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Student understanding of the kinematic quantities of angular speed and angular acceleration Rankin, Graham W. 1993

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STUDENT UNDERSTANDING OF THEKINEMATIC QUANTITIES OFANGULAR SPEED AND ANGULAR ACCELERATIONByGRAHAM W. RANK INB.Sc., The University of British Columbia, 1972M.Sc., The University of British Columbia, 1976A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCENTER FOR THE STUDY OF CURRICULUM AND INSTRUCTIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993© Graham W. Rankin, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatuDepartment of  Cu Cr t: vikteo■^NA OAThe University of British ColumbiaVancouver, CanadaDate  .SLA-^Z1/1 S DE-6 (2/88)ABSTRACTThis study describes first and second year universityphysics students' understanding and reasoning of the con-cepts, angular speed and angular acceleration.The analysis was based on student responses to varioustasks presented to them during one hour long interviews.These responses were characterized from a phenomeno graphicresearch perspective developed by Marton (1981) and hiscolleagues at Gothenburg University in Sweden.The findings of the study are described by categories ofdescription and by categories of reasoning. Categories ofdescription characterize; students' conceptualizations ofangular speed from different frames of reference, and theways in which students make comparisons of the angularspeeds of two objects. Categories of reasoning characterizethe ways in which students were thought to reason about theconcepts of angular speed and angular acceleration in sev-eral task settings.Interpretation of these findings are discussed with ref-erence to the role a typical introductory physics textbookmay have had in shaping the way in which students thinkabout these angular kinematic concepts. Finally, instruc-tional implications and directions for future research aregiven.iiiiiTABLE OF CONTENTSABSTRACTLIST OF FIGURESACKNOWLEDGEMENTS^ viCHAPTER 1: THE PROBLEM1.10 Introduction and background to the study^11.20 The problem investigated^ 41.30 Specific research questions 51.40 Rationale for the problem 51.50 Overview of dissertation chapters^ 7CHAPTER 2: LITERATURE REVIEW2.00 Introduction^ 92.10 A review of motion studies^ 92.11^Cognition and rotational motion studies^102.12^A problem solving study of rotational motion^112.13^Kinematic studies^ 112.14^Frame of reference studies^ 162.15^Situated Cognition 182.20 Phenomenography 242.30 Constructivism^ 262.40 Instructional strategies^ 28CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY3.00 Introduction^ 313.10 Subjects 313.20 The interview setting^ 323.30 Methodological framework 333.40 Categories of reasoning 343.50 Trustworthiness^ 353.60 Generalizability 373.70 The Tasks^ 383.71^The oval path task 393.72^The semi-circular path task^ 403.73^The elliptical path task 423.74^The irregular path task 44CHAPTER 4: ANALYSIS4.00 Introduction^ 474.10 The structure of the analysis^ 484.11^Dialogue format 494.20 Summary of findings^ 494.30 Introduction to research question #1 514.31 Overview of findings of research question #1 524.32 Discussion of findings of research question #1 644.40 Introduction to research question #2 674.41 Overview of findings of research question #2 674.42 Discussion of findings of research question #2 714.50 Introduction to research question #3 734.51 Overview of findings of research question #3 734.511 Word association 744.512 Symbolic association 784.513 Contextual association 854.52 Discussion of findings of research question #3 924.60 Introduction to research question #4 964.61 Overview of findings of research question #4 984.62 Discussion of findings of research question #4 107CHAPTER 5: CONCLUSIONS, IMPLICATIONS AND RECOMMENDATIONS5.00 Introduction^ 1105.10 Conclusions of the study^ 1105.20 Discussion of findings 1145.30 Implications for instruction 1225.40 Implications for future research^ 127BIBLOGRAPHY^ 130APPENDIX I A sample of a complete interview transcript 137APPENDIX II Diagrams of paths used in tasks^155APPENDIX III The Physics of the Task Settings 160ivLIST OF FIGURES1. Speed Comparison Task I 122. Speed Comparsion Task 2 133. Zebra 244. Diagram of oval path 395. Diagram of semi-circular paths 426. Diagram of elliptical path 437. Diagram of irregular path 458. Reasoning paths for angular speed 959. Reasoning paths for angular acceleration 109ACKNOWLEDGEMENTSI would like to express my sincere thanks to my researchsupervisor, Gaalen Erickson and my committee members, GarthJones, and Ron MacGregor for their guidance in preparationof this thesis.I would also like to acknowledge the students and facultyin the Department of Education at U.S.C., and my profes-sional colleagues at Kwantlen College, with whom I talkedabout some of my ideas presented in this thesis.Finally, I would like to dedicate this dissertation to:my parents Alex and Georgina Rankin, and my wife, Sumako,who have always encouraged me to continue my education.viCHAPTER ONETHE PROBLEM1.10 Introduction and background to the studyThis chapter provides an overview of the research inphysics education as it relates to this study, an introduc-tion to the problem investigated by this study, a descrip-tion of the study's specific research questions and a ratio-nale for the study.A large amount of work to date has been done in the fieldof physics education by researchers who have been primarilyinterested in student conceptualizations of physical con-cepts. These investigations have ranged over such diverseareas as; heat and temperature (Erickson, 1979), motion(Saltiel & Malgrange, 1980), gravity (Gunstone & White,1981), statics (Minstrell, 1982), electricity (Shipstone,1984), vectors (Aguirre & Erickson, 1986), optics (Goldberg& McDermott, 1987), sound (Linder & Erickson, 1989), andquantum mechanics (Fischler & Lichtfeldt, 1991).These studies and others like them have provided evidencethat many students before, during or after instruction holdconceptions of physical phenomena at variance with currentlyaccepted principles or theories of physics. Some of thesehave been found to be similar to those of scientists of thepast (Nussbaum, 1983; McCloskey, 1984; Baxter, 1989; and1Steinberg, Brown, & Clement, 1990). Efforts to design in-structional strategies to change these conceptions have meetwith mixed success (Champagne, Klopfer and Anderson, 1980;Clement, 1982; Minstrell, 1984; Happs, 1985, Brown andClement, 1989, Gunstone, 1991).The lack of success of these efforts may in part be at-tributable to the following beliefs: a belief that a concep-tion (metaphorically or literally speaking) is a stablemental entity inside the head of a student; a belief thatstudent reasoning is theory-like; and a belief that thecontext of a situation plays a minor role in influencingstudent's thinking.Some of these beliefs will be discussed later, however,with respect to the later belief, that context plays a minorrole in influencing students' thinking, the early work bythe cognitive psychologist Jenkins, (1974) is germane. Inhis numerous investigations of memory in higher mental pro-cesses Jenkins concluded, as some educators have yet to rec-ognize, that:What is remembered in a given situation depends on thephysical and psychological context in which the event wasexperienced, the knowledge and skills that the subjectbrings to the context, the situation in which we ask forevidence for remembering, and the relation of what thesubject remembers to what the experimenter demands. (p793)A philosophical and methodological foundation which re-flects in part this contextualized nature of student think-ing, as expressed by Jenkins is described by Marton's (1981)research perspective of phenomenography. This perspective2provides a framework by which to characterize the ways inwhich students understand a scientific concept which isembedded in the context of a task setting.These ways of understanding a scientific concept may alsobe influenced by the historical and cultural context as-sociated with the individual as it relates to the task envi-ronment, (see for e.g. Hewson's, 1985, notion of conceptualecology). What is less apparent, however is the influenceof the "mental context" generated as a result of one's ownthinking, as one processes; images, sights, and sounds, thatappear and disappear from consciousness, from moment tomoment during the interview.Concepts are not viewed (from the perspective of this re-searcher) as stable entities in the minds of students. Con-cepts, like memories, are viewed "..not like filed lettersstored in cabinets...(but rather)..like melodies realized bystriking the keys on a piano" (Wechsler, 1963 p151). Thecontext of the task situation plays a critical role in theway these "keys" are struck. Thus the term student concep-tualization is preferred to the term student conceptionbecause the former term tends to infer a mental activityrather than a static mental entity as the later term does.This study by drawing partly on the research in cognitivepsychology, artificial intelligence, and science education,may offer some new directions in the investigation of stu-dents' beliefs about scientific concepts. However, any newdirection is not without risk, as "One must make a start in3any line of research, and this beginning almost always hasto be a very imperfect attempt...We are almost always con-demned to experience errors in order to arrive at truth"(Diderot, D. circa 1750).1.20 The problem investigatedThis study investigated students' understanding of thekinematic quantities, angular speed and angular accelera-tion. Both the products and the processes of students'thinking about these concepts were characterized in terms ofa set of categories of description. These categories weredeveloped, in a manner to be discussed in Chapter Three, byMarton (1981, 1984).In general terms this study investigated four areas.These were:1. A student's ability to recognize the importance ofestablishing a frame of reference from which tomeasure the angular speed of an object.2. A student's ability to make comparisons between theangular speed of two objects.3. A characterization of students' reasoning about theconcept of angular speed.4. A characterization of students' reasoning about theconcept of angular acceleration.These four areas are framed in terms of the specificresearch questions outlined in the next section.41.30 Spenific research questions1. What are students' conceptualizations of the angularspeed of an object from different frames of reference?2. What is the nature of students' conceptualizations asthey make comparisons of the angular speed of two objectsmoving in semicircular paths of differing radii of curva-ture?3. What is the nature of students' reasoning as theyaddress several task problems involving the concept ofangular speed?4. What is the nature of students' reasoning as theyaddress several task problems involving the concept ofangular acceleration?1.40 Rationale for the problemThe objective of this study was to better understand howstudents think about and use the concepts of angular speedand angular acceleration. This type of study is importantfor the following reasons.Firstly, from a pedagogical point of view, for a physicsteacher to be more effective in communicating the conceptsof rotational motion such as angular speed it is importantto have a better understanding of how students are likely tointerpret present instructional attempts at communicating5these concepts by way of the classroom and physics textbook.Such an understanding helps in designing new instructionalstrategies.Secondly, this study addresses students' understanding ofconcepts which themselves constitute a very important areaof physics. Anyone planning a career in physics or engi-neering must have a good background knowledge in the area ofkinematics and dynamics of rotational motion.Thirdly, this study fills a gap in the literature ofmechanics as it pertains to students' conceptions of rota-tional motion. Both a survey of the literature and a reviewof papers presented at two major conferences at CornellUniversity (1983, 1987) dealing with students conceptions inscience show a lack of research into students' conceptual-izations of angular speed and angular acceleration.Finally, it is hoped that some new directions in thefield of science education will follow from studies such asthis. Such new directions may result in studies which notonly offer a description of what students understand aboutscientific concepts, but offer a more detailed descriptionof how such understandings develop, and how within a partic-ular context, students reason about such scientific con-cepts. This is necessary if more effective instructionalstrategies are to be developed.61.50 Overview of dissertation chaptersThis study is described over a span of five chapters.Chapter Two, which follows provides; a literature reviewwhich enables the reader to situate this study with otherrelevant studies in the field, a context by which the de-velopment of an appropriate methodological framework waschosen for this study, and a background for the reader inwhich to better understand the discussion of the findings ofthe study that follows.Chapter Three sets forth the methodology used in thisstudy, part of which is based on the phenomenographic per-spective developed by Marton (1981). The writings of Marton(1981, 1988), MaCracken (1988), and Lincoln & Guba (1987)have contributed to the development of a specific method-ological description which addresses the issues of trustwor-thiness and generalizability of the findings of the study.Chapter Four describes the analysis of the data. Thedata which consists of segments of interview transcripts arearranged and presented in such a manner so as to permit thereader to validate the characterization of various studentresponses which have been constructed from the data. Suchcharacterizations serve to describe both the content of stu-dent thinking and the processes of student thinking in thecontext of the task setting.Chapter Five presents a summary of the findings of thisstudy, followed by a discussion of the findings. Some ofthe findings which describe the ways in which students think7about angular speed and angular acceleration are related tothe treatment of these angular kinematic concepts as pre-sented in a typical introductory physics textbook.The contribution and implication of the findings to thefield of science education are also reviewed in ChapterFive, and the implications for instruction and for futureresearch are discussed.8CHAPTER TWOLITERATURE REVIEW2.00 IntroductionThis chapter describes the literature review from threedifferent contexts. The first context consists of a reviewof studies that have investigated students' understanding ofmotion as it relates to this study. This review will helpsituate this study in terms of other similar research work.The second context consists of a review of the literaturewhich has informed the perspective of this researcher as tothe way in which individuals acquire and construct theirknowledge.Lastly, the third context comprises a review of the lit-erature as it relates to instructional strategies. This re-view will serve to inform the discussion of instructionalrecommendations that appear in Chapter Five.2.10 A review of motion studies An extensive amount has been written about students'ideas of physical phenomena that exist prior to, during andafter instruction in the physical sciences. Some of thetopic areas that have been studied are listed in the intro-duction to Chapter One. Other areas that relate more di-rectly to the study of rotational motion, however, are fewin number, and as yet no study has been done of student's9understanding of rotational kinematics after instruction atthe first and second year university level.The studies of motion have been grouped for referenceunder various headings. These headings are: cognitive andcognitive development studies, problem solving studies,kinematic studies, frames of reference studies, and"situated cognition" studies. Each of these groups of stud-ies will be discussed in the order aforementioned.2.11 Cognition and cognitive development studies The age at which children begin to use linear and rota-tional terms to describe motion has been investigated byLevin and Gardosh (1987). Students by the age of ten werefound to be able to describe motion in rotational or linearterms and by the age of twelve could describe motion in bothterms at the same time for the same object. The choice ofdescriptors for the motion whether in linear or rotationalterms was however dictated by the feature of the motionwhich was most salient to the student.Students enrolled at the University of California weretested by Ehri & Muzio (1974) for cognitive style in solvinga rotational problem. Cognitive style was defined in thisstudy as being field dependent or independent as measured byan embedded figure testi. Students in the study were asked1. This test asks students to identify specific figuresthat are hidden in a more complex figure or field. Theability to separate the required figures from the differ-ent fields is a measure of field independence.10questions about the relative speeds of horses turning on amerry-go-round. The results revealed that nearly all thestudents who were strongly field dependent could not solvethe merry-go-round problem while those who were stronglyfield independent could. The researchers interpreted theirresults as providing evidence consistent both with cognitivestyle distinctions and with Pascual-Leone's (1970) informa-tion processing model.2.12 A problem solving studyBarowy and Lochhead (1981) conducted a study with aboutthirty university students shortly after they had completedthe section on rotational kinematics of a non calculus basedphysics course. These researchers found that, in many prob-lems involving torque, students' attention was focused onpreserving the superficial structure of the problem and assuch were unable to disregard or modify perceptually unim-portant features of the problem situation. It was concludedthat: "many students who have reasonable intuitions aboutqualitative physics are unable to apply them correctly be-cause they cannot isolate critical features of the problem"(p15).2.13 Kinematic studies Studies of the kinematic concepts of velocity and accel-eration are of particular importance to this dissertation asthese concepts are directly related to the concepts of angu-11lar speed and angular acceleration. Hence they will be dis-cussed in more detail than the preceding studies.Trowbridge (1980) investigated student's understanding ofthe concept of velocity2 . Students taking an introductoryphysics course at the University of Washington were asked tomake a comparison of the speeds of two objects in three dif-ferent contexts. A description of these three contextswhich follows, will be later contrasted to the responses ofstudents in this study with respect to questions about thecomparative angular speeds of two objects. The three tasksused by Trowbridge were: the speed comparison task 1, speedcomparison task 2, and a task consisting of written exami-nation questions.In the speed comparison task 1, a ball, A (see diagrambelow) was made to roll horizontally at a constant speed(from left to right) while at the same time a similar ball,B, was made to roll up an inclined plane starting with aninitial speed greater than ball A.Ball A ^o Figure 1: Speed Comparison Task 1. Motion is fromleft to right.2. The discussion to follow unless otherwise indicated isbased on Trowbridge's (1979) Doctoral Dissertation.12The horizontal component of the motion of the two balls(each ball having different starting positions) were juxta-posed such that the balls would appear to pass each othertwice. At the points of passing the balls did not have thesame speed, yet Trowbridge found that a significant numberof students responded to the points of passing as the loca-tion where the balls had the same speed.In the speed comparison task 2, a ball, C was made toroll down an inclined plane, starting from rest (see figure#2), while another ball, B was made to roll up an inclinedplane as was done in task 1.Figure 2: Speed Comparison Task 2: Motion is fromleft to right.In this task the motion of the two balls was such thatthey never passed each other (note: the balls have differentstarting positions). However, the balls did at one pointhave the same speed. A number of students stated that theballs could not have the same speed because they never metor lined up with each other. This response was similar tosome of the responses obtained in the speed comparison task1. One such response which illustrates a possible source13in every day experience of such a conceptualization as de-scribed is illustrated by the following excerpt from theinterview transcripts.[Transcript notation I: interviewer, S: Student]I: Do they ever have the same position along thistrack? (Balls are released)S: There are two positions, right around here andhere. So does that mean they have the samespeed then?I: What do you think?S: Well, it's hard to say. It seems like they would.....Yeah, they are, because like when I'm drivingon the freeway...You know how you hate to havesomeone directly on the side of you...I assume thateven though he might have been behind me, that he'scaught up, and now he's neck and neck with me for alittle while, so he must be going the same speed,even though he keeps on passing me. So at the timewhen we're together, we're probably going the samespeed.^ (p114)A third task given the students was a set of examinationquestions. The written answers to these questions showedthat students had difficulty with the concept of speed at aninstant. One of the exam questions asked whether it was al-ways true if: when two objects both reach the same positionat the same clock reading, then they must have the samespeed. Explanation, including an example (if the statementwas false) was required of the student. One response tothis question quoted below illustrates the student's inabil-ity to conceptualize the motion of the position of an objectat an instant in time.S: Objects can not really have speed for an instant;14for speed to be calculated, there must be an intervalof time. For an instant the objects have no speed,just a location.^ (p 127)Trowbridge maintained it is essential to be able to dis-criminate between instantaneous and average speed if thestudent is to understand the numerical meaning of Av, whichis crucial for an understanding of acceleration. Using asimilar task set as previously described Trowbridge investi-gated student understanding of the concept of acceleration.Trowbridge found that a common (incorrect) procedure fordeciding which of two balls had a greater acceleration wasto conclude that ball A had a greater acceleration thanball B because it was catching up to ball A. This wouldindicate a failure to differentiate between velocity and ac-celeration. Another (incorrect) procedure used by studentsin deciding which ball had the greater acceleration was tomake judgments based on the final speeds of the balls alone.From observations of these procedures used by students,Trowbridge concluded that even though most students coulddefine acceleration in an apparently acceptable manner afterinstruction, they could not use the definition to determinea procedure for comparing the accelerations of two movingobjects.Like Trowbridge, Reif's (1987) investigation of students'understanding of acceleration found that a significant num-ber of students lacked the procedural knowledge of how tointerpret the formula that defines acceleration (a = dv/dt)in problems that went beyond those ordinarily discussed in15physics courses3. Most students could not for example fig-ure out the acceleration of a pendulum bob at various pointsin its swing or the acceleration of a car that slows downwhile moving in a curved path.2.14 Frame of reference studies Frame of reference studies have particular relevance tothis study because students being interviewed in this studywere sometimes asked to respond to questioning about the mo-tion of an object from a particular frame of reference.A study by Saltiel and Malgrange (1980) of fifty firstyear university students found that students had difficul-ties with problems involving frames of reference. Suchproblems would, for example, require the student to thinkabout the distance travelled and velocities of two ob-servers, themselves in relative motion in three differentcontexts. These contexts involved a comparison of the mo-tion as seen by an observer on a river bank versus an ob-server in a boat moving parallel to the river bank; an ob-server in a boat crossing a river versus an observer in aboat crossing the same river at a different orientation; andin a different setting, an observer on the ground versus anobserver moving along on a moving pavement (as might befound at an airport terminal).3. Problems involving motion along a straight line or ofcircular motion at a constant speed.16From these problems the researchers found that studentswould freely combine velocities in different frames withoutany special consideration for the frame of the observer,even though students acknowledged the existence of differingframes of reference. It was believed by the researchersthat students thought not in terms of reference frames be-cause they saw "real" motion as being an intrinsic propertyof the moving body, independent of the observer, and havinga physical cause, whereas "apparent" motion is an illusiondevoid of physical reality. The researchers hypothesizedthat the use of such terms as absolute or relative velocityin the classroom may have given the students the idea thatsome velocities have a deeper, more physical meaning thanothers.Erickson & Aguirre (1985) in their study of students'perceptions of motion from differing frames of referencefound that grade ten students experienced difficulties de-scribing such motion. In one of the tasks students wereasked to describe the paths traced by a moving cart as seenfrom two frames of reference, one moving relative to theother. A significant number of students described the pathof the moving cart as being the same irrespective of the lo-cation of an observer in either of the two frames. And, thespeed of the moving cart was considered to be a constantindependent of the observer's motion with respect to thecart.17The second context that is germane to this study, con-sists of the literature which focuses on describing the man-ner in which individuals construct their knowledge. Thiscontext is described in two parts. First, a review of stud-ies is undertaken which have characterized learning or cog-.nition as dependent on the situation in which it occurs.And second, a review of studies is undertaken so as to pro-vide a philosophical basis for the assumptions being madeabout the way in which individuals construct their knowl-edge.2.15 Situated CognitionClaxton (1981) questioned those that believed: "..thatonce you are 'in possession' of some information, or have' acquired' an ability, they may be retrieved, in principle,whenever they are subsequently relevant. Once in your head,information and ability take on a life of their own, unre-strained by the incidental details of the context in which,and the purpose for which, they were originally registered"(p68). Claxton claimed as did Hills (1988), that the way inwhich children learn to understand and make sense of theirown everyday experiences is inherently contextualized withrespect to its purpose and occasion of use.Rogoff and Lave (1984) have likewise argued that cog-nitive skills and the ability to orchestrate them is not anabstract, context-free competence. They suggest that themanner in which a person interprets the context in any par-18ticular activity may be important in facilitating or block-ing the application of skills learned in one context to an-other. While it is possible that a level of generality ofskills or knowledge may be achieved in certain instances andmust be achieved in other instances if we are to function,it should not be assumed from the outset, say Rogoff andLave, that such generality is usually the case.Perkins and Salomon (1989) in answering the question:"Are cognitive skills context bound?", have suggested a syn-thesis in which a rich contextualized knowledge base oper-ates in conjunction with generalized heuristics to enhanceone's ability to solve problems.Gallagher (1992) in a discussion of the role of hermenu-tics in education puts forth the argument made by Husserl(the German philosopher) that everything comes to be knownwithin a context and never in isolation. It is the context,he argues, which enables one to make sense out of the"unknown" thing.Brown, Collins and Duguid (1989) claim that all knowledgeis like a language. "Its constituent parts index the worldand so are inextricably a product of the activity and situ-ations in which they are produced" (p33). In support oftheir argument they cite such diverse areas of study as:Miller and Gildea's (1987) work on vocabulary teaching andSchoenfeld's work on problem solving in mathematics (1985).Research in physics education, (Saltiel, and Malgrange,1980, Peters, 1981, Maloney, 1984, Halloun, and Hestenes,19201985, Clough and Driver, 1986) support the view that reason-ing is contextualized as evidenced by student responses ontasks and problems which differ only in surface features orperceptual features (Driver, Guesne, and Tiberghien, 1985).This situation specificity of knowledge has been commentedupon by Roncato and Rumiati, (1986) in their study of theconcept of equilibrium, by Reif (1986) in his study of grad-uate students' and professors' attempts at solving non-standard textbook physics problems, and by Erickson andAguirre (1987) who found that the: ".. task context appearedto play an extremely important role in terms of the way inwhich a student both frames and responds to a problem" (p17).A perspective from which to view knowledge as inherentlycontextualized is offered by phenomenography. Rather thanviewing students as holding concepts or other mental enti-ties (as schemata) in their heads which have been abstractedfrom the context from which they arose, a more dynamic, con-textualized, relational view of knowledge between the ob-server and that which is observed has been adopted by Marton(1981). However, while recognizing the importance of con-text, Marton (1988), believes that we do not have to "buy"the whole context; rather, it is our task he says to discernits most significant aspects. Which aspect is relevant inan interview setting (for example) will depend on the rela-tionship of each aspect of the context to a particular in-terviewer's question as judged from the viewpoint of theinterviewee.Not only do some researchers in science education viewknowledge as inherently contextualized but also some in thefield of artificial intelligence (A.I.). Shank and Seifert(1985) have tried to represent knowledge as a structure com-posed of scripts, where a script represents a sum of our ex-periences, as the restaurant script for restaurants. Thecontext of the situation would cue an appropriate script.Some of this work with scripts has informed the debate overwhether or not student knowledge is theory-like.Research with scripts was not able to answer some ques-tions. One such question is: how is it possible that thereexist many situations in which we as humans know what to doin novel contexts without having any situation specificscript for that situation? "There are many situationswhich, as individuals, we approach without any clear beliefsor sets of expectations." (Driver and Erickson, 1983, p42).Therefore, what kind of cognitive system can exhibit thisdegree of flexibility without being straight jacketed by aspecific context?The answer according to McClelland and Rumelhart (1987)is that the knowledge base of the human system does not con-sist of scripts, but a set of atomistic bits of knowledgethat configure themselves dynamically in each context toform tailor-made scripts. These elementary bits of knowl-edge can be actively constructed by the mind to form acoherent assembly called schemata. "Schemata are not ex-plicit entities" say McClelland and Rumelhart, "but rather21are implicit in our knowledge and are created by the veryenvironment that they are trying to interpret--as it is in-terpreting them" (p21).Descriptions of the details of the knowledge system asdescribed by Rumelhart and McClelland are unfortunately at amicrolevel of neuronal functioning or processing that is notappropriate for the study of pedagogy. There have been manydescriptions to characterize student knowledge at an appro-priate level of detail, such as: misconceptions (Helm, 1980;Strike, 1984), alternate conceptions (Clement, 1984), con-ceptual frameworks (Driver and Erickson, 1983), alternateframeworks (Watts, Gilbert and Pope, 1982; Nussbaum andNovick, 1982), alternate theories (Hanson, 1984), intuitivetheories (McCloskey, Kaiser, & Proffitt, 1986), mini-theo-ries (Claxton, 1982), situation-specific knowledge (Roncatoand Rumiati, 1986; Reif, 1986), and "knowledge in pieces"(DiSessa, 1988). One reason for such differences in the wayin which student's knowledge has been portrayed may be dueto the differing methodologies that have been used (Driver,1989). Nevertheless, even if one accepts the methodologyand research data of others there is still the problem ofinterpretation.DiSessa (1986), for example has strongly argued againstcharacterization of knowledge elements in terms of theories,claiming that it is possible to accept McCloskey's strongestdata without accepting his conclusions. DiSessa has choseninstead to characterize the intuitive elements of student's22sense of mechanism as being minimal abstractions, phenomeno-logical primitives, of common events.The characterization of student knowledge as being eithertheory-like or not theory-like has been debated by other re-searchers. For example, in a case study of a student learn-ing about the structure of matter and the way in which thestudent's ideas developed during instruction, Scott (1991)claimed that any prior conceptions the student had about thestructure of matter did not exist in a coherent, generaliz-able, theory-like form.However, Hills (1989) drawing on the work of the philoso-pher Churchland (1979), supports the theory-like view ofcommon sense knowledge. Hills claims that students' theo-ries represent an organized collection of propositions,based on experiences, which allow the student to categorizeobjects and events, form explanations of events, and predictfuture events. In other words, students' "untutored" be-liefs about natural phenomena are common sense theorieswhich share much in common with scientific theories. How-ever, McClelland (1984) states that: "To suppose that chil-dren are scientists of a sort when they think about suchphenomena (as heat conduction) seems to me to misconstruetotally the meaning and purpose of science" (p1).It could be argued that a significant amount of whatchildren do know about the world, as for example theirknowledge of the concept zebra cannot be represented by atheory consisting of only propositional elements as: a zebra23zEBRAIs ft HERetVoRous81.-AcK AND WHiTEAFRIcAN fiNiMALis a four legged animal with black and white markings. Theproblem with propositional knowledge is illustrated by thefollowing figure.Figure 3: "Zebra", from Lefrancois, (1975)This figure is meant to suggest that we clearly know moreabout the concept zebra than can be specified by a set ofpropositional elements which might be described by some the-ory.The foregoing argument is only one part of a much largerdebate over the status of the nature of student's knowledgeabout the world. Clearly such a debate is important and inthe next section two important philosophical perspectivesthat have informed this debate, phenomenography and con-structivism will be discussed.242.20 PhenomenographyPhenomenography is a view of how people understand phe-nomena. Rather than describing a phenomenon which Marton(1981, 1984) called a first order perspective, phenomenogra-phy characterizes the way in which individuals may under-stand and experience a phenomenon, called a second orderperspective.Phenomenography as a research perspective has drawn upontwo philosophical principles. First, the principle of in-tentionality, whereby all mental acts are directed towardssomething beyond ourselves; we cannot disconnect ourselvesfrom the world "out there". And a second principle, an on-tological principle of constitutionality where the individ-ual and the world are considered as one, a unity, in a rela-tionship together. It is this relationship which the phe-nomenographer seeks to describe, as opposed to the con-structivist, who according to Marton (1989) views the indi-vidual and the world as two distinct entities, a subjectiveworld created by the individual and the objective world "outthere".Conceptions are not seen by the phenomenographer as indi-vidual mental constructs residing in the mind of the indi-vidual by which we understand and experience the world outthere. It is the ways the individual experiences the livedin (not "out there") world which become the elementary unitsof analysis. The analysis of these relationships, ways of25perceiving the world of which we are a part, are recon-structed and characterized as categories of description.Which category of description is ascribed to an individ-ual is not fixed. Individuals may move from one category toanother in any one context or across contexts. The sum ofall possible categories form a set (an "outcome space" inphenomenography) which is considered by the phenomenographerto be stable and generalizable to other settings.Besides phenomenography, another perspective from whichto view how individuals experience and understand theirworld is offered by constructivism.2.30 ConstructivismIt is not an easy task to define what constructivism is.There is no agreed upon definition. To quote Kelly (1970):Personal construct theory (constructivism) has...beencategorized by responsible scholars as an educationaltheory, a learning theory, a psycho-analytic theory..alogical positivistic theory,..a reflective theory, andno theory at all! (p10)Noddings (1990) claims that to say knowledge is con-structed tells us nothing about truth, knowledge, the jus-tification of belief or the nature of evidence. And, VonGlasersfeld (1990), in the last few years has taken to call-ing his constructivist orientation a theory of knowingrather than a 'theory of knowledge'.Despite the lack of clarity as to what precisely the termconstructivism should encompass, there is general agreementon one central principle. Personal knowledge is not pas-26sively received but is formed by active construction by theindividual of information that comes to one's senses.For some the "..suggestion that people are active in pro-ducing knowledge seems. .to be 'trivially true'" (Strike,1987 p483). And for Von Glasersfeld (1987), this principlealone represents a trivial notion of constructivism.A second principle of constructivism that is neededaccording to Von Glasersfeld, is that the function of cogni-tion is adaptive and serves to organize our experientialworld, rather than the "discovery of ontological reality".This is to say that we are not trying to discover the real-ity outside ourselves as much as we are trying to make senseof experiences as we perceive them.A consequence of adopting the first (central) principleof constructivism is that one's understanding of construc-tivism is itself a construction. And, while this construc-tion may be idiosyncratic and therefore private it does notmean that a public meaning of constructivism is impossible.Constructivists do not deny the possibility of the existenceof shared meanings amongst individuals. They merely assertthat for a shared meaning to exist to the maximum degreepossible, one cannot simply ignore the personal constructsof others when engaged in the act of communicating.Despite the difference in language between the construc-tivist term conception and the phenomenographer's use of theterm categories of description they can share similar mean-ings under one condition, in which the individual's re-27sponses across a variety of task settings are identifiablewith one and only one category of description (i.e. a stableconceptualization). Under such a circumstance the con-structivist notion of conception comes close in meaning tothat of the phenomenographer's category of description, de-spite the differing ontological viewpoints of each perspec-tive. Or, to put it another way, a stable conceptualizationmay be thought of as the Constructivist notion of concep-tion.Three instructional models compatible with this construc-tivist perspective are reviewed below.2.40 Instructional strategies Three differing instructional models will be briefly in-troduced followed by a discussion of each approach. Oneapproach to instruction is based on (or similar to) the con-ceptual change model of Hewson (1981, 1982, 1989). Thismodel requires students to modify existing conception(s) byeither a process of: conceptual capture, by recognizing thatthe new conception fits into an existing conception (similarto Piaget's notion of assimilation); or by conceptual ex-change, where the new conception replaces another alreadyexisting conception (similar to Piaget's notion of accommo-dation).A second approach to instruction, is based on Niedderer's(1987) model which postulates that students need not neces-sarily replace their existing conceptions but recognize the28difference between everyday thinking and scientific think-ing.A third approach to instruction emphasizes strategieswhich build upon existing student intuitions in order tohave the student adopt a scientific viewpoint. Such anapproach is exemplified by Brown and Clement (1989,1991).The first approach, based on Hewson's conceptual modelrequires students to experience dissatisfaction with theirexisting conception(s). Dissatisfaction may be achieved intwo ways. One is by creating conceptual conflict where anindividual fails to explain or predict an event; or in thesecond way, the individual finds a present conception some-how inadequate.If an individual is to understand the new (scientific)conception being "presented" during instruction it must beintelligible, that is comprehensible. And if the conceptionis intelligible then it should at least be plausible, namely(in the case of a theory or proposition) it could be true.Lastly, the conception should be fruitful for the individ-ual. The acceptance of the new conception should in someway benefit the person, as for example it might contributeto a more coherent theory of physical events or phenomena.In the second approach to instruction, Niedderer's modelproposes that students should come to view their own expla-nation of phenomena (which are usually found to be localizedor contextualized), as being qualitatively different to whata scientist seeks of his or her own theories. The student29and scientist may, and usually do, have different objectivesin arriving at or developing their own "theories". Thescientist, unlike the student, seeks a greater generality.Thus, while it is desirable to have the student move towardsa scientific conception it is acceptable for the student tohave or hold alternate (non-scientific) conceptions of phe-nomena or events concurrently.In the third approach to instruction based on Brown andClement's (1987) model, an attempt is made to build onstudents' existing valid physical intuitions in order tochange their existing conceptions. Students are helped tomake intuitive sense of scientific theory by the use ofspecial types of analogies called bridging analogies. Abridging analogy helps the student "see" that a teachergenerated analogy is valid in viewing a phenomena or eventfrom a scientific perspective.From these three instructional models a background ispossible from which to discuss instructional implications asthey arise from this study. The instructional implicationswill be dealt with in Chapter Five following the analysis inChapter Four, and a discussion of the methodology, which isdiscussed next, in Chapter Three.30CHAPTER THREERESEARCH DESIGN and METHODOLOGY3.00 IntroductionThis chapter describes in order: the subjects, the inter-view setting, the methodological framework, and the tasksthat were used in this study.The basis from which to examine students' understandingof the angular concepts, angular speed and angular accelera-tion was provided by a set of interviews with fifteen uni-versity students. The subjects of these interviews are de-scribed next.3.10 Subjects The fifteen subjects that volunteered to participate inthis study were first and second year physics students allof whom had prior instruction in the concepts of angularspeed and angular acceleration at the time of the inter-views. Of the 15 students interviewed, 4 were Physics 115students, 8 were Physics 120 students and 3 were Physics 216students.Physics 115 students when interviewed were in the lastterm of a first year calculus based physics course in waves,electricity and mechanics. Physics 120 students were alsotaking a similar course to physics 115 in its coverage oftopics; however the depth of coverage of mechanics is31greater in physics 120 than in physics 1151. Physics 216students were taking a second year course in mechanics.3.20 The interview settingEach of the interviews lasted approximately one hour andwas audio and video recorded. The interviews were conductedin a small room having a table and two chairs. The studentwas seated at one chair and the interviewer at the other.The table was used to support the various task equipment ormaterials that were presented to the student during thecourse of the interview. Interviews were held on the campusof U.B.C. during the months of February and March, 1988.The interviews with the students were designed aroundfour tasks. Each task interview required a student to viewthe motion of one or more objects moving in a path from aspecified frame of reference. The interview protocol wassemi-structured consisting of some questions called focusquestions which were asked of each student. These focusquestions served to keep the discussion between the studentand interviewer centered on a particular research question.Other questions asked by the interviewer functioned in twoways.One function was to seek simple verification of the re-searcher's understanding of a statement made by the studentin response to a focus question. And a second function was1. Physics 120 is restricted to those students who have donewell in their previous high school grade 12 physics andgrade 12 math.32to probe issues that could not be anticipated beforehand,but which were deemed relevant by the student as to the wayhe or she thought about some aspect of the task situation.3.30 Methodological frameworkFollowing Marton's (1981, 1984) approach for analyzingqualitative data, categories of description were generatedto describe ways in which students conceptualize or under-stand the concepts of angular speed and angular accelera-tion. These categories of description together form part ofthe "outcome space" which collectively describe the ways inwhich angular speed and angular acceleration were concep-tualized.The process by which the researcher constructs categoriesis not "linear", that is the process of creating categoriesof description does not proceed by a mechanistic, rule basedprocedure. Categories of description are tentatively formedafter several readings of the transcripts only to be modi-fied or changed upon further reading or reflection upon whathas been read. This process is not unlike the following de-scription of the hermeneutical cycle:When we consider the task of interpretation...we haveto say that our increasing understanding of each sen-tence and of each section [of a text], an understandingwhich we achieve by starting at the beginning and mov-ing forward slowly is always provisional. It becomesmore complete as we are able to see each larger sectionas a coherent unity. But as soon as we turn to a newpart we encounter new uncertainties and begin again, asit were, in the dim morning light. It is like startingall over, except that as we push ahead the new materialillumines everything we have already treated..(Schleiermacher, 1977 p198)33Our understanding is never complete, however. "Not onlydo we never understand an individual view exhaustively, butwhat we do understand is always subject to correction"2.And, the process of forming categories of description likethe hermeneutic cycle, becomes a spiral, which is never in away complete: "We can never claim that there are no otherways of seeing a phenomenon than the ones we know of"(Marton, 1989, p37). However, at some point during the pro-cess a set of categories does emerge which serves to estab-lish a framework from which to describe and interpret stu-dent responses.In a like manner a similar discussion could also apply tocategories of reasoning as outcomes of the study.3.40 Categories of reasoningMarton created the term categories of description tocharacterize how students conceptualize or understand con-cepts or ideas. This researcher has created the term cate-gories of reasoning to characterize some of the forms ofreasoning students engaged in. Categories of descriptionthus serve to characterize the content of what students havesaid while categories of reasoning characterize what arethought to be their thought processes. For example, duringthe interview a student might respond to a question aboutthe angular speed of an object in terms of the object's(linear) speed. This response, of associating the word2. Schleiermacher (1977), Hermeneutics, p14934angular speed with speed was characterized as a form ofreasoning, a word association.The phrase "word association" as used in this study doesnot mean what a psychologist understands by this term. Themost important difference is that in psychology word associ-ation usually requires students to list words they thinkabout when given a target word and this is not (as thereader will note later) the same procedure employed in thisstudy. Word associations in this study are inferred fromstudent responses. Other categories of reasoning have alsobeen characterized as forms of association, namely: symbolicassociation and contextual association, both of which willbe discussed later.Finally, how is the reader who has not participated inthe process of creation of these categories of descriptionor categories of reasoning to have trust in such descrip-tions? The answer to follow is given for categories ofdescription but applies equally well for categories of rea-soning.3.50 Trustworthiness There are three distinct elements which serve to "trian-gulate", or establish trust in findings of studies likethese. These elements are signified by three words: pro-cess, product and person, which are described respectively.The process of documenting a description of; the tasks,the task setting, the subjects, the manner in which the in-35terviews were conducted, excerpts from the transcripts plusan entire interview transcript, and the process by which thecategories of description were created, all serve to estab-lish trust in the findings of the study.A product of this study is the categories of description.To the extent that this product, these categories "ringtrue" to the reader, that the reader may find a recognizabletruth by virtue of his or her own experiences, serves toestablish a trust in the outcomes of the study.Lastly, the person, as researcher provides yet anothercheck on the trustworthiness of the study. Who did the re-search, not just the content of the research itself, may beviewed as a criterion for judging knowledge claims. "Inother words, scientists often regard it as proper to judgethe man rather than the knowledge claim" (Mulkay, 1985,p67). But there is also a sense in which the person as re-searcher may be judged other than perhaps by previously es-tablished work in the field."It is precisely because the qualitative researchers areworking in their own culture that they can make the...inter-view do such powerful work. It is by drawing on their un-derstanding of how they themselves see and experience theworld that they can supplement and interpret the data theygenerate" (McCracken, 1988, p11-12). In this way this re-searcher may claim to have access into the world of the sci-ence student and thereby establish a degree of trustworthi-3 6ness in a way that the foregoing elements, process, andproduct do not.Even though conditions for trustworthiness of the studymay be established, there still remains a question as to theextent the outcomes of this study may be transferred toother contexts. This is the issue of generalizability.3.60 GeneralizabilityIn the physical sciences the justified belief that thelaws of nature have a universality of application provides arationale to generalize to other places and times. In thefield of statistics, a priori assumptions, such as normalityof population distributions, provide a rationale to general-ize from the sample to the population. But, in the case ofthe individual interviews in which the object of study is todescribe the way in which students conceptualize theirworld, such a rationale as is used in the physical sciencesor statistics is not possible. The purpose of qualitativeresearch is: "not to discover how many, and what kinds ofpeople share a certain characteristic...It is the categories...not those who hold them, that matter. In other words,qualitative research does not survey the terrain, it minesit." (MaCracken, 1988, p17).In qualitative research a different type of generaliz-ability is possible, a naturalistic generalizability, wherethe outcomes of the research are seen to be: "..in harmony37with the reader's experience and thus to that person a natu-ral basis for generalization" (Stake, 1978, p5).However, the issue of generalizability need not be framedas making a choice between two polar opposites, between ascientific view and a specific, idiographic view. Instead,"..we are dealing with a continuum, the two ends of which donot begin to encompass all of the possibilities that exist"(Lincoln and Guba, 1985, p122-3). Where on this continuumof transferability do the outcomes of this study lie? Thisdepends on both the researcher and the reader.It is the obligation of the researcher to describe thecategories of description and the context in which the in-vestigation was conducted in such a manner as to reasonablyallow anyone else interested in transferring the findings ofthe study to another situation to have a basis upon which tomake appropriate judgments. And, it is the role of thereader to judge the degree of fit that might exist betweenthe context of the study and the context to which the readerwishes to apply the findings of the study. Together the re-searcher and the reader provide the elements necessary toachieve a transferability or a naturalistic generalizabilityof the study's findings.3.70 The Tasks The tasks used in this study are identified by the shapeof the path of the moving object shown to the student duringa particular task. The tasks to be described in order are;38the oval path task, the semi-circular path task, the ellip-tical path task, and the irregular path task.A discussion of how a physicist might respond to thesetasks is provided in Appendix III. This will provide thereader who may be less familiar with these concepts a basisfor better understanding some of the students'conceptualiza-tions outlined in Chapter Four.3.71 The oval path taskIn this task a toy train travelled around an oval trackat a constant linear speed. The student who was seated at atable upon which the train moved was asked to respond toquestions from the position of a toy observer located at thecenter of the oval and from a position close to where thestudent sat.The track was approximately one meter across at its fur-thest points. The following diagram of the oval track showsthe points of the track which were used as references pointsduring the interview.A39XFigure 4: Diagram of oval pathA,B,C,D are reference points used in the interviewto help identify a point or section of the track.X, is the position of the student relative to the track.Focus questions Having identified to the student the position from whichhe or she was to act as the observer, the focus questionsasked were:1. Does the train have an angular speed betweenpositions B and D ...(and other parts of thetrack)? Why?2. Does the train have an angular acceleration?If so, where and why?As a reminder, these focus questions are only some of thequestions that were asked during the task. Other questionsasked depended in part on the responses given by a particu-lar student. Such questioning (see dialogue given in Chap-ter Four or in the Appendix) was directed at investigatingunexpected or unclear responses, or investigating a new di-rection in a student's thoughts that might offer new in-sights.The student was then asked questions about the motion ofobjects moving in semi-circular paths.3.72 The semi-rircular ._ath taskIn this task pingpong balls were made to travel down tworail tracks shaped into semicircles of differing radii of40curvature. The inner track had a radius of curvature of 46cm, and the outer track had a radius of curvature of 56 cm.The starting point of each track was elevated 4 cm above theend point.First, one ball was released from the top position whichis labelled the "starting point" of the outer track in thediagram to follow. After the ball had travelled a shortdistance it triggered a photogate which in turn triggered asolenoid causing a second ball located 15 cm from the top ofthe inner track to be released from its position on thetrack.The release position of the second ball was so designedthat both balls arrived at the same time at the end of eachtrack. A video recording was made of the motion of theballs on the track. This was done because it was difficultto consistently reproduce the movement of the balls. Thisvideo was shown to the student after the student saw severaldemonstration runs. Another advantage of the video record-ing was that the motion of two balls could be shown in slowmotion. Thus making it easier for the student to makecomparative judgments of the balls angular speeds. Despitethis visual aid some students still had difficulty in makingjudgments of the relative angular speeds. Due to thisproblem it was decided not to include in the task a questionabout the relative angular acceleration of the balls.However, a question regarding the angular acceleration ofone ball was asked.41 start point42X^ end pointFigure 5: Diagram of semi-circular pathsA-G: are used as reference points.X, is the position of the student as the observer.The focus questions The focus questions were:Do the two balls ever have the same angular speed?If so where and why?Does this ball (pointing to one of the balls) have anangular acceleration? Why?As with the previous task additional questions were askedwhich sought to enhance the student response(s) that weregiven to the focus questions.In the task that is described next the motion of an ob-ject moving in an elliptical path was used to further inves-tigate the ways in which students understood the concepts ofangular speed and angular acceleration.3.73 The elliptical path taskThe elliptical path made by a moving object was repre-sented by a diagram to the student. The object itself wasindicated by a dot located on the elliptical path. The stu-dent was informed that the object was to be thought of asmoving around this path at a constant linear speed. If dur-ing the interview it appeared as though a student might haveforgotten or not have adequately grasped the motion of theobject then the object's motion was again explained.The following diagram of the path (not the one actuallyshown the student) is shown here for reference. The diagramshown to the student appears in the appendix.A43XFigure 6: Diagram of elliptical pathA-E; A'-E' t are reference points.X's are observer positions.Focus questions The focus questions for this task were:1. Does the ball have an angular speed? Why?2. Is the angular speed of the object a constant?Why?3. If the angular speed is not a constant then whereis the angular speed a maximum and where is it aminimum as seen by someone standing at the:i) centre of the ellipse?ii) outside, near the edge of the ellipse?As with previous tasks additional follow up questionswere asked. The last task to be discussed was used toinvestigate students' understandings of the motion of anobject moving in an irregular path.3.74 The irregular path taskIn this task a diagram of an object moving in an irregu-lar path was shown to the student. It is assumed that mostif not all of the students interviewed would not have seenphysics problems which required them to analyze the motionof an object moving at a constant speed in such an irregularpath3.The diagram of the path which follows is provided forreference. As before, the diagram of the actual path shownto the student is illustrated in the appendix.3. As judged by the problems in physics textbooks and fromtalking with physics instructors, the students would nothave done kinematic problems in which an object moves in anirregular path.44Figure 7: Diagram of irregular pathA-H: are reference points used during the interview.X, is the position of an observer.Forps questions:The focus questions were1. Does the object moving in this path have an angularspeed? Why?2. If yes (to question #1), then does the object havea constant angular speed? If not, where is themaximum and minimum angular speed? Explain.As before, additional questions were asked of thestudents as was done in the other task interviews.In summary four different tasks were presented to stu-dents during the interviews. Each of these tasks involvedan object moving in a particular path. These paths were theoval, semi-circular, elliptical, and irregular.45Focus questions sought to elicit student reasoning aboutand conceptualizations of the concepts angular speed andangular acceleration. Other questions served to clarify aresponse or explored an interesting or unexpected responsewhich was thought to be important as to how the studentviewed the task situation.By a process not unlike the hermeneutic cycle, categoriesof description and reasoning emerged from repeated analysisof the interview transcripts. The particular interviewer-student dialogue which led to particular categories beingformed is part of the analysis that is described in the nextchapter.46CHAPTER FOURANALYSIS4.00 IntroductionAs discussed earlier the purpose of this study has beentwofold. First, to investigate the ways students concept-ualize angular speed and angular acceleration. And second,to investigate the nature of student reasoning as inferredby their responses to questioning about these angularconcepts.With respect to the first purpose, the analysis of thefirst two research questions from a phenomenographic per-spective has led to the construction of categories ofdescription that serve to describe students' conceptualiza-tions of angular speed and angular acceleration. This anal-ysis was based on student responses to questions about: anobject's motion from differing frames of reference (researchquestion one) and to comparisons of the motions of two ob-jects from a fixed frame of reference (research questiontwo).With respect to the second purpose, the analysis of thelast two research questions used similar analytical tech-niques to those used for the first two research questions.However, these techniques led to the construction of cat-egories of reasoning which characterize the reasoning used47by students in responding to questioning about angular speedand angular acceleration.The analysis of all four research questions (as describedin chapter one) is organized into a structure which is ex-plained below.4.10 The structure of the analysis The analysis of the interview data has been organized intothe following format:1. Research QuestionThe research question is stated.2. IntroductionAn introduction to the analysis is presented.3. Overview of findingsAn overview of the research findings is given.4. Category titleA category is identified by a label called a categorytitle. A category may be further broken down intosubcategories.5. Category descriptionA description of the characteristics that identify aspecific category or subcategory.6. Category examplesExcerpts from the transcripts that support the inter-pretation of a group of responses as belonging to aspecific category.7. Discussion of findingsA discussion of the findings of the analysis of eachresearch question.484.11 Dialogue formatAny dialogue quoted in the analysis will use the letter"I" to represent the interviewer. A student will be repre-sented by another letter. Square brackets, [ ] at the endof the quote contain information that is used to referencethe student identity and the physics course he or she wasenrolled in at the time of the interview. For example,[120-5] refers to student "#5" who was enrolled in physics120, a first year honors physics class.4.20 Summary of findings The analysis of student responses to the first researchquestion on reference frames revealed five conceptualiza-tions. These conceptualizations fall into two broad cate-gories in which the student's response is either associatedwith the dependence or independence of the observer's posi-tion or of some preferred position, and the path of theobject.The analysis of the second research question resulted infinding two major methods that students use to make compar-isons of the angular speed of two objects moving in semicir-cular paths. These methods were: to compare the arc lengthswept out by each object in a fixed time, or to compare therelative position of the objects at an instant in time.Analysis of the third research question focused on a de-scription or characterization of the forms of reasoning stu-dents use in responding to questioning about the angularspeed of an object. Student reasoning in this study was49best characterized by the concept of association. An asso-ciation characterizes a way of thinking about the concept ofangular speed. Several forms of association were identifiedas being: word, symbolic and contextual, each of whichwill be discussed, respectively.Word association is the inferred association of the words:angular, speed or other words, with the word: angular speedthat influenced a student's thinking about the angular speedof an object.Symbolic association is the inferred association of thesymbolic form of angular speed, w in terms of other sym-bols, namely formulae, which influenced a student's thinkingabout the angular speed of an object.Finally, contextual association is the association ofsome aspect of the context of the task setting with someother non-task context which influenced the student's think-ing about the angular speed of an object.The analysis of the student responses to questioningabout an object's angular acceleration (the fourth researchquestion) has led to findings similar to that of researchquestion three. The influences upon student reasoning wereagain characterized by forms of association. However, theparticular details of each form was different as the con-cepts being investigated (angular speed versus angular ac-celeration) were different. The details of these forms ofassociation for the last two research questions are dis-cussed in the analysis to follow.50Research Ouestion 41What are students' conceptualizations of the an-gular speed of an object from different frames ofreference?4.30 Introduction to research question 41 When measuring any kinematic quantity such as angularspeed it is necessary to have an understanding of the needfor establishing a frame of reference. Angular measurementsrequired for measuring the angular speed of an object arenormally taken from an agreed upon frame of reference. Oth-erwise, it is possible to arrive at apparently conflictingor ambiguous statements about the angular speed of an ob-ject. Thus it was important to investigate student thinkingabout reference frames.During the course of the interviews it was judged thatnot only the position of the observer but the shape of theobject's path was relevant to the way the student conceptu-alized the task. Thus the conceptualizations are dividedinto two major groupings labeled: POSITION and PATH. Eachgrouping has several identifiable conceptualizations whichare described later.In the dialogue and the discussion which follows, theorigin of the coordinates of a reference frame for an objectmoving in an elliptical or oval path is sometimes describedas being located at the geometric centre. The term "geo-metric centre" is meant to refer to or indicate the midpoint51of a line segment connecting the two extrema points (i.e.the distance between these two points is a maxima) of anelliptical or oval curvel. For some students this geometriccenter held special significance, and is discussed in thefollowing overview of the findings.4.31 Overview of findings of research question fl From an analysis of student dialogue five different cate-gories of description were constructed to reflect the vari-ous ways in which students responded to the task questions.These conceptualizations were grouped under two major head-ings depending on whether the particular conceptualizationcould be characterized as being associated with the positionfrom which to measure angular speed or characterized interms of the object's path . The following are the fiveconceptualizations grouped by the two headings of positionand path.POSITION:1. The angular speed of an object is independent ofthe position of the observer.2. An object may not have an angular speed if the po-sition of the observer is not enclosed by the pathof the object.3. The angular speed of an object should be measuredfrom a preferred position.PATH:4. An object moving in a rectilinear path doesn't havean angular speed or it doesn't make sense to talk1. While this is not a mathematical definition it does serveas an operational definition.52about the angular speed of such an object.5. An object moving in an irregular path doesn't havean angular speed or it doesn't make sense to talkabout the angular speed of such an object.These five different conceptualizations will be referredrespectively as first, second, third, fourth and fifth"beliefs" in the analysis which follows. A description ofthese beliefs are organized according to the format as de-scribed in the introduction.Category title:POSITIONCategory description:The measurement of angular speed is independent of theposition of the observer, dependent on whether or not theobserver is enclosed by the path of the object, or dependenton some preferred position.Subcategory title:Position: First beliefSubcategory description: The angular speed of an object is independent of theposition of the observer.Category examples Example #1: Position: First belief^(elliptical path)P: I think you would get the same (angular speed) forboth (observers) because actually when it comes downto it even if I can still only see the balls motionalong, within one plane I can still see the amountof time it takes for it (the object) for instance toreach one maximum of the ellipse to the other.I: Suppose you're looking down at it (the ellipse) justa little?53P: Just a little, that makes things even more simple.I: just the way you are right now, looking down at it.P: Right.I: Now, as opposed to you and a person standing here(at the center) would you both report back the sameangular speed?P: The same angular speed, I would say so, yeah.I: Ok.P: Assuming not so far away the speed of light makes nodifference.I: So it doesn't matter where you're standing or whereyour sitting where you are?P: That's right.^ [120-5]The angular speed of an object moving in an ellipticalpath is thought by this student to be independent of the ob-server position because (as suggested by the dialogue), thetime for the object to move from one position to another isthe same for both observers irrespective of the observer'sposition.Example 12: Position: First belief^(elliptical path)I: . .what would you reply if you were that observerstanding at the outside (edge of ellipse) there?Would the ball have an angular speed?F: Yes it does, it is still moving in an angle, angulardirection, circular direction.I: Now would the angular speed that you would give mefor some part of the path be the same as if you werestanding there (pointing at the "geometric center" ofthe ellipse)? Would it make any difference?F: It's the same because it's travelling over the sameangle for the same amount of time it travels over thesame angular distance.I: Angular distance being?54F: A full revolution. Travels that revolution in thesame time whether or not I am in the middle or outside(of the elliptical path).^ [115-8]Again, this dialogue suggests that the angular speed ofan object moving in a (uniform) curvilinear path is believedto be independent of observer position. The next exampleillustrates a different position related belief for motionin which the observer is enclosed by the path of the object.Subcategory title Position: Second beliefSubcategory description An object may not have an angular speed if the positionof the observer is not enclosed by the path of the object.ilhcategory example Example 11: Position: Second belief (elliptical path)I: You are standing at the centre and this object(indicated by a red ball moving in an elliptical path)is going around you at eye level--then does thered ball have an angular speed?D: Yes.I: Because?D: Because it is moving around me. [115-1]The student is then asked what his response would be ifhe were standing on the outside of the elliptical path.I: Does it (the object) have an angular speed as seen fromyour position?D: Uh -no.I: Ok, because it is not going around you? (possiblya leading question but it was meant to repeat thereasoning given by the student a few lines earlier)55D: Uh-huh.In a similar task setting a student became somewhat con-fused as to how to analyze motion in an elliptical path.Consequently, it was decided to ask this student a questionabout the angular speed of an object moving in a simplerpath, a circle, from the view point of an observer standingoutside the circle. The task is still however referencedwith respect to the elliptical path.Example 12: Position: Second belief^(elliptical path)H: I don't know if it has angular (speed)--it is movingthrough the angle but--(long pause).I: What bothers you the most about it?. The fact thatthe person (observer) is not at the center, or thefact that he is outside the circle?H: The fact that he is outside the circle. Like withthe last one (elliptical motion at a constant speed),I can't refer it to anything I've done.I: So, not too clear that you can talk about an objecthaving an angular speed from an observer outside (thecircle)? That's not too clear?H: Yeah.I: But (observer) inside, no problem?H: Inside you can see it moving around the angle likethat, but if you are outside I don't know.I: So in the inside you said you can see it is movingaround an angle. What happens if your are here,still inside the circle but half way between thebottom and the top, does it (the object) have anangular speed?H: Yeah, it is still-(pause)-still going through anangle.^ [120-9]56This dialogue suggests that the student had difficulty indeciding if an object moving in a circular path would for anobserver standing outside the circular path have an angularspeed. Other students may have "avoided" this difficultybecause they believed the angular speed of an object shouldbe measured from a preferred position. The existence ofsuch a preferred position characterizes the third belief.Subcategory title Position: Third beliefSubcategory description The angular speed of an object should be measured from apreferred position.Subcategory examples Example 11: Position: Third belief (elliptical path)I: So the angular speed depends on your position.G: Uh-huhI: So what does it mean to say that something has anangular speed if we can say its different— differentdepending on where you we are standing--so how can wetalk about an angular-?G: We have to have some fixed reference point to whereyou are referring to-where it is moving from.I: You often see in a text the angular speed is-or giventhe angular speed and-G: Yeah, usually you can just assume where the positionis they're referring to.I: Suppose we were talking about an ellipse. An objectgoing round an ellipse and I said the angular speedWas-G: Yeah.57I: Two radians per second or something. What would yousay?G: I would just assume that they were talking from themiddle.I: The middle - the geometric "center" ?G: Right.^[216-3]In the next example a student thought that in order todetermine the angular speed of the toy train moving in anoval path it would be necessary to measure the "radius" fromthe "center" of the oval path (the center being a preferredposition).Example #2: Position: Third belief (oval path)I: Does the train have an angular speed between points Aand B?A: Yes.I: Because?A: Because, eh it is a change in the effective radius ofthe train...I: Where would you measure the radius you mentioned-isthat what you said?A: Yeah, we would have to measure from the center of thetrack (student indicates the "geometric center" asdefined earlier in the introduction of the analysis)[120-12]The need to measure the angular speed from some preferredor "standard" position is again illustrated in a differentcontext in the following example.Example #3: Position: Third belief (elliptical path)J: You have to measure things according to a certainstandard. It is all relative. If you are moving at 30km/hr and you see a car coming at you at 30 km/hr itseems to be coming at you at 60 km/hr. You would be58fine if you measured everything according to you movingat 30 km/hr, but you just measure everything relativeto the earth.I: Ok, so it is to some standard: the standard being thecenter of--?J: Of the circleI: What about an ellipse, where is the standard there, doyou think?J: Uh, it would be the center of rotation. Uh, there aretwo centers of rotation of an ellipse isn't there? Eh,just the center where the observer is here.^[115-2]Another student during a discussion of how to measure theangular speed of an object moving in an elliptical pathbriefly made reference to the preferred observer positionas being the most logical point from which to measure theangular speed.Example #4: Position: Third belief (elliptical path)I: If an observer were standing outside the ellipse doesthat ball have an angular speed?R: Yes.I: Because?R: Because the angle is changing.I: Ok.R: But there are times when I guess it would be zero.I: Would it be the same--would the angular speed be thesame as a person who is in the middle?R: No.I: Ok, what you're saying is that depending on where theobserver is the angular speed will differ.R: Yeah.I: So there are many, possibilities for the angular speeddepending on where the observer is located. So whatdoes it mean then to talk about the angular speed of59an object if there are so many possibilities, perhapshundreds of possibilities?R: Well, whenever you're using it to figure something outyou would have to specify the most logical point thecenter here.I: But, why is it logical?R: Well I'd guess that is premature. You would have toknow what you were trying to do.^[216-10]The next two beliefs characterizes those responses wherestudents decide on the basis of the object's path whether ornot it is possible to say an object has an angular speed.Subcategory title Path: Fourth beliefSubcategory descriptionAn object moving in a rectilinear path doesn't have anangular speed or it doesn't make sense to talk about theangular speed of such an object.Subcategory examples Example #1: Path: Fourth belief^(oval path)I: Does the train from your position, from where yourlooking have an angular speed between points A and B?(A to B is a straight section of the track)D: NoI: No. Ok, and why?D: It's not rotating through an angle it is movingstraight.^ [115-1]The student was then asked without changing his positionif the train had an angular speed between another straightsection of track, D-C which runs parallel to A-B.I: Does the train have an angular speed between D and C?60D: No.I: Why?D: Same as AB moving straight.Again, the response is the same as before, that an objectmoving in a straight line whether or not the observer is inthe same line as the motion of the object, cannot have anangular speed.Example #2: Path: Fourth belief (oval path)I: Does the train have from the position where you aresitting; from where you are-does the train have anangular speed between C and A?L: Yes,I: Because?L: Because there is a radius and it is moving incircular motion.I: Does the train have an angular speed between points Aand B?L: Uh, no it does not.I: Because?L: Because it is not moving in a circular motion.I: And does it have an angular speed between B and D?L: Yes, it does.I: Because?L: Because it is moving in a circular motion.I: And finally between D and C does it have an angularspeed?L: No, because it is moving in a straight line. [120-11]61As with the previous example, an object moving in astraight line is believed by the student not to have anangular speed.Example 13: Path: Fourth belief (oval path)I: Does it (the train) have an angular speed between Aand B?P: No.I: Because?P: Because it is moving in a straight line.I: Between B and D?P: Yes.I: Because?P: Well, because it is not moving in a straight line.It's curving.I: Between D and C does it have an angular speed?P: No.I: Because?P: Because it is moving in a straight line. [120-5]The next example of the fourth belief is illustrated foran object moving in an elliptical path.Example #4: Path: Fourth belief (elliptical path)I: That's your position right there at the center. Youcan imagine your the observer.(The investigator further describes the task pointingout that the ball is moving at a constant speed in anelliptical path)I: Does the ball have an angular speed from your ,positionthere in the middle? (Student is aided by thinking ofhimself in the middle of a football field with somebodyrunning around the outside track).62J: Yeah (there is an angular speed).I: Again, why would you have said that - there is anangular speed - if you were to explain to somebody?J: Again it is not travelling in a straight line it istravelling through an angle. [115-2]These foregoing examples illustrate the belief that theangular speed of an object moving in a rectilinear path doesnot exist. However a similar belief was also evident by thestudent responses to motion in a closed irregular path in-stead of a rectilinear path.Subcategory title Path: Fifth beliefSubcategory descriptionAn object moving in an irregular path doesn't have anangular speed or it doesn't make sense to talk about theangular speed of such an object.Example #1: Path: Fifth belief (irregular path)I: Does it make any sense to talk about that objecthaving an angular speed?M: Where am I?I: Suppose you were right here (pointing to a positioninside the path).(The student then explains with some lack of clarityhis thinking. And eventually concludes by saying:)M: So, I am trying to say that if you are discussingangular speed, I don't think you should. [115-15]Example #2: Path: Fifth (irregular and elliptical paths)I: Can we talk about that ball (moving in an irregularpath) having an angular speed?K: We'd have a very hard time-in relation to the centerit would be rather difficult.(and in reference to the ellipse)I: I am interested in a person on the outside of theellipse. Can you talk about the angular speed? Doesit make a difference (where the observer is)?K: I'd think he would have trouble calculating theangular velocity of the object because he wouldn'tbe really sure where the center is..^[120-6]Example #3: Path: Fifth belief (irregular path)I: How about an irregular shape. Suppose I asked youabout the angular speed there. What would it meanif I said the angular speed is--?R: It would mean nothing. [216-10]Example #4: Path: Fifth belief (irregular path)I: Does it make sense to talk about it (the ball movingin an irregular path) having an angular speed?K: I don't know if it makes sense. [120-7]These foregoing examples illustrates the difficultystudents have with the concept of angular speed for motionof an object in an irregular path. A summary of thisfinding and previous findings so far discussed follows.4.32 Discussion of findings of research question fl Five important conceptualizations ("beliefs") about angu-lar speed have been categorized. These have been grouped interms of whether or not the angular speed of an object wasconceptualized in terms of position: angular speed is depen-dent or independent of the position of the observer, ordependent on some preferred position. Or in terms of path:the angular speed of an object is dependent on the shape of64the path of the object. The context of the task settingvis-a-via the position of the observer and the shape of theobject's path played an important role in the manner inwhich the student responded to questioning about the angularspeed of an object.The belief that the motion of an object is independent ofthe observer's position has been observed by Saltiel andMalgrange (1980) in their work on student's ways of rea-soning in elementary kinematics. They noted that studentssee motion as being an intrinsic property of the moving bodyand not dependent on the position of the observer. In addi-tion Saltiel and Malgrange noted that students would some-times seek to measure the velocity from some preferred posi-tion.Velocities are only very loosely defined with respectto frames. This probably corresponds to the fact that,in everyday life, a particular frame is very oftenhighly favoured for practical reasons.. (p75)While these findings of Saltiel and Malgrange have paral-lels to this study, other findings of this study have noparallel with other research work in the domain of kinemat-ics. These (other) findings are, the belief that an objectmoving in a rectilinear path doesn't have an angular speed,and the belief that an object doesn't have an angular speedif the position of the observer is not enclosed by the pathof the object.Besides characterizing the ways in which angular speed isconceptualized by students from different frames of refer-65ence or observer positions it was also of interest to char-acterize the ways in which students make comparisons betweenthe angular speeds of two objects. This is the focus of thenext research question.66Research Ouestion 12 What is the nature of students' conceptualizationswhen making nomparisons of the angular speed of two ob-jects moving in semicircular paths of differing radii ofcurvature?4.40 Introduction to research question 12 The purpose of this question was twofold. Firstly, tofind out how students compare the angular speed of twoballs, each of which is made to roll down an inclined semi-circular track from the perspective of an observer locatedat the "center" of this (semi) circular path. And secondly,to compare the findings of this task with the work of Trow-bridge's (1980) study on the ways students compare the speedof two balls rolling down inclined planes.4.41 Overview of findings of research question 12Most students were able to effect a comparison of the an-gular speed of the two balls rolling down inclined semicir-cular tracks. Students frequently stated that at some time,or position, or over some part of the track, the two ballsdid have the same angular speed. From their responses twomajor ways of comparing the angular speeds of objects movingin semicircular paths were categorized. These categories tobe discussed respectively are entitled: Arc Length Equiva-lence and Position Equivalence beliefs.67Category title Arc Length EquivalenceCategory descriptionTwo objects (balls) are believed to have the same angularspeed if they travel in the same arc in the same amount oftime over some part of their motion.Category examples Example #1: Arc Length EquivalenceI: Do the two balls have the same angular speed?K: I think at point C they have the same angular speed...sorry, they meet at point C and during the timebetween point C where they meet first and point E -(student meant point G and corrects himself) - theyactually meet again at point G when they drop off..Between point C and point G they cover the same arcin the same time so they have the same angular speed(emphasis added)^ [120-6]Another physics 120 student quoted below (identified bythe same letter "K" but not the same student [#6] as above)replied in a similar manner.Example #2: Arc Length EquivalenceI: Do the two balls ever have the same angular speed?K: Yeah, from the formula they should.I: The formula?K: D-theta: dt, from that point of view..the same arc-length is cut out in the same time.[120-7]While the arc length equivalence belief was used by somestudents in deciding whether or not the two balls had thesame angular speed another form of comparison, positionequivalence, was also used.68Category title Position EquivalenceCategory descriptionTwo objects are believed to have the same angular speedif they have the same position at the same time on some partof their paths.Category examples Example #1: Position EquivalenceI: Do the balls ever have the same angular speed? Takeyour time--if it is not clear as to whether at anypoint or at any time they have the same angular speed--tell me.J: I think the angular speed would have to be the same,the angular speed is the same.I: For?J: For both of themI: Over what period, over the whole track, over part ofthe track? I'm just trying to clarify.J: Over the whole track.I: Because?J: It's mainly because they both end up at the same-theyboth drop off the track at the same time.The student is asked later how he would explain to some-one else that the angular speeds of the two balls were thesame in order to clarify the student's earlier remark.I: How would you explain to someone that the angularspeed is the same, because--? What was it abouttheir motion..?J: Its mainly because they end up at the same time. Theouter one's velocity was larger but they did end upat the same time.^ [115-2]69Another student (example #2) refers to the place wherethe balls are close as being the position where they havethe same angular speed.Example 12: Position EquivalenceI: Do the two balls ever have the same angular speed?Y: They doI: Over what part of the motion?Y: Near the end.I: Why?Y: They are close to each other.^[120-14]Both statements by this and the previous student wouldindicate that since the two balls arrived at the end of thetrack (the same position) at the same time then they musthave the same angular speed. The student in the followingexample further illustrates this belief.Example #3: Position EquivalenceI: Do the two balls ever have the same angular speed?M: Yes. .from the point of the observer the balls havethe same angular speed when they are in line witheach other with respect to the observer.I: And that would be at? Where does that happen?M: That happens right at the end at point (interviewerinterrupts and says point G to confirm this is thepoint being mentioned by the student) point G.I: They have the same angular speed because they arein a line?M: Because they are in line.I: Any other points?M: It looks like (looking at the video replay)- it looks70like as if, around C. They're in line at C and G.[115-15]Again, this student like the previous students believedthat the two balls have the same angular speed if the ballshave the same position (are in "line") even if this is justfor an instant in time. In the discussion to follow it isclear that the "position equivalence" belief is not justisolated to the comparison of angular speeds.4.42 Discussion of findings of research question #2 The analysis of the data revealed two forms of conceptu-alizations used to effect comparisons between the angularspeed of two objects moving in semicircular paths.One form of conceptualization, Arc Length Equivalenceis characterized as follows: two objects have the same angu-lar speed if they move in the same arc over a certain timeperiod.The other form of conceptualization, Position Equiva-lence is characterized as follows: two objects have thesame angular speed if they have the same position at thesame time. In Trowbridge's (1980) study in which two ballswere made to roll down inclined planes at slightly differingaccelerations students were asked if the two balls ever hadthe same speed. To this question a substantial number ofstudents responded that they did at the "instant of pass-ing". To quote Trowbridge:..failure on the speed comparison tasks was almostinvariably due to improper use of a position criterionto determine relative velocity. (p1027)71This observation parallels the one in this study in whichsome students were found to use a form of position equiva-lence in making comparisons, not between the speed of theobjects (as in Trowbridge's study) but between the angularspeed of the objects.Trowbridge also observed that in another task where theballs do not pass each other but at some point have the samespeed that:..logical arguments did not necessarily influence stud-ent responses. Even after satisfactorily describing thespeed of ball B as decreasing to zero and the speed ofball C as increasing from zero, some students observingthe demonstration would still claim that the speeds werenever the same since the balls never passed each other.(p 1024)While no such logical arguments were presented by theinterviewer to any student in this study, Trowbridge's ob-servation suggests that some students are strongly committedto a belief of "position equivalence".72Research Ouestion 13What is the nature of students' reasoning as theyaddress several task problems involving the conceptof angular speed?4.50 Introduction to research question 13 Previous analysis of the content of student responses asdescribed in research question #1 has led to categories ofdescription. The bringing of a different type of analysisto research question #3 (as described earlier in chapterthree) has led to categories of reasoning being used tocharacterize student thinking.Research question three was designed to investigate theways in which students reason or utilize various strategiesin thinking about the concept of angular speed. These rea-soning strategies which collectively are called categoriesof reasoning, are individually described by the concept ofassociation. Association (as used in this study) representsa simple mental link made by the student with words, sym-bols, events or experiences to some aspect of the task situ-ation. The concept of association is not meant to describeall aspects of student thinking, nor is any one particulartype of association (to be discussed) assumed to be repre-sentative of the reasoning of any one student.4.511 Overview of findings of research question 13 The categories of reasoning that were constructed tocharacterize students' thinking about an object's angular73speed were described by three forms of associations. Eachform of association represents either a weak or a stronglink in the student's memory to: the words angular speed,the symbol for angular speed, or to the context of the taskenvironment. These three forms of associations have beencategorized as: word associations, symbolic associations,and contextual associations, and are discussed respec-tively.4.511 Word associationA word association is characterized by the words "angularspeed" being associated with the word angular or speed. Astudent in responding to a question about the angular speedof an object might think for example in terms of the speedof the object (instead of its angular speed). This form ofreasoning is described as an association and is labelled forreference first by the type of association, "word associa-tion", followed by the words which are being associated,angular speed with speed to give the label: "word associa-tion::angular speed:speed". This particular form ofassociation, is illustrated next.Subcategory title Word Association: :angular speed:speeduhcategory descriptionBecause the word speed is associated with the termangular speed, a question about the angular speed of anobject moving in a curvilinear path is interpreted by a74student as if the speed of the object is being asked aboutrather than its angular speed.Example #1 Word Association::angular speed:speed(oval path)I: Ok, is the angular speed a constant?(4 train is moving around the oval track)J: It would be - wait (pause) - yeah.I: Because?J: Because it is going around at the same speed. [216-4]Example #2: Word Association: :angular speed:speed(oval path)I: Is the angular speed between A and C the same asbetween B and D?P: Uhm, well, its speed would be because assuming uni-form velocity of the train eh, its velocity would bedifferent because its going in a different directionof course. Velocity is continuously changing whenyou're going around the corner, its acceleration--ithas acceleration right, but its speed, its angularspeed would be the same, I would assume. [120-5]Example #3: Word Association::angular speed:speed(irregular path)I: Does it make any sense to talk of angular speed?L: I don't think so. Because I think about justmagnitude. So it is a constant speed (referringto the object moving in an irregular path).Since the student referenced the magnitude as influencinghis thinking it was decided to investigate this student'sresponse to a question about the angular velocity of theobject.I: Suppose I had talked about angular velocity?L: Yeah, I'd say it has an angular velocity.75I: Because?L: Uh, it is moving through these different anglesand it varies. Right?^ [120-11]For this student, different angles like the differentdirection of the velocity vector are associated with achange in angular velocity.Not only did the word speed influence the way somestudents thought about angular speed but so did the wordangular.Subcategory title Word Association: :angular speed:angularSubcategory descriptionThe word angular is associated with the term angularspeed. The word angular suggests to the student that anobject must move in a curved path in order to have an an-gular speed.Subcategory examples Example #1: Word Association:: angular speed: angular(irregular path)A physics 120 student when thinking about angular speedwas reminded of an angular momentum problem in her textbook.While the word angular momentum was mentioned it is clearfrom the dialogue that follows that it is the word angularand her interpretation of this word that guided her thinkingabout the angular speed (or angular velocity as used by thestudent) of an object moving in an irregular path.H: I know that anything can have angular momentum dependingon the reference point..somebody explained (it) to me-I76was having some trouble understanding the idea of theangular momentum-cause a little diagram in the book (adiagram in her physics text book)..there was an object....I couldn't understand how something that just happenedto be moving along (in a straight line) had an angularvelocity when it wasn't actually rotating or going roundin a circle.^ [120-9]This comment by the student revealed a barrier to the in-terpretation of angular speed as a consequence of the asso-ciation of the word angular to the student's conception ofwhat the term angular means to her, namely something that ismoving in a path that is not straight. Two other examplesof this form of association follow.Example #2: Word Association::angular speed:angular(oval path and irregular path)I: Ok, does it (the train) have an angular speed betweenpoints A and B? (a straight section of track)P: No.I: Because?P: Well, because it is moving in a straight line. It hasgot a linear velocity.Later, when shown an object moving in an irregular paththe student response was:P: Well, uhm-all things aside, if all I could observe isthis ball as it moves along this path, I could saywith certainty that it has an angular speed becauseany time it is moving in a curved path in which it ischanging direction it has an angular speed.^[120-5]Example #3: Word Association:: angular speed: angularsemicircular path)I: Let me talk about speed and angular speed, howwould you compare the two?J: Ok, well when I think of speed I think of it more asin a linear term, like straight forward in a line, andangular speed would be like a wheel turning. [216-14]77Not only do words themselves influence student thinkingbut so do symbols. This influence is characterized as asymbolic association, by which students link the symbol forangular speed, the Greek letter omega, Co to formulas thathave the term, w in them.4.512 Symbolic associationIn general the use of formulas by students in solvingvarious task problems may not be surprising (see for eg.Reif, 1986 and Trowbridge, 1980). But it is not the use offormulas per se that is of interest but which formulas (evenif incorrect) are used while others are not.The use of formulas by these students as they think abouttheir responses to questioning about the angular speed of anobject is indicated by the following brief quotes.H: I am trying to think in terms of formulas [120-9]J: I was thinking about the formula [115-2]G: I am trying to think in terms of an equation [216-3]The most commonly cited formula (or equation) that astudent referenced was w = v/r for uniform circular motion(v vtangential) which along with other forms of symbolicassociation are illustrated by the examples that follow2.2. The use of a particular formula by a student for w isstated as reflecting what the student said, even though theformula may not be appropriate.The reader should note that in the special font beingused,w is a scalar, andw is a vector quantity.78Subcategory title Symbolic Association: :FormulasSubcategory descriptionThe angular speed, w is interpreted via a symbolicassociation with a formula(e) in which the symbol w is aterm.Subcategory examples In the example to follow student "J" (quoted above) madeseveral references to the use of formulas. One such refer-ence is illustrated.Example #1: Symbolic Association:: Formulas: w = v/r(elliptical path)J: I was thinking in terms of the formula.I: The formula being?J: Velocity equals r times w. So, w would be v over rso if the radius was greater the angular speed wouldactually be less. [115-2]Example #2: Symbolic Association: Formulas: w = v/r(elliptical path)I: Is the angular speed (of the ball) a constant?P: Eh, no it's notI: Why?P: Because its radius isn't a constant. It also changes.I: Where is its (the ball's) maximum and minimum (angular speed) then? It must have a maximum and a mini-mum, if not a constant.P: Yeah, uhm I'll just start getting the-eh-formulahere. Its angular speed will say omega because thatis a convention, is equal to, proportional to v oris equal to v over its radius, so its nice to have aconstant speed.^ [120-5]79Of those students that used the formula, w = v/r few in-dicated an awareness that "v" in the formula is the tangen-tial component of the velocity of an object moving in acurvilinear path (see the kinematics of curvilinear motionin the footnote below 3). The awareness of this fact is rep-resented as aspect #1 of symbolic association and is illus-trated below.Subcategory title Symbolic Association::Formulas:w = v/r:aspect #1Subcategory descriptionA distinction is made between the velocity, v and itscomponents: v tangential and vradial when using the formulaw = v/r.Suhcategory examples Example #1: Symbolic Association:Formulas:w = v/r:aspect #1(oval path)I: Does the train have an angular speed betweenpoints A and B? (a straight section of track)D: From here?I: Where you are sitting.D: Well, what angular speed is--is the velocity per-pendicular to a radius from where I am standingand from where I am standing here this is myradius, and there is no velocity perpendicular toit.3. For an object moving in a curvilinear path the velocityvector can be broken up into two vector components. Onecomponent, the tangential component, has magnitude vt. Theother component, the radial component, has magnitude vr.For uniform circular motion the speed, v is equal to themagnitude of the tangential component of the velocity sincethe radial component is zero. Under such a circumstance theangular speed, w may be written as v/r rather than vt/r.80I: flow about between D and C? (a curved section oftrack)D: Yeah, because from where I am standing it does.It is going this way which is partially a velocitythis way and partially a velocity that way andthis times the radius here is its angular speed.I: And how about between C and A? Same type ofreason.D: Sure, yeah.^ [120-13]This student was aware of the need to take components ofthe velocity and not the velocity of the object into accountwhen computing the angular speed. This form of reasoningwas also illustrated when this student attempted the ellipsetask:I: Is the angular speed a constant?D: (long pause)-huh-emI: Well you can tell me some of the considerationsthat are going through your mind.D: Some of the considerations that are going throughmy mind: when you are at point A here its all itsvelocity is perpendicular and it is very close-uhm-when you are over here it is taking a sharperturn (pause)-did you ask me if it was constant?I: (nods head)D: No, it's probably not constant too many thingschanging, the radius changes, the angle of thevelocity compared to the radius changes becauseit is not a circle it's not perpendicular all thetime.Another student (in the example to follow) also made aneffort to use velocity components, in order to compute theangular speed of the object moving in an elliptical path.81Example #2: Symbolic Association:Formulas:w = v/r:aspect #1(elliptical path)M: And going back on that ellipse when you asked how Iwould measure the angular speed.I: Ok, let's go back.R: At B the angular speed would have to be the perpendicu-lar component of velocity divided by the radius, so notjust the speed divided by the radius. [120-10]The use of the term "perpendicular component" by the stu-dent is interpreted as being v tangential. Again an aware-ness by the student of the need to distinguish between var-ious components of the velocity and the velocity itself isevidenced in this response. However, some students failedto make a distinction between the velocity and its compo-nents. This failure to make such a distinction is labeledas aspect #2 of symbolic association and is illustrated bythe following examples.Subcategory title Symbolic Association::Formulas:w = v/r:aspect #2Subcategory descriptionNo distinction is made between the manitude of thevelocity, v and the magnitude of the object' s vradial orvtangential components when using the formula Co = v/r.Subcategory examples Example #1: Symbolic Association:Formulas:w = v/r:aspect #2(elliptical path)I: ..can we talk about angular speed (of an object goingaround in an ellipse)?K: Yeah.82I: And why?K: From definition of v = w times r so w is v/r, andthere is change..radius is changing, the angular speedis slower at E and faster at A (corresponding respec-tively to points labeled on the elliptical path where ris at its maximum and minimum).I: Why faster at A?K: Because r is smaller, Co is faster.^[120-7]This response by the student suggests that the speed ofthe object is equated with its velocity. No distinction ismade by this student between v radial or vtangential whencomputing the angular speed of the object via the formula w= v/r. The following example of a student quoted earlieralso illustrates this lack of distinction.Example 12: Symbolic Association:Formulas:m = v/r:aspect #2(elliptical path)P: I'll just start getting the-eh-formula here. Its an-gular speed will say omega because that is a conventionis equal to, proportional to v, or is equal to v overits radius, so it's nice to have a constant speed. Theradius however changes continuously, so at the pointwhere it is furthest away from the center, where itsradius is the greatest, its--I: E to E'?P: Yeah, from observer to E or E'. Its semimajor axis,uhm, it will have its smallest angular velocity becauseagain its inversely, angular velocity is inversely pro-proportional to the distance, the radius.[120-5]Again no distinction is made by the student between thevelocity, v and its components. Sometimes a student wouldassociated the angular speed, co with a formula for theangular momentum or energy of a rotating body (since bothformulas have the symbol, w). This form of associationwhereby another formula other than w = v/r was used by the83student is considered to be a subcategory of symbolicassociation and is referenced as: "other" formula(s).Subcategory title Symbolic Association:: Formulas: "other"Subcategory descriptionThe use of a formula other than w = v/r to "measure" theangular speed of an object.Example 11: Symbolic Association::Formulas: "other"(elliptical path)I: So you are not so clear how you might calculate theangular speed of an object moving in an ellipse.K: I should be though, because that was what our lastmidterm was about...normally I use angular momentumand energy considerations.I: To calculate?K: Eh, to calculate omega (the angular speed).[120-7]Example 12: Symbolic Association::Formulas: "other"(elliptical path)I: flow would you measure the angular speed at point C..?S: (a long pause)I: What are some thoughts going through your mind?S: .. energy conservation and the fact that "I", theinertial moment would be changing over here..linearvelocity would be constant so "I" times w would beconstant, so if calculate change in "I", (a pause) mr2that would probably give you the change in w, I think.[120-8]Not only did the association of angular speed with formu-las influence student thinking but so did the context of thetask also influence student responses. This influence is84characterized as a contextual association and is discussednext.4.513 Contextual associationIn this form of association some aspect of the contextprovides a cue by which the student is influenced in his orher thinking about the angular speed of an object. This cueis usually interpreted in the form of some analogy as theexamples to follow illustrate.Subcategory title Contextual Association: analogySubcategory description,The context of the task setting cues an analogy for thestudent. Students were found to use a variety of analogies.The analogies that were used by students for analyzing theangular speed of an object were: planetary motion, circularmotion, simple harmonic motion, and the coriolis force, eachof which will be respectively discussed.The analogy of elliptical motion to planetary motion forone student evoked the use of Kepler's laws. This student(quoted in the next example) tried to use Kepler's, "equalarea law" for deciding where the maximum angular speed of anobject moving in an elliptical path would occur.Subcategory examples Example 11:Contextual Association::analogy:planetary motion(elliptical path)L: I am just using one of Kepler's laws.85I: Which is?L: Which is in a certain time interval the area that theobject sweeps out-the sector (?) area-the area from thecenter is equal at all time intervals...[120-11]Some other students who made an analogy to planetary mo-tion found the recasting of elliptical motion into one ofplanetary motion perplexing. Because, the object in the"ellipse task" was moving around at a constant speed unlikethat for planetary motion.Example #2:Contextual Association::analogy:planetary motion(elliptical path)M: You made those restrictions (referring to the constantspeed of the object)..it shoots out what I know about-about-eh-space, elliptical orbits, foci, speeds andeverything. Now you are saying the speed is a con-stant and now you're telling me I am in the middle.So, fine, so--eh.I: Ch, because you were thinking of planetary motion?M: Right.^ [115-15]Another student during the interview also had difficultyin evaluating the use of his analogy of comparing ellipticalmotion to circular motion.Example #3:Contextual Association::analogy:circular motion(elliptical path)I: Is the angular speed (of the object moving in theelliptical path) a constant?K: No, because the radius varies in each case.(A few moments later the interviewer returns to thequestion of how to measure the angular speed at acertain point on the ellipse.)I: If you have to measure the angular speed at point C.K: Need location of foci.86I: One here and here.K: The instantaneous angular speed?I: Yes, what are some of your thoughts?K: I am trying to think of it in terms of a circle asin sweeping out an area. .but the strange thing is theradius varies. Ellipses are not really my speciality.[120-6]For an object moving in an irregular path some studentsalso made use of the analogy to circular motion. These stu-dents when asked how to measure the angular speed (havingpreviously acknowledged that it made sense to do so) for anobject moving in an irregular path proceeded to approximatethe radius of curvature at various points on the object'spath by drawing circles of varying radii (a diagram of thisis illustrated in the example to follow). These attemptsare interpreted as being an effort by the student to find avalue for the radius, r in the formula, co = v/r. Themagnitude of the radius vector that was used was not (as itshould have been) the distance from the observer to the ob-ject but rather the radius of curvature at some point on thepath where the object was located.Example #1:Contextual Association::analogy: circular motion(irregular path)L: Yeah, I'd say it has an angular velocity.I: Because?L: Because, uhm it is moving through these differentangles and well they vary right?I: And what does that make change?87<---irregularpathr: indicates the r, the stu-dent was trying to measure.L: It changes the radius.I: What radius?L: The centre of motion.I: Ok, so is the radius at this position different thanthe radius at another position?L: Yeah, I think you have to--(interviewer puts down atransparent overlay on the object's path with variouspoints marked for reference)I: (interviewer having interjected)--you can give me anexample.L: Ok, for example if it is moving through (the point) C,the radius will probably be somewhere around here.I: So you would construct what?L: You kind of construct--can construct-(interviewerinterjects)I: You can draw on that (the overlay).Student then proceeds to draw a circle as shownbelow.(The circle drawn by the student was either within oroutside the irregular path depending on the sign of theradius of curvature of the irregular path at a particularpoint.)I: So at what points will its angular velocity begreater?L: It will be greater at the sharp ones, because theradius (of the circle) will be shortest.[120-11]88The student went on to indicate that point "F" (see dia-gram of irregular path) has a minimum angular velocity be-cause the curve is almost straight here (and the radius ofthe circle would therefore be very large).Example #2:Contextual Association::analogy: circular motion(irregular path)P: Well, again linear speed, you would try and find someway to look at its radius of movement and that's notan easy thing to do. These are completely irregularcurves but basically at a given instant you can find acircle which approximates its curve.I: Ok, let's take a point C or D or E.P: Well, at point C you can look at a very small circlehere.I: Which is inside the path (interview describing in wordswhat the student has indicated by hand on the overlay).P: Which is inside the path, right. Because it is a smallcircle it has a very small radius of curving. Again,because angular speed is inversely proportional to theradius and whenever the radius is the smallest that'swhere your going to have your greatest angular speed.[120-5]Example #3:Contextual Association::analogy:circular motion(irregular path)Q: I think we can say that at any point the ball has aninstantaneous angular speed that-uhm-would be-we'dhave to extrapolate a circle from a small arc.I: Suppose you want to measure the angular speed of thatball. How would you go about that?Q: We'd have to extrapolate. Take the arc there and putit into a circle, and.. (interviewer interjects)I: And what circle would that be? Could you just show?Q: In this direction.I: Ok, in that direction (student indicates an arc on theoutside of the irregular path)89When the student was asked to indicate where the maximumand minimum angular speed would be the student response was:Q: The highest angular speed would be achieved at thepoints that have very small radius of this virtualcircle that I've being talking about. [120-12]Besides the analogy to circular motion sometimes ananalogy would be made to simple harmonic motion.Example #4:Contextual Association:analogy:simple harmonic motion(elliptical path)P: ..If I know that, if I know it's an ellipse and I knowit's going to hit its lowest velocity at the two semi-major axises. And when it does that it will appear tokind of slow down and speed up and slow down again.It's like simple harmonic motion again. So I cansimply time it (the object) takes for it to go fromone extreme to the other.^(emphasis added) [120-5]While simple harmonic motion (SHM) along a straight lineis related to uniform circular motion it is inappropriate tolink SHM to elliptical motion.Finally, another analogy which was cued by the context ofthe task setting was to the coriolis force. In the exampleillustrated below a student was shown a diagram of an objectmoving in a circle after having experienced difficulty inresponding to a question about an object's angular speedduring the "elliptical" path task.Example #5:Contextual Association:analogy: coriolis force("elliptical" path)I: Let me go back to something that might be a bit morefamiliar. (Interviewer shows the student a diagram ofa ball moving around a circular path)90H: Oh, yes that is lovely.I: Here is a circle, and you're in the middle. Does it (theball) have an angular speed?H: Yes.(The same question was later repeated for an observerstanding outside the circle)I: If that observer stands outside (the circle) can thatobserver still talk about that object having an angularspeed?H: Well--I: Or (to reword the question) does he (the observer) haveto be inside (the circle) to talk about it (the angularspeed), or can he be outside "looking in".H: I think-(pause)-, ok when you asked me that (question)the first thing that came into my head was the corioliseffect. Because, uhm, when something is moving on theearth that is rotating around it--it appears to have anextra component to the way it moves...I: When did you take coriolis? just recently, or--?H: A few weeks ago.I:...and this reminds you of coriolis?H: I am just thinking of coriolis because the frame of ref-erence is changed, like the frame from which you arelooking at is changed.(emphasis added)^[120-9]The interpretation of this student response is that thechanging of the position of the observer from one frame ofreference to another acted as a cue. This in turn remindedthe student of something discussed a few weeks ago in aphysics class about (rotating) reference frames (as theearth) and the coriolis "effect".91The different types of contextual association and otherforms of student's reasoning so far discussed are now summa-rized in the following review of the research findings.4.52 Discussion of findings of research question 13 The concept of association was found to best characterizethe categories of student reasoning as judged by student re-sponses to questioning about the angular speed of an objectmoving in different paths. The associations students madewere described as: word, symbolic, and contextual; associ-ation, each of which will be reviewed respectively.A word association is an association of the word angularspeed with other words. These "other" words were found tobe: angular and speed. A student for example in respondingto a question about the angular speed of an object moving ina curvilinear path might respond as if the question askedwas about the speed of the object and not its angular speed.Likewise, a student might respond by associating the wordangular with non linear motion. Consequently an object mov-ing in a straight line is not seen as having an angularspeed (irrespective of the observer's position) by the stu-dent. Even though the words speed and angular influencedstudent reasoning it is in principle possible that otherword associations may have influenced student reasoning.Not only did word associations influence student's thinkingabout angular speed but so did the symbolic association ofw, the angular speed with formulae having this symbol.9293The most common formula that students explicitly men-tioned was co v/r for determining an object's angularspeed. A common error in the use of this formula was thefailure to distinguish between the speed, v and the magni-tude of the velocity components (vradial and v tangential) ofan object's motion. This error was found among students atall levels, physics 115, 120, and 216. And while such er-rors are not the central interest of this research questionit is nonetheless of interest from a instructional point ofview.Momentum or energy formulae having the symbol, a) as a termwere also recalled by some students. Such formulae, wereless commonly referred to, perhaps because no information orcue was given about the mass, moment of inertia or energy ofthe object.Finally, the third form of association that was identi-fied was contextual association. In this form of associa-tion some aspect of the context of the task setting cues amemory of an event, some piece of knowledge, or experiencethe student has had. The context may for example remind thestudent of a picture in a physics text book, a physics lec-ture, or a personal experience. And (as noted in the liter-ature review) Erickson and Aguirre (1987) have observed thatthe "..task context appears to play an extremely importantrole in terms of the ways in which a student both frames andresponds to a problem" (p17).Analogies, a form of contextual association were found tobe made to circular motion, planetary motion, simple har-monic motion and the coriolis force. While some studentsrecognized the appropriateness of some of their analogiesothers did not. The failure by some students to evaluatethe appropriateness of an analogy has also been observed byClement (1987) in his work on analogical reasoning in thefield of mechanics.In summary these three associations word, symbolic orcontextual serve to describe the form of reasoning "strat-egies" students are thought to engage in, at an appropriatelevel of detail (for pedagogical purposes). While it mayappear that such associations represent a description of"surface" level thinking it is important to emphasize thatsuch forms of thinking as contextual association can be veryeffective. A great deal of what experts are able to do saysSimon (1985) is because of an ability to recognize cues infamiliar situations. However, experts unlike novices usu-ally are able to correctly evaluate the appropriateness oftheir initial response(s) to such cues.The diagram on the next page summarizes the relationshipamongst the forms of association which have been discussed.94Student Reasoningof angular speedbyASSOCIATIONWORDspeed^angularSYMBOLICformulaeCONTEXTUAL events/episodesanalogiesFigure 8: Reasoning paths for angular speedThe question as to whether student's reasoning about theconcept of angular acceleration is similar to that for an-gular speed is the subject of the next research question tofollow.95Research Ouestion 44 What is the nature of students' reasoning as theyaddress several task problems involving the conceptof angular acceleration?4.60 Introduction to research question 44 Reif (1987) investigated the reasoning processes of ex-perts and novices as they interpreted the concept of accel-eration in various task settings. He described the knowl-edge of being able to find the acceleration of a particlemoving in an arbitrary path as being procedural knowledge.The procedural knowledge necessary to find the accelerationof a particle is represented as five steps:1) Identify the velocity v of the particle at the time ofinterest.2) Identify the velocity v' of this particle at a slight-ly later time t'.3) Use vector subtraction to find the velocity change, Av= v' - v of the particle during the short time inter-val,At = t' - t.4) Divide Av by At to find the ratio Av/At.5) Imagine that the time t' is chosen sufficiently closeto the time t so that the time interval At becomesinfinitesimally small. Find the limiting value dv/dtof the ratio Av/At and call it the "acceleration a ofthe particle at the time t".Reif observed that student novices in their reasoningabout kinematic quantities as acceleration, rarely use anyform of procedural knowledge, instead relying heavily onvarious pieces of knowledge. These knowledge pieces statedas propositions are for example: if the speed of a particle96changes then its acceleration cannot be zero; if a particlemoves in a circle its acceleration is directed towards thecenter (only true if the particle is travelling at a con-stant speed), and if a particle has a velocity that is zeroor a constant then its acceleration must be zero (incor-rect). It was from observations such as these that Riefconcluded that student knowledge was fragmented, a descrip-tion echoed by DiSessa (1986) who characterized the intu-itive physical dynamical knowledge of students novices as"knowledge in pieces".Reif's investigation of student understanding of theconcept of acceleration is extended in this study to thedomain of angular acceleration. Before proceeding with anoverview of the findings to research question #4 it will beuseful to review some elements of the concept of angularacceleration.Angular acceleration is symbolized by the Greek letteralpha,a and is for the purpose of this study treated as ascalar (non-vector) quantity4, a. Angular acceleration isdefined as the limiting change in the vector, angular speed,co with respect to time, t which expressed symbolically isdo),/dt. Again, for the purpose of this study only themagnitude, Idwidt1 or a is of interest.4. Physicists use the name speed for the magnitude of thevector quantity velocity. Unfortunately, there is no spe-cial name for the magnitude of the vector quantity accelera-tion or angular acceleration.974.61 Overview of findings of research question 14 The reasoning students used in answering questions aboutthe angular acceleration of an object moving in a variety ofpaths is again characterized by the concept of association.As before, three forms of association were found: word,symbolic and contextual, each of which will be describedrespectively.Word association is a simple or a complex association ofwords. The words associated by students with the words an-gular acceleration were inferred from their responses. Forexample, a student might state that an object moving in anoval path at a constant speed does not have an angular ac-celeration because it is never increasing its speed. Whilethe student never used the word acceleration (by itself), itwas nevertheless assumed that a mental reference to thisword was made by the student, as such a reference provides acoherent explanation of what was said.A student may go through a series of mental steps in ar-riving at a response to a question about the angular accel-eration of an object. These "steps of reason" may at timesbe represented by a sequence of inferred associations. Sucha sequential order of steps is indicated by an arrow, "-->"and the non verbalized association is placed in brackets,"( )" to give for example:angular acceleration->(acceleration)-> change in speedwhich symbolically can be rewritten as:a ---> ( a )^As98Again, this sequence is meant to represent a hypothesizedordering of mental steps characterized by associations, ofthe student's reasoning process. Using this approach theanalysis of other response(s) to the interviewer's questionswhen appropriate to do is summarized as a sequence of word,symbolic, and contextual associations. As before, symbolicassociation manifests itself in the use of formulae, whilecontextual association may manifest itself in the use ofanalogy.The following illustrates possible sequential associa-tions of word, symbolic and contextual associations.WORD associations:1. a ---> ( a ) ^> As2. a ---> ( a ) or a ^> AvSYMBOLIC associations:3. a ---> formulaCONTEXTUAL associations:4. a ---> analogySometimes a multiple of these sequences are inferred tohave occurred during a student's response to a question.Beginning with word associations the following sequencesof word associations and the detailed findings and support-ing evidence for each sequence follows.99Word AssociationsSubcategory title Word Association: angular acceleration->acceleration---->A speedSubcategory descriptionThe term angular acceleration is associated with theword acceleration. In turn the word acceleration isassociated with a change in the speed of the object.Subcategory examples In the examples to follow (unless otherwise indicated) theobject (a toy train ) moving in an oval path.Example #1:Word Association: angular acceleration->(acceleration)--->A speedI: Does the train ever have an angular acceleration?K: (pause) No, because during the track you are neverincreasing the speed. [120-6]Example #2:Word Association:angular acceleration->(acceleration)--->A speedI: Does the train have an angular acceleration?A: No, because the speed doesn't change.I: Angular speed?A: The angular speed doesn't change. [120 - 12]Example #3:Word Association:angular acceleration->(acceleration)--->A speedI: Does the train ever have an angular acceleration?G: (after a false start)..it is clear to me that itdoesn't have an angular acceleration. And why?100It is just not-it is not like increasing-goingaround faster and faster. [216-3]For some students a change in angular speed was associatedwith a change in angular acceleration.Example #4:Word Association:angular acceleration->acceleration-speed)-->6. angular speedI: Does the train have an angular acceleration?D: Sure.I: Where?D: Uh, most places except right along here (section ABof track), it is kind of obvious because its angularspeed is zero here and it has an angular speed hereso it must have accelerated somewhere. [120-13]Example #5:Word Association: angular acceleration->acceleration--->(i speed)-->A angular speedI: Does the train have an angular acceleration?F: Acceleration is the change in velocity and angularspeed. So, here it has angular speed around thecurve, but on the straight track there is no angularspeed. I would say the angular speed is changingobviously.^ [115-8]Acceleration was also associated with a change in veloc-ity, rather than a change in speed and is represented as asubcategory of word association.Subcategory title Word Association: angular acceleration->acceleration--> A velocitySubcategory descriptionThe term angular acceleration is associated with the wordacceleration. In turn the word acceleration is associatedwith a change in the velocity of the object, however it was101usually a change in the direction of the velocity vectorthat led some students to conclude that an object had an ac-celeration and thus an angular acceleration.Subcategory examples Example 11:Word Association: angular acceleration->(acceleration)--->A velocity-change in directionI: ..does that train have an angular acceleration?K: Yeah, because it is a vector. .magnitude and direction.There is a change in direction. [120-7]Example #2:Word Association: angular acceleration->(acceleration)--->A velocity-change in directionI: Now, does the train ever have an angular acceleration?J: It would have an angular acceleration because thevelocity vector is changing as it is going around frompoint B to D, and C to A.^ [216-4]Sometimes a student would recall a formula in helping todecide if an object had an angular acceleration. The use ofa formula as described earlier is a form of symbolic asso-ciation.Symbolic associationSubcategory title Symbolic Association: :angular acceleration:formulaSubcategory descriptionThe angular acceleration, of an object is associated withits symbolic form, namely a formula (or formulae). The for-102mula used by a student may at times have been inappropri-ately used in the context given.iihcatpgory examples In the subcategory example, example (#1) to follow a stu-dent cites a formula for the angular acceleration of an ob-ject moving uniformly in a circular path of radius r, namely,a = a/r (only true if it is understood that a = atangentialin the context given). The subsequent example (#2) a stu-dent cites (an incorrect) formula a = w/r to find the angu-lar acceleration of an object.Example #1: Symbolic Association:Formula-> a = a/r(semicircular path)I: How about angular acceleration which you were justabout to mention.K: Yeah, because alpha is a over r so if there is a changein acceleration there has to be a change--there has tobe a change in angular acceleration. There has to bean angular acceleration.^ [120-7]Example 12: Symbolic Association:Formula->a = w/r(semicircular path)I: Let's talk about angular acceleration. What thingscome to your mind when I talk about angular accel-eration?L: I think about the angular speed and the radius andformula.I: Which is?L: Angular acceleration equals co over r.[120-11]Another form of association that was identified was con-textual association. This form of association is charac-terized by the student relating an event, experience or some103piece of knowledge with some aspect of the context of thetask setting.Contextual associationSubcategory title Contextual Association: circular motionSubcategory description The task context cues some physical knowledge that anobject moving in a circular or curved path accelerates.Subcategory example Example #1: Contextual association:circular motion(oval path)[the dialogue below of student L, is the same student inthe foregoing example #2 but in a different task context]I: Does the train have an angular acceleration? Or, isit hard to tell?L: (a pause) uhm, it is hard to tell. I think it doeshave angular acceleration because it is moving in acircular motion depending on which points from C toA and.. (emphasis added).^ [120-11]Example #2: Contextual Association:circular motion(semicircular path)I: Does the ball have an angular acceleration down thetrack?P: Uh, yeah it does continuously because the track iscurved.^ [120-5]Another form of contextual association was an associationbetween acceleration a kinematic quantity and force, a dy-namical quantity, for an object moving in a circular likemotion, as the following examples illustrate.104Example #1:Contextual Association: :circular motion:force(semicircular path)I: Now, how about angular acceleration?M: Always has (angular acceleration), because it iskeeping within the course of the track in orderto do that it must have a force to allow the ballto stay in.^(emphasis added)^[115-15]Example #2:Contextual association: :circular motion:force(irregular path)I: Let's go back to the irregular path for a moment.If I were to ask you does the ball have an angularacceleration--you said it would be very difficultto judge, would that be correct? Does it make senseto talk about the ball having an angular speed?K: Well, it makes sense to talk about the ball havingan angular speed, but to talk about the ball havingan angular acceleration it implies a force beingacted upon it and there is nothing pushing the objectaround the "circle" at a constant speed.(emphasis added)^[120-6]Sometimes the context cued a personal analogy for thestudent. Such a form of contextual association was evi-denced in the response to a question about the angular ac-celeration of a train moving in an oval path.Example #3Contextual association:: (personal) analogy:forceM: I think the key factor for angular acceleration isthat the train--how the train feels...I: By the train feels, you would be referring to what?M: Uh.I: Do you think about yourself inside the train or--?M: Yeah, basically. If you are inside the train I wouldconsider that the same thing.I: And how does this feeling translate into--?105M: Let's say it wasn't a train, let's say it was anotherperson and I was to run around the track. I wouldhave to--if I wanted to run around at a constant speedI would always have to make my left foot-like-put moreforce on my left foot to turn right.Another student (cited earlier) made a specific referenceto the coriolis force.Example #4: Contextual association:coriolis force(semicircular path)I: Does that ball (rolling down the inclined path)have an angular acceleration?L: Yes.I: And because?L: Because, again its moving in circular motion and becauseof the coriolis force, eh the coriolis force is angularacceleration times radius, so-- (emphasis added)I: Therefore?L: Angular speed times radius (student "correcting" previousstatement) and therefore angular acceleration.Even though the student's formula for the coriolis forceis incorrect, it is the association (cued by the context ofthe task) to the coriolis force that is noteworthy.Some hint as to the reason for the student thinking aboutthe coriolis force (or acceleration) is given by the follow-ing dialogue.I: Do you want to explain that to me a little bit?L: Coriolis acceleration?I: Yeah.L: How I understand it, I understand it to be, eh-uhm,when an object is on a frame that's rotating.I: When an object is on a frame that's rotating(repeating student's statement), uh-ha.106L: For example, the ball on the earth, on the table andthe table is on the earth and well the frame would besay the earth, well initial frame--the bigger framethat's the earth and the earth is rotating, so thatcauses a coriolis force.The ball's motion on the "rotating' table was seen by thestudent to experience a coriolis force. This suggests thatthe student was reminded (by the context of the task set-ting) of a physics lecture in which rotating referenceframes were discussed. While this is an inappropriate anal-ogy, it does nevertheless represent an interesting exampleof contextual association.Finally, a student response may exhibit more than oneform of association as the following example illustrates.Example of a multiple association:Contextual association: circular motion-->(acceleration)Word association:(acceleration)-->ang4ar accelerationSymbolic association: formula-->a = 17'/r(oval path)I: Does it have an angular acceleration between C and A?(the semi-circular end of the oval track)L: I think it does.I: Because?L: Because it is moving in a circular motion and angularacceleration is v squared over r. So if that is radiusand that's velocity then I think it does have angularacceleration.^(120-11)While it is possible to represent the flow of reasoningdifferently than shown above, it is reasonable to character-ize such a flow as some sequence of associations.This and other characterizations of student reasoning aresummarized in the next section.1074.62 Discussion of findings of research question 14 Student responses to questioning about the angular accel-eration of an object have been described in terms of asso-ciations. Three forms of associations have been character-ized. These are: word, symbolic and contextual, each ofwhich will be discussed respectively.Word association is the association of particular wordswith the term angular acceleration. Angular accelerationwas associated with the words: acceleration, force, a changeof velocity, or a change in speed. A consequence of usingthis form of reasoning is that a student might for exampledecide that an object had an angular acceleration on thebasis of whether or not the object was accelerating or if aforce was acting upon it.Symbolic association is the association of the symbol a,angular acceleration with other symbols, namely formulae.Formulae that were recalled by students were: a = a/r, a =co/r, anda = v2/r. A consequence of this form of reasoningis that a student may decide that an object has an angularacceleration on the basis of a particular formula, eventhough the formula may be incorrect or incorrectly inter-preted.Finally, contextual association, is the association ofsome event, experience, or personal knowledge that is cuedby the context of the task setting. A student may reasonfor example on the basis of analogy to: planetary motion orthe coriolis force, the force exerted by one's foot, in108IWORDacceleration_Student Reasoningof angular accelerationbyASSOCIATIONSYMBOLICformulae1CONTEXTUALanalogyvelocityspeed^WORD^SYMBOLIC^CONTEXTUALI Iforce formulae analogydeciding whether or not an object has an angular accelera-tion. Clement (1987), in reference to personal analogies(as the force exerted by one's foot) noted that: "approx-imately half of the analogies (made) were personal analogiesreferring to some sort of body action" (p6).It should be remembered that neither contextual asso-ciation or any other form of association uniquely character-izes the reasoning process of any one student. A studentmay use several forms of reasoning even though on a particu-lar occasion only one form of reasoning may be exhibited.The possible pathways of student reasoning by associationare illustrated by the diagram below.109Figure 9: Reasoning paths for angular accelerationCHAPTER FIVECONCLUSIONS, IMPLICATIONS and RECOMMENDATIONS5.00 IntroductionThe conclusions of this study are presented in this chap-ter for all research questions. Each research question willbe presented in the order in which the questions were ana-lyzed in Chapter Four followed by a brief summary of thefindings of each question. After the findings for all re-search questions have been presented a discussion of thesefindings, instructional recommendations, and recommendationsfor future research will be given respectively.5.10 Conclusions of the StudyThe following is a short summary, without comment, of thefindings of each research question.IleaearrIL_Queation_11.What are students' conceptualizations of the angularspeed of an object from different frames of reference?The findings to research question #1 can be expressed interms of five categories of description which were con-structed to describe student conceptualizations of angularspeed from different frames of reference. These conceptual-izations were grouped in terms of whether or not the ob-ject's angular speed was independent of the object's path or110observer position. Accordingly, the summary of the findingsto research question one are:Position.1. The angular speed of an object is independent of theposition of the observer.2. An object may not have an angular speed if the po-sition of the observer is not enclosed by the pathof the object.3. The angular speed of an object should be measuredfrom a preferred position.Patti:4. An object moving in a rectilinear path doesn't havean angular speed or it doesn't make sense to talkabout the angular speed of such an object.5. An object moving in an irregular path doesn't havean angular speed or it doesn't make sense to talkabout the angular speed of such an object.The findings of research question one were based on taskswhich involved the angular speed of one object. The find-ings of the next research question were based on a taskwhich involved the angular speeds of two objects.Research Ouestion 12 What is the nature of students' conceptualizations asthey make comparisons of the angular speed of two objectsmoving in semicircular ,paths of differing radii of curva-ture?Students were characterized as using two different meth-ods in comparing the angular speeds of two balls, each onerolling down separate semicircular inclined planes. Thesemethods were: i) Arc Length Equivalence and ii) PositionEquivalence. The student using a method of Arc Length111Equivalence believed that the two balls had the same angularspeed if they traveled the same arc (angular) distance inthe same amount of time. While the student using a methodof Position Equivalence believed that two balls had the sameangular speed if they had the same (radial) position at thesame time.Later, the findings of this research question will becommented upon as it relates to the work of Trowbridge(1980) upon which this research question was developed.ReaearciLDileatian_iaWhat is the nature of students' reasoning as they addressseveral task problems involving the concept of angularspeed?The findings of research question three (based on a simi-lar set of tasks as described in research question one)characterizes student reasoning as a form of association.Three forms of association were constructed to describe stu-dent responses, these were: word association, symbolic as-sociation or contextual association.Word association is characterized by an association ofthe words angular speed with angular, speed or other words.For example, a student in responding to a question about anobject's angular speed might respond as if the questionasked was about the object's speed rather than its angularspeed. Consequently, a student might conclude that an ob-ject moving in a curvilinear path had a constant angularspeed if it had a constant speed.112Symbolic association is characterized by the associa-tion of the symbol, w ,with other symbols, which is ex-pressed as some formula containing the symbol w. Theseformulae as given by the students were either kinematic inform, Co = v/r or dynamic in form, as Iw equals someconstant. Sometimes a no distinction was made between v,vtangential and vradial of the object in the use of some ofthese formulae.Lastly, contextual association is characterized by theassociation of circular motion, planetary motion, simpleharmonic motion and the coriolis "effect", to some aspect ofthe context of the task setting.The fourth and last research question of this studysought to extend the investigation of students' reasoningabout angular speed to angular acceleration.Ileaearsch_aaleatian_14.What is the nature of students' reasoning as they addressseveral task problems involving the concept of angular ac-celeration?The categories of association which were developed as aresult of student responses to the above question werecharacterized in the same way as were the findings to re-search question three. These categories of association be-ing: word, symbolic and contextual.Word association in the context of the fourth researchquestion is the association of the words angular accelera-113tion with the words acceleration, force, and a change in,velocity or speed. For example, a student might concludethat an object moving in a curvilinear path had a constantangular acceleration if it had a constant acceleration basedon a form of word association between the term angular ac-celeration and the word acceleration.Symbolic association is the association of the symbol,a with other symbols, as formulae, which as given by thestudents were: a = air, a = w/r and a = v2/r. For example,a student might decide, based on the (incorrect) formulaa =v2 /r, if an object had or had not an angular acceleration.Finally, contextual association is the association ofphysical analogies as planetary motion or of personal analo-gies as the force one feels in one's foot in going around acurved track, to some aspect of the context of the task set-ting.5.20 Discussion of Findings The contributions of this study to the field of scienceeducation and the manner in which the findings of this studyare situated with respect to the work of others in the fieldwill be discussed with respect to each research question.The findings of research question one are a uniquecontribution to the field of science education, as no otherstudy has yet documented students' conceptualizations of an-gular speed. The findings of research question one have114also extended Saltiel and Malgrange's (1980) work of stu-dents' reasoning in the domain of elementary kinematics.Saltiel and Malgrange observed that students regardedvelocity as being loosely associated with reference frames,"Almost never do students think in terms of frames" (p79).This independence of the observer's position parallels oneof the conceptualizations characterized in research questionone, that the angular speed of an object is independent ofthe position of the observer. Such a conceptualizationmight be intrepreted as a belief that angular speed is anintrinsic property of the object. If this were so it wouldconcur with Trowbridge's interpretation of his findings thatstudents conceptualized velocity as being an intrinsic prop-erty of the object independent of the observer's motion orposition.Finally, a conceptualization as the angular speed of anobject should be measured from a preferred position, could(in part) be explained by Saltiel and Malgrange observationthat in everyday life a particular frame is very oftenhighly favoured, because for practical reasons it is oftenunnecessary to think explicitly about reference frames.The findings of research question two have also extendedTrowbridge's (1980) findings in his investigation of themethods used by students to compare the relative speeds oftwo objects. The finding that some students state that twoballs have the same angular speed if they have the same po-sition at the same time parallels Trowbridge's finding that115according to some students, two objects have the same veloc-ity when when they have the same position.The findings of research question three and four repre-sent a contribution to the field of study called situatedcognition. The words and images used in the context of atask setting (part of the situatedness of the task) appearedto influence the way in which a student conceptualized theangular speed and angular acceleration of an object. Suchinfluences were interpreted as arising from some aspect ofthe task setting being associated with some prior knowledgeor experience the student might have had.For example, the construct word association, offers a wayof viewing student reasoning as being influenced by thewords used in the task dialogue. The word angular for exam-ple was sometimes associated with an arc or curved path.This form of association is hypothesized to be responsiblefor the conceptualization that an object moving in a recti-linear path does not have an angular speed.Another construct, symbolic association, offers anotherway of viewing student reasoning as being influenced by theassociation of the symbol, co, with a group of symbols, aformula. It is to be expected that students might have dif-ficulties with concepts such as angular speed if they usethis form of reasoning without being able to operationalizel1. The use of this term is to be understood as being theprocedural knowledge which allows a student to find theangular speed by geometric construction in a similar mannerto that defined by Reif (1987) for acceleration (see ChapterFour).116the concept of angular speed as defined by the formula w =IdO/dtl. This interpretation would be consistent withReif's (1987) finding that students often lack the procedu-ral knowledge of how to interpret the formula that definesthe concept of acceleration, a = dv/dt.Finally, the construct contextual association offers away of viewing student reasoning as being influenced by thecontext of the task setting (excluding those that may be de-scribed by word or symbolic association). Reasoning by con-textual association, as for example the association of theelliptical path with planetary motion, contrasts sharplywith "spontaneous analogical reasoning" defined by Clement(1987).Clement, in defining this concept of analogical reason-ing, excluded the use of trivial strategies involving sur-face similarities of a problem. The characterization ofanalogical reasoning in this study does not exclude suchstrategies, nor are such "strategies" considered trivial.Being reminded of something, whether or not it occurs bywords (as in word association) or by symbols (as in symbolicassociation) or by surface similarities (as in contextualassociation), with some aspect of a problem situation isthought to be an important way in which we frame, thinkabout, or understand a problem.Schank and Seifert (1985) assert that our understandingof a situation means being reminded of the thing in memory117that is closest to the experience we are currently having.To illustrate this principle Schank (voicing a construc-tivist theme) recalls as a public lecturer the followingepisode:I used to be infuriated with the person in the audiencewho said, "Isn't this just like the work of X?" I'd bemad. It wasn't like the work of X. How could they notsee that? Now I say, "Ha! You've just proved my theory.You've understood what I said in terms of the thingsclosest to it in your memory."^ (p73)This view of understanding supports one characterizationof reasoning as a form of contextual association. Such acharacterization of reasoning is consistent with the viewthat how a person acts and responds in a situation will beinherently contextualized, as posited by Brown, Collins andDuguid (1989).Reasoning, whether by word, symbolic or contextual asso-ciation, provides a powerful way in which to view how stu-dents are able to understand both familiar and novel situa-tions without the need to theorize, or even the desirabilityof theorizing about what to do or how to act next. For ex-ample, if a very large rock is suddenly thrown towards you,you might, given the context of the situation, quickly moveaside to avoid injury without "theorizing" about what to do.Usually this action would be considered appropriate eventhough later you might find out it was a "Styrofoam" rockand you were not in danger of being injured.Or, if as a physics student you are shown a picture of anobject moving around an ellipse you might remember planetary118motion, this being the "closest" thing available to you inyour memory to make sense out of the situation, even thoughlater you may realize this was an inappropriate analogy tomake.While theorizing by individuals does occur, and needlessto say some of us spend a great deal of time doing this, itis nevertheless possible to explain a great deal of our ac-tions, especially the actions of students, without positingthat reasoning is theory-like. In this regard the findingsof this study support and extend to student reasoning aboutkinematic quantities the position held by DiSessa (1987)that student reasoning about dynamical quantities need notbe theory-like.Besides relating the findings of this study to the workof other researchers in the field, these findings can be re-lated to the way in which concepts of angular speed and an-gular acceleration have been treated by authors of physicstextbooks. It is hypothesized that some of the findingsarising from this study can partly be explained by the treat-ment of kinematic and dynamic quantities as exemplified inthe first year physics text Fundamentals of Physics, by Hal-iday and Resnick, 2nd edition.In the introductory chapter to angular kinematics, theconcept of angular speed is first examined for a rigid bodyin simple planar rotational motion2. From examination of2. A rigid body is one in which the relative positionsbetween the particles that make up the body remain at aconstant distance.119this introductory chapter and subsequent chapters in thetext certain observations about the content have been made.These observations are as follows:i) The introductory chapter's title is RotationalKinematics.ii) Angular speed, w is often measured in rotationsor revolutions per second'.iii) Textbook problems are given of rotating wheels,rotating phonograph tables and other rotatingobjects revolving about a central axis.iv) While some mention is made of reference frames,there are no problems (or examples) which require(or show) the calculation of the angular speed, wof an object from differing frames of reference.v) The motions of the planets are portrayed as movingin elliptical orbits about the sun in which the sunis taken to be at rest at one focus.vi) A special emphasis in the section of the text whichdescribes the relationship between linear and angu-lar kinematics is given to the relationship vwhere v is the speed of a particle in circularmotion.vii) The foregoing section of the t,ext (vi) gives, therelationships at = ar, ar = v'lr and ar = w'r.All the problems given in the text (except for two)are examples of objects moving at a constant angularacceleration. The two exceptions to this are prob-lems which require the student to find the angularacceleration of an object from a functional rela-tionship which expresses the object's angular dis-placement as a function of time.viii) Simple harmonic motion, (SHM) is represented as aprojection of uniform circular motion.3. At least eight of the review problems (on p176-177) referto Co having units of rev/sec. For example problem #13 be-gins with the following words: " A uniform disk rotatesabout a fixed axis starting from rest and accelerating withconstant angular acceleration. At one time it is rotatingat 10 rev/sec. After completing 60 more complete revolu-tions its angular speed is 15 rev/sec...".120From these observations it is possible to infer that stu-dents may be influenced in their understanding of angularspeed and angular acceleration in the ways given below.Each way is referenced with respect to the research finding(r.f.) of the corresponding research question (#1, #2 or #4)and to the specific text observation or observations (i,...viii) so that the reader may judge as to the validity ofthe inference being made.a) Objects must revolve around something to have anangular speed. [r.f.#1: i, ii, iii]b) Objects moving in a rectilinear path do not have anangular speed. [r.f.#1: i, ii, iii]c) Angular speed is to be measured (if possible) from apreferred position, the geometric centre or in thecase of elliptical motion from a foci.[r.f.#1, iii, v]d) Angular speed is independent of the observer's frameof reference. [r.f.#1, iv]e) No distinction is made between v, and v tangentialwhen students use the formula, v = wr [r.f.#3, iv]f) Inability to state the correct expression that definesthe angular acceleration, a of an object. Or, an in-ability to find the angular acceleration of an object.[r.f.#4, vii]g) Inappropriate analogies are made to simple harmonicmotion and planetary motion. [r.f.#4, v, viii]If, as inferred by the foregoing discussion, the influ-ence of the textbook does help shape students conceptualiza-tions of angular concepts as described, then there exists abasis upon which to make instructional recommendations as tothe way the textbook presents these concepts.121Also, conceptualizations as described in parts a and bas: objects moving in a rectilinear path do not have anangular speed, can have instructional implications beyondthe study of kinematics to the study of dynamics. To quotea student cited earlier (in Chapter Four):H:..I was having some trouble understanding the idea ofthe angular momentum. .1 couldn't understand how some-thing that just happened to be moving along (in astraight line) had an angular velocity when it wasn'tactually rotating or going round in a circle. [120-9]And, even when a student recognizes a difficulty with howthey understand a situation, he or she may be unable to pro-ceed. Again, to quote a student cited earlier (in ChapterFour):M: You made those restrictions (referring to the con-stant speed of the object in an elliptical orbit)..it shoots out what I know about-eh-space, ellipticalorbits.. .and everything.Such difficulties as this including the various conceptu-alizations of angular speed and angular acceleration whichare at variance with accepted scientific interpretations canonly be addressed by instruction.5.30 Implications for InstructionBased on the findings of this study and a reading of thetextbook it is possible to make certain instructional recom-mendations. However, the recommendations should be consid-ered as being tentative, as it is not certain that any rec-ommended change will result in overcoming the difficultiesso far mentioned or in altering inappropriate ways in which122a student thinks about or conceptualizes the concepts of an-gular speed and angular acceleration.Some of these recommendations which follow may not applyto the reader's particular situation. For example, a rec-ommendation to change some feature of the textbook may notapply to a physics teacher who is presently using a textbookwhich is in some important respects different from the oneupon which these recommendations are based. Or, an in-structional recommendation may not be compatible with an in-dividual's teaching practice. It is for the reader to judgethe appropriateness of the recommendations to his or her ownparticular situation.Recommendation 41 In the introduction to the concept of angular speed, theunits of angular speed, w should be measured in radians persecond exclusively4, and the symbol v or f should be usedfor revolutions, cycles or vibrations, per second (thus wwould equal 2nv or 2nf). Even though v, f and w do have thesame dimensions, the use of revolutions per second for wmay give the impression to some students that the angularspeed of an object moving in a closed curvilinear path canbe determined solely from the time it takes an object tocomplete one or part of one revolution.4. Units of angular speed as measured in radians per unitof time need not have as the unit of time the second. How-ever, it is standard practice to use the second since it isa basic unit of the cgs, mks and English systems.123Recommendation 12Students need more practice in differentiating betweenthe average angular speed and the instantaneous angularspeed. In many problems the average angular speed and theinstantaneous angular speed have the same value. This maylead to a difficulty for the student who may associate thesymbol, (1), with the average speed as well as the instanta-neous speed.It may help if a number of problems are given to the stu-dent in which the instantaneous angular speed (dO/dt) of theobject is not equal to the average angular speed (6.0/At).This would for example, prevent a student from simply find-ing the angular speed of an object moving in a circle (at anon-uniform speed) by measuring the time for the object tomake one revolution.Recommendation 13 It is recommended that the student be shown how to find,using explicit geometric procedures, the angular speed of aparticle. Problems which require the student to use such aprocedure should be given. For example, a problem might beto find with reference to an observer, "B", the angularspeed of some particle at point "A" moving at a constantspeed along a path which is not definable by any analyticexpression.This recommendation parallels that given by Reif (1987)in his study in the domain of kinematics. Reif suggestedthat one should teach explicitly the formal knowledge speci-124fying a concept, by teaching both the descriptive knowledgeof the concept and an explicit procedure for interpretingthe concept.Recommendation 14 Students should have ample opportunity to compare the an-gular speed of a particle from differing frames of refer-ence. Since the treatment of reference frames is sometimesgiven little emphasis in the topic area of kinematics itwould be advantageous to have more examples throughout thispart of the course (as judged by the emphasis given in thetextbook) which illustrate the procedure of making measure-ments from differing frames of reference. Otherwise, stu-dents may think about reference frames in isolation, partlybecause it is seldom an issue that needs to be explicitlyconsidered in most problems.Recommendation 15 The axis of rotation of a rigid body is usually "made" topass through its geometric center, as illustrated by text-book examples of: rotating grindstone wheels or phonographrecords. In the study of particle motion a student mayextend this idea to think that the angular speed for an ob-ject's motion in a closed path is to be taken to be at some"geometric center" (assuming such a center is identifiable).Thus it is recommended that during the study of kinematicsthat the student be presented with a number of problems in125which the axis of rotation of a rigid body does not passthrough the geometric center.Recommendation 16 Extending the discussion in recommendation #5, it is ofnote that some students view either the focus or geometriccentre of an ellipse or a circle as being a preferred posi-tion from which to make measurements of the angular speed ofan object. This may be attributable to the manner in whichplanetary motion and rigid body rotation is treated in thetextbook.Planetary motion in physics is often used to illustrateKepler's Laws in conjunction with that of conservation ofangular momentum. Since the mass of the sun is very muchgreater than the mass of any planet, the position of thecenter of mass (of the sun-planet system) is very close tothe position of the center of mass of the sun. This makesit difficult for the student to differentiate the physicalsignificance between the position of the center of mass ofthe sun-planet system and the center of mass position of thesun. Furthermore, the student may think that the focus ofthe elliptical path traced out by the planet has specialsignificance, only ^it is a special geometric posi-tion and not because it is special as a consequence ofgeneral physical laws, and a central force which variesinversely with the square of the distance between the sunand the planet.126Therefore, it is recommended that students be made awareof this distinction between some special geometric positionof an orbit (either a focus or the "center" of the ellipti-cal orbit) and how, if appropriate to do so (because ofphysical laws) it relates to the center of mass position.5.40 Implications for future researchFour recommendations for future research based on thefindings of this study are as follows:1) A subject for future research is the investigation ofthe frequency distribution of conceptualizations of angularspeed. It would be useful to know (given the task context)the relative frequency of students (based on a properly cho-sen sample from the general student population) exhibiting agiven conceptualization as for example, an object moving ina rectilinear path doesn't have an angular speed. Knowingthe relative frequencies of various inappropriate conceptu-alizations helps design curriculum materials which aim toaddress the most frequent of these conceptualizations in aparticular instructional setting.2) One subject for future research might be to test the ef-fectiveness of any recommendation suggested in this study ina classroom setting. Any one recommendation by itself maynot be effective in the classroom unless applied in conjunc-tion with another recommendation or instructional strategy.127A recommendation to make explicit the procedural knowl-edge by which to find the angular speed of a particle, forexample, is not in itself sufficient for students to achievean understanding of the concept of angular speed. Because"..it is the interrelationships (within a much larger scien-tific knowledge structure) which provides the concept with arich 'contextual meaning' extending far beyond its formaldefinition" (Reif, 1986 p27).Thus it is for future researchers to decide, with due re-gard to these interrelationships and to the context of theclassroom setting, the effectiveness of any specific recom-mendation.3) In an investigation of high school physics teachers,Berg and Brewer (1991) observed that few teachers were awareof student alternate conceptions about rotational motion andgravity. It thus might be of interest to compare whatphysics teachers believe are students' conceptualizations ofangular speed and angular acceleration with the findings ofthis study.4) Finally, a language needs to be developed which charac-terizes the reasoning activities students engage in when an-alyzing problems or tasks presented to them. The appropri-ate level of detail conveyed by such a language should re-flect the purpose for which it is to be used, in this casepedagogy.128For example, the use of the expression reasoning by asso-ciation in this study serves to characterize student reason-ing at a sufficient level of detail without the need to useterms that describe the neurological functioning of thebrain.In conjunction with the need to describe reasoning at anappropriate level of detail is the need to describe memory(again at an appropriate level of detail) associated withthe act of reasoning.What do students retain of their formal learning expe-riences? And why do some learning experiences get recalledin one situation but forgotten in another? The answers tothese questions are incomplete, despite the thousands ofmemory experiments that psychologists have done, "It isdifficult to find even a single study, ancient or modern, ofwhat is retained from academic instruction" (Neisser, 1982P5).^It is suggested that psychologists alone should notbe doing memory experiments; some researchers in science ed-ucation should be, especially if a comprehensive account(and hence a language) of student's reasoning processes isthought to be a desirable goal.129BIBLIOGRAPHYAguirre, J. (1980) Identification of student's conceptions about static forces in a classroom and related teachingstrategies. Paper presented at the anual meeting of C.S.S.E.Montreal, Quebec.Aguirre, J. & Erickson, G. (1986). Problem context andstudent's thinking about vector quantities. 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BEN Report No. 6470,Cambridge, MA: BBN LaboratoriesAPPENDIX IA SAMPLE OF A COMPLETE INTERVIEW TRANSCRIPTSubject [120-5]The Oval Path Task'(Student is shown a toy train moving around an oval trackand is asked to imagine that he is located at the positionof a toy observer placed at at the center of the oval track)I: If I were to ask that observer at the center of the trackdoes the train have an angular speed, what would yourresponse be?P: At point A or just in general?I: Ok, does it have an angular speed between points A and B?(The train).P: No.I: Because?P: Well, because it is moving in a straight line. It has gota linear velocity.I: Ok, and secondly, does the train have an angular speedbetween points B and D as seen by that observer?P: You (observer) can turn around?I: YesB. Uh-yeh, yes it would because its heading is changing evenif its actually instanteous velocity is not-its headingis changing so it would appear to have an angular veloc-ity.I: It would appear to have an angular velocity?P: Well, it does have an angular velocity.I: Because, its heading?C. Its heading is changing-its changing direction its on anarc.1. This task is again discussed in review: see p.149137I: Is there an angular speed between points D and C?P: Uhm-relative to his position-no. The train has in thecourse of going around this course has changed its direc-tion, you know continuously while its on the curves andnot so continuously while its on the straight a ways.I: So its not changing?P: No, its not really changing between D and C. So ineffect you can say it has a linear velocity and that noneof its velocity is angular.I: And between C and A?P: It has angular speed.I: For the same reason. (a reference to an earlier studentresponse re: angular speed between B and D)P: Yes.I: Between B and D because its--changing?P: Changing direction continuously.I: Continuously?P: Right.I: Is the angular speed between A and C the same as betweenB and D?P: Uhm, well, its speed would be because assuming uniformvelocity of the train eh, its velocity would be differentbecause its going in a different direction of course.Velocity is continuously changing when going around thecorner its acceleration--it has acceleration right, butits speed, its angular speed would be the same, I wouldassume.I: Does the train ever have an angular acceleration?P: Between points B and D and between points C and A it hasangular acceleration.I: Because?P: Uhm, well because it is constantly changing eh-angularacceleration isn't dependent necessarily on whether it ischanging, increasing or decreasing its linear velocity.It has an acceleration because it is continually changingits direction and acceleration is defined to be a changein velocity and since velocity is dependent both on head-138ing and on speed on magnitude-its a vector quantity therefore acceleration can also occur when you are simplychanging your direction.I: So, therefore?P.It has an angular acceleration.I: Ok, good.(A toy observer is now placed at the edge of the track,close to where the student is sitting)I: If I were to place the observer just about where you aresitting but your head is lined up with AB (section of thetrack). If you were at that position and I were to ask:does the train have an angular speed between A and B,your response would be?P: No.I: Because.P: It doesn't appear to be moving laterally or any otherway.I: And, between B and D does it have an angular accelera-tion?P: Well actually you you can argue that it wouldn't appearto be. But it does. It would necessarily appear to be,because as I am looking at it would appear to accelerate.It would appear to accelerate linearly-uhm say I have nodepth perception then eh..(Interviewer interupts and refocuses the questioning bystarting again with the question of the train's angularspeed.)I: Does it have an angular speed between A and B?P: NoI: Because?P: Because it is moving in a straight line.I: Between B and D?P: Yes.I: Because?P: Well, because it is not moving in a straight line. Its139curving.I: Between D and C does it have an angular speed?P: No.I: Because?P: Because it is moving in a straight line.I: And between C and A?P: Yes, because it is not moving in a straight line itscurving.I: Is the angular speed between C and A the same as betweenD and C?P: Yes it is.I: And that's because?P: Because in spite of the change in direction its got thesame magnitude of speed.I: Ok, good.The Semicircular Path Task(Interviewer releases one ping ball down the track. Beforeasking about angular speed the objects speed and accelera-tion are investigated first.)I: First question I want to ask you, is that ping pong ballaccelerating down the track? (ping pong ball is made toroll down the track)P: Yes.I: And the reason it has an acceleration down the track.P: Well, because it is at a slight slope so it is also beingsubjected to the acceleration due to gravity-and eh-wellthat's basically it-you can break it into a couple ofcomponents, but its effectively on a slope this way andit actually eh there is some lateral--well you can breakit down into gravitional acceleration downwards but theslope actually changes because of the curve but that'ssort of trivial.I: Now, does it also appear to have an acceleration. Sup-pose it was dark and all I could see was a flourescentball.140P: Sure you can, there is a distinct increase in its roll-ing. It seems to be rolling at a faster and faster rate.I: Ok, same would be true if I roll the ball on the outertrack, it has an acceleration? (ball is shown rollingdown outer track)P: Yeh, it does.I: Same reason?P: For similar reasons, yeah.I: Does the ball have an angular acceleration down thetrack?P: Uh-yeh it does continously because the track is curved.I: Ok.(Student is shown a video recording of each ball goingdown the track separtely then together. The mechanism ofrelease of one ball is explained as being triggered by thepassage of the other ball through a photogate.)I: First thing I would like to talk about is speed. If youcould compare the speed of these two balls. You canrefer to inner (ball) and outer one so I know which oneyou are talking about.P: Are you speaking of linear speed, now?I: Speed.(Interviewer has cards on which are marked speed,angular speed, acceleration, angular acceleration.Interviewer shows student these cards, and selectsthe one written "speed" placing it on the table)P: Well, in terms of this video tape it appears that--(pause)I: I'll play (video tape) again.P: Yeah play it one more time. They (the balls) are bothreaching the end at the same time I notice-uh-what thatindicates to me is that obviously they can't be travel-ling the same speed because one ball is farther along thetrack than the other but, uh-at the same time because theinner track has a smaller radius-uh-the linear velocitieswill be different so- (student watchs replay of tape)P: Yeah, it is sort of the instance in which you get when141you watch speed skaters on the ice in which one appearsto start further ahead on a track and one may look to beahead but in fact their are effectively neck and neck andthis becomes apparent when they hit a situation of beingat a parallel point, perpendicular to the tracks, thetwo tracks.I: Let me ask this: does one always have a greater speed ora less speed or at times do they have the same speed?.Does the outer ball--P: Uh, let's see, in terms of just motion the angular accel-eration is independent of the other accelerations so thespeed of the balls will be--the outer ball has more timein which to build up speed.I: When you look at the video tape do the two balls everhave the same speed?P: No.I: Ok, therefore one must have a greater speedP: Right, I would assume the outer ball did.I: And the reason? Because it appears, or because youworked it out somehow, is there some other factors notjust appearance?P: Just based on appearances is appears to have a fasterspeed simply because it was rolling for a longer periodof time.I: Ok.P: I may be completely over looking some factor.I: What is the strongest factor that you say it has?P: Well, it has been rolling longer and basically that iswhat I am basing it on; It has a greater distance toroll and its acceleration does vary with distance thelonger you roll the more quicker you are going to begoing.I: Ok, now what I would like to ask you--do the two ballsever have the same angular speed? (Interviewer rewindsthe video tape so student may see again)I: Do the two balls ever have the same angular speed andwhy? (tape is replayed again for the second time)P: Uh-again I would say no, uh, never simultaneously thesame speed.142I: The same angular speed?P: The same angular speed, because effectively both ballsare gaining their angular speed from like convertingpotential energy of their height into kinetic energy.And some of that goes into translational and some goesinto angular. And I would say not because again thefirst one you can relate angular speed to linear speed.The first ball has been rolling for a greater distance.Again I am trying to figure this out in terms of what Iknow of theory and the fact that one track is of asmaller radius than the outer track indicates that theremight be an instance in which they happen to be--but Ican't observe any point which they have the same speedsimultaneously.I: And for angular speed, you're speaking?P: Both (speed and angular speed).I: And for them to have the same angular speed, for them toappear to have the same angular speed what would have tohappen in your mind, is it clear to you?P: It isn't. They would have to have the same linear speedat the same time because obviously you are dealing with asmaller radius. One ball can have a higher velocity onthis track and this ball can have a smaller velocity onthis (outer) track. They can still have the same angularspeed. It is not necessarily apparent from the videotape, judging by just watching it. But, even if theynever had the same velocities simultaneously it isconceivable there will be a point in which uhm the totalangular speed determined by the linear velocities at acertain point of the track will be the same between thetwo because they have different radii. But I can'tobserve that, it is not easy for me to see.I: Is it easy to observe under any conditions the angularspeed of the two balls going down.P: In effect you could, you are dealing with acceleration ona curve you could do the old simple harmonic motion ex-periment from observing it from a two dimensional pointof view, and eh-one dimensional point of view and watch-ing the two balls on their plane as they move you couldsimply eliminate all this mismatch of going around corn-ers and things and would be seeing the balls as theytravel straight. But it would be difficult to observetheir angular speed because they are going to be in dif-ferent places at different times.I: Now, do the two balls have the same acceleration?143P: Yes, well-yes they do they have the same downwardacceleration.I: Ok, because?P: Well basically because 9.8 m/sec squared is uniform forall objects-uh-lets see-because they might not necessar-ily be at the same angle then you are not guaranteed theyare going down the ramp at the same speed but in effectthe components of their motion indicate that downwardsthey are going to be pulled at 9.8 m/s.I: So what you are saying is that the force acting on boththe balls is the same.P: Is the same.I: Therefore?P: Therefore the acceleration is going to be the same,assuming they have the same massI: Let me talk about angular acceleration. Do they have thesame angular acceleration?P: They wouldn't because their radii are different-uh,however there could be instances, again as I said before,instances in which they have the same angular accelera-tion because you know-I: So they can have the same acceleration but they won'thave in this case the same angular acceleration.P: Not necessarily they might but-there could be instancesin which that was the case but it would be an instance.I: Why would it be the case that they might as opposed tonot?P: Well-again it is a situation where it is not clear, be-cause uhm-angular acceleration is dependent on theradius and on the linear velocity of the two balls whichmeans if at any point the two balls had velocitities suchthat with their different radii they got the same accel-eration it would only be for an instant, because ehm-their velocitities are linearly are changing as theyaccelerate due to gravity. Right. See their velocitieseither way are not staying the same they are in a con-stant state of downward acceleration plus they are in aconstant state of angular acceleration. Right. So theangular acceleration depending on linear velocity alsodepends on their acceleration due to gravity so theywould never consistently have the same angular accelera-144tion. They might for a brief instant have the same ang-ular acceleration simultaneously.I: And, it is hard to say where?P: It would be very difficult to say where. I can't judge.I: Because they are several things you have to think about?P: There are several things you have to watch at once-yeahbecause they are going through two different types ofaccelerations at the same time.I: One is being--the two acceleration being? To repeat.P: One due to gravity and the other due to the angularmotion about the trackI: And where does that point? Where does the angular--due to the track? One due to gravity is down.(student digresses a bit from the question being asked)P: They are going around the track. Right. Which meanstheir direction is constantly changing which means theyhave an angular acceleration. Because again, velocitybeing a vector quantity depends on the direction as wellas magnitude.I: And where does that acceleration point?P: Uhm, oh I see what you are saying, ok. Inwards along theradii of the track.I: Ok, so the one due to gravity points--?P: Points down.I: And, the one due to--F: the angular motion points inwards towards the centeralong the radius.I: Good, and that's the factors you consider?P: Yeah.I: And which it is difficult to decide--not easy tocalculate?P: It would be easy enough to calculate but I am just tryingto judge by observation-and-I can say with relativelycertainty their is probably a point but not necessarily apoint at which they have the same angular acceleration145I: Ok.The Ellipitical Path TaskI: Here I have an ellipse and this ball. This blue ball isgoing around this ellipse at a constant speed of onemeter per second.P: Ok.I: And assuming here you are an observer standing in themiddle.P: Uhm-huh.I: Now, does the ball have an angular speed? Does it makesense to talk about the ball having an angular speed?And why?P: Yes, again because it is continually changing its direc-tion throughout the entire path it follows.I: Is the angular speed a constant?P: Eh, no it's not.I: Why?P: Because its radius isn't constant. It also changes.I: Where is its maximum and minimum (angular speed) then?It must have a maximum and a minimum if not a constant.P: Yeah, uhm I'll just start getting the-eh-formulae hereIts angular speed will say omega because that is a con-vention is equal to, proportional to v or is equal to vover its radius, so its nice to have a constant speed.The radius however changes continuously so that at thepoint where it is farthest away from the center, whereits radius is the greatest, its--I: E to E' ?P: Yeah, from observer to E or E'. Its semimajor axisuhm, it will have its smallest angular velocity becauseagain its inversely, angular velocity is inverselyproportional to the distance, the radius.I: Does the ball ever have an acceleration-just accelera-tion?P: Just plain acceleration. Uhm, yeah because again--well146it maintains a constant speed its direction changes-likeI said velocity is a vector quantity, so it is continu-ally changing.I: Does it have an angular acceleration?P: Well it must because its angular speed is never the same.I: And that's throughout the entire--?P: And that's throughout the entire ellipse.I: And the same as with acceleration?P: Yeah.I: Now what would you do if I asked (you) to measure theangular speed at point C ( on ellipse)? What infor-mation--I've told you that it (the object) is travellingat a constant speed at one meter per second--given anyother information that you would like.P: Just the radius.I: Suppose the radius was 2 meters between the observer and(point) C.P: Well I'd know that omega, the angular speed would beequal to the constant speed over 2. So it would be onemeter per second over two, so that's eh--see is thatexpressed in radians yeah I think that's right that wouldbe half a radian per second.I: Ok, that's fine.The Irregulat Path TaskI: I'll show you another picture of an object travelling ata constant speed. Does the ball have an angular speed?Or, does it make sense to talk about a ball travellingin such an irregular path as having an angular speed?P: Well, if it is on a track, I am assuming it is on a trackthen it can certainly have an angular speed.I: What happens if not on a track? Why does it have to beon a track?P: There has to be some force, either friction or somethingwhich keeps it on a track or keeps it on this path.Otherwise, it wouldn't do that, nothing behaves this wayin and of itself.147I: That's important for you that it has a force acting, inorder for you to determine whether it has an angularspeed?P: Well, uhm-all things aside-if all I could observe is thisball as it moves along this path, I could say with cer-tainty that it has an angular speed because any time itis moving in a curved path in which it is changing direc-tion it has an angular speed. The key question if youare to determine its angular acceleration you would needto know what sort of force is acting on it.I: Is the angular speed the same as it goes around thattrack?P: Not a chance.(Interviewer puts a plastic overlay on the irregularpath figure. The overlay has marks for reference on it.)I: Point X is where you are standing. If the angular speedis not a constant, if I were to ask that person at pointX if the angular speed is a constant, or if ask does ithave an angular speed, what would his response be? Itdoes have an angular speed?P: Yeah.I: And would it be a constant?P: No.I: And where abouts given the letters (of reference) I'veshown you, where would you think the maximum and minimumangular speed would be?P: Oh-(pause)-again its hard to get away from this intuitivesense of the ball rolling due to gravity-because eh.I: Just imagine this is in outer space, and there could beforces as you said acting on it to make it move in thisdirection, but you have no sense--P: So, no linear acceleration. Uhm, as long as there is nolinear acceleration. Uhm, well again linear speed, youwould try to find some way to look at its radius of move-ment and of course that's not an easy thing to do. Theseare completely irregular curves but basically at a giveninstant you can find a circle which approximates itscurve.I: Ok, let's take a point C or D or E.P: Well, at point C you can look at a very small circle148here.I: Which is inside the path?P: Which is inside the path, right. Because it is a smallcircle it has a very small radius of curving, againbecause angular speed is inversely proportional to theradius and wherever the radius is the smallest that'swhere your going to have your greatest angular speed.I: So, point C is greater (the angular speed) than point D?.P: Yeah, certaintly, because obviously D is a very shallowcurve. And it's probably got a radius some where aboutthe observer.. (inaudible)I: How would you compare G and F?P: Well, G would have-again its weird, the forces are nowthis way -it would be outside the curve and G would havea greater angular speed than at F.I: Ok, good.(This completes the task set. The interviewer nowasks "review" questions and some further exploratoryquestions.)ReviewI: Back to the train. Here is the observer at the center.and you said the angular speed of the train as it wasgoing around, what was it between A and B?P: Angular speed would be zero.I: Because?P: Because it is going in a straight line.I: Then I asked you, the angular speed betweed B and D.Is it a constant?P: Pretty well, this looks a pretty close semicircle to me.I would say so.I: And does the train have an angular acceleration betweenC and D or A and B?P: No.I: Does the train have an acceleration between A and B?P: No, not as far as I can tell, the magnitude of its149velocity is a constant.I: Ok. And when it (the train) is over on the trackbetween B and D, does it have an angular acceleration?P: Angular acceleration?I: Sorry, acceleration.P: Yeah, it would because its heading is changing. Themagnitude of its velocity would remain the same, itsconstantly changing direction, so--I: So the train does have an acceleration?P: Does have an acceleration.I: And it doesn't have between A and B because it is goingin a straight--P: going in a straight line.I: And between B and D it points towards, the acceleration?P: Towards, inwards along the radii.I: So, it has an acceleration, does it have an angularacceleration?P: No, it wouldn't.I: It doesn't have an angular acceleration because?P: Well, because its rate of going around this curve (B toD) is uniform.I: So it doesn't have angular acceleration?P: No.I: Now, so it has an acceleration but not an angularacceleration?P: Yeh-uhm-right because again its radii-its velocityremains the same, I mean its linear velocity remains thesame, its angular speed-well uhm-its angular speed isthe same because it is a uniform curve so if its angularspeed is the same then it wouldn't really have an angularacceleration. But although that doesn't make altogether alot of sense anyway, because if it didn't have an angularacceleration it wouldn't be going in a curve-uhm-its gotan acceleration inwards along the radii.I: Well, let me ask you (student interjects).150P: But that's just its acceleration in effect. It's whatwhat keeps it on the curve its not speeding up, itsactual rate of going around isn't increasing. Right.I: What is your definition of angular acceleration?P: Well let's see, it's the rate at which angular velocityis changing.I: So if the angular velocity changes then?P: Then you have angular acceleration. In this case thethe angular velocity changes at B and at D as itstraightens out. But along this track-I: So does it have at any point on the track have an angularacceleration?P: Well right at A,B,C,D-because it suddenly goes into acurve.I: Ok.(Interviewer returns to the task where an objectmoves in an elliptical path. Diagram of ellipticalpath is again shown to the student).I: What happens if you were standing outside the ellipsejust like where you are over here looking at it. And,your at one edge of this ellipse, and this ball is goingaround. Can you still talk of angular speed? Or, doesit make any difference whether you are at this point atthe center of the ellipse or there (at the outside edgeof the ellipse)?P: As long as I can perceive the ball is changing directionand going along a path that is not simply not one dimen-sional then I can speak of angular velocity as long as Ian accurately measure the radius or at least the rela-tive radii that I can see.I: If I were to ask you, and all you could see is thisflorescent ball going around this path, for the angularspeed, and if I were to ask an observer standing at thecenter for the angular speed, would you and the otherperson report back the same answer for the angular speed?Or, would they get different values?P: I think you would get the same for both, because actuallywhen it comes down to it even if I can only see the ballsmotion along within one plane I can still see the amountof time it takes for it for instance to reach one maximum151of the ellipse to the other.I: Suppose your looking down at it just a little.P: Just a little, that makes things even more simple.I: Just the way you are right now, looking down at it.P: Right.I: Now, as opposed to you and a person standing here (atthe center) would you both report back the same angularspeed?P: The same angular speed, I would say so, yeah.I: Ok.P: Assuming not so far away the speed of light makes nodifference.I: So it doesn't matter where you standing or where yoursitting where you are?P: That's rightI: And would both of you (two observers) measure the angularspeed in the same way? Would that be correct?P: Well, probably. Again like depending on what the circum-stances are if you can only see the ball and really can'tsee how its track is-uh it doesn't make a lot of differ-ence because he can still see how much time it takes tocomplete one cycle for the ball to return back to thesame position. And if you can time that you can effec-tively time the number of revolutions per second, orwhatever.I: And so you would figure out time of revolution for this,for example, for this ball to go around once and then--?P: I would simple time it. I'd see the ball at a given spot.I: And that, so that would be a certain number of degrees orradians per unit of time?P: Yeah, but also simply, it would appear at a certain pointto hit its lowest velocity. I know that, if I know it'san ellipse and I know that its going to hit its lowestvelocity at the two semimajor axes and when it does thatit will appear to kind of slow down and speed up, andslow down again. Its like simple harmonic motion again,so I can simple time it takes for it to go from one ex-treme to the other.152I: Now, you talked about angular velocity. Is angularvelocity a vector?P: Yeah, well yeah, velocity is always a vectorI: Then, where does the angular velocity vector point atpoint C (a point marked on the ellipse)?P: Uh-damn-uhm-well actually that's interesting I think withangular velocity you are dealing with vectors that aremultiple-cross multiple-I am trying to think this is thepart I am a bit shaky on-I think-(pause)-it's not-becausea vector has to -it's straight-this is moving in a curvedpath. I don't know whether you actually employ a righthand rule here or not-uhm-something tells me from my rec-ollection it's not-it's nothing that I can say intuit-ively but something is telling me that it actually pointsdownwards-uhm-in effect along the axis.I: Points down perpendicular to the plane of the table?P: Yeah.I: So where the angular velocity (vector) points and wherethe velocity points is not the same?P: No.I: Is that what your saying?P: Yeah.I: And the direction of the velocity is changing but thedirection of the angular velocity is not?P: That's rightI: Ok, I'm just seeing what that consequence of it changing(the vector changing direction), but the magnitude--P: The magnitude of the angular velocity changes.I: As it goes around.P: Yeah.(Interviewer asks about definition of somekinematic concepts so far discussed)I: How would you define speed?P: Speed is merely the magnitude of velocity, independent ofdirection.153I: And velocity is?P: Velocity is a vector quantity that describes the motionof an object in terms of its speed and direction-trulya textbook definition.I: And angular speed?P: Angular speed, uhm-would simply be the rate at which anobject completes a number of cycles, like an object in--when something is in angular motion it is in rotation orrevolving around something. It is the rate, the amountof arc it can complete in a certain amount of time.I: Angular velocity?P: Uhm, would be a measure again of its motion in terms ofboth angular speed and in terms of its direction.I: What is the definition of acceleration?P: It is the rate of change of velocity.I: Angular acceleration?P: The rate of change of angular velocity.I: Ok. Just the rate of change. So you just take thedifference. So if I want to know the acceleration,acceleration is the change in velocity?P: Oh, linear velocity.I: Rate of change?P: The rate of change of linear velocity.I: Or velocity?P: Yeah, and angular acceleration is the rate of changeof angular velocity.I: I just want to know how you think about-what comes toyour mind when you say angular.P: Well, angular I simply mean curved motion. Curved motionin which you have a vector that pulls it in towards thecenter of some curve.I: Ok, good.154(END of INTERVIEW).APPENDIX IIDIAGRAMS OF PATHS USED IN TASKSThe path diagrams are illustrated in the following order.1. The oval path path^ p 1552. The semicircular paths^ p 1563. The elliptical path^ p 1574. The irregular path^ p 158155OVAL PATHSide View --- Track1SEMI-CIRCULAR PATHSELLIPTICAL PATHIRREGULAR PATHAPPENDIX IIITHE PHYSICS OF THE TASK SETTINGSThe following is a description of the responses a physicistmight give to the focus questions (as presented in ChapterThree) of each task. The responses given are descriptiveavoiding where possible the use of formulae.The Oval Path TaskQuestion: Does the train have an angular speed between B andD and other parts of the track? Why?Response: From the position of the observer (X in the dia-gram of Figure 4) the train does not have an ang-ular speed along the straight section of track.Because, from the viewpoint of the observer theangle the train makes with respect to the observeris not changing with respect to time. However,when the train moves along the other straight sec-tion of track CD the train will have an angularspeed because, the angle with respect to the ob-server is changing with respect to time.On both curved sections of track the train willhave an angular speed, as seen from the observer'sposition since the angle the train makes with re-spect to the observer is changing with respect totime.Question: Does the train have an angular acceleration? If sowhere and why?Response: Along the straight section of track, AB the an-gular speed is not changing with respect to time,it has a constant value of zero as measured by theobserver, hence the angular acceleration is zerosince the angular speed is a constant.At all other places the train will have an angularacceleration which is not zero, because the angu-lar speed is continually changing.160The Semi-circular Path TaskQuestion: Do the two balls ever have the same angular speed?If so where and why?Response: Yes, the two balls appear to have approximatelythe same angular speed somewhere between C and G.Because, for a short moment in time they seem tosweep out the same angle in the same time from myposition as observer (X in diagram Figure 5).Question: Does this ball (pointing to one of the balls) havean angular acceleration? Why?Response: Yes, the ball does have an angular acceleration.The ball appears to be steadily increasing its an-gular speed as it rolls faster and faster down theinclined plane. At first the angle swept out asmeasured by me in say a second is very small, butas the ball gets closer towards the end of thetrack the angle swept out in a second is largerthan before. Thus the angular speed is increas-ing, therefore the ball has an angular acceler-ation.(Note: other more formal arguments are possible)The Elliptical Path TaskQuestion: Does the ball have an angular speed? Why?Response: Yes, the ball does have an angular speed. Becausethe ball from my position as observer either in-side or outside the elliptical path (as indicatedby the X's in Figure 6) is changing its angle withrespect to me as it moves around the ellipticalpath.Question: Is the angular speed of the object a constant?'Response: No, the angular speed is not a constant. Fromthe observer at the center of the elliptical paththe point A is closer than the point E. In a veryshort time interval the ball will move through agreater angle as seen by me when it passes point Acompared to point E in the same time interval.1. "Is the angular speed a constant?" is meant to be inter-preted as asking: "Does the object have the same angularspeed no matter where you measure the angular speed?"161Question: If the angular speed is not a constant then whereis the angular speed a maximum and where is it aminimum as seen by someone standing at the:i) centre of the ellipse?ii) outside, near the edge of the ellipse?Response: The angular speed will be maximum near points Aand A' because these points are closest to thecentre of the ellipse. Since the ball is travell-ing at a constant speed then in a small time pe-riod the ball will as seen by an observer at thecentre of the ellipse sweep out a greater angle ina given time period than when the ball is at anyother point(s).For the observer outside, near the edge of theellipse (as indicated by the X in the diagram).Since the ball will travel from E to A', and fromE' to A in the same time and since the angle EXA'is greater than the angle E'XA, then the angularspeed will be greater (and will reach a maximum)between EA' than between E'A as seen by the ob-server.The Irregular TaskQuestion: Does the object moving in this path have an an-gular speed? Why?Response: Yes, the object has an angular speed. Becausefrom the position of the observer (X in Figure 7)the ball sweeps out an angle with respect to time.Question: If yes, then does the object have a constant an-gular speed? If not, where is the maximum andminimum angular speed? Explain.Response: The object does not have a constant angular speed,because the angle swept out per unit time is notthe same. Since the object has a constant speed,in a small time interval the object will movethrough a greater angle when it is close to pointsA or G as compared to points points B or F. Italso would appear without doing actual measure-ments that at points A or G the object will have amaximum angular speed, and a minimum angular speednear points B or F.162The FomulaeFor the reader's reference some of formulae that pertainto circular motion are given below.Angular Speed, cow = limit 120/Atl = IdO/dt12At->0In uniform circular motion, i.e. motion around acircle at a constant speed, then:V = vtangential = wr vradial = 0Angular Acceleration, aa = limit lAco/Atl = Idw/dt13At->0In uniform circular motion:atangential^0^aradial = CO 2r = v 2/raradial is the centripetal accelerationIf the object is increasing its speed at a constantrate around a circle, (and because it is a circle)the angular speed will be increasing at a constantrate, i.e. a = constant, then:atangential = ar^aradial = wr where w = w(t)(The previous example of uniform circular motionis a special case, where a = 0)2. For planar motion (as described in this study), IdO/dt1 =dO/dt.3. For planar motion (as described in this study), Ida)/dt1 =dw/dt163


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