Using Technology to Transform Mathematics Teacher Education: An Example from Graphing Instruction

According to Pea (****), technology can change education in two basic ways: it can amplify, and it can transform. When a new use of technology allows people to do the same things they had done before, but more quickly and accurately, it is amplifying. When the technology allows or encourages people to do different things, or to do the same things in profoundly different ways, it is transforming. Pea contends that the effects of transformation are less predictable than those of amplification, but likely to be more significant.

A study (Murphy, 1997) of microcomputer-based laboratory (MBL) work to teach students about line graphs, particularly those representing kinematics, provides an example of a use of technology in mathematics education. In the present paper, I examine how this use of technology fits Pea's framework, and look for general principles that may be extracted from this example for use in mathematics teacher education programs.

Before planning instruction, one must consider the initial knowledge states of the students. The present paper and the previous one (Murphy, 1997) examine two common misinterpretations of line graphs, graph-as-picture and slope/height confusion, which have been found in students from middle school to college. In the present paper, these misinterpretations are used to illustrate students' naive conceptions and ways that mathematics teacher education programs can prepare teachers to recognize and address these conceptions.

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Defining Terms: Assimilation, Accommodation, Amplification, and Transformation

Assimilation and Accommodation
Piaget (****) described learning through the processes of assimilation and accommodation. In assimilation, a learner who has some general framework for understanding a subject fits a few more details into that framework. For example, when a student who has a good general understanding of kinematics and line graphs finds that the velocity-time graph of projectile motion is linear, the student assimilates the information. Most of our daily learning is assimilation.

Sometimes a learner finds that some new information does not fit the previously established framework and, as a result, modifies the framework or discards the old framework in favor of a new one. That change in the framework is accommodation. For example, many students expect a distance-time graph to resemble the path traveled by the moving object. When such a student is confronted with contradictory examples, the student may accommodate this information by changing his or her concept of line graphs. Accommodation is more difficult, and learners tend to resist it. As Svec (1995) writes:

Challenging the students' understanding does not guarantee that conceptual change will occur. When faced with a discrepancy, students can change their own beliefs, rationalize the data away or they can become apathetic. Indeed, in lab, students would ask the instructor if they "should write down the correct answer or what the graph said," thereby attempting to rationalize away the data and maintain their current beliefs. (p. 21)

A great deal of attention is given to finding ways to facilitate the difficult process of accommodation. Indeed, the present paper will give more attention to accommodation than to assimilation. However, one should not conclude from this that assimilation is not important, or that accommodation is somehow better. Both are essential.

Mathematics teacher education programs should help pre-service teachers become familiar with both of these learning processes. Throughout the program, as various examples of teaching and learning are considered, pre-service teachers should consider the knowledge states of the learners and discuss which aspects of the lessons call for assimilation and which call for accommodation. They should also consider ways to facilitate these learning processes in different contexts, including being sensitive to the learner's resistance to the changes involved in accommodation.

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Amplification and Transformation
Pea (****) writes: "I take it as axiomatic that intelligence is not a quality of the mind alone, but a product of the relation between mental structures and the tools provided by the culture. Let us call these tools cognitive technologies" (p. 91). Cognitive technologies include written language, systems of mathematical notation, and more recently, computers and software. Pea discusses Bruner's idea of cognitive technologies as amplifiers of the mind (Bruner, 1966), and quotes Cole & Griffin (1980), who note that amplification, in the scientific sense, "refers specifically to the intensification of a signal (acoustic, electronic), which does not undergo a change in its basic structure" (p. 349) In the context of education, amplification is often interpreted to mean doing more easily, quickly, neatly, and accurately the same things that were done before the new technology was introduced. Pea argues that this is the least of the benefits new technologies can provide. In his view, new technologies can radically restructure education, changing both methods and goals in a way that extends far beyond amplification. Pea refers to such uses of technology as transformation.

The ideas of amplification and transformation discussed by Pea are analogous to Piaget's ideas of assimilation and accommodation, but Pea's terms apply to instructional methods rather than to an individual's construction of mental structures. Amplification refers to using technology to do essentially the same things that were done in the past, more easily, neatly, and quickly. For example, consider a teacher who draws graphs on a chalkboard to illustrate a lecture. If the teacher instead projects computer-generated graphs onto an overhead screen to illustrate the same lecture, that is amplification. The graphs are neater and more accurate, but the general instructional technique is the same: the teacher produces graphs to illustrate a lecture. Like assimilation, amplification involves minor alterations, additions, and improvements, but does not change the general structure.

Transformation refers to using technology to do new things, substantially different from what was done in the past. The teacher in the previous example may assign students to work in pairs at computers, producing their own graphs to explore the concepts formerly covered in the lecture. Such a change is transformation. The technology has facilitated a change from teacher-driven lecture with teacher-produced graphs to student-driven exploration with student-produced graphs. Such dramatic changes in methods of instruction may produce dramatic changes in student learning. Like accommodation, transformation is difficult, and is often resisted.

The naive observer may assume that all uses of technology are transformative. One sees the computers and other paraphernalia, and assumes that the classroom must be very different from what it was in the past. Beneath the surface appearance, the classroom may be substantially the same as it was before the computers arrived. Teachers have their own habits and expectations, and may simply incorporate the new technology into their old patterns, rather than changing their patterns. A teacher may use a spreadsheet to compute grades, a word processor to produce worksheets and quizzes, and a projector to draw graphs in lecture, but still teach in essentially the same ways as before.

My conception of amplification and transformation is a little different from the one Pea describes. He writes of "the roles played by cognitive technologies as reorganizers rather than amplifiers of mind" (p. 89, italics added). This seems to involve a binary concept of amplification and transformation; a use of technology either amplifies or transforms. Later, he contrasts his emphasis on transformation with other ways of examining the use of technology in education:

One could approach the question of technologies for math education in quite different ways than the one proposed. One might imagine approaches that assume the dominant role for technology to be amplifier: to give students more practice, more quickly, in applying algorithms that can be carried out faster by computers than otherwise. One could discuss the best ways of using computers for teacher record-keeping, preparing problems for tests, or grading tests. In none of these approaches, however, can computers be considered cognitive technologies. (page 98, italics in the original.)

Again, I see a dualistic perspective. A particular use of computers is either a cognitive technology or it is not, according to whether it transforms or amplifies.

I see transformation and amplification as the ends of a continuum, with most uses of technology somewhere in between these two extremes. It may be useful to think of a particular example as being primarily amplification or primarily transformation, but closer examination will often reveal elements of both. For example, when I started college, I wrote papers on a manual typewriter. Today, I use a word processor. From my point of view, the primary difference is that is it much faster and easier to make changes to my text when I am using a word processor. On the surface, this looks like a clear-cut case of amplification--faster and easier, but otherwise the same. A deeper look reveals something more. When writing was a laborious chore, I avoided it; now, I enjoy writing. When reversing the order of two paragraphs or rewording a sentence required retyping the whole page, I didn't revise. Nearly everything I turned in was a first draft; revision was too tedious and time-consuming to appear practical. Now I revise continuously throughout the writing process, continually improving content and form. In the process, I return repeatedly to my sources, trying to get the subtle shades of meaning exactly right. As a result, I gain a deeper understanding of my subject. I refer repeatedly to my thesaurus, trying to find the right words to express my points, and I reconsider the structure and punctuation of my sentences, trying to achieve a desired writing style. This has given me a whole new understanding of what it means to write, an understanding I did not develop by turning in first drafts and going on to the next assignment. All of this is the result of making editing faster and easier. It is classic amplification, but it has transformed writing for me.

Seeing amplification and transformation as points on the same continuum, rather than separate entities, affects my view of cognitive technologies. Unlike Pea, I would not say that uses of technology that are near the amplification end of the continuum are not cognitive technologies, although I do recognize that they are different from those that are more toward the transformation end. However, I do agree with Pea that exclusive attention to amplification overlooks some very important benefits of the new technology. As Pea writes: "Although quantitative metrics, such as the efficiency and speed of learning, may truly describe changes that occur in problem solving with electronic tools, more profound changes . . . may be missed if we confine ourselves to the amplification perspective" (page 92).

To broaden their perspective, pre-service teachers should read some of the literature on the benefits possible from using technology in the mathematics classroom, including literature on the concepts of amplification and transformation. They should also observe or participate in mathematics classes using technology in a various ways. Through reflection and discussion with peers, pre-service teachers should relate these examples of actual instruction to their readings about theories of instruction. This will make the theory more meaningful for them, by tying it to concrete personal experiences. In particular, pre-service teachers should identify aspects of the instruction that amplify or transform, and examine the effects of this instruction. This will help them develop their understanding of amplification and transformation, and learn to discriminate between uses of technology that are significantly different despite being outwardly similar. They should be asked to write about their conclusions, giving reasons why certain uses of technology are or are not desirable in particular contexts.

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Transformation from Whose Perspective?
Although Pea did not discuss perspective in his description of amplification and transformation, I think it is an important issue. When I consider whether instruction has been transformed, as compared to some prior state, I am thinking of a transformation in the learner's experience. If an innovation radically alters the teacher's understanding, and yet has no effect on classroom practice,(Footnote 1) then from the students' point of view nothing appears to have changed. I would not call this a transformation. My criteria for whether instruction has been transformed is whether or not the change makes a significant difference to the student.

However, I do not want to neglect the teacher's point of view. On the contrary, I see the teacher as the key to changes in the student's experience. A teacher who understands the potential of a technological innovation and wants to use it to transform instruction for the benefit of the students is likely to do just that, whereas a teacher who does not understand the potential of the technology or the idea of transforming instruction is likely not to make good use of the innovation. For this reason, interest in the student's experience compels attention to the teacher's knowledge, attitudes, and beliefs.

Teachers' knowledge, attitudes, and beliefs can be greatly influenced by their pre-service education. As various uses of technology are considered, mathematics teacher educators should ask pre-service teachers to consider the student's experience, especially how it is similar to, and different from, the student's experience before the technology was introduced. Where feasible, the pre-service teacher should experience using the technology to learn mathematics, and then reflect on how this experience compares with his or her previous learning experiences. This will help the pre-service teachers understand what it means to change instruction in ways that are meaningful to the student.

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Learning to Transform Mathematics Education
Just as using new technology permits transformation, but does not require it, so transformation permits improvement, but does not guarantee it. A profound change may, of course, be a change for the worse. This possibility justifies some of the caution (also characterized as conservatism) that teachers may show toward proposed changes. Teachers need to know how to evaluate technological innovations and the transformations they make possible, so that the teachers can select the forms of amplification and transformation most beneficial for their purposes.

In the present paper, I consider particular examples of technology being used both to transform and to amplify, in an effort to produce both assimilation and accommodation in students who are learning to interpret and produce line graphs. In the process, I hope to illustrate how a teacher or curriculum designer might examine a variety of apparently similar uses of technology and select the most effective for the problems at hand.

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Common Misinterpretations of Line Graphs: Graph-As-Picture and Slope/Height Confusion

In a variety of disciplines, students are asked to produce or interpret line graphs showing the value of some variable over a period of time. For many students, these graphs are not intuitive. Researchers have documented two common misinterpretations of graphs: graph-as-picture and slope/height confusion. Consideration of these examples may shed some light on the implications of students' naive conceptions.

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Graph as Picture
In the graph-as-picture (GAP) misinterpretation, the graph is seen as a picture of the event or a map of the path of a moving object. GAP has been observed in middle school students (Barclay, 1985; Berg & Phillips, 1994; Clement, Mokros, & Schultz, 1985; Mokros & Tinker, 1987), high school students (Berg & Phillips, 1994), and college and university undergraduates enrolled in physics courses (McDermott, Rosenquist, & van Zee, 1987). Barclay (1985) noted that GAP was also common in middle school teachers. In all of these studies, GAP has been found only in the context of kinematics. Mokros & Tinker (1987) suggest that GAP may be used in other areas, and was not found beyond kinematics in their study simply because they did not include any non-kinematics questions that might reveal a GAP misinterpretation. It is possible that uses of GAP in other areas have been overlooked in the research literature, due to the heavy reliance of many graph-interpretation studies on examples drawn from kinematics. It is also possible that GAP is specific to kinematics, and not used in other contexts. This is a question to be resolved by future research.

A typical instance of GAP misinterpretation is illustrated in figure 1. The student is shown a sketch of a ball rolling along a track, or perhaps observes an actual ball rolling on a track, and is asked to draw a graph of the velocity of the ball over time, or to select the graph from a collection of graphs provided. Students with a correct understanding will show the velocity increasing when the ball rolls downhill, and decreasing when the ball rolls uphill. The velocity-time graph will resemble the track, but will be inverted. Students using GAP will draw the velocity-time graph as a copy of the track shape, without inverting it. These students may also show the acceleration-time graph of the ball's motion as a copy of the shape of the track.

Students also may be given a description of a motion event, such as a person walking away from home and then returning, and be asked to produce or select a graph of the distance from home with respect to time, as illustrated in figure 2. Students using GAP may show the graph as a closed loop, forming a map of the person's movements. 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. This indicates a lack of understanding of the significance of the time axis, since such a graph shows the person moving backwards in time. Rather than considering the meanings of the axes, the student using GAP is considering the graph to be a picture of the event. Changing the way the student interprets graphs, so that graphs are seen as more abstract and less pictorial, is a major act of accommodation.

To learn to help students with this accommodation, pre-service teachers need some understanding of students' prior knowledge states. Pre-service mathematics teachers should read some of the literature on student's naive conceptions in mathematics and attempt to relate this to their observations of particular students. When naive conceptions, such as thinking of a graph as a map, are discovered, pre-service teachers should consider ways to address these conceptions and foster accommodation. Ideas may come from the pre-service teachers, cooperating teachers, mathematics teacher educators, and the literature. Pre-service teachers should discuss the likely effects of the various suggestions, and write about their reasons for preferring particular ones. If feasible, they should have the opportunity to work with students to get first-hand experience helping a student work through accommodation and emerge with a new understanding.

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Slope/Height Confusion
The second common misinterpretation is slope/height confusion (SHC). Students using SHC will find the point with the steepest slope when the correct answer involves the highest point, or vice versa. Like GAP, SHC has been observed in a variety of populations, including middle school students (Barclay, 1985; Clement, Mokros, & Schultz, 1985; Mokros & Tinker, 1987), high school physics students (Brasell, 1987a, 1987b), and college and university undergraduates enrolled in physics courses (McDermott, Rosenquist, & van Zee, 1987; Svec, 1995; Thornton & Sokoloff, 1990). Barclay (1985) noted that SHC was also common in middle school teachers. Unlike GAP, SHC has been observed in contexts other than kinematics. Clement, Mokros, & Schultz (1985) showed students graphs of temperature vs. time, and asked when the temperature was rising most rapidly. More students selected the point of highest temperature than selected the correct point. Clement, Mokros, & Schultz classified this as slope/height confusion.

In kinematics, it is difficult to distinguish SHC from confusion among distance, velocity, and acceleration, since the highest point on the velocity-time graph corresponds to the point of steepest slope on the distance-time graph, and so on. When students select or draw an incorrect acceleration-time graph that resembles the correct velocity-time graph, the researcher may interpret this as SHC, confusion between acceleration and velocity, or lack of understanding of acceleration. Unless one has more information about the student's understanding of graphing and kinematics, it is impossible to be certain which of these explanations is the most accurate.(Footnote 2) I have argued (Murphy, 1997) that SHC may be a manifestation of a deeper confusion between the value of a quantity and the amount of change of the quantity. When the quantity and its change are displayed graphically, the student appears to have problems interpreting the graph. However, if the same concepts are represented in different ways, the student's problems may remain. Instruction limited to graph interpretation may not be sufficient to address this situation.

If SHC is indeed a manifestation of a deeper confusion, then the task of remedying SHC is both more difficult and more important than would be the case if SHC were simply a graphing problem. In addition to mastering an abstract system for representing quantities and changes in quantities, students have to learn the meanings underlying the representations. This presents an instructional challenge, the solution of which could pay dividends in many areas. With this in mind, I suggest that instruction should give particular attention to the distinction between a quantity and a change in that quantity, and that this should be addressed in a variety of ways, including, but not limited to, graphical representations.

What at first glance appears to be a problem of graph interpretation may in fact be a conceptual problem with quantities and changes in quantities. As in the example of this so-called slope/height confusion, students' errors sometimes indicate a deeper misunderstanding than a casual examination would suggest. Mathematics teacher educators should encourage pre-service teachers to consider a variety of explanations for patterns in students' errors, and to devise ways to probe further, to determine which explanations are closest to being correct. A small research project in which the pre-service teachers interview students could be very helpful in this regard.

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Amplification and Transformation in Microcomputer-Based Laboratory Instruction

Several studies have used some form of computerized instruction to teach graph production and interpretation. Although the forms of instruction in the various studies may appear on the surface to be quite similar, some have been very successful while others have not. I will attempt to account for these differences by using Pea's ideas of amplification and transformation. When the computer is used for amplification, the resulting instruction is not substantially different from traditional instruction, although it may be more efficient. In these cases, one would expect the experimental method to produce results similar to those produced by the traditional method.

When the computer is used to transform instruction, the resulting instruction is quite different from the traditional instruction, and thus may produce quite different results. The results of transformative uses of technology are more difficult to predict than the results of amplification. A transformation may produce a great improvement, a different but equally effective method, or even a turn for the worse. Transformation may also change the goals as much as the methods, so that it can become difficult to compare the previous outcomes to the new ones.

Several researchers have used microcomputer-based laboratory (MBL) instruction in an effort to improve students' graph interpretation skills. Microcomputer-based laboratories (MBLs) involve a computer attached to a probe that 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).

Studies comparing MBL instruction with motion sensors to traditional instruction have shown significant differences in graph interpretation ability, favoring the MBL instruction (e.g. Brasell, 1987a, 1987b, 1987c; Svec, 1995; Thornton & Sokoloff, 1990). This supports the assertion that MBL instruction with motion sensors can be a transformative use of technology; if it were merely amplification, one would not expect the large differences that have been found.

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Reasons for the Success of MBL Instruction
Several writer have speculated about which attributes of MBL instruction are most responsible for the beneficial transformative effects. 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. (page 8)

Beichner (1990) writes:

The VideoGraph technique [developed by Beichner as an alternative to MBL instruction] 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. (page 812)

Linn. Layman, & Nachmias (1987) write:

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. (page 245)

Mokros & Tinker (1987) write:

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. (page 369)

Thornton & Sokoloff (1990):

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. (page 859)

These writers and others have stressed: a) the strong connections established between concrete motion and abstract graphs; b) the ability and motivation to experiment; c) the real-time display of graphs; d) the freedom from the more tedious aspects of data collection and graph production; e) direct student control over the data (motion) used in the experiment; f) the use of multiple learning modalities, especially the kinesthetic; and g) the variety of activities allowed by the same set of flexible tools.

The motion sensor work does seem to encourage experimentation, and to help students connect the concrete physical experience of motion to the abstract graphs. However, I put these attributes into a different category from the other five attributes of MBL instruction listed above. The other five attributes are all characteristics of the equipment or the instructional design--causes, if you will. The propensity to experiment and the strong connection formed between motion and graphs are effects of the type of instruction. I take these effects as indications that MBL instruction is transformative, since traditional instruction is not as successful in encouraging creative experimentation or in linking the concrete to the abstract. In what follows, I will attempt to determine what aspects of the equipment or instructional design are most instrumental in creating the transformation that produces these effects.

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). This is a dramatic change from the traditional laboratory, where the students take careful measurements, make tables of data, and then plot points to create the graphs. In traditional instruction, the event is long over before the students ever see a graph. The immediacy of real-time graphing can be considered transformative, when compared to traditional, pencil-and-paper methods.

MBL use frees students from the more tedious aspects of graph production, such as making tables of data and plotting points. This allows students to produce more graphs, more easily, quickly, neatly, and accurately than was possible with pencil-and-paper methods. This is a classic example of amplification. However, it is also an example of the connection between amplification and transformation. If the graphs were only a little easier to produce, or were produced only a little faster than previously, this would not significantly transform instruction. Since the MBL can produce dozens of graphs in less time than it would take a student to draw even one graph, and since the effort required to produce a graph by hand is often sufficient to dissuade the student from making the attempt, this amplification is so substantial as to become transformative, in comparison to traditional methods of producing graphs. A student who formerly would have made only the few graphs that were required will choose to experiment further with the MBL, in part because this is so quick and easy to do. This makes the freedom from drudgery one of the two aspects of the MBL experience that produce the propensity to experiment.

The other aspect of the MBL experience that encourages experimentation is the control the students have over the motion. Students using the motion sensor can vary the motion in any way they choose. They can spontaneously experiment with moving toward or away from the sensor, moving quickly or slowly, speeding up or slowing down, or any other motion that occurs to them. In one class using MBL motion sensors, I observed a student walking in a circle at a constant speed, with the intention of producing a sine wave. It was entirely her own idea; it took me a while even to understand what she was doing. This is quite a change from teacher-directed instruction in which students use prepared data from a textbook or take measurements of some process that the students simply observe. The control allowed by the MBL lessons can be considered transformative, in comparison to traditional instruction on graphing that do not allow the student any control over the data.

The MBL lessons using motion sensors bring multiple learning modalities into play. Most notable among these is the kinesthetic component of the lessons. The data the students are using is produced by their own physical motion. Several researchers have suggested that the kinesthetic aspect of the MBL experience is vital to linking the abstract graph to the concrete motion. Since this kinesthetic component is not present in most traditional instruction on graphing, it can be considered transformative.

Since most of the studies have involved only brief periods of instruction, the flexibility of the tools in supporting a variety of experiments and levels of sophistication has not been explored in the research literature. This flexibility is likely to be quite useful to a teacher with limited resources, or one who is concerned that an endless parade of new toys could be distracting. The tools, once purchased and introduced to the students, can be reused endlessly in a variety of contexts. When the students have become accustomed to the capabilities of the tools, they may begin to think of new ways to use them to explore new questions. This could be transformative, but I won't discuss it further in this paper due to the lack of available data.

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Research into Reasons for the Success of MBL Instruction
Heather Brasell (1987a, 1987b) hypothesized that the real-time display of graphs was the most important aspect of MBL instruction. She used pre- and post-tests to study the effects of one hour of instruction on the graph abilities skills of 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 produced about 20 to 30 seconds after the motion was completed.

Brasell found that the Standard MBL treatment group significantly outperformed all other groups, including the Delayed MBL group, on the post-test. Most of the difference was found in the test 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 statistically significant. Brasell estimates that the real-time feature accounted for about 90% of the improvement that MBL offered over pencil-and-paper instruction. This indicates that the real-time aspect of the MBL instruction is transformative, with respect to traditional instruction, and that this transformation substantially improves student understanding of distance graphs.

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, that produces graphs of motion shown in a 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 or not the students viewed an actual motion event. (In any case, the students did not view the event that had been videotaped; 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. (pages 811-812)

He adds: "The kinesthetic sense is a strong one and appears to make a difference in kinematics MBL's" (page 813).

The VideoGraph instruction may be transformative; the graphs are displayed in real time as the videotape is played, which cannot be done with pencil-and-paper or chalkboard methods. Even so, it certainly was not a transformation that produced noticeable results. It could also be argued that the VideoGraph instruction was not transformative at all. In traditional instruction, students watch as their instructor draws graphs to which the students have no personal connection. In the VideoGraph instruction, students watched as the computer drew graphs of motion to which they had no connection. There are strong similarities between the experiences of the students in the two types of instruction, which could justify the assertion that the VideoGraph instruction did not represent transformation when compared to traditional instruction.

Brungardt & Zollman (1995) followed up on 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.(Footnote 3) 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.(Footnote 4) 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 two studies, one can infer that the real-time effect can be substantial when the students actively produce the motion under study, but that delay is less important, perhaps even completely unimportant, when the work involves prerecorded motion.

Given Beichner's study, one might have predicted Brungardt & Zollman's result. Since Beichner found that the outcomes of videotape-based study with real-time display of graphs are not significantly different from the outcomes of traditional instruction, it seems clear that videotape-based study without real-time display also will not be noticeably better than traditional instruction (unless one wishes to hypothesize that real-time display is a detriment, which is implausible in light of Brasell's study). Assuming that it is also not significantly worse, one has to conclude that videotape-based instruction, with or without real-time display, is likely to have about the same effect as traditional instruction.

This appears to support the idea, discussed above, that the videotape-based instruction is not transformative when compared to traditional instruction. Brasell found that the benefits of the transformative use of MBL motion sensors are realized only when the students have the immediate feedback provided by the real-time display of the graphs. Beichner found that his videotape-based instruction lacked the transformative effect of the motion sensor instruction. In the absence of a transformation, there is no reason to worry about whether or not the feedback is prompt enough for the transformative effect to be realized, so Brungardt & Zollman's result is not surprising.

I see student control over the motion and immediate feedback as being twin keys to the effective transformation observed by Brasell and other researchers. If the student can control the motion, then he or she can spontaneously perform many little experiments in the course of the lesson: What happens to the graph if I walk toward the sensor? Away? Quickly? Slowly? In a circle? and so on. These mini-experiments can be varied or repeated according to the student's curiosity and interest, to create a customized lesson for each individual student. Also, the student is likely to get a sense of mastery by being in control. For these reasons, I think student control is the most transformative aspect of the MBL experience.

However, the student has to be able to see some results of his or her chosen motions in order to learn from the experience. When the graph is displayed immediately, it is clear to the student which features of the graph were caused by which features of the motion. When there is a delay, the student may forget some details of the motion, and may have more trouble matching those details that are remembered with the appropriate features of the graph. Thus, the real-time graphing is necessary for the student to be able to take full advantage of having control of the motion.

It may be that the kinesthetic experience is also important, particularly for students who are more kinesthetic learners. Since I can find no research involving student control without the kinesthetic component, the relative importance of these two factors remains a mystery.(Footnote 5)

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Implications for Mathematics Teacher Education
Pre-service mathematics teachers should read articles examining uses of technology in mathematics education, as is done above for MBL graphing instruction, and discuss these ideas with one another. They should focus on critically critically evaluating claims made about various innovations, and considering which aspects amplify and which transform. They teachers should learn to distinguish between speculations about the technology, such as the quotes above from various authors about why MBL instruction is successful, and the conclusions drawn from research, such as Brasell's study of the effects of delayed vs. real-time display of graphs.

After examining a few such articles, each pre-service teacher (or pair or small group of pre-service teachers) should select some example of technology used in mathematics instruction and research this example in detail, as I have done here with microcomputer-based instruction in graphing. By doing this case study, the pre-service teachers will gain experience in considering how and why a particular implementation is more useful than another in helping students to learn mathematics. Teachers use this kind of analysis when they decide how to best use the limited resources that are available for them to use in their classrooms.

The pre-service teachers should present their work to their classmates, who should ask questions and provide suggestions. After the group has discussed a project, the author(s) should reflect on this discussion and improve the project. In this way, the whole class will learn about each of the various technologies that have been examined in the case studies. In addition, the pre-service teachers will learn how to critique arguments made for and against various uses of technology, and the individual authors will be able to identify weaknesses in their reasoning and deepen their understanding.

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Graphing and Cognitive Development: Which Comes First?

Both McKenzie & Padilla (1984) and Adams & Shrum (1990) found graphing ability, as measured by the Test of Graphing in Science (TOGS)(Footnote 6) , to be significantly correlated with cognitive level, as measured by the Group Assessment of Logical Thinking (GALT). Similarly, Berg & Phillips (1994) found that ability to produce and interpret graphs was correlated to development of specific mental structures. While this comes as no surprise, it has launched something of a chicken-and-egg debate. Should we work on improving cognitive level and building mental structures, in hopes of gaining graphing proficiency as a side benefit, or should we work on graph interpretation directly, with the possible side benefit of raising cognitive level?

Many researchers start with an interest in the more applied area of instruction on graphing interpretation, find that MBL works well, and speculate that there may be cognitive benefits that extend beyond developing an ability to interpret the graphs used in the instruction. Linn, Layman, and Nachmias (1987) found that MBL instruction in chemistry and temperature that focused on improving graphing skills also improved students' ability to interpret graphs representing motion, although motion was not included in the instruction. Apparently, the students' graph interpretation abilities had improved in ways that were not dependent on the context of the instruction. Was there an underlying advance in cognitive development as well? Different researchers have advanced different theories about the effects on cognitive development.

As mentioned earlier, 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 concrete motion and the abstract graph, and thus "transfer the event-graph unit (already linked together) into long-term memory as a single entity" (Beichner, 1990, p. 804).

The apparent success of MBL instruction in helping students to link the concrete and the abstract inspired Mokros (1985) to write:

The power of the intervention stems partly from the fact that it reinforced many learning modalities. The kinesthetic experience of using one's own movements as "data" was linked with the visual experience of seeing graphs of these movements on the screen. By linking the concrete and the abstract, MBL may be providing a bridge that facilitates the development of formal operational thinking. (p. 0)

and:

The study also suggests that by linking the concrete and the abstract, the computer may serve as an important "carrier" of problem solving skills. Viewed within Piagetian theory, the value of MBL may be as a bridge between concrete and formal operations. It is possible that intensive juxtaposition of these concrete and formal operations could facilitate the development of formal operational thinking. (p. 3)

Berg & Phillips (1994) approach the question from the perspective of cognitive theory, and take issue with Mokros & Tinker's (1987) suggestion that MBL instruction may significantly aid cognitive development, serving as a "bridge between concrete and formal operations" (p. 381). Berg & Phillips believe that there are no "quick shortcuts to cognitive development" (p. 324). With reference to the work of Mokros & Tinker and others, Berg & Phillips ask: "We might expect some growth in being able to identify or correctly draw the right graph from recalling the experience, but does a permanent and transferable interpretive ability exist?" (p. 339) They appear to doubt that it does. Rather than using MBL instruction to develop graphing skills, with an expectation of corresponding advancement in cognitive structures, Berg & Phillips believe instruction should be designed to build (gradually, since presumably there is no other way) the cognitive structures from which graphing abilities are expected to follow.

In Berg & Phillips's view, to focus on graphing before the mental structures are in place puts the cart before the horse. They assert that "Evaluating a student's ability to construct or interpret graphs is certainly punitive if students have not yet developed the logical thinking structures necessary to make sense out of the material" (p. 340). While this view would allow instruction on graphing construction and interpretation, it would rule out assessment until the logical thinking structures are judged to be in place, effectively keeping graphing out of the curriculum for younger students.

Berg & Phillips are unfortunately vague on ways to build logical thinking structures. I think that Mokros & Tinker are right about MBL instruction helping to do that. I don't share Berg & Phillips's interpretation of Mokros & Tinker's statement about MBL instruction serving as a "bridge between concrete and formal operations" (p.381). Berg & Phillips characterize this as a quick shortcut to cognitive development, which they are certain is impossible. Mokros & Tinker did not say that the student receiving MBL instruction would move from concrete to formal operational overnight. There is room in what Mokros & Tinker have written for the idea that cognitive development is indeed gradual, with no "shortcuts," but that some forms of instruction can encourage and enhance the student's development more than others, and that MBL instruction is one of the more valuable.

I have never liked chicken-and-egg debates anyway. The truth is seldom one or the other; usually it is both. Appropriate instruction (including MBL instruction) can build graphing skills and help advance cognitive development, advances in cognitive development can make possible improvements in graphing skills, and the cycle of learning and growing continues.

This connection between mental structures, which cannot be seen, and observable behaviors, such as correctly drawing or selecting a graph to represent a given motion, is a key point in understanding the theory and practice of education. Mathematics teacher educators should frequently ask pre-service teachers to explore the cognitive implications of observed behavior, as well as the behaviors that might accompany the cognitive processes under discussion. Pre-service teachers should learn to read the literature critically, noticing when a writer emphasizes behavior but slights cognition, and vice versa. For a complete view of an area, such as graph interpretation, pre-service teachers should read both theory about cognitive processes and research into behaviors predicted by the theory. Pre-service teachers should see examples of debates, such as the one between Mokros & Tinker and Berg & Phillips, so that they will learn to critically evaluate and compare theories, rather than accepting the opinion of the first author they read.

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Summary and Conclusions

Just as Piaget's ideas of assimilation and accommodation provide a useful theoretical framework for the study of learning, Pea's concepts of amplification and transformation provide a useful theoretical framework for examination of applications of technology to mathematics education. Close examination of most applications will reveal that both amplification and transformation are involved to some extent, although a particular example may lean more toward one or the other. Pre-service teachers should be familiar with these theoretical frameworks, and use them to analyze their observations of mathematics instruction.

Students often misinterpret line graphs, taking graphs to be pictures and confusing slope with height. Instruction in graph interpretation must take these naive conceptions into account, and facilitate the accommodation that is necessary for the student to construct a better understanding of line graphs. Similarly, to design appropriate instruction, teachers need to know about the initial knowledge states of their students. Pre-service teachers should learn to identify students' prior conceptions and plan instruction accordingly.

Researchers have used several different applications of technology, including videotape-based instruction and MBL instruction with motion sensors, to teach students how to interpret graphs. Some of these have been more successful than others. Key features of the successful applications are real-time display of graphs and (kinesthetic) student control over the data. Both of these represent transformation, when compared to traditional instruction, because they are significantly different from what has been done in the past. Pre-service teachers should learn to read the literature on available instructional technologies and determine what factors are most significant in a particular context, as was done here for technology used in graphing instruction. In the process, they will learn about particular technologies. It should be noted that knowledge of specific technologies is no substitute for knowing how to examine new information, because new technology is continually being introduced.

There is some debate over the extent of the gains achieved with MBL instruction. Some researchers believe that the experience serves as "bridge between concrete and formal operations," in a way that actually stimulates cognitive development. Other researchers believe that the improvement in graphing skills is more shallow and short-lived, and that real cognitive growth is not produced. Unfortunately, much of this debate is speculative, since long-term comparative studies of the effects of MBL and traditional instruction on graphing skills and cognitive development are scarce.(Footnote 7) Pre-service teachers need to understand the connections between observable behaviors and cognitive structures. They also must be able to read the literature critically, distinguishing between theories and established facts, and to compare theories to one another.

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References

Adams, D. D., & Shrum, J. W. (1990). The effects of microcomputer-based laboratory exercises on the acquisition of line graph construction and interpretation skills by high school biology students. Journal of Research in Science Teaching, 27, 777-787.

Barclay, W. L. (1985). Graphing Misconceptions and Possible Remedies Using Microcomputer-Based Labs. (Technical Report Number TERC-TR-85-5). Cambridge, MA: Technical Education Research Center.

Beichner, R. J. (1990). The effect of simultaneous motion presentation and graph generation in a kinematics lab. Journal of Research in Science Teaching, 27, 803-815.

Berg, C. A., & Phillips, D. G. (1994). An investigation of the relationship between logical thinking structures and the ability to construct and interpret line graphs. Journal of Research in Science Teaching, 31, 323-344.

Brasell, H. (1987a). The effect of real-time laboratory graphing on learning graphic representations of distance and velocity. Journal of Research in Science Teaching, 24, 385-395.

Brasell, H. (1987b, April). The role of microcomputer-based laboratories in learning to make graphs of distance and velocity. Paper presented at the Annual Meeting of the American Educational Research Association, Washington, DC.

Brasell, H. (1987c, April). Sex differences related to graphing skills in microcomputer-based labs. Paper presented at the 60th Annual Meeting of the National Association of Research in Science Teaching, Washington, DC.

Brungardt, J. B., & Zollman, D. (1995). Influence of interactive videodisc instruction using simultaneous-time analysis on kinematics graphing skills of high school physics students. Journal of Research in Science Teaching, 32, 855-869.

Clement, J., Mokros, J. R., & Schultz, K. (1985). Adolescents� Graphing Skills: A Descriptive Analysis. (Technical report number TERC-TR-85-1). Cambridge, MA: Educational Technology Center.

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McKenzie, D. L., & Padilla, M. J. (1984, April). Effects of laboratory activities and written simulations on the acquisition of graphing skills by eighth grade students. Paper presented at the 57th Annual Meeting of the National Association of Research in Science Teaching, New Orleans, LA.

McKenzie, D. L., & Padilla, M. J. (1986). The construction and validation of the Test of Graphing in Science (TOGS). Journal of Research in Science Teaching, 23, 571-579.

Mokros, J. R. (1985). The Impact of Microcomputer-Based Science Labs on Children�s Graphing Skills. (Technical Report Number TERC-TR-85-3). Cambridge, MA: Technical Education Research Center.

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Thornton, R. K., & Sokoloff, D. R. (1990). Learning motion concepts using real-time microcomputer-based laboratory tools. American Journal of Physics, 58, 858-867.

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Footnote 1
I know this is unlikely; I am speaking hypothetically here.