Chapter 2: Review of the Literature

Graph Interpretation and Kinematics Knowledge

Researchers have identified two ways in which students commonly misinterpret1 graphs, particularly graphs of motion events. The first is the graph-as-picture (GAP) interpretation, in which students expect the graph to be a picture of the phenomenon described. In kinematics (the study of motion), this can result in the student interpreting a graph of distance vs. time as if it were a road map, with the horizontal axis representing one direction of the motion rather than representing the passage of time. In problems dealing with balls rolling in tracks or people riding bicycles over hills, students using GAP will often draw velocity vs. time graphs resembling the shapes of the tracks or hills, rather than showing the velocity of the ball or bicycle. The second common misinterpretation is slope/height confusion (SHC), in which students use the height of the graph at a point when they should use the slope of the line tangent to the graph at a point, and vice versa.

Both of these misinterpretations have been reported in a variety of populations, including middle school students (Barclay, 1985; Clement, Mokros, & Schultz, 1985; Mokros & Tinker, 1987), and college and university undergraduates enrolled in physics courses (McDermott, Rosenquist, & van Zee, 1987). Barclay (1985) noted that these misinterpretations were also common in middle school teachers. Berg & Phillips (1994) observed GAP in middle school and high school students. Other studies have found indications of SHC in high school (Brasell, 1987a) and college (Thornton & Sokoloff, 1990; Svec, 1995) students enrolled in physics courses.

Graph-as-Picture in Kinematics

It appears that there is a strong connection or interaction between students’ conceptual difficulties in kinematics and their difficulties in interpreting the graphs that are so often used to represent kinematic phenomena and concepts. McDermott, Rosenquist, & van Zee (1987) reported several difficulties that college physics students experienced in connecting graphs to physical concepts. They alluded to GAP, saying that the students had trouble “separating the shape of the graph from the path of the motion” (p. 509). This problem was evident whether the students were being asked to sketch graphs representing motion of balls on given tracks, or to set up tracks on which balls would roll as represented in given graphs.

Mokros & Tinker (1987) reported that middle school students could easily interpret graphs that resembled pictures of the phenomena,but had more trouble when the graph did not resemble a picture. They write: “Students scored the lowest--and subsequently made the greatest gains--on items where the mental image of the phenomenon and the graph of the phenomenon were discrepant. Interestingly, all of these items involved distance and velocity graphs” (p. 377); and “Unfortunately, all of the items where graph-as-picture rules did not work involved distance and velocity” (p. 378).

Mokros & Tinker appear to consider the absence of GAP misinterpretations in areas other than kinematics to be an artifact of their test, which used kinematics in many questions. In fact, all of the references to GAP that I have been able to find have involved kinematics (Barclay, 1985; Clement, Mokros, & Schultz, 1985; McDermott, Rosenquist, & van Zee, 1987; Mokros & Tinker, 1987). Three of these studies involved other types of graphs as well, but GAP was reported only in relation to graphs of motion. This may be caused by the popularity of kinematics items on graphing tests, or it may be due to a link between the way students think of motion and the tendency to use a graph-as-picture misinterpretation. Perhaps this misinterpretation should be more precisely named graph-as-trajectory, or graph-as-map-of-path.

Prior to reading literature describing GAP, I had identified the same misinterpretation in my own calculus students. In addition, I observed this misunderstanding in my interview with “Ann.”2 I asked Ann to “draw a graph, representing my distance away from home as it relates to time” during a trip when “I walked from my house to the cash machine, stood for a while at the machine getting out money, and then walked back home.” Early in the interview, Ann drew a graph similar to the one in figure 1. From her comments as she drew the graph, it was clear that she was representing the physical act of returning to the starting place by drawing the graph returning to the origin, which was its starting place. Later, I drew her attention to the fact that she had labeled the horizontal axis “time,” and asked her what that meant. After some thought, she revised her graph to resemble figure 2. At this point she became much more confident of her work, and repeatedly told me that the graph couldn’t go back to the origin, as she had originally drawn it, “because time doesn’t go back, time only goes forward.”

Until encountering the term in the literature, I did not think of this phenomenon as graph-as-picture. I referred to “the road-map misinterpretation,” since that was how I understood what I had seen with Ann and other students. Students seemed to think of a line graph representing motion as a road map, with the vertical axis representing the north/south component of motion and the horizontal axis representing the east/west component. Students generally have far more experience reading maps than interpreting line graphs, which may cause them to interpret the relatively unfamiliar graph as a more familiar map. If so, this is not a misconception so much as a misapplication of an otherwise correct schema. The students can correctly interpret maps, but then incorrectly apply this interpretation to other, more abstract, representations of motion. If this is the source of the misinterpretation, then one would not expect to find it in areas other than kinematics. I concur with Mokros and Tinker’s recommendation that further research should be done to determine whether the GAP misinterpretation is used in areas other than kinematics.3

Slope/Height Confusion in Kinematics

McDermott, Rosenquist, & van Zee (1987) considered straight-line graphs separately from curved graphs. Even in the simple case of straight lines, the college physics students in their study confused distance, represented by the height of the graph, with velocity, represented by the slope of the graph. Although McDermott, Rosenquist, & van Zee appear to have interpreted this as SHC, calling it a difficulty in “discriminating between the slope and height of a graph” (p. 504), it is not clear to me how much of this confusion is due to problems with graph interpretation and how much is due to lack of understanding of distance and velocity. The situation becomes even more complicated when the graphs are curved, making confusion of slope and height (or of distance and velocity) more common. McDermott, Rosenquist, & van Zee term this a difficulty in “interpreting changes in height and changes in slope” (p. 504), but it could also be a difficulty in interpreting changes in distance and changes in velocity. Since slope on the distance-time graph gives the same information as height on the velocity-time graph, it appears that, in kinematics, confusion between distance and velocity is closely related to SHC.

Thornton & Sokoloff (1990) report that university students in introductory physics courses commonly draw or select graphs for velocity vs. time that resemble the correct graphs of distance vs. time. Svec (1995) observed that prior to instruction, 25% of the undergraduate physics students in his study “used the height of curve as the criteria for determining the magnitude of the velocity from a distance-graph” (p. 14), where slope would be the appropriate criterion. Many of Svec’s subjects interpreted both distance-time and velocity-time graphs as if they were all distance-time graphs. Although neither Thornton & Sokoloff nor Svec call this slope/height confusion, it appears to be closely related to what McDermott, Rosenquist, and van Zee (1987) found. In all three cases, it is clear that these students had problems relating graphs of distance and velocity to one another and distinguishing them from one another.

Similarly, the high school physics students studied by Brasell (1987a) often drew erroneous velocity graphs that resembled the correct distance graphs. This could mean that these students were confused about the difference between distance and velocity, or that they were confusing slope and height. Brasell writes: “In kinematics, it is difficult to separate the slope/height confusion in interpreting graphs from the confusion between distance and velocity which appears to be prevalent among students from middle school through college” (p. 386). She seems to attribute these errors to confusion about kinematics, rather than graphing, as indicated by her attempts to use the nature of students incorrect responses to separate “graphing” errors from “conceptual” errors. She describes this difference:

The alternative answers students select on the posttest provide information about conceptual difficulties. Most of the items dealing with velocity included options that represent errors predominantly with either conventions of graphing (graph errors) or concepts of distance and velocity. Graph errors occur when students do not fully understand how to represent the direction of movement, and they select the alternative answer for an event with positive velocity instead of negative velocity or vice versa. Conceptual errors occur when students confuse graphs of distance and velocity, and select either a flat distance graph or a sloping velocity graph for an event of constant velocity. (p. 391)
The higher proportion of concept errors with velocity graphs than with distance graphs indicates that velocity is conceptually more difficult than distance. The lack of improvement in concept of velocity was anticipated because of the very brief treatment period and the prevalence and stability of confusions between distance and velocity. (p. 393)

Whether the researcher considers these errors to indicate SHC in graph interpretation, as opposed to confusion among distance, velocity, and acceleration, may depend in part on whether the researcher is studying primarily graphing or kinematics. In the introduction to their paper, McDermott, Rosenquist, & van Zee (1987) write that they examined “some of the graphing errors made by students” (p. 503, italics added), as part of ongoing research in student understanding in physics. They later identified confusion between distance, represented by the height of the graph, and velocity, represented by the slope of the graph, as a difficulty in “discriminating between the slope and height of a graph” (p. 504), without giving any reason why it might not as easily be difficulty in discriminating between distance and velocity. In a parenthetical addition to a description of student responses to a particular item, they write: “(It is also possible that some students correctly interpret the crossing of the graphs . . . as a time when the objects have the same position, but then incorrectly infer that the objects have the same speed as well.)” (p. 504). Despite recognizing this possibility, they clearly prefer the explanation related to confusion between slope and height.

McDermott, Rosenquist, & van Zee later discuss difficulties students have in “relating one type of graph to another” and write: “Many [students] are unable to translate back and forth from a position versus time ( x vs. t) graph to a velocity versus time ( v vs. t) graph” (p. 505). They discuss this entirely in terms of graphs, without suggesting that the students may actually be confusing the concepts of position and velocity, rather than the graphs. In a similar vein, they write of difficulties students have “distinguishing among different types of motion graphs” (p. 510):
When students are asked to sketch x vs. t, v vs. t, and a vs. t graphs for a motion demonstrated in the laboratory, they often draw three graphs that have basically the same shape. Even when they make measurements of a motion and obtain data to plot, we have found that students often try to make the shapes of the graphs match one another. . . . Some students seem to find it very difficult to accept the idea that the same motion can be represented by graphs of very different shape. (p. 510)
In contrast to McDermott, Rosenquist, & van Zee’s graph-centered interpretation of student errors, Thornton & Sokoloff (1990) focus on problems with kinematics. In their paper titled “Learning Motion Concepts using Microcomputer-Based Laboratory Tools,” Thornton & Sokoloff deal with kinematics instruction and the impact of MBL tools, but say very little about graphing skills. They interpret students’ initial errors and subsequent improvement as indicating a lack of understanding of kinematics, and subsequent learning about kinematics. Thornton & Sokoloff administered a pre-test asking their subjects to select velocity-time graphs showing objects moving toward and away from the origin “at a steady (constant) velocity” (p. 862). Although the correct graphs were horizontal lines, the most popular choices were diagonal lines. They write:
The most common error is the choice of the “distance analogs” . . . This is consistent with previous studies, in which students confused position and velocity graphs. . . . It should be noted that the students did not miss the questions because they were simply unable to read graphs. More than 90% could answer questions involving distance graphs correctly. (pp. 862-863)
Although it is difficult to estimate what proportion of students’ misunderstandings are due to confusion among distance, velocity, and acceleration, it is clear that this confusion does exist. Both Svec (1995) and Thornton & Sokoloff (1990) found that students are more likely to confuse acceleration-time graphs with velocity-time graphs than to confuse velocity-time graphs with distance-time graphs. It seems that this must stem from problems understanding kinematics, rather than graphing, since the graphs were of similar complexity. This is in agreement with Trowbridge & McDermott’s (1981) study of college physics students, which revealed that students have more trouble with the concept of acceleration than with the concept of velocity, even when graphs are not used. From this it seems clear that at least part of the problem lies in students’ understanding of kinematics concepts, rather than graph interpretation.

Microcomputer-Based Laboratory Instruction

Several studies have used microcomputer-based laboratory (MBL) instruction in an effort to improve students’ graph interpretation skills. Microcomputer-based laboratories (MBLs) involve some sort of probe which detects a physical quantity. The probe most often used in the literature is the motion sensor, which detects the distance between the sensor and the nearest object, usually a student. The sensor is attached to a computer, which creates a graph of the student’s distance from the sensor over a period of time, often about ten seconds. The graph is displayed in real time, as the motion progresses. The student can walk back and forth in front of the sensor and watch the graph appear at the same time, which is expected to help the student understand the abstraction of the graph by connecting it to the physical reality of the motion. This arrangement has been used in several descriptive (e.g. Barclay, 1985; Mokros, 1985; Mokros &Tinker, 1987; Thornton, 1985) and comparative (e.g. Brasell, 1987a, 1987b, 1987c; Svec, 1995; Thornton & Sokoloff, 1990) studies. Other sensors, measuring temperature, force, current, voltage, light intensity, and sound pressure, are also available, and have been used in studies examining the effects of MBL instruction in contexts other than kinematics (e.g. Adams & Shrum, 1990; Linn, Layman, & Nachmias, 1987). MBL equipment is described in more detail by Thornton & Sokoloff (1990).

As noted above, some of the problems students have in correctly interpreting graphs of motion events may be due to faulty or incomplete concepts of motion as well as to difficulties with graphical representations. Motion sensor work appears to have the potential to address both of those issues simultaneously. This possibility was explored by Svec (1995), who writes: “Because the use of graphs to learn content has important classroom implications, it is important to document what the students are learning when using MBL labs and how they are learning those topics” (p. 3).

Svec studied university students enrolled in two physics courses. One class used MBL equipment with motion sensors, while the other used more traditional motion laboratories. Two instruments, the Motion Concept Test and the Graphing Interpretation Skills Test, were administered at the beginning and end of the semester. Svec broke his research questions down into fifteen testable hypotheses about specific graph-interpretation abilities and motion concepts that he predicted MBL instruction would help students develop. Scores on the pre- and post-tests were used to test these hypotheses. Tests of two of those hypotheses revealed no noticeable difference. In one area, there was a small difference favoring the control group, which had received traditional instruction. In the twelve remaining areas, there was a difference in favor of the treatment group, which had received MBL instruction throughout the semester. Four of these differences were characterized as significant: determining the direction of motion from a motion graph and qualitatively interpreting distance-time, velocity-time, and acceleration-time graphs. Svec concludes that: “The study showed that the [MBL treatment group] students learned more about graphing interpretation skill, more about motion graphs and more about conceptual understanding of motion that did the [traditionally instructed control group] students. That learning was made possible by effective use of MBL activities” (p. 22).

Thornton & Sokoloff (1990) used MBL instruction with motion sensors to teach kinematics to 1500 students in several college and university physics courses. Through pre- and post-tests, they compared the kinematics knowledge of students who received traditional lecture instruction in kinematics to those who also participated in MBL instruction. Their post-tests included testing soon after the kinematics instruction was completed and again at the end of the semester, seven weeks after the completion of the kinematics instruction. On velocity questions,4 post-test error rates for students who did not have the MBL instruction were similar to pre-test error rates for all students. For students who did have MBL instruction, post-test error rates were substantially lower. Thornton & Sokoloff write: “There is strong evidence for significantly improved learning and retention by students who used the MBL materials, compared to those taught in lecture”5 (p. 862).

Brasell (1987a, 1987b, 1987c) used pre- and post-tests to study the effects of one hour of instruction on the graph interpretation skills of eighth-grade students. The group receiving the standard MBL instruction significantly outperformed the control and other groups. Brasell’s study is examined in greater detail in the next section.

Explaining the Success of the MBL

Several researchers have attempted to explain the success of MBL instruction in improving students’ abilities to interpret (and in some studies to produce) graphs. Barclay (1985) suggests:

Attributes of the MBL science units that seem important in contributing to [learning graphing skills] include:
a) The grounding of the graphical representation in the concrete actions of the students.
b) The inclusion of different ways of experiencing the material: visual, kinesthetic, and analytic.
c) The fast feedback that allows students to immediately relate the graph to the event. (p. 8)
Linn. Layman, & Nachmias (1987) state:
MBL offers one major advantage. The graphs in MBL are formed as the experiment is carried out and are immediately related to an experience that the students may have designed or set up themselves. Thus they are less likely to be seen as static pictures and more likely to be seen as dynamic relationships. (p. 245)
Mokros & Tinker (1987) suggest:
Four features of MBL seem to contribute to its success in facilitating graphical communication: MBL uses multiple modalities; it pairs, in real time, events with their symbolic graphical representations; it provides genuine scientific experiences; and it eliminates the drudgery of graph production. (p. 369)
Thornton & Sokoloff (1990) write:
The following characteristics of these [MBL] tools [including several types of data collection probes] are important to student learning:
1) The tools allow student-directed exploration but free students from most of the time-consuming drudgery associated with data collection and display.
2) The data are plotted in graphical form in real time, so that students get immediate feedback and see the data in an understandable form.
3) Because data are quickly taken and displayed, students can easily examine the consequences of a large number of changes in experimental conditions during a single laboratory period. The students spend a large portion of their laboratory time observing physical phenomena and interpreting, discussing, and analyzing data.
4) The hardware and software tools are general--independent of the experiments. the variety of probes use the same interface box and the same software format. Students are able to focus on the investigation of many different physical phenomena without spending a large amount of time learning to use complicated tools.
5) The tools dictate neither the phenomena to be investigated, the steps of the investigation, nor the level or sophistication of the curriculum. Thus a wide range of students from elementary school to the university level are able to use this same set of tools to investigate the physical world. (p. 859)
A common thread in these suggestions is an emphasis on the immediacy of the graph production as a key point in the MBL experience. The MBL equipment allows the student to watch the graph appear in real time, as the experiment progresses. The student’s own physical movement is very concrete, and appeals to the kinesthetic sense. The graph, on the other hand, is abstract, and appeals to logical thought. Several researchers (e.g. Adams & Shrum, 1990; Beichner, 1990; Brasell, 1987a; Mokros, 1985; Mokros & Tinker, 1987) have speculated that experiencing the movement while watching the graph appear helps the student to form a link between the two, and thus “transfer the event-graph unit (already linked together) into long-term memory as a single entity” (Beichner, 1990, p. 804).

Following this idea, Brasell (1987a, 1987b) hypothesized that the real-time nature of the graphs produced by the MBL equipment was critical to the success of this instruction. She used pre- and post-tests to study the effects of one hour of instruction on four groups of high school students: Test Only, Control, Standard MBL, and Delayed MBL. The Test Only group received no instruction. The Control group received pencil-and-paper instruction. The Standard MBL group used the MBL with motion sensor as it is normally used, with the graph of the student’s motion produced in real time as the motion progresses. The Delayed MBL group used a modified version of the MBL software, in which the graph was not produced until after the motion was complete. The delay was about 20 to 30 seconds. Brasell found that the Standard MBL treatment group significantly outperformed all other groups, including the Delayed MBL group. Most of the difference was found in the items related to distance. This effect was more pronounced for the female students than for the males (Brasell, 1987c). There was a smaller difference in favor of the Standard MBL group on the velocity items, but it was not significant. Brasell estimates that the real-time feature accounted for about 90% of the improvement that MBL offered over pencil-and-paper instruction. She writes (1987b): “The appearance of these differences after such a brief treatment, one class period, suggests that there is a fundamental difference in information processing generated by the immediate display” (p. 4).

Following on Brasell’s work, Beichner (1990) hypothesized that it was not necessary for the student to actually produce the graph by using a motion sensor to measure his or her own motion. Since Brasell had shown that the key point was the real-time nature of the graphing, Beichner hypothesized that the students could learn as well from prerecorded videotapes of motion not produced by the students, so long as the graphs were displayed in real time along with the display of the motion on the videotape. Beichner developed a system, called VideoGraph, which produces the graphs of the motion shown in the videotape. Like Brasell, Beichner used a single class period for instruction, preceded and followed by diagnostic tests. Beichner’s study used a simple two-by-two design. One factor was the type of instruction, either VideoGraph or traditional methodology. The other was whether the students viewed an actual motion event or not. (In any case, the students did not view the event that had been videotaped, but two of the groups viewed a similar event.) A fifth group took the diagnostic tests, but received no instruction. A total of 165 high school and 72 college students participated in the study.

Contrary to his expectations, Beichner found no significant differences among the groups in his study. He speculates that differences might be found with interventions lasting longer than one class period, but has no evidence for that. Since significant differences were found after an intervention of only one class period in Brasell’s study, length of intervention is not responsible for the differences between the two studies. Beichner writes:
The VideoGraph technique can present replications of motion events while generating graphs, but other than determining the rate of animation, students cannot control the motion. This ability to make changes--and then instantly see the effect--is vital to the efficacy of microcomputer-based kinematics labs. The feedback appeals to the visual and kinesthetic senses. A simple visual juxtaposition of event images and graphs is not as good as seeing (and “feeling”) the actual event while the graph is being made. (pp. 811-812)
He goes on to suggest that technology resembling VideoGraph might be useful in other areas, such as titrations and heating, and adds: “The kinesthetic sense is a strong one and appears to make a difference in kinematics MBL’s” (p. 813).

Brungardt & Zollman (1995) pursued Beichner’s suggestion that a longer period of instruction might reveal advantages of the videotape method that were not apparent after only one class period of instruction. The thirty high school physics students in Brungardt & Zollman’s study were given four class periods of instruction. Brungardt & Zollman used videotaped interviews, as well as written tests, in an effort to gain both quantitative and qualitative information about the effects of the instruction. They also interviewed eight of the students three weeks after the instruction was completed, to gauge the long-term retention effects.

The students in Brungardt & Zollman’s study worked with a computer program to generate graphs from videotapes of motion events. The videotapes all showed motion involved in sports, presumably a familiar context for the students.6 The students used acetate sheets placed on the video screen to record the position of the object or person at various times during the motion. They then measured the position of the object at each time, and entered the information into a spreadsheet, which was used to produce the graphs.7 Half of the students then saw the graph displayed as the videotape of the motion was replayed, so that the two were synchronized. The other half watched the motion replayed, and then saw the graph displayed several minutes later.

Brungardt & Zollman found no significant effects of real-time versus delayed display of the graphs. This seems to contradict Brasell’s results, which showed a great effect from a delay of only 20 to 30 seconds. The key difference appears to lie in Brasell’s use of motion detectors, so that the students were studying their own motion, as opposed to Brungardt & Zollman’s use of videotaped motion, which the students did not produce. From these three studies, one can infer that the real-time effect can be substantial when the students are actively producing the motion under study, but that delay is less important, perhaps even completely unimportant, when the work involves prerecorded motion.

Brasell found that microcomputer-based instruction using student-generated motion was dramatically more effective than traditional instruction, although this effect was largely lost if the graphing display was delayed. In contrast, Beichner found that microcomputer-based instruction using prerecorded videotapes of motion was not significantly more effective than traditional instruction. (Brungardt & Zollman did not use a control group with traditional instruction.) Considering both Brasell’s study and Beichner’s, it appears that Mokros (1985) and Barclay (1985) were correct in suggesting that both the student-controlled kinesthetic experience and the real-time graph production are important aspects of the MBL experience.

1 The vast majority of available studies have involved tests of students’ ability to interpret graphs, rather then their ability to produce graphs. I have chosen the word “misinterpretation,” rather than “misconception,” with the intention of focusing on how the student interprets a particular graph on a particular occasion, rather than on a stable, internal cognitive structure. I use “misinterpretation” to mean that the interpretation is not correct, since it does not correspond to the phenomenon which is the subject of the graph, without commenting on whether it is based on a misconception or not.

2 I presented this interview in Prof. David Brown’s constructivism class in the fall of 1997.

3 I don’t know of any other contexts in which a graph would be likely to be incorrectly interpreted as a picture, but Mokros & Tinker appear to think that such contexts exist.

4 Only velocity results were given in detail.

5 I would have liked to see some comparison to students using traditional laboratory instruction, but that was not included in this study. Thus, it is not clear whether the gains are due to laboratory instruction, as opposed to lecture only, or whether they are due specifically to the MBL instruction and would not have been achieved with conventional laboratory instruction.

6 I am interested to know whether the context of sports was as familiar to the girls as to the boys, and whether any differential effects were observed based on sex or previous involvement in sports. Unfortunately, Brungardt and Zollman do not address gender issues at all. Brasell (1987c) discovered, to no one’s surprise, that the female students in her study had weaker graph interpretation skills, but gained considerably from MBL instruction. Other researchers generally paid little or no attention to attention to sex differences. Given the common belief that boys are better with computers than are girls, greater consideration of sex differences in MBL studies could be enlightening.

7 One might expect the students to learn more from this experience than from simply watching the computer generate graphs, since the students had taken the measurements themselves. This might combine an important aspect of traditional instruction--active involvement in the production of the graph--with an important aspect of computerized instruction--real-time display of the graph. Unfortunately, this study did not involve any other methods, traditional or computerized, so no comparison can be made.

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