.” Acompeting academic definition of “complexity” views systems with more variables as

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that produce chaotic, unpredictable behavior. This has been known since the 1960s, andwas identified in Lorenz’s investigations of heat conduction. It can also be seen ingraphical form in the Logistic Map.6

Hypothesis H4, “deterministic chaos”: The computer simulation will allow students tosee examples of deterministic chaos (Prigogine, 1997). When two starting positions areinfinitely identical, trajectories will be the exactly same, even where arbitrarily closestarting points produce drastically different results.

Placing the chaotic pattern in multiple embeddings, students will see thatdeterminism arises in the computer simulation although it cannot be detected in students’investigations of the real world pendulum.

Hypothesis H5, “phase space”: Students will construct and understand graphicalrepresentations in phase space.

Phase space is a graphical representation in which velocity (or angular velocity) isplotted on the y-axis and position (or angle) is plotted on the x-axis. Below is a phasespace plot of a chaotic pendulum:

more complex. Do complex linear systems exist in nature? Is a computer a complexlinear system?6 See http://en.wikipedia.org/wiki/Logistic_map andhttp://mathworld.wolfram.com/LogisticMap.html for educational resources.

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Picture 1. A screen shot from http://www.myphysicslab.com/pendulum2.html shown tostudents on January 13th, 17th, 18th, and 19th.

Assessments

In the pre-assessments, students were asked the questions, “What is a physicalsystem?” (A1) and “What is a system in physics?” (B1). In the post-assessment, studentswere asked, “What is a system?” (E7,F6). In hindsight, it was an error to omit the world“physical” in the post-assessment. The purpose of pre-assessment questions A1 and B1was to identify students’ incoming notions of a physical system. The concept of a systemis fundamental to understanding nonlinear systems. Questions A1 and B1 were necessaryto determine if students were formulating concepts of a “system” throughout theintervention, or if they came in with clear notions of what a system is in physics.

Pre-assessment items A2, B2, A3, B4, A4 and B3 were designed to testhypothesis H1, students’ epistemological views on predictability prior to the intervention.In A2, students were given the question: “True or False: Some physical systems are, bytheir nature, unpredictable,” and asked to give the fraction of physical systems that areunpredictable by circling “None,” “A Few,” “Some,” “About Half,” “Most,” “AlmostAll,” or “All.” In pre-assessment B2, the statement was changed to: “True or False:According to physics, everything is predictable. If you answered “false,” what fraction ofphysical systems are unpredictable? Circle one.” The survey instruments were designedwith the impression that students tend to give affirmative answers to difficult questions.Since on B2, a “false” response was the same as a “true” response to A2, the strength ofsuch an effect could be documented. If an equal percentage of students answered “false”on B2 as answered “true” on A2, one could be reasonably sure that students did not tendtowards giving positive responses.

In pre-assessment questions A3, students were asked, “Can you think of a systemin which a small change in a variable could produce a completely different futureoutcome? If possible, provide an example.” In question B4, the question was rewordedas, “Can you think of a system in which a very small change in the values of the givens(the “initial conditions”) could result in a completely different answer?” In the A4 andB3, students were given the situation: “Say you solved a physics problem using a set ofgivens. (For example, v = 1.0000000000 m/s.) Then, you solved the same physicsproblem using a new set of givens in which the numbers are only slightly different. (Forexample, v = 1.0000000001 m/s).” Then, students were asked if it is mathematicallypossible to get a very different answer if initially given values are only slightly different.

Pre-assessment questions A2 and B2 were also designed to test sub-hypothesisH2a (limited prediction) in conjunction with the identical post-assessment items E2 andF2, respectively. A greater number of students viewing some systems as unpredictablewould support the sub-hypothesis. Qualitative data from post-assessment questions E5(“What was the most important thing you learned from the chaos project?”), F5 (“Whatwas the most interesting thing...”), E9 (“What is chaos?”), and F8 (“What is chaos?”)were also obtained to test H2a (limited prediction).

Pre-assessment questions A3 and B4, and A4 and B3, were designed to test sub-hypothesis H2c (sensitivity to initial conditions) in conjunction with post-assessmentitems E12 and F11. In the post-assessments, students were asked, “What happens when a

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system’s behavior is very sensitive to initial conditions?” (F11) and to provide anexample of “sensitivity to initial conditions” (E12). Sub-hypothesis H2c (sensitivity toinitial conditions) would be supported by pre-assessment responses indicating that smallchanges cannot produce different outcomes together with post-assessment commentsstating the opposite. Post-assessments E8/F7 (“What is an initial condition?”) were alsodesigned to provide data that might support H2c.

In a follow-up question, students were asked to give examples of chaotic systemsthat were not studied during the project and to explain why they are chaotic (E13, F12).These follow-up questions were designed to test sub-hypothesis H2d (examples of chaos)in conjunction with pre-assessment questions A3 and B4. A lack of examples ofscientifically chaotic systems in the pre-assessments combined with student examples ofchaotic systems in the post-assessment would support the sub-hypothesis. Post-assessment items E9/F8 (“What is chaos?”) and E12 were also designed to provide datathat might support H2d (examples of chaos), or to aid in the interpretation of otherresults.

Five Likert questions were identical in the pre- and post-assessments. A 9-pointscale was used, in which 1 represented “Strongly Agree” and 9, “Strongly Disagree.”

Likert questions A5/E14 (“If enough information is known about a system, it ispossible to predict everything that will happen in it”) and D6/F15 (“Nature has “built-in”limits to what it is possible to know about some physical systems”) were designed to testsub-hypothesis H2a (limited prediction). More disagreement in E14 compared to A5 andmore agreement in F15 compared to D6 would support this sub-hypothesis.

Likert questions C8/E17 (“If you analyzed a coin toss with scientific instruments,you could predict the outcome every time”), D8/F17 (“It is always possible to knowsomething with absolute certainty, if one has the right tools or measuring devices”), andC6/E15 (“Anything in nature can be accurately modeled with computers”) were designedto test sub-hypothesis H2b (modeling uncertainty). A greater degree of disagreementwith these statements in the post-assessment would support this sub-hypothesis.

Likert items C7/E16 (“Small influences in a system, such as air currents in aroom, cannot produce large changes in the future behavior of the system”) and D7/F16(“Small influences in a system, such as air currents in a room, can produce large changesin the future behavior of the system”) were designed to test sub-hypothesis H2c(sensitivity to initial conditions). More disagreement with E16 compared to C7 and moreagreement with F16 compared to D7 would support this sub-hypothesis.

Likert items C9/E18 (“In some systems, the system’s behavior cannot bemodeled by studying the system’s parts. The system must be studied as a whole”) andB5/F14 (“In physics, the universe is analyzed by breaking it down into its componentparts, just as one can figure out how a machine works by finding the purpose of each ofits parts”) were designed to test sub-hypothesis H2e (holistic view). More agreementwith E18 compared to C9, and more disagreement with F14 compared to B5 wouldsupport this sub-hypothesis.

Likert items D9/F18 (“In some systems, it is only possible to know the probabilitythat something will happen”) were designed to test sub-hypothesis H2g (limited toprobability) along with data from true or false questions E4 and F4. Question E4 asked,“Sometimes, physics can only predict the probability that something will happen, no

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matter how accurately things are measured.” Question F4 stated, “Probability is theresult of the laws of nature.” Affirmative answers would support the hypothesis.

Post-assessment true or false questions E1 and F1 were designed to test sub-hypothesis H2f (few variables). Post-assessment question E1 stated, “For a system to bechaotic, it must have many variables,” and F1 read, “Can a system with only a fewvariables exhibit chaos?” Answers of “false” for E1 and “true” for F1 would support thesub-hypothesis, although comparison data was not obtained in pre-assessments.

True or false questions E3 and F3 were designed to test hypothesis H3(complexity) by asking students if a chaotic system can be ordered or periodic.Affirmative answers would support the hypothesis. A better phrasing would have askedif a nonlinear system can be ordered or periodic, but the term “nonlinear” was notintroduced to students.

Qualitative data (students interviews, oral and written comments) were used totest hypotheses H4 (deterministic chaos) and H5 (phase space) and to supplementquantitative data for all other hypotheses and sub-hypotheses.

Curriculum Description

January 11, 2006

On Wednesday, January 11, 2006, pre-assessment surveys were collected anddiscussed in class. Students brought up weather and chemical reactions as examples ofan unpredictable systems. The instructor posed the question, “Will we ever be able toaccurately predict the weather?” Some students felt that some systems, like weather, areinherently impossible to predict. For example, BS said that since there are so manyconstantly changing variables, there is no way we will ever be able to accurately predictit. Other students thought that increasingly accurate predictions will be possible astechnology improves. When the instructor asked if a small change, like moving one’shands, could cause a tornado, MSt brought up the example of a bomb in which “a smallthing triggers a big explosion in a chain reaction.” Seven students voted that theydoubted if a small change could produce a large effect. One said that it depends on thesituation. The instructor acknowledged that it depends on what kind of system is beingstudied. QP said that something relatively minor, like the temperature, might influencethe outcome in a race.

The dueling calculators activity sheet was distributed. The instructor asked for arandom number; a student volunteered 510. Half of the class—those with oddbirthdays—was instructed to enter 0.510 into their calculators. The other half was told toenter 0.50999999999, and to keep typing 9s until they could not enter any more digits.They wrote this number on their worksheets as the “initial number.” In degree mode,students multiplied this number by 180 and hit the SIN key to take the sine of theproduct. Students were asked to make a hypothesis about what will happen when theselast two steps are repeated 25 times. Students did so and recorded their results.

Students’ hypotheses varied. Six students indicated that the numbers willdecrease, three that they would increase, and two said that they would change. Three saidthat the 25th iteration would be the same as the initial number. Eleven students gave

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hypotheses that compared the two initial numbers. Of these, three wrote that both sets ofresults would be the same, three that they would be slightly different, and four indicatedthat there would be two different sets of numbers. One student, MSt, predicted, “Thenumbers will differ more and more every iteration, the 25th iterations will be different.”MSt’s comment is noteworthy because he was able to provide a somewhat accurateprediction of the results of the activity.

Students were told that all results should be in between 0 and 1. When finished,students compared their results. One group found that their results were similar until theeighth iteration; then they began to diverge rapidly. AB said that this was because ofrounding. Through questioning, students concluded that the calculator has finite (limited)memory. CP said that the calculator has to round the numbers; the instructor explainedthat this is true—since each number is infinitely long, the calculator must round. MR andQP pointed out that all of their numbers were the same. The instructor pointed out thatthey used the same calculator, and asked, “What’s different between two calculators?”LR said that one has more memory, and the instructor acknowledged that the number ofdecimal places to which the number is rounded varies from calculator to calculator. Theinstructor pointed out that two students with the same brand of calculator got the same setof answers. Students were asked to write down the make and model of their calculators.

Students did not appear to be very surprised by these findings, even thought theyhad been rounding their answers in traditional physics problems throughout the semester.The concept of “sensitivity to initial conditions” was introduced. Students then answeredthe question, “Did your neighbors get similar results? Record any similarities ordifferences.” Five groups of students found that results were the same until the 3rd, 8th,13th, 15th, and 23rd iterations, respectively. Students with the same make and model ofcalculator found that their results were the same.

Students also wrote their conclusions. Out of 28 students, twelve (43%) indicatedthat the results depended on the kind of calculator used, in keeping with sub-hypothesisH2b (modeling uncertainty). Ten students indicated that the calculators’ roundingprocesses caused an effect, and seven explicitly stated that a slight difference in the initialnumber would change the final outcome; thus, 61% of students provided evidence forH2c (sensitivity to initial conditions).

When asked, “What did you learn from this activity?”, twelve out of 28 studentsmentioned that different calculators round differently, and three said that calculators areimperfect. Thus, 54% of students provided evidence supporting H2b (modelinguncertainty). Seven (25%) stated that small initial differences can result in bigdifferences in the end, providing evidence for H2c (sensitivity to initial conditions). It isnot surprising that more students mentioned differences in calculator rounding algorithmsthan mentioned initial differences in seed values, because even two identical seed valuescould produce divergent outcomes in two different calculators.

The final question on the worksheet was, “Write down another example of‘sensitivity to initial conditions.’” Two students mentioned a “weather forecast,” andanother stated that if one person on a sports team has a bad day, the whole team can havea bad game. Interestingly, one wrote, “Where your TV antenna is at. My sister has betterreception than me and our rooms are not that far.” Another stated that two differentprocesses can produce the same result using an example of hitting a tennis ball. Fivestudents suggested using another mathematical function such as cosine instead of sine.

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Taken together, these responses provide qualitative evidence for sub-hypothesesH2a (limited prediction), H2b (modeling uncertainty), H2c (sensitivity to initialconditions), and hypothesis H4 (deterministic chaos). In keeping with H2a, studentswere able to see that the outcome of the algorithm was unpredictable. In keeping withH2b, students realized that any given calculator was imperfect. The comments of manystudents explicitly mentioned the concept of sensitivity to initial conditions, or alluded toit through stating that a rounding can effect results, in keeping with H2c. Qualitativeevidence of students’ seeing that the same make and model of calculator gives the sameresults provides tentative support for H4.

January 12, 2006

On Thursday, January 12, 2006, each of the four groups of four students wasdivided in two. Two students from each group completed the bowl activity, and engagedin the faucet activity. Students conducted inquiry based investigations and wereencouraged to explore. These activities were only marginally successful because studentshad difficulties seeing their connections with the other activities in the intervention.

At one point during the bowl activity, the instructor asked, “What would be asmall external force on the marble?” Students could not answer the question, anunsurprising result given that people remain consciously unaware of the many smallforces about us all the time. (A common example of this given in physics instruction isthe Earth’s magnetic force field.) While pointing at an air vent, the instructor offered,“the wind currents in the room?” Students understood that air currents could exert smallforces, because this topic had been covered in lessons about air resistance and dragforces.

During the faucet activity, LR said that as the handle is turned, “the morepressure, it fills up the hole, [pointing to the faucet]- I’m not sure how to say it.” Shewrote, “when you gradually turn the knob more, the shape of the water is more defined.”The activity instructions were unclear as to whether the pattern involved the initialcreation of the water flow in addition to the way in which the flow changes as the handleis turned. Students were meant to focus on the latter effect, but often did not. Thequestion of “cut-off time” has been an issue in the Patterns research group on severaloccasions. When identifying a pattern, it is reasonable to ask, “when does the patternbegin, and where does it end?”

January 13, 2006

On Friday, January 13, 2006, Instructional materials were passed out to groups1,2,3 and 4 (16 students) with pictures of fractals from Burger and Starbird (2000). Theinstructor set up ring stands with materials for a simple pendulum. Students began byanalyzing the mechanical energy dynamics in the simple pendulum, a tie-in with a topicthat had just been covered in the class curriculum. Next, students were given the problemdrawing the path of the pendulum bob in phase space, provided with graph paper andlabeled axes. At first, instructor scaffolding was required. When the instructor explainedthe negative and positive directions and origin (0,0) on the position and velocity axes,Group 1 (QP, MR, AP, and JM) grasped the concept of phase space:

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MR: It’s a circle.AP: it’s going to be a shape/ a circle?Primary Investigator: [nods] yes, but it eventually stops, so what’s happeningAP: It gets smaller.MR: So it’s really a spiral.PI: yes!At this point, MR expresses confusion. His prior idea that the y-axis represents y-displacement shows robustness:MR: All this is x-axis? There is no y-axis?QP explained why it is a spiral. By this point, the group of four students could see as itgoes back and forth, it goes from + to - in position and velocity.

This is qualitative evidence that changing the axis labels confuses students. Forsome reason, students have difficulties understanding graphical representations whenaxes are defined in new or unfamiliar ways. It was observed in almost all other studentgroups, with some notable exceptions. It is a strong argument for further research inteaching students literacy skills in graphical representations. Even in a UC Berkeleyresearch group, participants could not see the spiral without some hints, so it is notsurprising that the high school students needed scaffolding. But significantly, since theanswer requires no advanced knowledge in physics, students eventually identified thecircle and spiral attractors: “it’s going to be a shape/ a circle?”

Next, JM said: “It starts from negative. It’s going around the wrong way.” Now,students debated whether the direction of the spiral should be clockwise, orcounterclockwise, a non-trivial question.7

Next, students were given magnets to place under their pendulums. Inquiry-basedstudent investigation was facilitated, with students writing hypotheses prior toexperimentation. 8 9 Students placed one magnet under their pendulums, creatingnonlinear systems. Before releasing the pendulum’s bob, all students’ hypothesespredicted linear behavior in keeping with hypothesis H1 (prior predictability).Significantly, no responses indicated anything about possible chaotic or random behavior.“Unpredictability” was not mentioned by any students.

7 The answer depends on whether the bob’s initial velocity (after its initial angulardisplacement, a debatable part of the pattern) is to the right (positive) or to the left(negative). If the right is negative and left is positive, it is the opposite sign.8 When facilitating this lab in the classroom, physics instructors should note that chaoticbehavior is the most obvious when the distance between the bob and the magnets is thesmallest. Magnets should be firmly attached to the table or ring stand base with tape or,if necessary, glue, so that they do not jump up and stick to the bob. Round circularmagnets work best, and the north and south sides should be labeled with paint orcorrection fluid. Attaching the magnets to the base of a ring stand changes the physics ofthe system, since the ring stand can resonate. Two or more magnets are necessary to seeobviously chaotic effects.9 See http://www.exploratorium.edu/snacks/strange_attractor.html for an educationalresource.

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In their hypotheses, nine students predicted the pendulum bob would slow down,but five wrote that it would speed up, and one that it would “not slow down.” Four saidthat the magnet would attract the bob; in contrast, four predicted that the bob would notstop moving. Three students indicated that the “pattern will change,” and another twothat the bob would stop in the middle. Other individual responses included “no effect,”“the shape created by the bolt becomes flatter,” and “the magnet will stick to the bob.”Each of the n=33 responses was coded into one of these ten categories. In future work,students could be given a more specific question, such as one asking if the bob’s motionwill become less predictable.

Before students placed a second magnet below the pendulum, they were asked togive another hypothesis about what would happen.10 Again, no students indicated that“chaotic,” “unpredictable,” or “random”11 behavior would occur in the system, evidencefor hypothesis H1 (prior predictability). Only one student’s (MSt’s) answer gave anyindication of there being limits to what is possible to know: “stops at one of the magnets,don’t know which.” One other student (JCh) hinted at strangeness, writing that themagnets give the bob “two jerks” when it passes over them. DS acknowledged, “If thereare two magnets, then the magnets will attract the bob with the same force.” Ten studentsstated the modal response, that the bob will slow down.12 The remaining 20 students gavea wide range of answers, such as “stick together” (n=3), “speed up” (n=2), move “backand forth” (n=2), “swing between both” (n=2), “stay at constant speed” (n=2), “patternwill stay but the distance will decrease” (n=1), “orbit” (n=1), and “elipse” (n=1).Significantly, 30 out of 33 student responses (91%) were descriptions of predictable,linear behavior—strong evidence for hypothesis H1 (prior predictability).

For example, in Group 1, AP wrote: “It will continue in orderly fashion back andforth from magnet to magnet.” Her group then observed the motion of their chaoticmagnetic pendulum (0:50):JM: It’s out of control / Chaos / I think it’s going to stay like that for a long time.MR: It will never stop / like metal balls.PI: Is it repeating the same-?Ss: No.JM: maybe it’s random.PI: Could you predict what it’s going to do?Ss: No.PI: but now it’s going back and forth / so it is periodic?[Then, the pendulum stopped acting periodically. Everyone sees this.]PI: now, it’s chaotic

10 When two magnets are placed below the pendulum, the system’s dynamics are suchthat a tiny change in the placement of a magnet can radically change the bob’s trajectory,as can two nearly identical releases from arbitrarily close starting positions.11 Random behavior is not chaotic, although chaotic behavior may appear to be randombecause it is unpredictable. Students who acknowledged randomness were correct innoting the unpredictable nature of the system, so their responses were interpreted assupporting H2a (limited prediction).12 This is not an incorrect answer, because energy dissipates in a non-linear pendulum justas it does in a simple pendulum.

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It is a property of nonlinear, chaotic systems that, under certain ranges of theessential variables, they can exhibit ordered, periodic behavior. But the students whoobserved the magnetic pendulum behaving periodically did not write answers supportinghypothesis H3 (complexity), which states that students were able to understand that non-linear systems can exhibit order. In future work, further scaffolding could facilitateunderstandings of this concept. However, JM did write “It’s random,” in keeping withH2a (limited prediction).

JM’s comment “I think it’s going to stay like that for a long time” is interesting.The instructor was not able to get JM to defend this initial hypothesis, because it wasquickly disproved by direct observation. However, JM’s assertion may be anacknowledgement of the way that certain non-linear systems can sustain ordered, periodicbehavior, far from equilibrium, for long periods of time. Such “dissipative structures”were discovered by Ilya Prigogine, and examples range from weather and chemicalsystems13 to living organisms themselves. 14

In Group 2 (KC,DS,PA, and FR), students realized that the simple pendulum isnot sensitive to initial conditions, noting that “there is no difference” when the bob isreleased from similar starting positions. After placing down two magnets, students hadvarying hypotheses. DS’s hypothesis, that the two magnets “will attract the bob with thesame force,” acknowledged the unstable equilibrium in the system. PA wrote, “I thinkthe bob will move around.” Upon observing the bob’s chaotic trajectory, DS wrote: “Thebob revolves & swings uncontrollably because the bob is looking for which magnet has astronger attraction.” KC wrote: “it moved out of control when the bob and magnite [sic]were close together but it’s still swinging.” FR wrote something similar: “The pendulumattracts to either one of the sides of the magnet and moves out of control.”15 Theresponses of DS, KC, and FR are in keeping with H2a (limited prediction).

After observing chaos, students in Group 2 performed the experiment of releasingthe bob multiple times from the same starting point. After one release, they recordedtheir hypotheses regarding the next release. DS did not seem to understand that thequestion was asking about how the bob’s swing would be different in consecutive trials:“it will move in two directions, back & forth.” PA wrote, “I think the bob will follow thesame path.” KC wrote that as well, but added, “As it gets closure [sic] to the magnitude[sic] it will go chaotic.” FR wrote, “The same pattern will keep on repeating because ofthe magnet.” 16

In his observations, PA acknowledged that his hypothesis was disproved: “Thebob did not follow the same path and continued to move,” qualitative evidence for sub-hypothesis H2c (sensitivity to initial conditions). KC wrote, “As the ball got closer to the

13 See http://people.musc.edu/~alievr/BZ/BZexplain.html for an educational resource.14 See http://www.prototista.org/E-Zine/OriginsofOrder/OriginsofOrder-TOC.htm for aneducational resource.15 The use of the phrase “out of control” to describe chaos is conceptually interesting. Ischaos an uncontrollable pattern? In some ways it is, although contemporary research inchaos theory does focus on ways to control chaotic systems by modeling themmathematically.16 Future work could involve further thinking about exactly what students meant whenthey wrote these statements.

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magnitude [sic] it did go chaotic.” Significantly, KC was comfortable enough with thephysics definition of “chaotic” to use it to accurately describe a chaoticsystem—evidence for sub-hypothesis H2d (examples of chaos). FR recorded: “The samepattern did occur again b/c of the magnet’s attraction.” Interestingly, FR chose toconceptualize chaos as a “pattern,” which is also how the UC Berkeley Patterns ResearchGroup sees it. He also noticed that the magnets played a role in the pattern’s existence.In a follow-up experiment, FR wrote that the overall “pattern” of the bob’s motion didnot change when a small wind force was exerted on it. He did not pay attention to theintricacies of the bob’s trajectory, and therefore wrote that a “small external force doesnot affect it significantly.” Although his conclusion was opposite to the one studentswere supposed to come to, from the perspective of the Patterns Research Group, it is notsurprising given that FR viewed chaotic motion as a “pattern.” Certainly, chaotic motionhas a recognizably different structure than harmonic oscillation or linear motion. FR’scomment indicates that he was aware of some of these differences.

January 17, 2006

Dr. Martin Luther King, Jr.’s birthday was celebrated on Monday, January 16,2006. On Tuesday, Groups 5,6,7 and 8 were given handouts with instructions similar tothose given on January 13, but with minor corrections and revisions. First, students builtsimple pendulums and drew the bob’s trajectory in phase space. One group of students(Group 5: CC, TL, IC, and SS) were able to draw a circle in phase space with noassistance from the instructor:PI: Why did you say it was a circle?TL: symmetryPI: Over the long term, what happens?CC: It’s a spiralAs expected, these students observed no differences in the trajectory of the simplependulum when it was released from two similar positions. However, their responsesindicate that they did not understand the meaning of the phrase “sensitivity to initialconditions” at this point in the intervention.

Later, these students wrote hypotheses predicting what will happen when twomagnets are placed under the bob. All four indicated that the “magnets will slow the bobdown.” These students did not predict chaos or random behavior, evidence supportingH1. From the perspective of the Patterns Research Group, these students gave amechanical rather than a “patterns level” type explanation.

When the group’s magnetic pendulum began acting chaotically, IC appeared to beamused by its unpredictable trajectory, evidence that its behavior acted as a discrepantevent to facilitate conceptual change. Her words and emotions indicated her surprise thata simple nonlinear system can behave strangely: “Look, it twists / oh its going to thisway, see, it’s turning around, now, look at it! I’m so proud of it...now it stopped.” 17 Forher observations, she was unfortunately epigrammatic, writing, “The magnets slightlypush the bob.”

17 IC’s comment may be an example of “magical thinking.” To her, the pendulum seemsto take on a life of its own.

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This group devised their own experiment, placing two magnets one inch apart andreleasing the bob from a height of 5.5 cm directly above a point on the table. Prior torelease, two students hypothesized that the bob would end up in the middle, between thetwo magnets. IC wrote, “I think that small forces can significantly change the behavior,”and SS indicated that the bob will stop at a “random” magnet depending on “when it runsout of energy.” The latter two responses support sub-hypothesis H2c (sensitivity to initialconditions). The students observed that the bob “randomly chose one magnet to stop at,”and when the experiment was repeated, “it chose the other magnet.” CC and SS calledthis phenomenon “random selection,” TL wrote “arbitrary selection,” and IC stated that“there isn’t a ‘constant’ answer, it’s unpredictable and random.” These experimentalconclusions clearly indicate that these students underwent conceptual change as predictedby sub-hypotheses H2a (limited prediction) and H2g (limited to probability).

Another group of students (Group 6: CP, JCh, MT and IP) hypothesized that afterplacing one magnet below the pendulum bob, “The bob’s swing will be interrupted by theforce of the magnet and make the swing different.” They noticed that when the bob isclose to the magnet, it “gets a little jerk and the swing is bumpy.” They predicted thattwo magnets will produce two jerks. Then, these students observed chaos:PI: Do you think you could predict which one it ends up at?JCh: No.PI: This is chaos, because it’s very sensitive to initial conditions. We cannot predictwhich magnet it will end up at.MT: that’s cool!This group of students, bored and frustrated until this point, immediately became amusedand excited by physics. Interestingly, students with dueling hypotheses began to bet eachother about which magnet the bob would end up over. As energy dissipated, the bobcircled around one magnet, then moved to the next, with a student advocating for it tostop at the magnet he bet on. Students clearly realized that they each had 50/50 odds; aprobability was all they could know. All four students in this group wrote that the bob’smovement was “unpredictable.” One also wrote that it was “chaotic.” This is qualitativeevidence for sub-hypotheses H2a (limited prediction), H2d (examples of chaos), and H2g(limited to probability).

Immediately afterward, these students predicted what would happen if the bobwere to be released from approximately the same place twice. Interestingly, studentswrote, “it will follow the same path as before,” indicating robustness in the view thatsmall changes cannot have divergent effects. After doing the experiment, students sawthat their hypothesis was disproved, as evidenced by the group’s dialog:PI: Even if you released from the same point in space, is there any way of knowing whereit’s going to land?JCh, MT, IP: [together] no!Thus, students were led toward sub-hypothesis H2c (sensitivity to initial conditions).

In Group 7 (JG, CS, BS and MS), CS predicted that the bob would stop at thesame magnet it stopped at the last time it was released. When it did, BS noted: “But itdidn’t follow the same path.” Then, hypothesizing what will happen if the bob isreleased from approximately the same place, BS wrote, “It will go to either of themagnets, can’t tell which,” in keeping with H2a (limited prediction) and H2c (sensitivityto initial conditions). His partner MS hypothesized that “it will do the same thing,” but

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observed that the bob followed a different path, concluding, “initial determines final,” inkeeping with H2c. On videotape, MS and BS appear surprised and confused by thependulum’s unpredictability. At the same time, they can see that unpredictability resultsfrom the trajectory’s sensitivity to the precise initial conditions of its two approximatelyidentical starting locations.

In Group 8, students (AB, JQ, ED and JC) observed no sensitivity to initialconditions in the simple pendulum, as expected. This group predicted that placingmagnets under the pendulum would make it “slow down.” Students in this group weredivided; one, JQ, had difficulties accepting sensitivity to initial conditions andunpredictability: 18

PI: If you were to do that experiment over again...do you think it would end at magnet Bagain?AB: noJQ: yesPI: So what’s going to determine if it ends up at A or B?JQ: Where you release it fromPI: Do you think it would be easy to determine where it ends based on what point yourelease it from?JQ: yesAB: noPI: Do you think if you release it from a point around this one [pointing at one of themagnets] it will end up at that one? [releases the pendulum]JQ: yesAB: no, because this one will slow it down, and it will go to this one [pointing at theother magnet. As if on que, the bob does this.]The above dialog shows that students had differing views on predictability, offeringinconclusive evidence for H1 (prior predictability). JQ robustly held on to the view thatthe bob would follow the same path as before, but AB was more willing to acceptunpredictability as natural. The extent to which students like AB’s epistemological viewswere altered is unclear, but JQ’s surely were, as his observations disproved hispredictions. Since his observations contradicted his epistemological beliefs, conceptualchange occurred, evidence for H2a (limited prediction).

Later, this group of students released their bob from approximately the same setof initial conditions (0:52):PI: Did it follow the same path?AB: kind ofPI: kind of?AB: noIP: try it againPI: Is this motion it’s doing now the same as it was before?AB: It looks kind of the same, but not reallyJQ: it didn’t do that (before), [pointing to the bob going around in circles]ED: I guess because it’s not launched from high

18 See videotape 1/17B starting at 0:36:00 for further reference.

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For their conclusions, all four students indicated that the bob’s trajectory was differenteach time it was released, which supports sub-hypothesis H2a (limited prediction). EDand AB wrote, “It depends how you release it,” evidence for sub-hypothesis H2c(sensitivity to initial conditions). JQ could not let go of the notion that there must besome predictability in the system, incorrectly assuming that the bob “will always go tothe opposite magnet” based on two trials, evidence for hypothesis H1 (priorpredictability). However, there is also significant qualitative evidence that JQ’s viewswere altered during the intervention (“it didn’t do that”), so his comments support H2a aswell. 19

January 18, 2006

On Wednesday, Groups 2 (FR, PA, EZ, and KC) and 6 (MT, CP, IP, JCh) beganthe computer portion of “chaos project.” Students observed an evolving a phase spaceplot of a chaotic pendulum on the Internet.20 Answering the question, “Does thependulum’s motion ever repeat itself?” EZ wrote, “yes and no because it never actuallygoes on the same line twice.” CP observed, “There is a pattern, but it changes a little biteach time. It could never go through the middle or the corners.” Thus, students observedstrange attractors in phase space, and could see how they are bounded and ergodic.Because of time constrains, such technical terms were not introduced, but students gainedsome understanding of these concepts because they had already been introduced to phasespace graphs when working with the simple pendulum.

Next, students were shown a computer simulation of the magnetic pendulumwritten in the Boxer programming language by Dr. Andrea A. diSessa, Boxer’s inventor.The software helped students perform the experiment of mapping starting points (“initialconditions”) with the magnet at which the bob eventually ends up. To facilitate thisexperiment, one magnet was colored red and the other was colored green. Once the bobicon stops moving, a red or green dot is left on the screen showing at which magnet thebob ended up when it started from that point. Students were given a worksheet (seeAppendix) with a blank graphics box just like that in the Boxer pendulum simulation andasked to “make a hypothesis about what you think the graphics box would look like if allstarting positions (initial conditions) were tried.” JCh predicted “it’s random,” and KCdrew interwoven regions of red and regions of green throughout the box. 21

Using the Boxer software, EZ explored the boundary between a red region and agreen region. He found that the boundary has “hooks,” much like the pictures of fractalspassed out in class. The concept that zooming in on the border region produces a greaterand greater level of detail was facilitated through a magnifying tool in Boxer. Students

19 ED observed that the magnets did not seem to effect the bob when it was dropped froma large height because the bob had a large momentum by the time it came close to themagnets. In future work, instructions should clearly indicate that students should onlyrelease the bob from small initial displacements.20 See www.myphysicslab.com/pendulum2.html for this educational resource.21 Students should be given red and green pens to draw their hypotheses. Scaffolding wasrequired to get students to understand how to use the zooming in tool built into thesoftware.

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could click on a magnifying glass icon to see a magnified portion of the graphics box inanother box, as shown in the picture below:

Picture 2. The results of EZ’s experimentation with the Boxer simulation. The large redand green dots are the two magnets. The small red and green dots represent positionsfrom which the bob’s trajectory ends at the red or green magnet, respectively. On theright is a magnified view of what appears to be a mostly green square to the upper left ofthe red magnet.

Students commented on their findings:FR: As you look at smaller scales, you see more thingsPI: Recall the dueling calculator activity...KC: When you get closer, you see more detailPI: Do you see a connection between round off and...FR: it’s getting smaller as you go on...there’s something beyond thatPI: When you round it, you’re eliminating the small detailPI: EZ is seeing the shape of the fractal boundary...it’s a hook shapeWith the Boxer simulation, students could actively construct a visual map of the chaoticand non-chaotic regions of starting positions, a reproducible benefit of using computersto supplement the teaching of modern physics concepts in classrooms.

January 19, 2006

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On Thursday prior to class, a magnetic pendulum was set up for students toobserve. Groups 3 (WL, DN, BT, and AC) and 7 (JG, CS, BS, and MSt) were instructedto describe the behavior of the magnetic pendulum. 50% of students (MSt, DN, BS, andWL) wrote “chaotic and unpredictable,” with MSt adding “speratic [sic]”— evidence forH2a (limited prediction), and H2d (examples of chaos).

Students in Groups 3 and 7 then read a handout with five pieces of informationabout the project:1. As the system approaches equilibrium and stops moving, its mechanical energydissipates, becoming thermal energy.2. The boundaries between regions of initial conditions from which the pendulum reachesequilibrium at magnet “A,” and regions from which it reaches equilibrium at the othermagnet “B,” can also be fractals.3. Altering the system's parameters (bob mass, string length, friction) can make it eitherchaotic, periodic, or quasi-periodic on its way to equilibrium.4. When it is chaotic, "windows" of ordered, periodic behavior can occur (called"dissipative structures").5. The system becomes truly chaotic after repeated period doubling, or "bifurcation."22

Cycles of two become cycles of four, eight, 16, 32, 64, and on to infinity (true chaos), inless and less time. Chaos is when the same pattern never repeats. The way in whichperiod doubling occurs is similar in all chaotic systems.23

Below this information was a series of pictures of the Mandelbrot Set from Burger andStarbird (2000). A sequence of smaller scale, zoomed in images embedded within largerimages showed how increasing magnification power produces new intricate details onsmaller and smaller scales. 24

The handout also included a picture from Gleick (1987) of a possible hypothesisas to which initial conditions would end in the bob being suspended over the red magnet,and which would result in it ending up at the green magnet. In this picture, the “red” and“green” regions swirl around each other. Students were instructed to draw their ownhypotheses. Most drew something similar to the example hypothesis; some drew moresimplistic pictures. They then used the Boxer software to produce evidence supportingor disproving their hypotheses.

22 In the study of dynamical systems, a “bifurcation” occurs when a small smooth changemade to the parameter values of a system will cause a sudden qualitative or topologicalchange in the system's long-term dynamical behavior. This can be seen mathematicallyin the logistic map. Bifurcations in the pendulum’s trajectory could not be detected byvisual observation alone. However, it is noticeable that when released twice fromapproximately the same position, the pendulum starts out moving the same way beforetrajectories bifurcate and diverge.23 This information was probably not well understood by many of the students. In futurework, the information should be rephrased using fewer technical terms, and given out atthe end of the project.24 Mandelbrot (1977) wrote, “in the final analysis, fractal methods can serve to analyzeany ‘system,’ whether natural or artificial, that decomposes into ‘parts’ articulated in aself-similar fashion, and such that the properties of the parts are less important than therules of articulation.”

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When a bug in the software caused some problems, the instructor opened a filecontaining EZ’s work from the day before. Comparing this information with their ownexperimental explorations, students saw how the boundary between “red” initial positionsand “green” ones was very complex, and did not look anything like the picture fromGleick (1987). Since no students had predicted such a level of intricacy, students sawhow the Boxer representation provided evidence against their initial hypotheses. Adiscussion was then held about whether or not it is appropriate to apply evidence from asimulation to make conclusions about a real world system:PI: Are we disproving our hypothesis? Why do you think this simulation might bedifferent from what we found in the lab?AC: It’s on the computer, it’s not in real life.PI: Would it be possible to create a computer program where it would be the same?AC: No.PI: Why not?AC: It’s not random.PI: Are you saying that you can’t use a computer to model behavior that in nature israndom?AC: yeah, exactlyInterestingly, AC expressed an epistemological belief that computers cannot accuratelymodel nature, because nature has more inherent randomness than computers are able tohandle. This is qualitative evidence supporting sub-hypothesis H2b (modelinguncertainty). BS, however, expressed an opposing view:BS: The computer knows enough to calculate what’s going to happen, and so iteliminates some of the variables, because it knows how to compute them out and it cando itPI: Are you talking about round-off error like we saw in the dueling calculators?BS: Something like that.PI: Are you saying that you think it’s impossible to get a computer simulation to behavelike an actual system?BS: Not necessarily, I’m saying it canPI: Do you think you can see some of the same patterns that exist in nature using thecomputer simulation?BS: Yeah, you probably could program it so you couldPI: In this particular simulation, there is no gravity. Would this effect the shape of themap?BS: maybe...BS believes that computers can accurately model nature. He is unsure if the lack ofgravity in the simulation may have produced inaccurate results, a non-trivial question.

In their conclusions, students compared their results with their hypotheses. Sixout of eight students explicitly stated that their hypotheses were “disproved” based on theevidence that the boundaries (between the red and green regions) were “fractal.” Aseventh student (BT) described the fractal nature of the boundary without using the termexplicitly: “There were some points that were in between green and red.” Thus, 87.5% ofstudents successfully learned about the nature of fractal boundaries, regions where thependulum’s behavior is unpredictable and is extremely sensitive to its initial conditions.This is qualitative evidence for sub-hypotheses H2a (limited prediction) and H2c

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(sensitivity to initial condition). BS even wrote, “there are fractal points on the y axis,”showing an understanding that starting positions on the y-axis result in unpredictablebehavior.

January 20, 2006

On January 20, 2006, Groups 8 (ED, JC, AB, JQ) and 4 (MS, DS, MV, MP)began by observing a magnetic pendulum set up prior to class. Students wrote downdescriptions of its behavior and discussed their ideas:AB: It’s getting hyphy,25 going crazy, spinning in circles.PI: would you say it’s unpredictable?AB: yesMP: It’s not predictable / it’s chaotic.PI: what do you mean by chaotic?MP: It’s crazy, it’s moving around like everywhereMS noted, “It seems as if the situation of the magnets affect the bob,” a true statementacknowledging that bob’s trajectory is infinitely sensitive to the placement of themagnets. AB wrote, “The pendulum is not predictable because any small force canchange the pendulum.” MS’s and AB’s comments are in keeping with sub-hypothesisH2c (sensitivity to initial conditions). DS wrote, “there is chaos from the two forces,”recognizing that the two competing magnetic forces create an unstable system that givesrise to chaos. DS’s remark is in keeping with H2d (examples of chaos). Altogether, fourstudents or 50% (AB, JC, MP, and MS) described the system as “unpredictable,”qualitative evidence for H2a (limited prediction):PI: Do you think that if we start it from the same starting position twice, it will end up atthe same magnet every time?Ss: noPI: why not?ED: You can’t control how much push or pull you give it when you release itPI: So if we use the computer simulation and start it from the same position, would it bethe same?AB: yes, because all the numbers would be the sameDS: do they have the same magnet force?PI: yes, they do

Students then drew their hypotheses as to what a mapping of initial positions tofinal magnet positions might look like, shading in the region(s) from which the bob endsup at the left magnet. This day, two example hypotheses were provided.26 The instructormentioned that a third hypothesis: that the boundary could be fractal, showing students aseries of pictures of a fractal in which zooming in produces more and more detail.

Next, students moved to the computers and worked in Boxer, seeing how theprogram leaves a green or a red dot at the bob’s starting point depending on whether it

25 “Hyphy” is a slang term from hip hop meaning “out of control” or “crazy.”26 Pictures were taken fromhttp://dept.physics.upenn.edu/courses/gladney/mathphys/subsection3_2_5.html andGleick (1987), p. 235.

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ends up at the green or the red magnet. The instructor noted that the simulation couldeither support, disprove, or be irrelevant to their hypotheses. Students saw that the bobicon follows the same path if it is released from the same point multiple times. In theBoxer simulation, the bob’s path is deterministic because the initial conditions are exactlythe same. At this point, MP asked, “what will happen if you release it from the middle?maybe it will stay in the middle?” AB found that if started from the y-axis, the bob movesup and down and eventually stops at the origin rather than one of the magnets. Theinstructor explained that the middle is a line of unstable equilibrium like a pencilbalanced at its point; one cannot predict which way it will fall.

Students brought up a concern that consecutive trials might depend on previoustrials, and the instructor said that they do not. They attempted to find starting points forwhich the outcome is not deterministic. Due to the nature of the program, some pointsnear boundary regions appeared to do this. However, these points were not exactly thesame, they were just too close to be distinguished given the resolution of the graphics:MS: there’s a 50/50 probability which magnet it ends up atDS: in two trials, it went to two different magnets from the same pointPI: are you sure it was exactly the same point?DS: noThe instructor then explained how to use the magnifying glass feature to examine aregion in finer detail. Students saw that each unique point did yield a deterministictrajectory—until, after zooming in many times, the program’s numerical resolution wasexhausted.

In their conclusions, six out of eight students (75%) wrote that their datadisproved their hypotheses because the computer simulation did not produce a patternsimilar to their drawing. DS said that he had not thought that there would be regions ofmixed colors. MP had hypothesized that the bob would end up on the same side most ofthe time, and ED that bob would always go to the magnet near its starting point. Makingsensible inferences, these students realized their experimentation disproved theirpredictions. This is evidence for sub-hypotheses H2a (limited prediction) and H2g(limited to probability). 27

January 23, 2006

Monday, January 23, 2006 was the final day of the educational intervention.Groups 1 (QP, JM, MR, AP, and LR) and 5 (CC, TL, IC, and SS) began by observing amagnetic pendulum and writing down what they had already learned about it.MR: It moves in different ways.LR: not reallyMR: every time it gets on top of it [a magnet], it moves in chaosPI: ok, what is chaos?MR: it does that [pointing], I don’t know how to say it in words though

27 For an unknown reason, two students indicated that the simulation data did supporttheir hypotheses. Perhaps these students thought they would receive a higher grade forsuch a response.

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AP: It reminds me of a ouija board28

MR: every time it gets on top of it, it does its random thingsPI: why is it random?LR: the magnet attracts itMR: the magnet will just do things to itLR: the magnet attracts loose stringsPI: Say you release it from the same point twice?AP: so there’s no way to determine where it will go?MR: Unless we have a very precise way to do it the same every time...we have only ourhand and our hand is not that precise.PI: exactlyThese students’ written responses indicated an understanding of sensitivity to initialconditions in keeping with sub-hypothesis H2c, and the deterministic nature of chaos, inkeeping with hypothesis H4. For example, MR wrote, “unless you have a very preciseway of measuring, every time we drop the pendulum it will follow a different path b/cour hand [sic] are not that precise.” AP wrote, “Without precision to determine the exactposition to start from every time we only assume that the activity occuring [sic] is chaoticbecause our hands are not precise.” AP did not see that chaotic behavior can bedeterministic, but realized that a lack of precision in the experiment was the cause of thediscrepant results. LR wrote that “precision is the only way to determine the exactdirection of the pendulum,” indicating an understanding of deterministic chaos aspredicted by hypothesis H4 (deterministic chaos).

Students in Group 5 engaged in dialog:TL: when it’s done swinging, it favors one of the magnets. it’s arbitrary selectionPI: Say you released it from the same point twice, would it follow the same path?TL: noCC: yesTL: It’s a possibilityPI: Are the laws of physics based on probability?TL: you can’t say that things will be a certain way all the time; there will always beexceptionsSS: I think it’s sensitive...if there’s no air resistance, the motion could be very different.These students’ written responses provide evidence for sub-hypothesis H2c (sensitivity toinitial conditions). For example, CC noted, “It is sensitive to many things. If we blow onit, it will move a little or a lot,” and IC wrote, “at the end of the screw’s trajectory, itfavors one magnet and stays with it, not the same one b/c sensitive to the slightestchange.”

Students then moved over to the computer simulations. The instructor noted anexperimental advantage to using them: a far greater degree of precision than existed whenstudents were using their hands to release the bob. Students began the experiment ofstarting the Boxer simulation from the same initial position twice in a row. For herhypothesis, AP indicated that the bob will follow the same path “because it is startingfrom the exact same position; making the experiment precise and without the error of ourhands.” Her group’s results concurred with this hypothesis, and LR noted, “unless you

28 AP’s comment is more evidence for “magical thinking.”

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start at the EXACT same beginning point, it will not follow the same path.” Thus,students demonstrated an understanding of the concept of deterministic chaos in keepingwith hypothesis H4 (deterministic chaos).

Next, students in Group 5 were asked if there were any regions from which onecould be sure the bob icon would end up at a one particular magnet. Such regions werelocated:TL: I found a regionPI: How many data points do you think you would need to plot to be sure?TL: I think five.Then, students found regions from which one could not predict where the bob wouldland:(0:24) PI: is it like the real magnetic pendulum?AP: In a sensePI: why do you say that?AP: there are areas where you don’t know where it’s going or where it’ll end up...wefigured out that from up here, it will go to red. If it’s a little bit closer, it will go to greenPI: there’s got to be a boundary between the regions of the red dots and the regions of thegreen dots

Students used the magnifying glass feature to zoom in on a “chaotic region”:AP: we found a borderPI: what do you think is happening there?AP: we can’t zoom in anymorePI: you’re at maximum magnificationAt maximum magnification, resolution limits prevented students from exploring thefractal structure of the boundary any further. AP wrote, “at first we thought there wouldbe a definite border then there were no definite pattern.” CC concluded, “An area closerto the green magnet is more likely to be green, however it may be red too. You canpredict for some areas, however, there are some places that are undetermined. The closerto the center, the easier to predict.” Thus, students saw that probabilistic information isall that can be known, in keeping with sub-hypothesis H2g (limited to probability).

Students in Group 1 claimed to have found a clear-cut border:(0:40) PI: So you’re finding that there is a definite cut-off?AP: It should be like 7 million decimal placesPI: If you could keep on zooming in, you might find that the green dots and the red dotsare mixed together...you might not necessarily have found what you found in otherregions.AP wrote, “in our experiment we tried to find the exact point of the boundary. So far wefigured that 131.299 is green and 131.300 is red. Somewhere in between that regioncould be the boundary.”

Students in Group 5, however, noted that there was “really no defined boundaryline. There was a weird area where the dots could be a boundary, but not” (CC). SSwrote, “I found that there is something more complicated, there isn’t just a simpleboundary. There are green dots mixed in w/ the red dots and visa versa.” IC wrote, “Isaw that the results can vary. If it is closer to green it goes to green, red—goes to red. Ifit is in the middle, it is random, like a pattern. There are some regions where you canpredict where, there is some that have arbitrary selection.”

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At the end of the period, students compared their results with those displayed onother computers. Students in Group 5 were interviewed about their conclusions:(0:45) PI: There’s some regions for example here where it’s unclear whether it’s going toend up and the red or the green. it’s complicatedCC: TL and I were just talking about that.TL and PI pointing at the computer screen.PI: what if you’re right here where the red and the green dots are very closeTL: That’s why I called it random selection because you don’t know where itgoes...unless you measure it by something... i don’t know[dialog skipped]PI: [pointing at a region] So there’s regions like here where there is no arbitraryselection...TL: So, right, there is some arbitrary selectionPI: So there’s some starting points where there’s no arbitrary selection, but then there areother regions where there is?TL: um hum, yeah. because it’s not definite at all that’s why. There are exceptions.PI: So sometimes it’s definite or determined, and then for other conditions or otherlocations it’s not?TL: yeah. It’s not always definite.TL’s comments are in keeping with sub-hypotheses H2a (limited prediction), H2c(sensitivity to initial conditions), and H2g (limited to probability). Toward the end of theinterview, she states that there are some regions of starting points where outcomes are“arbitrary,” and some that are not, reflecting an understanding that nonlinear systems canbehave predictably or unpredictably depending on their parameters. This is evidence forhypothesis H3 (complexity).

Results

Hypotheses Likert True or False WrittenResponses

InterviewsandComments

H1: priorpredictability

n/a A2: nsB2: ns

pre:A3 & B4: 89%A4 & B3: 54%same, 67% sameor maybe

during:hypothesis1:100%hypothesis2: 91%

Jan. 13Jan. 17: JQ

H2a: limitedprediction

A5/E14: p=0.06D6/F15: p=0.007

A2/E2: nsB2/F2: p=0.05

during:observation1: DS,FR, KC, CP, JCh,MT, IPhypothesis2: BSobservation2: CC,SS, TL, IC, BS,ED, AB, JQ, JC

Jan. 11Jan. 13: MSt,JMJan. 17: JCh,AB, JQJan. 23: TL

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SS, TL, IC, BS,ED, AB, JQ, JCobservation3:MSt, DN, BS,WLJan. 19: 88%Jan. 20: AB, JC,MP, MS

post:E5 & F5: 8 SsE9/F8: 53%

H2b: modelinguncertainty

C8/E17: p=0.03D8/F17: nsC6/E15: ns

n/a calculator1: 43%calculator2: 54%

Jan. 11Jan. 19: AC

H2c: sensitivityto initialconditions

C7/E16: nsD7/F16: ns

n/a pre:A3 & B4: 11%A4 & B3: 46%

during:calculator1: 61%calculator2: 25%hpothesis2: SS,IC, BSobservation2: PA,BS, MS, ED, ABJan. 19: 88%Jan. 23: MR, LR,AP, QP, JM, CC,TL, IC, SS

post:E5 & F5: 5 SsE8/F7: 80%F11 & E12: 43%

Jan. 11Jan. 13Jan. 17: JCh,MT, IP, ABJan. 19Jan. 20: MS,ABJan. 23: MR,SS, TL

H2d: examplesof chaos

n/a n/a pre:A3 & B4: 7%

during:observation2: KCJan 19: MSt, DN,BS, WL

post:E9/F8: 83%F12 & E13: 77%

Jan. 13Jan. 16Jan. 19Jan. 20: DS

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H2e: holisticview

C9/E18: nsB5/F14: ns

n/a n/a

H2f: fewvariables

n/a E1: 33%F1: 75%Ave: 55%

n/a

H2g: limited toprobability

D9/F18: ns E4: 80%F4: 69%Ave: 74%

observation1: CP,JCh, MT, IPobservation2: CC,SS, TL, ICJan. 20: 75%Jan. 23: CC, TL,IC, SS

Jan. 17Jan. 20: MSJan. 23: CC,TL

H3: complexity n/a E3 & F3: 53% n/a Jan. 23: TLH4:deterministicchaos

n/a n/a Jan 23: MR, AP,LR, QP, JM

Jan. 11Jan. 23: MR

H5: phasespace

n/a n/a PA,EZ, FR, DS,KC, AP, MR, QP,JM, SS, TL, CC,IC, DN, MV, MP,MS, JG, LR, MS,CS, BS

Jan. 13: QP,MR, AP, JMJan. 17: CC,TL, IC, SSJan. 18: EZ,CP

Table 2. The results of quantitative and qualitative data analysis. All results listed areincluded in text of this thesis. It should not be taken as an exhaustive list.ns = no significant resultn/a = not asked

Hypotheses

Hypothesis H1 (prior predictability): “After learning traditional high school levelclassical mechanics and prior to the modern physics educational intervention, studentsbelieve that all physical systems are predictable.” Answers to true or false questions A2and B2 seem to disprove this hypothesis, but data from A1 and B1 indicate that studentsdid not have a clear concept of a physical system prior to the curricular intervention.Although students thought that small changes can produce different answers, when theywere actually presented with a chaotic system, no students hypothesized that themagnetic pendulum would behave unpredictably. Interestingly, in the act of doingscience, students did not think that unpredictability was likely, even though they hadadvocated for it in the pre-assessments.

Sub-hypothesis H2a (limited prediction): “Students will learn that in somesystems (“chaotic” systems), there are limits to what it is possible to know the system’sfuture behavior, no matter how much information one has about the system initially. Inother words, some systems are not predictable.” This sub-hypothesis was supported withstatistical significance by pre-assessment questions B2, A5, and D6 and post-assessmentquestions F2, E14, and F15, respectively (p=0.05, p=0.06, p=0.007). It was also

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supported by post-assessment questions E5, F5, E9, and F8, and by numerous studentinterviews and written responses.

Sub-hypothesis H2b (modeling uncertainty): “Students will have a greatertendency to disagree with the view that given the right measuring devices, all systemscan be accurately modeled with certainty.” This sub-hypothesis was somewhat supportedby the data. Given the specific example of a coin toss (C8 and E17) students movedaway from a belief in inherent predictability (p=0.06). However, questions D8/F17 (“It isalways possible to know something with absolute certainty, if one has the right tools ormeasuring devices”) and C6/E15 (“Anything in nature can be accurately modeled withcomputers”) produced average student responses of “unsure” both before and after theeducational intervention. During the dueling calculators activity, about half of classexplicitly indicated that different calculators do not always give the same results, thusplacing constraints on modeling.

Sub-hypothesis H2c (sensitivity to initial conditions): “During the intervention,students will learn the concept of “sensitivity to initial conditions,” and afterward theywill have a greater tendency to believe that small influences in a system can sometimesproduce large changes in the future behavior of the system.” Data from A3 and B4, andA4 and B3, in conjunction with E5 and F5, E8/F7, and E12 and F11, provide evidence forthis hypothesis. In pre/post question C7/E16, students remained unsure, and in D7/F16,students tended to agree that small influences can produce large effects both before andafter the intervention. However, numerous student interviews and qualitative responsessupport H2c.

Sub-hypothesis H2d (examples of chaos): “Before the intervention, students willhave difficulties providing examples of systems in which sensitivity to initial conditionsoccurs (A3, B4). After, students will be able to provide physics definitions of “initialconditions,” “chaos,” and examples of chaotic systems. Some students will be able todemonstrate their comprehension of these concepts in writing.” This sub-hypothesis wassupported by the data from pre-assessment questions A3 and B4 in conjunction with post-assessment data from E9/F8, E13 and F12, and by several student interviews and writtenresponses.

Sub-hypothesis H2e (holistic view): “The educational intervention will causestudents to move away from reductionist epistemologies and adopt a more holistic viewof physical systems.” This sub-hypothesis was tested by questions C9/E18 and B5/F14.Data did not support this sub-hypothesis. Students tended to agree with or were unsure ofa statement advocating reductionism (B5/F14) both before and after the intervention.Interestingly, students also tended to agree with or were unsure about a statementadvocating holism (C9/E18) both before and after the intervention. This hypothesisrequires further investigation.

Sub-hypothesis H2f (few variables): “Before the intervention, students believechaotic systems must have many variables. After interacting with physical systems withfew variables, students realize that chaotic systems with few variables can exist.” Thishypothesis was tested by post-assessment questions E1, F1. In these questions, 55% ofstudents indicated that many variables are not a prerequisite for chaos. The investigatorassumed that before the intervention, students would not be aware of the differencebetween high-dimensional systems and nonlinear systems, so this hypothesis was nottested on the pre-assessment, perhaps in error. However, pre-assessment question C10

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did show that students did not understand the meaning of “chaos” as it is used inscientific academic language, thus it is difficult to formulate the question pre-intervention. This hypothesis requires further investigation.

Sub-hypothesis H2g (limited to probability): “The intervention will show studentsthat sometimes in physics, it is only possible to know the probability that something willhappen.” Results from E4, F4, and F18 indicate that after the intervention, most studentsagreed with the epistemological view that, “In some systems, it is only possible to knowthe probability.” A comparison of pre- and post-responses to D9 and F18 show that theintervention did not make students more likely to agree. However, in E4 and F4, 74% ofstudents indicated that sometimes, a probability is all that physics can predict. H2g wasalso supported by student interviews and written responses.

Hypothesis 3 (complexity): “During the intervention, students will see how non-linear systems exhibit both order and chaos, with windows of key variable ranges thatresult in periodicity mapped within variable ranges resulting in chaos.” In post-assessments E3 and F3, 53% of students indicated that a chaotic system can exhibit orderor periodicity. An interview with TL on January 23 also provides evidence for thishypothesis.

Hypothesis 4 (deterministic chaos): “The computer simulation will allow studentsto see examples of deterministic chaos. When two starting positions are infinitelyidentical, trajectories will be the exactly same, even where arbitrarily close starting pointsproduce drastically different results.” This hypothesis was supported by qualitativeevidence collected on January 23.

Hypothesis 5 (phase space): “Students will learn about graphical representationsin phase space.” With scaffolding, most students (79%) were able to draw phase spaceplots of the simple pendulum. No conclusions could be drawn regarding the degree towhich students understood computer generated phase space plots of a chaotic pendulum.

True or False Questions

In pre-assessment question A2, all students circled true, indicating an awarenessof unpredictability in physics that was not suggested by this experiment’s primaryhypothesis. However, student answers to questions A1 and B1 indicated that manystudents did not have a good understanding of what a physical system is. It is possiblesome students thought that social interactions could be an example of a physical system.Social interactions appear to be unpredictable, but they are not physical.

Seven students indicated that “some” physical systems are unpredictable; onecircled “about half.” Five indicated “most,” one gave “almost all,” and no studentsindicated that “none” or “a few” are unpredictable. The average response was “abouthalf,” but the modal answer was “some.”

In pre-assessment question B2, four students answered “true” and nine indicated“false.” Of the latter group, three indicated that “a few” systems are unpredictable, twogave “some,” one answered “about half,” two indicated “most,” and one gave “all.” Theaverage response was coded as 2.67, in between “some” and “about half.”

In post-assessment item F2, students were given the same question. This time,fifteen students answered false and one student, DS, answered true. (DS also indicatedtrue in pre-assessment B2.) This change was statistically significant (t-statistic = -

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1.666; r=27; p=0.054 in lower-tailed test),29 giving support to sub-hypothesis H2a(limited prediction).

There were five false and ten true responses to E1, and four false and twelve trueresponses to F1, indicating that the wording of the question strongly effected the results;students were more likely to answer true in both cases. Altogether, 17 students indicatedthat systems with few variables can exhibit chaos (55%); 14 felt that many variables arenecessary. Thus, H2f (few variables) was supported by a thin majority of students.

In post-assessment questions E3 and F3, phrasing did not effect students’responses, as the class was divided in their answers to both positive and negativephrasings. 16 students indicated that chaotic systems can exhibit both chaos and order;14 said that they can never exhibit “ordered or periodic” behavior. Thus, a majority ofstudents (53%) understood that a “chaotic system” and “chaos” are not the same thing,and that chaotic systems can exhibit order—evidence hypothesis H3 (complexity). It isunclear if the remaining students were confused about the semantic similarities between“chaos” and “chaotic system,” or never realized that nonlinear systems can also exhibitorder.

To post-assessment question E4, twelve students answered true and threeanswered false; to question F4, eleven students answered true and five gave false. Inkeeping with H2g (limited to probability), 74% indicated that probability is inherent inphysics, since it is not always possible to give anything more accurate than a probabilisticanswer.

Likert Question Results

The Likert questions asked in pre-assessments were compared to the same Likertquestions in post-assessments E and F. In their responses to post-assessment F, fourstudents (JG, AC, MT, and EZ) gave responses of 4, 5, and 6 only, indicating that theydid not strongly agree or disagree with any of the questions. All other students in bothpost-assessments included at least one response greater than 6 or less than 4. In class,these students had English difficulties, and it is possible that they could not comprehendthe questions due to their poor English skills. MT answered 5, or unsure, for allquestions, and AC answered 6 for all questions. Because of these concerns, the post-assessment F data was reanalyzed with these four students removed. These results arereferred to below as the n=11 post-assessment responses. No students were removedfrom post-assessment E data sets.

Pre-assessment question A5 and post-assessment E14 were statistically comparedto test sub-hypothesis H2a (limited prediction). In the n=13 A5 responses, the averagescore was 4.15 with a standard deviation of 1.91. In the n=15 E14 responses, the averagescore was 5.20 with a standard deviation of 1.47. Thus, students tended to disagree morewith the view that predictability is always possible after the educational intervention, assuggested by sub-hypothesis H2a. This result was statistically significant with p=0.06 ina lower-tailed t-test.

29 The following websites were used for the calculation of p values:http://www.stat.sc.edu/~ogden/javahtml/pvalcalc.htmlhttp://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/pvalues.htm

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Sub-hypothesis H2a (limited prediction) was also test by contrasting pre- andpost-questions D6 and F15, respectively. In the n=14 pre-assessment responses (D6), theaverage score was 4.57 with a standard deviation of 1.16. In the n=15 post-assessmentresponses (F15), the average score was 3.67 with a standard deviation of 1.63. In then=11 post-assessment responses (F15), the average score was 3.09 with a standarddeviation of 1.51. Thus, after the educational intervention, students agreed more withthis statement, as suggested by sub-hypothesis H2a. This result was statisticallysignificant, with p=0.007 in a lower-tailed t-test. This result is similar to the one obtainedfrom the question’s analog, A5/E14.

Pre-assessment question C8 and post-assessment E17 were compared to test sub-hypothesis H2b (modeling uncertainty) using the example of a coin toss. In the n=13pre-assessment responses (C8), the average score was 4.69 with a standard deviation of2.18. In the n=15 post-assessment responses (E17), the average score was 6.27 with astandard deviation of 2.05. Thus, students moved toward strong disagreement after theeducational intervention, as predicted by sub-hypothesis H2b. This result wasstatistically significant with p=0.03 in an upper-tailed t-test.

Pre- and post-questions D8 and F17 were also statistically contrasted to test sub-hypothesis H2b (limited prediction). In the n=14 pre-assessment responses (D8), theaverage score was 5.79 with a standard deviation of 1.85. In the n=15 post-assessmentresponses (F17), the average score was 5.60 with a standard deviation of 2.03. In then=11 post-assessment responses (F17), the average score was 5.82 with a standarddeviation of 2.32. Students were largely unsure of this statement both before and afterthe intervention. In addition, pre-assessment question C6 and post-assessment E15 werecompared statistically to test sub-hypothesis H2b (modeling uncertainty). In the n=13pre-assessment (C6) responses, the average score was 5.69 with a standard deviation of1.44. In the n=15 post-assessment (E15) responses, the average score was 5.60 with astandard deviation of 1.55. Students remained unsure regarding whether computers couldaccurately model nature both before and after the intervention. The experimentalhypothesis suggested that students would move toward disagreement. However, sincethe chaos project itself involved computer modeling, this question was likely to confusestudents, and probably should not have been included.

Pre- and post-questions D7 and F16 were compared statistically to test sub-hypothesis H2c (sensitivity to initial conditions). In the n=14 pre-assessment responses(D7), the average score was 3.71 with a standard deviation of 1.20. In the n=15 post-assessment responses (F16), the average score was 3.73 with a standard deviation of 1.67.In the n=11 post-assessment responses (F16), the average score was 3.36 with a standarddeviation of 1.75. Students tended to agree that small changes can produce large changesboth before and after the intervention. These results are similar to those obtained fromthe question’s analog, E16. Pre-assessment question C7 and post-assessment E16 werealso compared to test sub-hypothesis H2c (sensitivity to initial conditions). The n=13pre-assessment responses (C7), the average score was 5.38 with a standard deviation of2.06. In the n=15 post-assessment responses (E16), the average score was 5.20 with astandard deviation of 2.24. Students remained unsure if small changes “cannot producelarge changes” both before and after the intervention. It is possible that students misreadthis question as “can produce large changes” rather than “cannot.”

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Pre-assessment question C9 and post-assessment question E18 were compared totest H2e (holistic view). In the n=13 pre-assessment responses (C9), the average scorewas 3.46 with a standard deviation of 1.76. In the n=15 post-assessment responses (E18),the average score was 3.57 with a standard deviation of 1.50. Thus, students tended toagree with the holistic paradigm both before and after the educational intervention.

H2e (holistic view) was also tested by comparing pre- and post- questions B5 andF14, respectively. In the n=13 pre-assessment responses (B5), the average score was3.54 with a standard deviation of 1.39. In the n=15 post-assessment responses (F14), theaverage score was 3.87 with a standard deviation of 1.36. In the n=11 post-assessmentresponses (F14), the average score was 3.36 with a standard deviation of 1.20. Ironically,students also tended to agree with the reductionist paradigm both before and after theeducational intervention.

Pre- and post-questions D9 and F18 were statistically compared to test sub-hypothesis H2g (limited to probability). In the n=14 pre-assessment responses (D9), theaverage score was 3.43 with a standard deviation of 1.50. In the n=15 post-assessmentresponses (F18), the average score was 4.53 with a standard deviation of 1.64. In then=11 post-assessment responses (F18), the average score was 4.27 with a standarddeviation of 1.79. Thus, the educational intervention made students more unsure or morelikely to disagree with this statement, the opposite of what is suggested by thisexperiment’s hypothesis, but this result was not statistically significant (p=0.22 in a two-tailed t-test). Perhaps students thought that in chaotic systems, it is not even possible toknow a probability.

Open Response Questions

Data from the pre-assessments A1 and B1 indicated that students could not cometo a consensus on the definition of a physical system, although the concept had alreadybeen covered in prior instruction during units on the conservation of momentum and theconservation of energy. At this time the instructor defined a system as “a definedcollection of objects.” He also stated that a “closed” system “does not gain or lose mass”and an isolated system is “a closed system where the net external force is zero.”

In the pre-assessment, two students used the “system” definition given in class.Five students’ responses were coded as “a group of objects,” and another five as thosementioning “forces” or “energy.” Three students wrote about objects “inside an area,”three mentioned “solid,” “matter,” or “Earth,” and three more defined physical systemsas “anything” or “everything.” Two said systems “change in time” and another twomentioned “equations.” Four students did not know or did not give a response.

Students’ vague concepts of a physical system should be taken into considerationwhen interpreting responses to the true or false questions that followed them aboutpredictability in physics and unpredictability in physical systems. One student’sdefinition of a physical system was a “living organism” and “everything” includes humanbeings. If students’ concepts of a physical system included themselves, it is of littlesurprise that most answered that some systems are unpredictable. No matter where theystand on the free will debate, many people feel that their lives are somewhatunpredictable. In future research, survey tools should ask students if they considerthemselves or their lives to be physical systems.

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In the post-assessment, seven students mentioned “a group of objects,” anincrease of two. Surprisingly, six students included “experimentation” in theirdefinitions. The only explanation for the appearance of this term post-intervention is thatthe students had recently conducted experiments of the magnetic pendulum. Fivementioned “forces” or “energy,” and another five wrote “inside an area.” Seven studentssaid they did not know or did not give a response. Overall, these responses indicate thatthe majority of students were aware that the computer simulation represented a system.Most likely, they also conceived the magnetic pendulum as a system, although they maynot have been aware that, by itself, it is not an isolated system. 30

In pre-assessment questions A3 and B4 (n=27), students were asked to provideexamples of systems in which a small change in a variable could produce a completelydifferent future outcome. Before the intervention, seven students mentioned somethingcovered previously in the physics course. Most examples students gave were notexamples of sensitivity to initial conditions at all, but two did mention unboundedmathematical operations like “squaring, cubing” and “exponents.” Two students (7%)did provide possibly chaotic examples, both meteorological: “earthquakes, hurricanes,and tsunamis – natural disasters” and “a tornado.” One student mentioned E=mc2: aninteresting answer, since the factor c2 is usually extremely large compared to the changein mass. Four students provided mathematical examples, two gave chemical examples,and single students mentioned examples from astronomy and biology. One studentmentioned a plane crash, and eight students did not provide an answer.

In one version of the pre-assessment, fourteen students were asked if they wouldalways get the same answer if they solved the same physics problem using “v =1.0000000000 m/s” and solved it again using “v = 1.00000000001 m/s,” (A4). Ninerespondents indicated yes, four gave maybe, and only one wrote no. In another version,thirteen were asked, “Is there any chance that the new answer could be very different?Why or why not?” (B3). Here, five indicated “no” and seven gave “yes.” A yes answerto the first phrasing corresponds to a no answer on the second phrasing. For bothphrasings, students tended to answer “yes,” which may be because survey respondentsare more likely to give affirmative answers to complex questions, a common problem insurvey design. However, the combined result of fourteen students indicating that answerswould be the same and eight that they may be different is significant. Out of 26 students,fourteen (54%) held this view; including the four “maybe” responses, 67% did,supporting hypothesis H1 (prior predictability).

30 Even the magnetic pendulum-Earth system is not isolated system. Since smallinfluences matter, one must also include the wind currents in the room and all minorforces as significant.

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In the post-assessments, students were asked, “What happens when a system’sbehavior is very sensitive to initial conditions?” (F11) and to provide an example (E12).In a follow-up question, they were asked to give examples of chaotic systems that werenot studied during the project and to explain why they are chaotic (E13, F12). Surveyversions E and F were coded separately. Thirteen students (43%) could answer thisquestion successfully, with eleven providing examples from meteorology like “windspeed & direction.” Eight students cited examples from the traditional physicscurriculum that are not chaotic, such as “rolling balls from a high surface” and simplependulums.31 Three gave chemical examples, such as “A flame of fire is chaotic; itflickers where it wants.” Two students mentioned “earthquakes.” These students mayhave been drawing on prior knowledge that the moment an earthquake strikes cannot beaccurately predicted.

Before the intervention, 54-67% believed that a small change in a variable couldproduce a completely different outcome, but only 11% could provide an example of sucha system. After, 43% of students could provide a real world example of sensitivity toinitial conditions. Students’ written and verbal comments provide many examples ofstudents understanding the concept behind H2c (sensitivity to initial conditions). Inconjunction with the data supporting hypothesis H1 (prior predictability), these resultsindicate that, overall, students moved toward an epistemological view in which there aresystems in which small changes can have large effects, the position taken in sub-hypothesis H2c. However, pre- and post- results from Likert questions D7/F16 andC7/E16 do not provide further evidence for or against this sub-hypothesis, since studentstended to either agree that small influences can produce large changes both before andafter the intervention.

Although the survey indicated that “answers will not be graded,” students’responses in E12, E13, F11, and F12 were also coded based on the grade they would havereceived if they were graded by the teacher and conductor of this research. Ten students’writing samples used modern physics terms appropriately in their answers (A=33%);eight students’ responses indicated that they could comprehend and adequately answerquestions about nonlinear physics concepts (B=27%); five students showed someunderstanding (C=17%); three students demonstrated misconceptions ormisunderstandings (D=10%); and four did not give answers (F=13%). This codingscheme indicates that roughly 23 out of 30 students (77%) showed understanding ofchaos theory concepts in their writing. Since only two students (7%) could giveexamples of chaos in the pre-assessment A3 and B4, these results clearly supporthypothesis H2d (examples of chaos).

In the post-assessment, fifteen students were asked for the most important thingthey learned from the chaos project (E5). Four responses were coded as “some things areunpredictable or not controllable,” evidence for H2a, and two as “small change can have

31 A somewhat surprising result was that twelve students’ examples were somewhatpsychological in that they included a physical agent: “We could...,” “...you don’t know.”For unknown reasons, no students gave physical agent responses in the pre-assessment.A possible reason for this finding is that since students had recently engaged inexperimentation, they were in a more active frame of mind, and thus more likely to writeabout situations in which they themselves could perform experiments.

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big effects,” in keeping with H2c. Six students mentioned a specific detail of the project,and three stated that pendulums cannot be predicted, which is not generally true. Threestudents wrote something wrong or irrelevant. Fifteen students were asked for the mostinteresting thing they learned from the chaos project (F5). Four responses were coded as“some things are unpredictable or not controllable,” in keeping with H2a, and three as“small change can have big effects,” evidence for H2c. Ten students mentioned specificdetails of the project. Since these results were difficult to understand, in future research,the “interesting” phrasing will be eliminated in favor of the “important” phrasing.Significantly, 13 out of 30 students (43%) volunteered one of the primary concepts ofchaos physics without scaffolding.

Post-assessment questions F7 and E8 were used to provide data for testing sub-hypothesis H2c (sensitivity to initial conditions). All 30 students were asked, “What is aninitial condition?” (F7/E8). 24 students (80%) provided an adequate definition,mentioning “beginning,” “starting,” or “before.” This is not that surprising, since theterms “initial position” and “initial velocity” had been used throughout the semester inthe traditional physics curriculum. Four students demonstrated misunderstandings, andthree did not answer, but these data provide evidence that most students understood thisconcept.

“What is chaos?” was also posed to 30 subjects in post-assessments E9 and F8.Sixteen, or 53%, of answers were coded as “unpredictability” – strong evidence insupport of H2a (limited prediction). Surprisingly, no students mentioned sensitivity toinitial conditions, but seven gave “no repeating pattern” and five indicated something thatis “not controllable.” Four students mentioned “going crazy,”32 and three, “randomness.”33 Altogether, 25 students, or 83%, gave responses that overlap with the scientificdefinition of chaos, strong evidence for sub-hypothesis H2d (examples of chaos). Of theremaining 19%, four did not answer, two wrote “confusion,” and one thought chaos was“a danger in the system.” 34

Discussion

This project was motivated by prior research indicating that high school studentscan learn modern physics concepts through their own experimentation. The question ofwhether such concepts are worth teaching is not worth asking, for there is no reason whystudents should only have the opportunity to learn about physics discovered before the20th century. It is the responsibility of physics educators to keep up with current trends inthe field. Of course, many modern physics concepts may be too difficult to teach at thehigh school level. The results of this research show that classroom experimentation innonlinear systems is quite doable, especially if done in conjunction with computersimulations.

32 “Going crazy,” though certainly not academic language, may be an example of studentsattempting to devise their own terminology for scientific phenomena. This has beenobserved before in the Patterns Research Group when a student accurately described theprocess of temperature equilibration using the phrase “freaking out.”33 Technically, chaos is not randomness, since is it usually deterministic.34 Chaos is not necessarily dangerous, and it is not a psychological state like confusion.

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Results clearly indicate that most students learned that chaotic systems areunpredictable and that small influences can have large effects. However, the degree towhich students came into the intervention believing that everything is predictable inphysics cannot be ascertained. Initially, a majority of students indicated that somephysical systems are unpredictable, but did not think the magnetic pendulum would beunpredictable. Perhaps survey instruments could not distinguish between students’notions of unpredictability in their daily lives and their ideas about the nature of physics.

It is clear that at the end of the intervention, most students realized thatunpredictable physical systems are possible. However, some students robustly held on topredictable epistemologies well into or throughout the intervention. Further research isnecessary to see the degree which advocacy of predictability can be correlated withbeliefs about the nature of physics, or a subject’s level of physics education. It is evenpossible that physics undergraduates may be more likely to believe in predictability andthat small influences can be neglected than high school physics students.

Unfortunately, analysis of the Likert questions showed that students tended toanswer them the same way on post- and pre-assessments. This may be due to apsychological effect observed in survey research: when presented with the same questiontwice, subjects tend to recall their old answer and provide a similar response (Feldman &Lynch, 1988). Despite this effect, three of the Likert questions did return statisticallysignificant changes. In support sub-hypothesis H2a, students tended to disagree morewith the view that predictability is always possible “if enough information is known”after the intervention (p=0.06), and agree more with the view that nature has limits towhat it is possible to know (p=0.007). These results are strong evidence that theintervention changed students’ epistemological beliefs.

Another statistically significant change was a greater tendency to disagree withthe view that scientific instruments could be used to predict the outcome of a coin toss(p= 0.03). Ironically, this statement is true; a coin toss is not a good example of anonlinear system. In an analysis of coin flipping, Stanford physicist Persi Diaconis(1988) found that a mechanical coin flipper that imparts approximately the same initialconditions for every toss has a highly predictable outcome. In other words, the phasespace is fairly regular, more like a simple pendulum than a chaotic one. Thisexperimental finding is not intuitively obvious, so it is unreasonable to expect thatstudents would know it. Since the practical effect of a coin toss is unpredictability, it isnot surprising that students tended to frame it as an example of chaos. Students probablyconnected the perceived unpredictability of a coin flip with the unpredictability of themagnetic pendulum and inferred that a coin toss is chaotic as well.

Conclusion

The chaos project is highly recommended to all high school physics teachers. Inhindsight, the dueling calculators activity was a good way to start the project, as it servesto introduce the concept that small changes—such as rounding off a number—canproduce large effects. However, it is basically mathematics, not physics. It would not beuseful as a stand-alone activity in a physics course.

The faucet and bowl activities did not work well. Students seemed confused as towhat they were supposed to be learning from the activities. In the bowl activity,

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exploring stable and unstable equilibrium conditions seemed too obvious to them; theydid not feel is if they were learning anything new. In future work, it should probably bereplaced with a Boxer simulation used in the Patterns Project. The faucet activity may beuseful if it is restructured with more instructor scaffolding. In hindsight, too littlescaffolding was provided. Students should have been told not to worry about the waterpressure in the faucet itself, but instead to focus on how the shape of the water streamchanges as the water pressure is increased. For unknown reasons, students did not tend topay attention to this. Perhaps students tend to connect physics with engineeringapplications like plumbing, so the nature of turbulence in the water stream wasoverlooked. However, these activities did serve to get students into the habit of simpleexperimentation by asking them to write hypotheses beforehand, make observations, andwrite conclusions.

The explorations with the real magnetic pendulum were the heart of this project.In hindsight, too much time was spent having students analyze the energy dynamics ofthe simple pendulum. Although this activity connected with the topic taught mostrecently in the regular curriculum of the course, it did not connect conceptually with thechaos concepts. An alternative activity suggested by Oliver (1999) is to analyze theenergy dynamics of the magnetic pendulum, including magnetic potential energy alongwith gravitational potential energy and kinetic energy. A drawback of such an approachis that detracts from students’ understandings of modern physics concepts likeunpredictability and sensitivity to initial conditions.

Students study graphing in mathematics courses, but seem to have difficulties inphysics when axes represent physical quantities. Students struggled with graphicalrepresentations throughout the semester. The difficulties students had in constructingphase space plots of the simple pendulum show how graphical representations canconfuse students again when the quantities represented on the axes are different fromthose to which they are accustomed. More research into fruitful ways to teach studentsabout graphical representations in physics should be conducted, starting with a literaturesearch, in order to figure out how the phase space activity could be better scaffolded. Theactivity should not be omitted from the chaos project. Graphical representations areimportant in understanding physics, and asking students to draw a phase space plot is anexcellent way to get students thinking critically. It also presents students with a differentkind of graphical representation, and an extremely powerful one.

Observation of phase space plots of a chaotic pendulum available on the Internetwas a good use of web-based resources, but it was unclear how well students couldconnect the phase space plots they made of the simple pendulum with the computergenerated chaotic plots. Students could see that the chaotic pendulum did not displayperiodicity, but the phase space representation may have been too abstract for them totruly understand. It is unclear whether this portion of the project should be included infurther work, or how it should be better scaffolded.

The Boxer simulation provided students with a means to conduct experiments thatcannot be conducted in with a real world pendulum. Students enjoyed this portion of theproject, expressing enthusiasm when they realized that the magnetic pendulum could bemodeled with a computer. The Boxer representation allowed students to explore achaotic pattern, and their results surprised them. Explorations of the fractal boundarywhere arbitrarily close starting positions produce radically different outcomes were

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especially interesting to students. Unfortunately, resolution limitations in thesoftware—or perhaps in the computer itself—were an obstacle to students’understanding. One of the main concepts students were supposed to grasp was that theboundary is not a line with finite length; rather, it is a fractal with an infinite length anddetail. When resolution limits were reached by some students, they falsely concludedthat there is a define boundary line. It is unclear if these students understood theinstructor’s comments about resolution limits. However, most students neverencountered the resolution limits, and thus correctly concluded that the boundary wascomplex.

The simulation also served to provide a deeper understanding into thedeterministic nature of chaos. In the physical experiment, it was impossible for studentsto see that chaotic trajectories are deterministic, because it was not possible to release thebob from the exact same position in multiple trials. The simulation enabled students todo such an experiment, and provided further insight into what students think is happeningin chaotic systems.

In hindsight, important questions were left out of the pre-assessments that couldhave shed more light on sub-hypotheses H1 (prior predictability) and H2f (few variables).To test H1 more accurately, students could have been asked, “Is it possible for apendulum to be unpredictable?” To test H2f in the pre-assessment, the question, “Can asystem with few variables be unpredictable?” could have been included.

More activities focused around hypothesis H3 (complexity) could have beenincluded in the curricular intervention. There are various web resources about theLogistic Map that show how ordered, periodic behavior can exist within chaos. Suchactivities were left out of this curriculum because they were too mathematical, and thegoal of this project was for students to explore chaos in physical systems. Unfortunately,it was difficult for students to see complexity in the magnetic pendulum. When it didoccur, students could have dismissed it as an anomaly. Periodic behavior is much moreobvious in other nonlinear systems like the Lorenzean water wheel and the doublependulum. A water wheel may be difficult to obtain or transport, but a double pendulumis not difficult to construct, and it can exhibit both periodic and chaotic behavior. Infuture work, students could explore the dynamics of a double pendulum—which, forsmall angular displacements, is clearly periodic—in addition to those of a magneticpendulum. Another avenue of investigation for future work is to see if students are able tounderstand that the unstable equilibrium created by two competing magnets is what givesrise to chaos. This approach was taken by Duit and Komorek (1997). In the chaosproject, one student, DS, did conclude that the conflicting pulls of the two magnets gaverise to chaos (January 20), indicating that high school students are capable ofunderstanding the concept.

There are also connections between chaotic behavior and time scales. One canpredict the weather a few seconds from now fairly accurately, but weather predictionbecomes less and less accurate the further into the future one chooses to look. Amagnetic pendulum behaves in much the same way: on short time scales, one can seewhere its trajectory is taking it, but the bob’s long term trajectory is a mystery. Furtherresearch into students’ views about the connection between time scale length andunpredictability in chaotic patterns is warranted.

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Survey instruments are never definite measures of students’ epistemological andconceptual views, so all results obtained in this project are preliminary. However, theinvestigator feels that results demonstrate that some concepts in chaos theory are possiblefor students to comprehend, and that the intervention altered some students’epistemological views regarding predictability in nature, and the nature of predictionitself. The significance of these findings should not be taken lightly.

In diSessa’s view, a pattern is the interaction among qualities (2006). In themagnetic pendulum, gravity, magnetic forces, and string tension all interact to produce achaotic pattern. But unlike any other patterns, the chaotic pattern is also influenced bythe multitude of small, even infinitesimally small, forces around us all the time, like aircurrents and the earth’s magnetic field. In most traditional physics problems, these smallforces can be safely ignored. But for nonlinear systems acting chaotically, unknownminor forces can produce dramatically different outcomes. Hopefully, students cameaway from their work in this project with a newly found respect for the unknown.

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