Physics students' understanding of relative speed: A phenomenographic study

JOURNAL OF RESEARCH IN SCIENCE TEACHING VOL. 30, NO. 9, PP. 1133-1 148 (1993)

Physics Students’ Understanding of Relative Speed: A Phenomenographic Study

E. Walsh

LaTrobe University-Carlton, Melbourne, Australia

G. Dall’Alba, J. Bowden, and E. Martin

Royal Melbourne Institute of Technology, ERADU, GPO Box 247W Melbourne. Victoria 3001, Australia

F. Marton

University of Gothenburg

G. Masters

Australian Council for Educational Research

P. Ramsden and A. Stephanou

University of Melbourne

Abstract

It is important that students of physics develop both quantitative and qualitative understanding of physical concepts and principles. Although accuracy and reliability in solving quantitative problems is necessary, a qualitative understanding is required in applying concepts and principles to new problems and in real-life situations. If students are not able to understand what underlies quantitative problem-solving procedures nor interpret the solution in physical terms, it is questionable whether they have developed an adequate understanding of physics. The research reported here is part of a larger phenomenographic study that is concerned with the assessment of physics students’ understanding of some basic concepts and principles in kinematics. In this article students’ understanding of the concept of relative speed is de- scribed. A variety of ways of understanding relative speed and of viewing a problem that dealt with this concept were uncovered. The results are used to suggest ways for teachers to proceed in assisting students to enhance their understanding of this concept. The teaching principles outlined concern both teaching relative speed, in particular, and teaching scientific concepts and principles, more generally.

Earlier investigations (Clement, 1981) show that students who are adept at using mathe- matical equations in the solution of physics problems are not necessarily able to demonstrate understanding of the underlying physics concepts, nor provide a physical interpretation of the solution. Indeed, this discrepancy has even been noted among physics graduates (Champagne,

0 1993 by the National Association for Research in Science Teaching Published by John Wiley & Sons, Inc. CCC 0022-4308/93/0901133-16

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Gunstore, & Klopfer, 1985, Lindner & Erickson, 1989). It highlights the distinction between quantitative and qualitative understanding in physics. It would appear that, although the under- standing of mathematical procedures is necessary to learning physics, such a quantitative understanding is not sufficient. If students are not able to understand what underlies the mathe- matical procedures nor interpret the solution in physical terms, it is debatable whether they have developed an adequate qualitative understanding of physics. Further, if methods of assessment preferentially focus on the reproduction of facts, formulas, and mathematical procedures rather than on understanding of the underlying concepts and principles, it is not surprising that students who successfully pass examinations may display a limited understanding of key concepts and principles within the subject.

The research reported here is part of a larger study that is concerned with the assessment of physics students’ levels of understanding of some basic concepts and principles in kinematics. Levels of understanding are identified for each concept or principle using a phenomenographic approach. (This approach is outlined in the section that follows.) In particular, this article describes Year 12 (final year secondary school) and first-year university students’ understanding of the concept of relative speed. The results are then used to suggest more effective ways for teachers to proceed in assisting students to enhance their understanding of this concept. The principles for enhancing understanding that are explored here are relevant to other physics concepts and contexts.

The Phenomenographic Approach

The phenomenographic approach to research is concerned with describing the “quali- tatively different ways in which people experience, conceptualise, perceive, and understand various aspects of, and phenomena in, the world around them” (Marton, 1986, p, 31). The present study is concerned with the ways in which physics students experience or understand selected concepts and principles relating to kinematics.

Categories of Description

In phenomenographic studies it has been found repeatedly that “each phenomenon, con- cept, or principle can be understood in a limited number of qualitatively different ways” (Marton, 1986, p. 30). The present study assumes a limited number of conceptions of the concepts and principles being explored.

In phenomenographic research the conceptions are presented in categories of description that constitute the main outcome of the research. These categories are drawn from the data; there is no attempt to fit the data into predetermined categories. The categories are based on the most distinctive features that differentiate one conception from another.

The categories of description arising from phenomenographic research are typically pre- sented in the form of a hierarchy of conceptions. The hierarchy reflects increasing levels of understanding and displays the relation between the conceptions. The focus on making explicit the relations between conceptions is one of the characteristics of phenomenographic research that distinguishes it from other approaches, such as alternative conceptions research. Although displaying the relations between conceptions, it is not claimed that the categories represent a developmental sequence; the way in which the development of conceptions occurs is an empiri- cal question.

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Relational and Experiential Descriptions

Phenomenography is not concerned only with the phenomena being investigated, nor with the people who are experiencing the phenomena. Rather, it is concerned with the relation between the two, that is, the ways in which people experience or conceptualize the phenomena. This means, for example, that the present study does not focus on relative speed, nor is it solely concerned with features of the students themselves (as might be the case in some traditional psychological studies). Rather, the focus is relational in the sense that it is concerned with the relation between the students and relative speed, that is, with the way in which relative speed is understood or experienced by the students. As Marton points out, “we try to describe an aspect of the world as it appears to the individual” (p. 33). Accordingly, the descriptions that arise from phenomenographic research can be described as relational and experiential.

This article reports the results of an investigation into students’ understandings of the concept of relative speed. Students were interviewed about their understanding of the concept, which was explored through applying the concept in a new situation. The results of the phenom- enographic analysis that follow may be contrasted with the Hoz and Gorodetsky (1983) study in which students responded in written form to two relative speed questions. In that study, the responses were analyzed in terms of the solution strategies. In the present study the analysis is concerned with students’ understanding of relative speed, with the solution strategy forming only part of the data. This study is relational and experiential; it is concerned with the ways in which the students understand relative speed. Hoz and Gorodetsky used their analyses of solution strategies to “characterize the knowledge structures that probably led to the use of the solution models” (p. 371). They inferred what the students’ understandings might be. In the present study, exploration of the students’ understandings was the focus, so that such inferences were minimized. This focus determined the way in which the data were collected and analyzed.

Qualitative Descriptions

Phenomenographic descriptions are based on ways of understanding particular phenomena or aspects of phenomena and are thus qualitative. This does not imply that quantitative content cannot be explored phenomenographically but, rather, the descriptions would deal with the qualitatively different ways in which the quantitative content is understood. In both the Hoz and Gorodetsky study and the present one, the results are presented in the form of qualitative descriptions. In contrast, the Trowbridge and McDermott (1980) study yielded quantitative descriptions of students’ performance on speed comparison tasks. The performance of the students was measured on a three point scale-0, 1, or 2-relating to accuracy.

Content-Oriented Descriptions

The content that is being thought about or conceptualized (such as principles relating to kinematics) is seen as being an integral part of the way in which it is understood; descriptions are presented in terms of the content. Thus, phenomenography can be referred to as content oriented. For example, in this study relative speed is explored in a context in which a runner is pursuing another runner. One of the conceptions is described as follows: “The individual runners are united in a system in which a new entity, relative speed, is formed. Relative speed or ‘catching speed’ is the rate at which the initial distance between them (or the initial gap) diminishes to zero.” (Further explanation of this and other categories is provided in the Results

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section.) The conception is described in terms of the content that is understood; that is, the description incorporates relative speed, the initial distance, and the relationship between them as seen by the student.

The categories of description arising from the present study may be contrasted with the Hoz and Gorodetsky descriptions, which were also based on students’ responses to questions about relative speed. Although segments of the Hoz and Gorodetsky descriptions were content ori- ented, there was an attempt toward abstraction away from the content. Consider, for example, an extract from one of their categories: “Model I has complete physical representation because (a) the physical aspects of the situation, the concepts, and their relationship are recognised, explicitly mentioned and fully considered, and (b) local and global aspects are clearly distin- guished” (p. 367). Without referring back to the content of the students’ responses, it is not clear from this description to what the “physical aspects of the situation, the concepts, and their relationship” refer. In an effort to make the descriptions general (and, perhaps, generalizable), the key features of the students’ understanding are not treated as having primary importance.

As they are content oriented, the categories of description in the present study are not framed in abstract terms that are generalizable across content, although they could be formu- lated in these terms. For instance, the example of a content-oriented description from our study that was outlined above could be reformulated in more general terms, as follows: “Relevant concepts and their relationships are explicitly stated and a new entity formed, leading to a correct solution.” This more general formulation would no longer be content oriented. Although it might be generalizable to other concepts (such as terminal velocity), it fails to convey the more significant aspects of the students’ understanding of relative speed, particularly in relation to other understandings of this concept (which are described in the Results section). For exam- ple, the distinction between conceptions that incorporate relative speed and those that deal only with the speeds of the individual runners is not reflected in this more general form. This distinction is of fundamental importance in uncovering students’ understanding of, or ways of seeing, relative speed. The distinction is also critical for decisions about how teaching can assist students to develop an understanding of this concept that is in accordance with the scientific community, at a level appropriate for the students.

The usefulness of framing the descriptions in terms of the content as it is understood by students is further demonstrated in the discussion of implications for teaching that appears later in this article. Teaching approaches and strategies are discussed in relation to the content in a way that would not be possible if the results did not focus on students’ understanding of the content. (For further discussion of the distinctions between phenomenographic research and other approaches see Marton & Neuman, 1990.)

Research Methods

Research Design

Thirty first-year students from two tertiary institutions and 60 Year 12 students from six schools were interviewed about their understanding of particular concepts and principles being investigated (including acceleration, relative speed, Newton’s laws, composition of velocities, and terminal velocity). The tertiary institutions and schools were selected to maximize the range of types of students in the sample (including socioeconomic background, likelihood of complet- ing tertiary studies, gender, and coeducational versus single-sex schooling). Accordingly, 15 students from each tertiary institution and approximately 10 from each participating school (usually comprising a large proportion or all of the physics students at that year level) ensured a

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broad representation of students. The tertiary students volunteered to participate, whereas the Year 12 students were nominated by their teachers to provide the broadest possible range of achievement in physics.

The students responded to a total of 14 questions. Prior to the interviews, the questions were randomly assigned among the 90 students and were not seen by the students until the interview. Each student answered 4-5 questions in an interview of approximately 1 hour. Hence, there were 25-30 responses to each question.

Interview Question: Arthur and Martha

The students were interviewed about a question that was presented in written form, as follows:

Martha and Arthur are running along a straight level road at constant speed. Arthur is ahead of Martha. Arthur’s speed is less than Martha’s speed.

How far must Martha run before she catches up to Arthur, and how long will this take her?

The situation may be depicted as shown in Figure 1. (Figure 1 was not part of the question presented to students.) Supplementary questions to be asked during the interview were

What do you need to know to answer the question? Explain how you would find “how far” and “how long”. Explain why you answered the question in this way.

Data Collection and Analysis

The question above was used as the basis for interviews about how students understand relative speed. The focus of the interviews was on exploring the students’ understanding through questions such as, “Could you explain that further?’, “What do you mean by that?’, “Why does that happen?”. The purpose of this form of questioning was to allow the student to interpret the Arthur and Martha question according to their understanding of relative speed.

All of the interviews about this question were audiotaped. The interviews were transcribed and the transcripts subjected to rigorous phenomenographic analysis. This process began with

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one member of the research team reading all of the transcripts with the purpose of identifying the ways in which the students understood relative speed in the question. The different student understandings were used to form a draft set of categories of description. Each transcript was tentatively classified against the draft categories. Other researchers in the group independently classified the transcripts against the set of draft categories. The classifications of transcripts were compared with each other. Any differences in classification pointed to the need to modify or further elaborate the categories of description. Through an iterative process, the categories of description were refined on the basis of evidence from the transcripts about the students’ understandings. During this process the student’s meaning was explored, taking the transcript as a whole rather than by matching particular statements with specific categories. The characteris- tic features of each category were described with reference to the transcripts that were classified against the category. The final descriptions reflect these characteristic features of each category and the differences between categories (see categories of description below). In so doing, they reveal the relationship of one category to another.

Results

Categories of Description: Relative Speed

The categories below are descriptions of the ways in which relative speed was understood in the context outlined above. The categories are empirically determined and may not be exhaustive. For example, very young pupils may display less developed conceptions than those described here.

The scalar terms speed and distance, rather than velocity and displacement, are used in the description of the categories. Students used the scalar and vector terms interchangeably. As this distinction was not the main focus of the question and did not affect the method of obtaining the solution to the problem, the scalar terms are considered to be more accessible to readers.

The categories of description and the focus of students’ responses are summarized in Table 1. The detailed categories of description are as follows.

Category Rs: Relative Speed as a New Entity: Initial Distance as a Fixed Quantity. The individual runners are united in a system in which a new entity, relative speed, is formed. Relative speed, or catching speed, is the rate at which the initial distance between them (or the initial gap) diminishes to zero. (Numbers in parentheses following the quotes refer to transcript numbers. )

You got to incorporate both of them into this problem, you take the difference between their speeds and that is the catching speed, that is how fast Martha is, that is the speed at which she is catching Arthur. Martha is running at ‘p’ speed. Speed at which Martha is catching up to Arthur is ‘p-q’. . . . Arthur is standing still and Martha is catching up at a certain speed. (75)

This fellow Arthur is ahead of Martha by some distance and he’s travelling slower than her by some amount so the speed with which she closes the gap is equal to the difference of their speeds. (90)

The motion of Arthur is treated as frozen; the frame of reference is made explicit. Speed, and then distance, are considered relative to Arthur, the latter being addressed in terms of closing the gap. Neither relative speed nor relative distance are derived mathematically, al- though they could be (see category Rd).

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Table 1 Summary of Categories of Description and Focus of Student Answers

~ ~ ~~

Category Summary of Categories of Description Student Focus

Rs (5;3)

Rd ( 1 ;O)

Di ( 1 ;6)

Relative speed as a new entity: initial

Relative distance as a variable quantity,

Distance, = distance, + initial distance

Runners as a system Relative speed is catching speed. Runners as a system. Relative distance diminishes with time. Individual runners run for same length

Individual distances related through ini-

Individual runners run for same length

distance as a fixed quantity

a function of time.

of time.

tial distance.

of time. D ( 1 2 ) Distance, : Distance,

Ut (O;2) Discontinuous perspective: unitizing time Individual runners. Distance each runner runs determined

in successive units of time. Ud (0;4) Discontinuous perspective: unitizing dis- Individual runners.

tance (Zeno’s paradox). Comparison of distances run by each in the same (decreasing) time interval.

Nore. The numbers associated with each category are the numbers of students interviewed from tertiary and secondary institutions, respectively, who were in that category.

The time taken to close the gap is determined through the relationship between relative distance and relative speed:

Initial distance (relative distance) = dM - d A

where dM and d A are the total distances run by Martha and Arthur, respectively, until they meet:

initial distance relative speed time interval =

Then the distance Martha runs to catch up is given by dM = vM X (time interval). Hence, to address the initial question, the focus returns to the individual. This solution relies on an intuitive grasp of the problem. The interview transcripts show that the students go immediately, unprompted, to considering the difference of the two speeds (the relative speed) and the mean- ing of this difference in physical terms, that is, the speed with which the gap closes. The solution is not obtained by a methodical working out from first principles, as in Rd.

Category Rd: Relative Distance as a Variable Quantig, a Function of Time. The individual runners are again combined as a system, but in this category the distance between them (the relative distance) is expressed as a function of time. The solution to the problem is given as a special case when this distance diminishes to zero. Relative speed is imported via the mathemat- ics into the expression for the relative distance but it is not the focus, although it is acknowl- edged as being the appropriate way to express speed in obtaining a solution to this problem.

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In this category (of which there is only one example) the solution to the problem is constructed from the equation of motion, taking into account the initial distance between the runners. The time t is measured from some arbitrary zero, such as the instant at which the initial gap is established up to the instant of time being considered.

The position of the runner can be obtained from the formal definition of velocity:

x( t ) = vt + x(0). x(0) is the position of the runner when the initial gap is established, at t = 0; x ( t ) is the later position of the runner at the instant of time under discussion; v is the speed of the runner.

For the runners in the problem, the expression for the position becomes

xA(t) and XM(t) are the total distances travelled by Arthur and Martha, respectively, up to the instant of time under consideration.

The initial gap between the runners is the difference in their positions at time t = 0:

The instantaneous gap between the runners is the difference in their positions at the time under consideration:

Using the individual expressions for positions of the runners to obtain this instantaneous differ- ence gives

This expression allows a determination of the distance between the runners at an arbitrary point in time and is therefore more general than the one achieved in Category Rs.

The right-hand side of this equation must be a decreasing function of time, in this case a straight line as the speeds are constant, in order for the gap between the runners to close as time goes on. Figure 2, the sketch from the student’s transcript, shows how the size of the gap, plotted on the vertical axis, reduces smoothly to zero as time goes on. This student comments on the nature of the function, saying, “So therefore that’s a decreasing function or straight line graph” (65). In so doing, the student suggests that the conditions set by this particular problem dictate the nature of the (va - vM)t term. In a related but different problem, for example, where one of the runners is accelerating with respect to the other, the equation would be altered by modifying this term. In the sense that the term (vA - vM)t can be varied, the solution is more powerful than that achieved in category Rs, although it is not clear in the transcript whether the student has understood this possibility.

At any instant of time chosen, the size of the gap between the runners can be calculated or, conversely, the size of the gap may be specified and the instant of time at which it occurs can be

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Relative Distance

(size of gap)

0 t

Time

Figure 2 . Graph of relative distance vs. time from student transcript.

calculated. The answer to the question posed is obtained by allowing the instantaneous distance to diminish to zero:

The solution (which is the same solution as that in category Rs) is given as

Finally, the student obtains Martha's total distance using her individual velocity and the value for time calculated above, as in category Rs.

Category Di: Distance, = Distance, + Initial Distance,,. Martha runs the same total distance as Arthur, plus the initial distance between them. In addition, it is recognized that Martha and Arthur are each running for the same length of time. Hence, a solution to the problem is possible. The primary focus is on individual runners each running a separate total distance, with the individual distances related through the initial distance.

The distance travelled by Martha in that time equals the distance travelled by Arthur

In the time 't' Martha will travel vMt and Arthur will travel vAt . . . But given that Martha has to travel an extra 'd' we'll take that away from the distance and then they'll be in the same place. (67)

+ x. (7)

The distance can be given in two ways:

and

dM = d , + initial distance

dM = vMt and dA = V A ~ ,

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so vMt = v,t + initial distance,

therefore initial distance t =

vM - “A

Finally, the total distance that Martha runs is given by v,t. The solution in category Di is the same as that in categories Rs and Rd, where the initial

distance corresponds to the relative distance, and vM - V, is the relative speed. However, in category Di neither relative distance nor relative speed is explicitly recognized.

Category D: Distance,:Distance,. The focus is on the total individual distances traveled by Martha and Arthur, without considering the relation between the two. As in category Di, it is recognized that each runs for the same length of time, thus allowing expressions for time to be equated.

‘t’ is constant so you put ‘t’ equals ‘d’ on ‘v’ for both of them so ‘d’ on ‘v’ equals ‘d’ on ‘v’, velocity for her and velocity for him . . . If you’ve got velocity that means you’ve got distance over time. If you take time as a constant it means your ratio of ‘v’ to ‘d’ will be the same, won’t it? So therefore you just find the ratio of their velocities and then that’ll be the distance he runs in comparison to the distance that she runs and it takes the time for her to catch up to him. (64)

Recognizing that time is the same for each individual,

the student obtains a ratio of velocities equal to a ratio of distances:

The relationship between individual distances via the initial distance is not recognized, that is, distance is not seen in two different form-as a measurable quantity and as related to velocity and time. Hence, there is no alternative way of calculating distance d, or d , and the problem cannot be solved.

Category Ut: Discontinuous Perspective: Unitizing Time. The focus is on how far Martha and Arthur individually run in a discrete unit of time, with numerical examples used to show that the initial distance diminishes after each time unit, a stepwise calculation. Depending on the numerical values chosen, it can be shown that if at one step in the calculation Martha is behind Arthur and after the next unit of time she is in front of Arthur, the solution is arrived at by iterating, that is, splitting the time unit and checking again to find where Martha is in relation to Arthur. In this way a solution may be obtained for a particular numerical example, but not necessarily an exact solution nor a general (algebraic) solution.

Just at the moment the only way I would think of working it out would be by trial and error . . . So after one second he would be going two metres. He is two metres ahead . . . And say if Martha is going at five meters per second . . . 1 would work out

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like how much the distance has decreased . . . You keep working it out until you come to a point where . . . like Martha was just behind and then say a second later she would be just in front, so you know it is going to be between it . . . If it was between five and six seconds I would use 5 .5 seconds as the time and work it out this way. (30)

There’s twenty metres between them and I’d just do it in sort of like a time line sort of thing and break it into each separate second ’cause that’d be the most basic way for me to do it. (24)

Category Ud: Discontinuous Perspective: Unitizing Distance (Zeno’s Paradox). The initial distance between Arthur and Martha is acknowledged; the focus is on the distance run separately by each runner in the same time interval. The time for Martha to run this initial distance is calculated using arbitrary numerical values for her velocity and for the initial distance. The distance that Arthur has run ahead during the same period is then calculated (using some arbitrary numerical value for his velocity). Arthur is, therefore, always ahead of Martha by some diminishing amount and Martha never catches up. Thus, the problem is insoluble.

Well she covers the distance D1 she takes some time and in that time Arthur will have covered some other distance D2 . . . the difference between them now will be D2 once Martha has reached that point where Arthur used to be before. (52)

The difference between Arthur and Martha will be D2 that is the distance Arthur covered while Martha was covering distance DI . . . I know the time required to cover that distance so I can work out while Martha is covering the second distance Arthur himself keeps running and covers another distance D3. (52)

Ordering Principles

The categories of description have been ordered in terms of the level of understanding of relative speed that is displayed. In all of the categories, speed, distance, and time are treated as significant elements, although their treatment varies. The identification of conceptions and ordering of the categories is based on the way in which these elements are treated, in particular, on what constitutes the focus of the conceptions. Categories Rs and Rd describe the highest level of understanding identified, with the understanding decreasing through Di, D, Ut, and Ud. The ordering principles are described below.

Conceptions Rs and Rd each consider the gap (the initial distance between Arthur and Martha) as an entity, combining the individuals into a system. In Rd the primary focus is on the instantaneous magnitude of this gap, whereas in Rs the initial gap is subsidiary to the relative speed. The key feature in Rs is an intuitive focus on the speed of the system, by treating the motion of Martha as relative to Arthur’s motion. In Rd this same notion of relative speed is imported through the mathematical derivation from the kinematic equation. An expression is chosen that describes the instantaneous gap and from which the particular solution emerges as the gap closes. In Rd relative speed is acknowledged but it is not the focus.

Conceptions Di and D focus on the total individual distances travelled from the initial positions in the same time interval. In D only the individual path lengths are examined and, therefore, a solution is not reached. In contrast, in Di the individual paths are linked through the initial distance between the runners, leading to a solution.

In Ut and Ud there is a focus, in a stepwise fashion, on the total distance covered by each individual but the two distances are not linked through the initial separation. In Ut time is unitized and in Ud distance is unitized.

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In categories Rs and Rd the initial distance is seen as a relative distance. In Di, D, Ut, and Ud the initial separation is not focused on as the initial distance between individuals. However, in Di the initial distance is inferred in order to link the total distances of the individuals. Categories Rd, Rs, and Di lead to algebraic solutions of decreasing generality and elegance. In category D, an attempt is made at an algebraic solution that is only partially successful. Category Ut leads to an approximate numerical solution. Category Ud does not lead to a solution at all.

Implications for Teaching

As discussed in the introduction, it is important that students of physics develop both quantitative and qualitative understandings of concepts and principles. Although accuracy and reliability in solving quantitative problems is necessary, a qualitative understanding enables students to apply those concepts and principles to new problems and in real-life situations. Van Heuvelen (1991) argues that physics teaching should encourage students to solve problems in the way that experienced physicists do. He points out that “a physicist depends on qualitative analysis and representations to understand and help construct a mathematical representation of a physical process” (p. 89 1). Accordingly, students should begin by qualitatively analyzing the problem, that is, by determining what the problem is about in a physical and qualitative sense. During the problem-solving process, questions can be posed that focus students’ attention on understanding the relevant factors. For example, in considering the motion of a parachutist who falls into a bank of snow, Van Heuvelen suggests that “the student might be asked how the magnitude of the average upward normal force of the snow on the parachutist compares to the magnitude of her downward weight as she sinks into the snow” (p. 892).

In the present study, students were required to articulate their qualitative understandings, even if they had obtained a correct mathematical solution by substitution in a formula. As noted in the results, a variety of ways of viewing the same problem were uncovered that, clearly, have implications for teaching. Those implications concern both teaching relative speed, in particu- lar, and teaching scientific concepts and principles, more generally. The implications that specifically relate to relative speed are explored by considering, in turn, each conception identified in this study. Following that, more general implications arising from the categories of description are discussed.

Implications for Teaching Relative Speed

In teaching the topic of relative speed, teachers would normally aim to teach so that the students achieved Rs or Rd understanding, after a course of instruction. Developing an Rs or Rd level of understanding can assist in developing a flexibility of outlook, that is, being able to change one’s perspective. For example, an intuitive understanding usually assumes ground as a frame of reference (Aguirre, 1988), whereas relative speed presents an alternative perspective, namely, the motion of one runner with respect to the other. [Results from two other questions in the present study (Bowden et al . , 1992) and from other research (Aguirre, 1988) also suggest that students generally have difficulty in establishing alternative perspectives.] Such flexibility is important in developing problem-solving competence in physics.

Rd is superior to Rs because Rd generalizes Rs so that the size of the gap can be calculated at any chosen point in time. This allows an answer to questions such as “After Martha has run for some specified time, by what amount has the gap closed?’, or “How far must Martha run before the distance between the runners is less than 30% of the original distance?”. (A student

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who gives a category Rs response would be able to answer these questions but would need to use more mathematical steps because the general expression is not employed.) The Rd response also allows application of the equation to a wider variety of initial conditions, as noted in the description of the Rd category. In order to generalize an Rs level of understanding to an Rd level, the student would be required to express the relative distance (gap) as a function of time, starting from the equations of motion, and would also be required to incorporate the initial conditions of the problem.

A category Di response is logically removed from Rs; a new idea is needed in order to achieve Rs, as it cannot be done by simple extension. In teaching, one approach might be to develop the concept of frame of reference within which the new idea of relative speed can then be introduced. It is reasonable to assume that students have experienced traveling in lanes of traffic (multiple cars passing one another); this may be a familiar place to start. Imagining what you would see if you were in the car that was overtaking and what you would observe from inside the car that was being overtaken, leads to the idea of frame of reference. It would be necessary to alert students to the fact that considering the question from the point of view of an observed standing at the edge of the road is to choose a frame of reference with respect to the ground.

When students have developed the concept of frame of reference, relative speed may be introduced. In category Di responses, the runners are linked via the distances that each runs. Students could be asked to treat the two speeds in a parallel manner, that is, to find the difference in the two speeds. It is important to make the distinction that even though the actual speeds of cars may be quite large, the difference in their speeds-the speed at which one car approaches or recedes from the other (as observed from inside one of the cars)-can be quite small. To consider only the difference in the two speeds promotes a category Rs solution strategy in terms of the mathematical procedures, but may sidestep the question of frames of reference, which is essential to a complete understanding of the question.

In category D the runners are not seen as part of a system. The attempt at an algebraic solution is superior to those in categories Ut and Ud, which are dependent on the use of numerical values, and so do not allow the procedures to be generalized. Unlike the Rs category, there is a failure to bring more powerful and sophisticated concepts to bear on the problem. Again, it would be necessary to introduce the concepts of frame of reference and relative speed.

Implications for Teaching Scientijc Concepts and Principles

In working with students to assist them to develop their understanding of scientific con- cepts, some teachers introduce a new topic with a discussion that draws on the students’ everyday experiences relevant to the topic and highlights for them the vividness and immediacy of these experiences. The purpose this discussion is to help the students to realize that their own experiences outside the classroom are a genuine contribution to their scientific knowledge, that in this sense they have already learned some science. Teachers might also use this initial discussion, where appropriate, to prepare students for an experimental investigation of the topic. Nelson (1986) provides an example of this type of introductory classroom discussion.

Such a discussion also provides an opportunity for teachers to identify in detail the levels of understanding with which students comprehend the new material, by stimulating them to ex- press their ideas, and their understanding, in their own words. Teachers who become aware of the range of ways in which their students are conceptualizing material can plan improved teaching strategies. Beginning with the conceptions drawn out initially, they can challenge the students’ incomplete understanding and direct the teaching toward the desired scientific under-

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standing. This might include discussing with students alternative ways of approaching a particu- lar problem, where the alternative ways mirror a variety of conceptions as starting points, and considering the relative merit in each approach. Such strategies, which focus on the students’ actual levels of understanding, are superior to strategies which assume that the students will naturally understand the material in exactly the same way as the teacher.

It is often the case that a problem in science can be presented in more than one guise. For example, problems about relative speed can take various forms. Alternative problems to the one of the two runners could be closing a sliding door on a moving van, or a crane lowering its load to the ground at constant speed. From the point of view of the students’ background knowledge, some guises may be more accessible than others. Research (for example, Chi, Feltovich, & Glaser, 1981) shows that novice problem solvers tend to concentrate on the surface features of a problem rather than on the underlying concepts and principles, so there may be educational value in having students imagine and discuss different settings (guises) for the same basic problem. Making explicit various understandings of a concept or phenomenon provides invalu- able information to subject teachers concerning the conceptions that their students bring to classes or are likely to develop following instruction. The physics students in this study had successfully passed examinations either in year 1 1 or 12 (the last two years of secondary schooling). Some had competitively gained entrance to physics classes in tertiary institutions. Despite the students’ previous experience of physics instruction and success in negotiating examinations in physics, some of their conceptions do not accord with a scientific understand- ing. In teaching, therefore, it cannot be assumed that the teacher’s meaning is understood by the students in the way intended.

The categories describe the different ways in which a concept or phenomenon is concep- tualized. In teaching, there may be value in making students aware of the diversity of concep- tions instead of focusing only on the correct one(s), so that the position of the students’ own current views in the spectrum becomes apparent to them. If a student does not use an appropriate concept or principle in the solution of a problem there may be several reasons. In the problem investigated in this article, if a student does not use the concept of relative speed (that is, displays an understanding at levels Di to Ud) then the data do not allow us to infer either that the student (a) has not yet acquired the concept of relative speed, (b) has the concept but does not see it as relevant to the particular problem, or (c) has the concept but is unable to apply it in this case. It should be noted that the difficulty of drawing such inferences is also encountered with current methods of assessment. These methods require sampling a student’s knowledge and understanding within selected contexts and situations in which particular concepts and princi- ples apply.

As noted earlier, there is no suggestion that the categories represent a developmental sequence, that is, that students must progress through the lower-level categories to the higher ones. The categories reflect different conceptualizations of the problem; the students understand the problem in different ways. (See also Chi et al., 1981 on different understandings of problems by experts and novices.) In teaching, it is generally most appropriate to foster a new way of understanding the problem. (See also Svensson, 1989.) For example, in the Arthur and Martha question, many students understood relative speed in a way that differs significantly from an accepted, scientific understanding. In teaching this concept, both frame of reference and relative speed should be introduced.

It is important to emphasize the distinction between students grasping a concept and students simply being given information about a concept in the form of a definition. The former encourages students to make the concept their own, understanding how and when to apply it,

STUDENTS’ UNDERSTANDING OF RELATIVE SPEED I147

and thus extending their problem-solving capacities. The latter relies on an effort of memory to recall the information and does not necessarily incorporate an understanding about when it is appropriate to use the concept, nor of how to apply it in new contexts. In addition to providing information relating to students’ understanding of a concept or phenomenon, the categories of description foster an awareness that a variety of meanings are likely to be attributed to other concepts and principles. Such an awareness can prompt teachers to look for understandings that differ from a developed, scientific view, and hence inform the teaching process. Furthermore, the process of seeking to explore student understanding by teachers has the potential to foster a view of teaching and learning that focuses on student understanding. (For further discussion of a phenomenographic view of teaching and learning see Bowden, 1986 and Ramsden, 1988.)

Other research (in particular, alternative conceptions research) has also promoted an aware- ness that students understand scientific concepts and principles in a variety of ways. Phenome- nography adds an additional dimension to that awareness in a manner that has potential to be of great benefit to teachers. The power of phenomenography in relation to teaching arises from two aspects of the categories of description. First, the categories are more than a map of the various responses to particular questions or phenomena, in this instance, understandings of relative speed. The categories represent attempts to explore what underlies the response in a way that does not merely point to differences in understanding but, rather, illuminates the nature of those differences. For example, what lies behind the solution strategies is explored; the ways in which speed, distance, and time are incorporated into the solution (or overlooked); and the treatment of the individual runners as separate entities or as part of the system reveals aspects of the students’ understanding.

Second, the categories of description make explicit the relation between the conceptions, for example, the combining of the individuals into a system in Rs and Rd. The relations between conceptions highlight for the teacher those aspects of a concept, principle, or phenomenon on which the students focus. In doing so, those aspects that have been overlooked are also high- lighted for particular conceptions or understandings. For example, a teacher who is teaching about relative speed can identify the way in which some students focus on comparing the distances traveled by two moving bodies, while overlooking their speeds (or distances) relative to one another; these students focus on two independent bodies rather than the system. For the teacher, this provides a starting point when thinking about what the teaching should involve.

In conclusion, meeting the aim of uncovering students’ understanding in order to inform the teaching process is best achieved through descriptions that explore that understanding from the students’ perspective, going beyond solution strategies or definitions; are qualitative; are framed in terns of the content that is being understood; and make explicit the common features and differences between the various understandings. The results of phenomenographic research give rise to such descriptions. In so doing, they provide a basis for decisions about teaching and open the way for research to inform teaching practice.

The authors acknowledge the assistance of an Australian Research Council grant.

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Manuscript accepted January 29, 1993.