Textbook treatments and students' understanding of acceleration

JOURNAL OF RESEARCH IN SCIENCE TEACHING VOL. 30, NO. 7, PP. 621-635 (1993)

Textbook Treatments and Students’ Understanding of Acceleration

Gloria Dall’Alba

ERADU, Royal Melbourne Institute of Technology, Melbourne Vic 3001, Australia

Eleanor Walsh

Department of Human Biosciences, Lincoln School of Health Sciences, La Trobe University-Carlton, Melbourne Vic 3053, Australia

John Bowden

ERADU, Royal Melbourne Institute of Technology, Melbourne Vic 3001, Australia

Elaine Martin

Royal Melbourne Institute of Technology, Melbourne Vic 3001, Australia

Geofferey Masters

Australian Council for Educational Research, Hawthorn Vic 3122, Australia

Paul Ramsden

CSHE, University of Melbourne, Parkville Vic 3052, Australia

Andrew Stephanou

Australian Council for Educational Research, Hawthorn Vic 3122, Australia

Abstract

A single science textbook often provides the syllabus for courses at upper secondary and tertiary levels, and may be used as a principal source of information or explanation. The research reported in this article challenges such practices. The ways in which the concept, acceleration, is treated in physics textbooks is compared with understandings of the concept demonstrated by final-year secondary (Year 12) and first-year university students. Some students’ understandings are shown to be incomplete in ways that parallel misleading or inaccurate textbook treatments of the concept. In addition to misleading or inaccurate statements, the limitations of some textbook treatments of acceleration were found to include: lack of attempts to make explicit relationships with other concepts, failure to point out when it is appropriate to use particular definitions or that an alternative definition might be more appropriate in specific situations,

0 1993 by the National Association for Research in Science Teaching Published by John Wiley & Sons, Inc. CCC oO22-4308/93/07062 1 - 15

622 DALL‘ALBA ET AL.

inclusion of operational definitions without conceptual explanations, and a focus on quantitative treatments while overlooking the development of qualitative understanding. Two principal aspects that distinguished the ways in which the students understood acceleration were identificd: (a) the relation between acceleration and velocity; and (b) the relation between acceleration and force(s). The results of the study have implications for teaching and, in particular, for the use of textbooks in teaching. These implications are discussed in the article.

In many upper secondary school and tertiary-level science courses a single textbook provides the syllabus for the course. Sometimes the textbook is also used as a principal source of information or explanation. In this article some of the limitations of using textbooks for these purposes are examined through an exploration of the ways in which the concept, acceleration, is treated in physics textbooks. The textbooks that were selected were those used by the students who participated in the study. We compare the textbook treatments with the ways in which final- year secondary students (Year 12) and first year university students understand this concept. We demonstrate that some students’ understandings are incomplete in ways that parallel misleading or inaccurate textbook treatments.

In other studies, incomplete understandings of concepts have also been demonstrated (see, for example, Johansson, Marton, & Svensson, 1985) but few studies have related these incomplete understandings to textbook treatments. A notable exception is a study that revealed the various ways in which chemistry students understand the mole (Lybeck, Marton, Stromdahl, & Tullberg, 1988). As the researchers in that study indicate, misconceptions and alternative understandings often are attributed in the literature to ideas developed outside school. However, students have few out-of-school experiences of the chemical concept, the mole. Those researchers report that confusions about the mole were found among teachers and textbooks, paralleling those found among students. The results of that research support the present study in pointing to the impact of teachers and textbooks on the development of students’ understanding.

The results of the present study, which compares the textbook treatments of acceleration and students’ understandings of this concept, have implications for teaching and, in particular, for the use of textbooks in teaching. These implications are discussed in this article.

Theoretical Framework

In this study the phenomenographic research approach developed by Marton (1986) is used to discover the “qualitatively different ways in which people experience, conceptualize, per- ceive, and understand various aspects of, and phenomena in, the world around them” (p. 31). This means, for example, that the present study does not focus on the acceleration of a falling object, 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 the acceleration of the object, that is, with the way in which the acceleration of the object is understood or perceived by the students. AS Marton points out, “we try to describe an aspect of the world as it appears to the individual” (p. 33). The present article is part of a larger study that describes the ways in which physics students experience or understand selected concepts and principles of kinematics.

In phenomenographic studies it has been found repeatedly that “each phenomenon, concept or principle can be understood in a limited number of qualitatively different ways” (Marton, 1986, p. 30). The present study assumes that a limited number of conceptions or understandings of acceleration can be found. Once they are identified, the conceptions are presented in categories of description that constitute the main outcome of the research. For example, the range of

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distinctly different ways in which students in this study understand acceleration (that is, the students’ conceptions of acceleration) are presented in categories of description. These categories are drawn from the research 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 or way of understanding from another. Typically, the categories are presented in the form of a hierarchy, reflecting increasing levels of understanding. The hierarchy of categories of description illustrates how the various conceptions are related to each other and provides a basis for decisions about teaching.

Methodology

This article reports findings of one part of a study of 30 first-year students in two univer- sities and 60 final-year high school students (Year 12) who were interviewed about their understanding of particular concepts and principles of kinematics. These students responded to a total of 14 physics problems. The problems were randomly assigned among the 90 physics students prior to interview, with each student answering four or five problems in an intervie,v of approximately one hour. Hence, 25 to 30 responses to each problem were obtained. One of these problems that dealt with acceleration is shown in Figure 1 . The problem was used as the basis for interviews about how students understand acceleration. The focus of the interviews was on exploring the students’ understanding through nondirective questions such as, “Could you explain that further?”, “What do you mean by that?”, “Why does that happen?’

All of the interviews about this problem were transcribed and the transcripts subjected to rigorous phenomenographic analysis. This process began with one member of the research team reading all of the transcripts with the purpose of identifying the ways in which the students understood acceleration in this problem. 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. There was agreement among the researchers that the draft categories essentially represented the range of ways in which acceleration was understood. Any differences in classification pointed to the need

Figure 1. resistance is negligible.

approaches the ground.”

“A small steel ball, thrown up in the air, follows the trajectory shown. Air

Discuss the acceleration of the ball from the time it leaves the hand until the time it


to clarify and 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 characteristic 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). In so doing, they reveal the relationship of one category to another.

Textbook Treatments of Acceleration and Projectile Motion

Before the students’ responses to this projectile problem are described, the ways in which the topics of acceleration and projectile motion are treated in the textbooks used by the students is examined. In particular, the assumptions that are made about the students’ grasp of the relevant basic concepts which underlie the textbook treatments are explored.

Exploration of textbook treatments of acceleration and projectile motion reveals several aspects that have implications for the development of students’ understanding. For example, related concepts are often treated in the textbooks investigated here as though they were unrelated, with no attempt to make the relationships explicit. Generally the textbooks do not point out when it is appropriate to use particular definitions, nor that an alternative definition might be more appropriate in specific situations. Often only operational definitions, without conceptual explanations, are provided in the textbooks. Development of qualitative understanding is frequently overlooked, with the textbooks focusing on quantitative treatments. These aspects of the treatments of acceleration and projectile motion are elaborated below, with reference to specific instances in the textbooks used by students in this study.

Although the situation depicted in the problem shown above is commonly found in textbooks, it is not generally used to focus on acceleration. Rather, it is used as a basis for numerical solutions to problems about projectile motion. Projectile motion-the motion of an object thrown in a gravitational field, including the inclined projectile-is considered an integral part of syllabi in mechanics. When projectile motion is treated in the textbooks investigated here, it is not integrated with the discussion of acceleration that appears earlier in the textbooks. Rather, the treatment of projectile motion is generally based on an assumption that students have adequately mastered the concept of acceleration in previous instruction, for example, about Newton’s laws of motion.

The textbooks present a variety of different definitions of acceleration but generally do not point out when it is appropriate to use a particular definition nor that there might be alternative definitions that are more suitable in other situations. The treatment of acceleration in one- dimensional motion in the textbooks varies from a definition of instantaneous acceleration as the limit of the ratio AvlAt as At approaches zero (Haber-Schaim, Dodge, & Walter, 1981, p. 52) to a more widely used definition of average accelerdtion as a = Av/At, where for the purposes of solving numerical problems Av = v(fina1) - v(initia1) (Storen & Martine, 1987, p. 38). The definition of average acceleration is also applied to both straight-line velocity-time graphs where acceleration is represented as equal to the “gradient of the velocity-time graph,” and to v-t graphs of variable slope where acceleration is represented as equal to the “gradient of the tangent” (see, for example, Storen & Martine, 1987, p. 38; Tait, 1980, p. 8). (The former expression gives the average acceleration for one-dimensional, uniformly accelerated motion and the latter, the instantaneous acceleration when acceleration is variable. Textbooks such as


Storen & Martine, 1987, and Tait, 1980, do not explain nor clearly distinguish the concepts of instantaneous and average acceleration as do others such as Haber-Schaim et al., 1981, although they use both concepts.) Finally, (average) acceleration is defined as change in velocity/time taken or as a = [v(final) - v(initial)]/t (Parham & Webber, 1986) with the assumption that t is understood as a time interval (p. 11). (The results reported later in this article and those in Trowbridge & McDermott, 1981, call into question such assumptions about students’ understanding of formulae and concepts.)

The definition of acceleration as the instantaneous rate of change of velocity emphasizes that the acceleration of an object can be different at each point along its path; that is, it implies the idea of acceleration at a particular instant and is central to an understanding of Newton’s second law. This definition of acceleration allows for the situation where acceleration is varying because the force is also changing. However, most examples of acceleration in textbooks used by students in this study are confined to cases where the resultant force does not change over time. (Instantaneous velocity, or instantaneous rate of change of displacement, can be measured on the speedometer of a car but rate of change of velocity is more difficult to demonstrate.)

Average acceleration (AvlAt) refers to the change in velocity over a time interval, that is, the idea of motion along a segmented path, with the acceleration constant throughout each small segment and changing stepwise from one segment to the next. Clearly, if the instantaneous motion is varying rapidly, average acceleration over a time interval can be very different from the instantaneous acceleration at each point within the time interval. In the textbooks that include the concept of instantaneous acceleration, the usual approach adopted is to consider successive path segments, each of which has its own characteristic average acceleration. As the segments become progressively smaller, which means that Av and At also progressively decrease, then the ratio AvlAt approaches a constant value that is the instantaneous acceleration at the point. This is the concept of a limit which is encountered also in differential calculus. Rosenquist and McDermott ( 1987) have noted the difficulties students have with this concept of the limit, as it is not accessible by direct experience. They have discussed methods of developing the concept of instantaneous velocity through experiments and exercises involving nonuniformly accelerated motion.

In terms of the velocity-time graph, the instantaneous acceleration can be seen as equal to the slope of the tangent. The average acceleration is equal to the slope of the secant joining two points on the curve. As the interval between these two points is diminished, the average acceleration approaches the acceleration at a particular instant (see Thomas & Finney, 1984, p. 40, for discussion of the relation between the tangent slope and the secant slope).

These different definitions of acceleration are not equivalent. The most complete understanding of one-dimensional, nonuniformly accelerated motion would involve being able to distinguish between instantaneous and average acceleration, in terms of the concept of a limit, or being able to explain these ideas with reference to a velocity-time graph. Where textbooks present only an operational definition-not a conceptual explanation-of instantaneous acceleration as equal to the gradient of the tangent to the v-t curve, this is likely to encourage a less complete understanding. (Note that textbooks such as Storen & Martine, 1987, and Tait, 1980 use the term gradient instead of slope.) Such an operational definition does not necessarily require a grasp of the concept of a limit and in some textbooks is presented in a way likely to cause confusion. For instance, Tait (1980) depicts a velocity/time graph and describes average acceleration as “the gradient of a chord” and instantaneous acceleration as “the gradient of a tangent” (p. 8). Weidner and Sells (1975) include similar statements, although explicit reference is made to the idea of a limit and the term slope is used rather than gradient (pp. 22-23). However, in both Tait (p. 9) and Weidner and Sells (p. 23), the tangent is drawn at a point


midway between the ends of the chord used to define the average acceleration. This does not represent the tangent as the limit of the secants.

If the velocity-time graph is a straight line so that the magnitude of the acceleration is everywhere the same, then acceleration is equal to the slope of the velocity-time graph. The commonly used definition of acceleration that has the form of a kinematic equation, acceleration = (final velocity - initial velocity)/time taken, also implies a constant acceleration, that is, uniformly accelerated motion over a finite time interval. In one-dimensional uniformly accelerated motion there is no distinction between instantaneous and average acceleration because acceleration is constant throughout the motion. If understanding of acceleration is limited to one-dimensional uniformly accelerated motion, it does not allow for situations in which acceleration can change.

It is important to note that qualitative explanation surrounding the definitions of acceleration is often minimal in the textbooks investigated. Exploring the qualitative meaning of acceleration is largely overlooked. In some of the textbooks it is stated that “a = -g,” this statement being derived, say, from experimental evidence of an object falling in a gravitational field. The relevant kinematic equations can then be rewritten with this substitution for a and solved to obtain particular values for velocity, displacement, and time. The textbooks usually offer an algorithm for solving quantitative problems. Consider, for example, the following recom- mended method of solution from Storen and Martine (1987):

a.

b. c.

Sketch a diagram-this will help with the sign convention and you should gain a clear idea of the physics involved. List the known (given) values in the equations. Solve the equations. (p. 70)

The most difficult aspect of solving the problems, namely, to “gain a clear idea of the physics involved” is left to the students, without any indication of what this entails.

The importance that should be attributed to developing qualitative as well as quantitative understanding is not evident in the textbooks. Worked examples in the body of the textbooks are typically limited to quantitative treatments. Problems found at the end of the chapter are, again, mostly quantitative. This suggests that the emphasis is on procedures for solving problems; rarely is there a requirement for a qualitative understanding to be demonstrated.

Furthermore, acceleration is typically treated within the context of straight-line motion, where the rate of change of velocity relates entirely to the rate of change in the magnitude of the velocity because direction does not change. Discussion of uniform circular motion, in which the rate of change of velocity relates entirely to the rate of change of direction, is usually considered later in the text and is not integrated with the earlier material. Similarly, there is usually no discussion of a situation where the acceleration arises through a rate of change not only in the magnitude but also in the direction of the velocity. This restriction of acceleration to treatment in narrow situations, together with the strong emphasis on a purely procedural way of solving quantitative problems, without a qualitative interpretation, is likely to inhibit students’ development of a complete understanding of the concept. In the categories of description that appear later in this article we see evidence of incomplete understanding of acceleration of the type that might be encouraged by such textbook treatments.

What Should a Response to the Acceleration Problem Include?

Considering the tangential and radial components of acceleration, the motion can be seen in terms of changes in speed and direction of motion as is done in circular motion. In this way it is

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possible to see that the speed does not change at a constant rate even if the acceleration, being the rate of change of the velocity vector, is constant. None of the students we interviewed treated the problem in this way. Generally, the students treated the horizontal and vertical components of the vectors separately. We are not aware of any introductory textbooks that consider the motion in terms of changes in both speed and direction. It seems that the priority in some textbooks is to use approaches that are most effective in solving problems, rather than in achieving higher levels of understanding both of physical quantities and phenomena.

In this section we describe an informed response that considers horizontal and vertical components so that it can be compared with the student conceptions described later in the article.

An Informed Qualitative Response

The ball is thrown upward at an angle to the vertical. At the instant the ball leaves the hand, it is moving at a certain velocity in a gravitational field of force. Since the problem states that air resistance is negligible, the resultant force acting upon the ball is the gravitational force. This force acts vertically downward on the ball and gives it a downward acceleration which is constant. That is, the vertical component of the ball's velocity changes from instant to instant and this instantaneous rate of change has a constant value, usually taken to be 9.8 ms-2. (The gravitational force has no component which can affect the horizontal component of the ball's velocity, and so this horizontal velocity does not change throughout the flight.) The effect of the gravitational force is to slow the ball at a constant rate on its upward trajectory until its vertical velocity (upward) reaches zero.

At this instant the ball is at the top of its path, and although its vertical velocity is zero (and the horizontal velocity remains constant), it is still experiencing the same constant downward acceleration due to the effect of the gravitational force. (At this point it is not in equilibrium, even though its vertical velocity is instantaneously zero.) Thus its vertical velocity continues to change because the gravitational force still acts upon it, but the ball is now moving downward toward the earth. The effect of the gravitational force is to move the ball faster and faster toward the ground. The rate of change of vertical velocity from instant to instant remains the same, as before. As the direction of the vertical velocity changes by 180" at the top of the path, the constant downward force of gravity may be seen to have a braking effect on the ball's motion in the first half of the trajectory and to cause an increase in vertical velocity at a constant rate on the downward path.

Analysis of the Elements of the Response

In simple projectile motion, the acceleration of the ball can be described as the instantaneous rate of change of vertical velocity. This change in vertical velocity is caused by the force of gravity which is the only external, unbalanced force acting (with air resistance taken as negligible). Hence, the necessary elements of an adequate understanding of the acceleration of the ball include instantaneous rate of change of vertical velocity and the causal relation with the force of gravity.

The ball is moving in a gravitational field of force: The acceleration experienced by the ball, that is, the instantaneous rate at which its velocity changes, is determined by the resultant force per unit mass acting upon it, as stated in Newton's second law of motion. An answer that would demonstrate a complete understanding of the acceleration problem must include both the kinematic and dynamic aspects. A kinematic treatment assumes a grasp of the kinematic concepts time, displacement, velocity, acceleration, and their scalar equivalents, as applied to


one-dimensional uniformly accelerated motion. (It is also important for the students to grasp that vector quantities in one dimension can be described with scalars if a sign convention is added.) Students must also understand what is meant by resolution of vectors, such as velocity, into horizontal and vertical components.

In addition, a dynamic treatment relies on an understanding of Newton’s laws of motion, particularly the second law, in terms of the causal relationship between the resultant force exerted on the object and the acceleration that results from this force. A dynamic approach allows a more complete understanding of the motion of a projectile because it describes the relation between forces and the motion. The kinematic approach is limited to a description of the motion showing the relations between the kinematic quantities and does not address the causes of nonuniform motion.

In summary, a complete answer to the acceleration problem should include:

the identification of the external force acting on the ball, here limited to the force of gravity as air resistance is negligible consideration of the effect of the external force on the motion of the ball an expression obtained from Newton’s second law causally linking the resultant force with the instantaneous acceleration of the ball and identifying the acceleration with - g the implication of identifying acceleration with -g for the rate of change of the vertical component of the velocity of the ball

Alternatively, the argument could be given in reverse, starting from the observation of the ball’s motion.

Categories of Description: Acceleration

The understandings or conceptions that students demonstrated when interviewed about the acceleration problem are described below. The most distinctive features of those conceptions are presented in categories of description and supported by extracts from the interviews with students. Each single extract has been chosen to illustrate one or more of the aspects that are described in the category to which it refers. However, the classification of transcripts against the categories was done on the basis of whole transcripts, not individual statements. (Numbers in parentheses below refer to transcript numbers.)

Category Cr: Caused by Gravity; Rate of Change of Velocity

Vertical velocity changes throughout the path, while horizontal velocity remains constant. Acceleration is described as the rate of change of vertical velocity, although the explanation is not based on instantaneous rate of change but varies from change of velocity in unit time to the v = at relationship. Acceleration is caused by the force of gravity. As the force of gravity is constant and downward, the acceleration of the ball is constant and downward.

The force of gravity is directly downwards . . . downwards velocity it’s changing sort of changing constantly so the acceleration is constant. (51)

The rate of change of velocity is acceleration. This ball is in the earth’s gravitational field, the field strength is g which is 9.8. Anything in the earth’s gravitational field, close


to the surface, is always under that same rate of change of velocity, with no air resistance. . . . From the time it (the ball) leaves the hand to the time it reaches maximum displacement (at the top of the path) . . . its rate of change of velocity, in other words every second it decelerates 9.8 m/s. Its velocity decreases by that much. (18)

Category R : Rate of Change of Velocity

Acceleration is rate of change of velocity and it has constant magnitude throughout the path of the ball. It is caused by forces but there is an incomplete understanding of the causal relationship between gravity and the acceleration of the ball; for example, on the way up acceleration is seen to be caused by the force of the throw and, on the way down, by the force of gravity.

Velocity changes at a constant rate . . . acceleration will be constant all the way up and be constant all the way down. . . . When it’s coming down the acceleration is resulted from gravity pulling it towards the earth, um, on the way up it’s just resulted from acceleration it’s given when it’s put up. (37)

Category G: Gravity is Closely Linked but not Causally

Acceleration is closely linked with gravity and so its magnitude is seen as constant but there is no evidence that the causal nature of the relation is recognized. The distinction between the acceleration of the ball and the acceleration due to gravity (i.e., between g and a ) is unclear. Acceleration is related to change in vertical velocity in a v = at relationship. The sign of the acceleration changes with the direction of motion.

It’s constant (acceleration) except that’ll be a positive acceleration, that’s got acceleration g (on the way up). And that’ll be a negative acceleration because it’s coming down from the other side in a different direction. . . . It’s the acceleration due to gravity that the ball is experiencing. (38)

Its speed’s going to change but at the same time as speed’s changing, time’s changing as well and as your time gets bigger, speed’s gonna get less. (43)

Category F: Acts as Force

Acceleration acts as a force affecting the motion of the ball. On the way up, acceleration opposes the motion of the ball and, on the way down, acts in the direction of motion of the ball. Acceleration is related to change in velocity but the distinction between the acceleration of the ball and acceleration due to gravity is unclear. Acceleration due to gravity is constant and downward.

Acceleration . . . would be always 10 newtons downwards . . . acceleration opposes the motion of the ball . . . on the way up . . . because the ball’s moving upwards and the acceleration is downwards so it’s opposing. (13)

The velocity will be decreasing in the initial part due to the acceleration due to gravity acting on the ball, slowing it down. . . . The acceleration acts on the ball and changes its velocity. (77)


Category D: Di$erences in Velocity

Acceleration is change in velocity and it is caused by the resultant of the forces acting at various stages during the path. The magnitude of the acceleration (or change in velocity) varies during the path of the ball. The changes in velocity are described in various ways ranging from differences in velocity at two positions in the path of the ball, differences over a specified time period, in some instances represented quantitatively by the relationship (v - u)/ ( t , - t l ) , to differences during a unit of time, for example, one second. The forces that operate include: on the way up, gravity opposing the force that causes the upward motion of the ball (the force of the throw, the velocity or acceleration of the ba11); on the way down, gravity alone, or the force resulting from the opposition between gravity and the upward force, or the force resulting from gravity and the downward force (velocity or acceleration) of the ball.

Acceleration is just a change of velocity. . . . If that’s got a velocity of zero (at the top of the path) and here (after leaving the hand) it’s got a velocity of whatever he’s thrown it with that’s a change in velocity so it’s got an acceleration. (74)

It would have a um initial acceleration aftcr lcaving the hand of urn greater than gravity , . . the acceleration becomes less and less until it becomes equal to gravity at the height of the motion . . . and then begins ah just it falls due to gravity. (41)

Category Fgb: Forces-Acceleration Due to Gravity and Acceleration of the Ball

Acceleration of the ball is distinguished from acceleration due to gravity; they are treated as two forces. Throughout the path of the ball the acceleration due to gravity remains constant and downward. On the way up, acceleration due to gravity opposes the acceleration of the ball, causing it to continually decrease, and the ball slows down. At the top of the path the acceleration of the ball is overcome by the acceleration due to gravity. On the way down, acceleration due to gravity contributes to the acceleration of the ball, causing it to move more quickly.

With the gravity acting downwards . . . the next time the acceleration . . . wouldn’t be as large because . . . the resultant of that is the acceleration of the ball . . . So you get a resultant force that’s smaller . . . Acceleration downwards due to gravity is like a resistance . . . as soon as g (acceleration due to gravity) becomes greater than the acceleration of the ball then it’ll start going downwards. (35)

Ordering Principles

The categories of description have been ordered in terms of the level of understanding of acceleration that is displayed. The order of the categories is based primarily on two principal aspects of the ways in which acceleration is understood: (a) the relation between acceleration and velocity; and (b) the relation between acceleration and force(s). Category Cr is regarded as describing the highest level identified, with the level of understanding decreasing through R, G, F, D, and Fgb. The principles on which this order was established are described below.

In category Cr, acceleration is seen to be the rate of change of vertical velocity, although this understanding is not based on the instantaneous rate of change but, rather, on the average rate. There is a causal relation between the force of gravity and acceleration. Hence, this category includes both rate of change and the causal relation, although the understanding of rate of change is not complete. In category R, the relationship between acceleration and velocity is

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understood in terms of rate of change, as in category Cr. While acceleration is seen to be caused by forces, understanding at this level is limited by inclusion of the effects of forces other than gravity (e.g., force of the throw) that do not affect the acceleration of the ball after release.

In category G, there is a limited understanding of the relation of acceleration to both gravity and velocity. There is a close relationship with gravity to the extent that the distinction between acceleration due to gravity and the acceleration of the ball (that is, g and a) is unclear. So, while acceleration is seen to be related to gravity, there is no evidence that the relationship is seen to be causal. Acceleration is related to change in vertical velocity (not rate of change) in a v = at relationship.

In category F, the understanding of the relationship between the force of gravity and acceleration is limited. Rather than being caused by the force of gravity, acceleration itself is treated as a force, acting to oppose or contribute to the motion of the ball. Acceleration is related to change in velocity but, as in category G, the distinction between g and a is unclear. In category D, acceleration is seen to be caused by the resultant force but the contributing forces vary at different stages during the path. Acceleration is related to velocity but is limited to changes in velocity. These changes are understood in various ways, as described in the category of description. In category Fgb, while acceleration due to gravity is distinguished from the acceleration of the ball, both accelerations are treated as forces. This limits the understanding of acceleration with respect to velocity.

It should be noted that in responding to the acceleration problem, students used positive and negative acceleration to describe both the direction of the acceleration (that is, upward or downward) and whether the acceleration was increasing or decreasing in magnitude (although, in the latter cases, acceleration was seen as related to change in velocity, as described in G , F, and D, and it was the velocity that increased or decreased). Warren (1979) noted that textbook writers are inconsistent in their use of positive and negative acceleration in the same way as students are in the present study. Trowbridge and McDermott (1981) also found confusion among students about the meaning of negative acceleration. As these expressions about positive and negative acceleration were not distinctive features of students’ understandings of the problem and were not attributable to specific conceptions, they have not been used in the categories of description. Rather, the meaning underlying the expressions was explored in defining the categories.

Relating Textbook Treatments to Student Conceptions

Examining textbook definitions and statements exposes how incomplete conceptions of acceleration might be encouraged and developed. In cases where students rely substantially on a single text in developing their understanding, misleading or inaccurate statements in textbooks may have considerable impact. For example, if students have been exposed to a noncalculus treatment, it is unlikely that they will be able to distinguish between instantaneous and average acceleration or see the point of doing so. None of the students in this study made such a distinction, even those displaying the most complete understanding that was observed (as described in category Cr). A similar finding was reported by Trowbridge and McDermott in their 198 1 study of students’ understanding of acceleration in one-dimensional motion.

The failure of some textbooks to explore physical situations dynamically and qualitatively does not encourage students to confront the limited understanding of what is meant by acceleration due to gravity that is evident in categories R and G . Furthermore, when textbooks use the kinematic equation v = at to define average acceleration, the understanding of acceleration as change in velocity, not rate of change, (as described in categories G, F, and D) may be


reinforced. In responding to the acceleration problem, some students claimed that acceleration must be constant because this is the necessary condition for using such kinematic equations. The students were unable to give a reason beyond the purely operational one.

Inaccurate textbook statements can be seen to blur the distinction between some terms and the relationships between concepts, for example, “When a body is thrown obliquely into the air, it changes from its otherwise straight-line motion because of the acceleration due to gravity” (Weidner & Sells, 1975; p. 41; our italics). The inaccuracy of the statement lies in the notion that change in the motion of a body is caused by acceleration rather than by the resultant force acting on it. The view that acceleration acts as a force is seen in categories F and Fgb and is likely to be encouraged by such textbook statements. Such inaccurate statements would not assist students to overcome the lack of clarity in distinguishing acceleration of an object and acceleration due to gravity that is evident in categories G, F, and Fgb. Statements such as the one quoted above may confuse students as they attempt to understand the relationships between acceleration and velocity and between acceleration and forces, such as gravity. It is the ways in which they understand these relationships that form the basis of the variation in students’ conceptions of acceleration.

Other textbook statements are likely to confuse students in similar ways. Consider, for example, the following description of an object’s motion as it rises and falls: “During both the rise and descent the friction force opposes the velocity, and thus reduces both the vertical and horizontal velocity components” (Noonan, 1987, p. 58; our italics). This statement suggests that the friction force and velocity act in opposition to each other. Would it be surprising if students interpreted this statement to mean that friction and velocity both behave like forces? Such an interpretation would parallel the understanding of acceleration as a force that is expressed in categories F and Fgb.

Storen and Martine (1987) state that “the force that causes (free fall) motion is the object’s weight” (p. 66). Such statements, while not inaccurate, are open to misinterpretation. In this instance, for example, students may conclude that there is a weight force residing in the object. In responding to the acceleration problem some students considered the upward motion of the ball to be caused by the force of the throw (as described in categories R and D) or by the velocity or acceleration of the ball which were treated as forces (as described in D). The notion of a force residing in an object and affecting motion has been reported earlier in several studies (see, for example, McDermott, 1984).

Since the acceleration problem involves forces that affect the motion of the ball, it can be seen as an application of Newton’s second law of motion. This approach to the problem is adopted in some mathematics textbooks which begin with the equation, F = ma (Newton’s second law), and use integral calculus to derive the kinematic equations (see, for example, Fitzpatrick & Galbraith, 1985, pp. 368-369). In the physics textbooks investigated, the problem is typically treated as a kinematic one, with a brief dynamic statement about why a in the kinematic equations must be replaced by -g (the acceleration due to gravity, directed downward) in this instance. Such explanations include the following: “The only force acting is vertically down. Thus the acceleration and change in velocity are vertically down also” (Noon- an, 1987, p. 53) and “a = -g since acceleration is downwards” (Storen & Martine, 1987, p. 68). When the kinematic approach is adopted in the textbooks there is no exploration of the nature of g, nor of its relationship to the universal gravitational constant G in Newton’s law of gravitation. Furthermore, the reasons that ,g can be assumed to be constant are not dealt with. Indeed it is common for projectile motion and gravitation to be treated separately in the textbooks without the relationship between the two made explicit. It is not surprising, then, that


some students are unclear about the distinction between a and g (as described in categories G and F), while others see them as two separate accelerations that behave as forces (see Fgb).

Implications for Teaching

The results presented in this article demonstrate that students who have successfully negoti- ated physics courses at upper secondary level show a range of understandings of acceleration. An accurate understanding of this concept is assumed for further work on topics such as projectile motion and Newton’s laws of motion. This assumption is called into question by the results presented here. Rather, the results provide evidence that teaching should focus explicitly on bringing about conceptual change, that is, change in students’ ways of understanding concepts. Such change often does not occur merely as a by-product of current approaches to teaching. (Similar arguments have been made by other researchers who have investigated students’ understanding of acceleration and also a broad range of other scientific concepts, although many of those studies were carried out from research perspectives that differ from that presented here. See, e.g., McDermott, 1984; Rosenquist & McDermott, 1987; Trowbridge & McDermott, 1981 .)

What is teaching for conceptual change? The different understandings that students have of particular concepts provide a starting point for teaching that focuses on conceptual change. For example, if physics teachers are aware that students in their classes understand acceleration in the ways described in this article, they can direct their teaching to bringing about changes in these ways of understanding. Having knowledge about both students’ current understandings of concepts and the desired understanding is likely to make teaching more focused and effective. Teaching is then a process of providing learning activities that assist students in changing their understanding from a less complete to a desired conception or way of understanding. However, it is not necessary that research studies be carried out in relation to every concept that is taught in an area of study. Questioning of students during classes and listening to the comments they make about the content during any interaction with them can provide valuable insight into their understanding. In an ongoing way, good teachers are informed about students’ understanding and use this information to direct their teaching.

As well as teaching approaches and methods, assessment can play a part in encouraging learning that brings about conceptual change. Assessment gives clear messages to students about what is important in the subject; for example, is it sufficient to reproduce material delivered in lectures or is understanding of concepts required? Most students will direct their efforts to what the subject demands, particularly as expressed in the assessment. Only when the assessment is based on understanding will students be rewarded for the efforts they make to understand.

The value of qualitative problems in establishing the extent to which students understand the content has been demonstrated in this article. It can be too easy in quantitative problems to substitute given values into rote-learned formulae to produce the correct numerical solution in a way that masks the understanding of the concepts (see, for example, Bowden et al., 1992; Clement, 1981). It is rarely the case that teachers would be satisfied with students developing only quantitative skills without an understanding of the underlying concepts. As qualitative understanding is desired, it should be developed and assessed; it cannot be assumed.

The treatment of concepts in textbooks and the ways in which that treatment is likely to encourage incomplete or inaccurate understandings of those concepts must be taken into account when using textbooks in teaching (see also Lybeck et al., 1988; Iona, 1991). In particular,


uncritical reliance on textbooks as though they embody the course is clearly problematic. Textbooks cannot take on the role of teacher. For example, seeking feedback from students about their understanding of the content, as described previously, is a necessary part of teaching that falls outside the part played by textbooks. Similarly, while textbooks may be useful references about key formulae and laws of physics, their lack of explanation in ways that would assist the development of understanding means that they are not suitable as a principal source of explanation.

In this study two aspects of acceleration, namely, the kinematic and dynamic aspects, were often not integrated in students’ thinking. A similar lack of integration was evident in the physics textbooks investigated in this study. In many instances, the textbooks do not make clear the relationships between aspects of a concept, several formulae relating to the same concept, and different concepts. It is understanding these relationships that often presents the greatest difficulty for students who lack the breadth of experience of physics that the teachers and textbook writers generally have. A key role in teaching should be to make explicit these relationships and assist students in developing understanding of them.

As indicated previously, some physics textbooks do not make clear what aspect of acceleration they are focusing on, while varying in terms of the approach to acceleration and the contexts they include. The approach and contexts within which definitions and concepts are treated in textbooks are likely to promote particular understandings. In a similar way, work problems in textbooks can either illuminate fundamental aspects of concepts or simply reinforce a narrow, incomplete understanding. The use of textbooks will only enhance learning if teachers use them with an awareness of the ideas that they are likely to promote. Textbooks should be used selectively to suit the approach and contexts that are of relevance to the course, as expressed in course objectives. Accordingly, the objectives of a science course should not simply indicate what topics or concepts are dealt with but also the type of understanding to be developed. For example, simply indicating that acceleration is included in a physics course is insufficient. Is student understanding of only one-dimensional uniformly accelerated motion to be developed? Alternatively, which other contexts are students expected to understand? The current lack of clarity in some textbooks is likely to confuse students on this issue.

There is an implication for teaching that is concerned with student expectations. It is important that students are aware that qualitative understanding of key concepts is expected in a course. Where changes are made to place more importance on students’ understanding in the ways outlined above, students must be informed about what these changes mean. For instance, examples of the kinds of assessment items focusing on understanding that students can expect to encounter should be provided to them so that they have clear expectations of the course.

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

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Manuscript accepted February 12, 1992.

Textbook treatments and students' understanding of acceleration

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