Rate of Change and Tangent

Preparing the Ground and Planting the Seeds:
Initial Knowledge States and Early Concept Development in Introductory Calculus Students

by Lisa Denise Murphy

Laying the Foundations for the Derivative
Summaries of Studies
Lessons Learned
References

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This page last revised Januray 19, 2000

Laying the Foundations for the Derivative

My dissertation research involves helping students to understand the concept of the derivative, which I believe is built upon a foundation laid by other, earlier concepts. Thus, I am interested in students’ understanding of those foundational concepts and the ways in which that understanding affects students’ efforts to learn calculus. In this review, I discuss students’ concept of rate of change and tangent, along with a few other concepts that are necessary for the study of calculus. The concept of limit is not included in this review, since it is the subject of another review (Murphy, 1998). Together, rate of change, tangent, and limit form the foundation for understanding the derivative.

My choice of topics for these reviews is based on the assumption that mathematical concepts are built one upon another, with students’ understanding of each concept resting on certain previous concepts, and in turn helping to lay the foundation for later concepts. In particular, I assume that the specific concepts of limit and rate of change are necessary for the successful study of derivative. This idea of mathematics as being in some sense linear, with concepts built up in a necessary order, is not universally accepted.

Some researchers¹ claim that this ordering of concepts is not due to the nature of mathematics, but is instead a quirk of our educational system. They maintain that this artificially induced ordering of concepts holds some students back, keeping them out of mathematics. A student fails to grasp some concept that we consider to be an important prerequisite for calculus, for example, and is thus kept out of calculus. According to this theory, such a student might be very successful in calculus if only we could give up our insistence that concepts be learned in a specific order. By this reasoning, the traditional ordering--or, indeed, any fixed ordering--of topics in the curriculum is inherently inequitable, because it is thought to contribute to preventing some students from learning mathematics. Racial and ethnic minority students are believed to be particularly likely to be injured by an insistence on prerequisites.

In deciding to approach students’ learning and understanding of the derivative by studying students’ conceptions of rate of change and limit, I am assuming that this theory--that there is no natural ordering of mathematical concepts--is false. However, this should not be taken as a wholesale endorsement of the status quo. The natural ordering, to the extent that such exists, may not always match the traditional² ordering. Indeed, the two cannot be entirely the same. In mathematical terms, a syllabus is a “total ordering”--for any two concepts in a particular syllabus, it is possible to say which is encountered first in that syllabus. By contrast, the natural ordering of mathematical concepts is a “partial ordering”--some pairs of concepts have a natural order, but other pairs may be learned equally well in either order.

All of the researchers whose papers I have read for this review either assume a natural ordering of mathematical concepts or find such an ordering in the concepts involved in their studies. For example, Paul White and Michael Mitchelmore (1996) conclude their paper “Conceptual Knowledge in Introductory Calculus” by writing:

Most students appeared to have an abstract-apart concept of variable that was blocking meaningful learning of calculus. . . . The inevitable conclusion is that a prerequisite to a successful study of calculus is an abstract-general concept of a variable at or near the point of reification. Even a concept-oriented calculus course is unlikely to be successful without this foundation. However, just as much experience manipulating and modeling arithmetical expressions must precede reification of algebraic expressions, students probably need to spend a considerable amount of time using algebra to manipulate relations before they can achieve a mature concept of a variable. It is not, therefore, realistic to attempt to provide adequate remedial activities within a calculus course for those students who have an abstract-apart concept of a variable. Entrance requirements for calculus courses should be made more stringent in terms of variable understanding, or an appropriate pre-calculus course should be offered at the university level. (page 93)

Clearly, White and Mitchelmore believe in a natural ordering of concepts, with an abstract-general concept of variable naturally preceding the central concepts of calculus. They also call for stringent enforcement of prerequisites, so that students entering calculus can be expected to have acquired the necessary understanding of variable. However, this does not mean that following the natural ordering of topics and enforcing prerequisites will solve all problems in the teaching and learning of calculus. There are problems inherent even in the natural ordering. David Tall states this explicitly (Tall, 1989):

The implication of Piagetian stage theory is that there are certain fundamental obstacles that occur for us all. . . . I postulate that the reason for the belief in fundamental obstacles arises from the fact that certain concepts have a degree of complexity that makes it necessary to become acquainted with them in a certain order. For example, fractions are, of necessity, more complicated than whole numbers, and experience with operations on whole numbers leads to the implicit generalization that “multiplication makes bigger,” which leads to a cognitive obstacle when the individual meets the multiplication of fractions less than one.

However, some topics that are traditionally taught in a certain order may not have the a priori property that one concept is essentially more complex than the other. For instance, fractions are usually met in traditional syllabuses before negative numbers, but there is no reason why, given an appropriate context, the two topics should not be taught in the reverse order. (page 88, italics added)

Tall goes on to suggest that traditional orderings should be reconsidered in an effort to prevent as many obstacles as possible, but recognizes that certain orderings are necessary and give rise to what he terms “fundamental obstacles,” which cannot be prevented. As Tall indicates, assuming that there is a natural partial ordering does not end the discussion. One must determine what specific concepts naturally precede other concepts. In some cases, there will not be sufficient difference in complexity or dependence of concepts to create a natural ordering, but there still may be advantages to one ordering over another for certain students or in certain situations. When an ordering has been determined, one must discover the obstacles inherent in this ordering, and consider how they may be surmounted.

The researchers whose work is reviewed here and in the earlier review on students’ conceptions of limit (Murphy, 1998) appear to have assumed that there exists some natural or inherent ordering of rate of change, tangent, derivative, and certain underlying concepts. Although not all papers address this assumption explicitly, this research investigates the relationships among these concepts, thus helping the reader to distinguish natural orderings from orderings imposed by custom or whim. In addition, this research aims to identify obstacles created by prior knowledge or concepts, many of which are partially correct but come with connotations that impede the learning of more formal concepts. Some of these papers also address ways to help students surmount these obstacles.

At this point I hope it is clear that none of these researchers are advocating inequity. Rather, they believe that the most effective methods for teaching mathematics to all students must take into account the inherent ordering of mathematical concepts. Under this assumption, eliminating relevant prerequisites can be expected to lead to failure for under-prepared students, not to increased access to mathematics. Thus, equity is advanced, not by rejection of order, but by improving our understanding of natural orderings and their consequences, and basing instruction on this understanding.

In addition to this assumption of a natural ordering of mathematical concepts, the studies described here have other features in common. All were intended to discover how students think about specific concepts. Some studied how students change their concepts, others simply took snapshots of students’ thinking at some moment in time, but in all cases the object of study was the student’s conceptions. These are studies of properties, since they concentrate on identifying and describing concepts, thought processes, and ways that concepts are changed.

All of these researchers treated the individual student as the unit of analysis. There exist studies (not related to calculus instruction, as far as I can find) in which researchers study the classroom culture, and consider what ideas are taken as shared and how these evolve through negotiation and other social processes. The studies discussed here do not have that perspective. All interviews are with one student at a time, rather than groups of students. To the extent that the classroom culture influences the ways that students learn and think about the derivative, this focus on the individual to the exclusion of the group restricts the researcher’s ability to fully understand the process of concept development. However, it has the advantage of narrowing the field of study to a manageable size. It would be interesting to compare and contrast these studies with studies of the development of the same mathematical concepts in groups of students working together, if such studies could be found. I suspect that examining concept formation in calculus from both the intrapersonal and the interpersonal perspectives would provide additional insight, which could be used to improve instruction.

Summaries of Studies

Hauger
Garnet Hauger has written papers on two different aspects of the same study of high school and college students’ knowledge of rate of change. The study involved hour-long interviews with 12 high school pre-calculus students, 15 college students enrolled in second-semester calculus, and 10 upper division college mathematics majors. Rather than focus on the knowledge that the students lack, which may impede their work on the interview tasks, Hauger focused on the knowledge that the students have, and how they bring that to bear on the problems at hand.

Hauger examined three categories of rate of change problems. Global rate of change is qualitative, and deals with the overall shape of a graph. For instance, one might look at a curve and say that it represents the motion of an object moving at a constant slow pace, gradually speeding up, or changing direction. Interval rate of change is quantitative. It corresponds to the average speed of a moving object over a specified interval. Instantaneous rate of change is also quantitative, but corresponds to the speed of a moving object at only one instant in time, rather than the average speed over an interval. Instantaneous rate of change is given by the derivative of the function evaluated at the point of interest. Interview items, representing functions both by graphs and by tables of values, were created to probe the students’ understanding of each of these three categories of problems.

Hauger: Interval and Instantaneous Rates of Change
The first paper (Hauger, 1995), an analysis of responses of all 37 participants to the fifth interview task, deals with resources students bring to the study of instantaneous rate of change. Hauger writes:

Of particular interest in the part of the study reported here are the resources students use to address instantaneous rate of change, specifically features of interval rate of change used to address instantaneous rate of change. . . . What interval rate of change quantities do students use in rate of change situations and how do these quantities support their thinking in these situations and advance their knowledge of rate of change? . . . Of interest here are the resources students use to get at instantaneous rate of change when given graphs and tables of values and what this tells us about how students construct knowledge of instantaneous rate of change. (pages 11-12)

This paper has significant qualitative and quantitative aspects.³ Responses were categorized, and the numbers of responses in each category were tabulated and compared across the three groups: pre-calculus students, calculus students, and post-calculus students. Excerpts from the interviews were presented to illustrate the thinking behind the answers summarized in the tables.

In this fifth task, subjects were shown a graph of the number of yeast cells in a culture over an period of eighteen hours. In part (a) of the task, subjects were asked to describe what was happening to the yeast population over the whole time period. Each student gave one of two general descriptions. Although both are reasonably accurate, they reveal different ways of thinking about the graph and the growth of the population. Some students divided the graph into three parts, reducing the problem to three approximately linear segments. Growth is slow in the first segment, rapid in the second, and slow again in the third. There is no talk of speeding up or slowing down, and so very little basis evident in these responses on which to build an understanding of concavity or inflection, important concepts in calculus.

Other students divided the graph into two parts, breaking it roughly at the point of inflection, and talked about the change of the growth rate. The population was growing slowly at first, but the growth rate was steadily increasing, so the population was growing “faster and faster” over the first part of the graph. In the second part, the population started out growing rapidly, but the growth rate was decreasing, so the population was growing “slower and slower.” When prompted, several of these students correctly used the terms “inflection point,” “concave up,” and “concave down” to describe features of the curve.

Eleven of the twelve pre-calculus students, ten of the fifteen calculus students, and seven of the ten post-calculus students gave the less sophisticated answer, dividing the graph into three parts. The remaining students gave the more sophisticated answer, dividing the graph into two parts. This reveals a difference between students who have studied calculus and students who have not. Only one student (less than 10% of that group) who had not studied calculus gave the more sophisticated answer, while about a third of the students who had studied calculus gave that answer.

In part (b), the students were asked to identify hours during which the population was changing rapidly and hours during which it was changing slowly. All were successful in doing this, but they gave two different types of explanations for how they identified slow and rapid growth. Some talked about the amount of increase in the population between two consecutive time points. For example, the population changed by only a few cells between midnight and 1 am, but changed by a large number of cells between 8 and 9 am. This reasoning indicates a focus on rate of change over an interval, which is accessible to pre-calculus students. As might be expected, six of the eight pre-calculus students who were asked for their reasoning on this item talked about the amount of change over an interval. The other students said the population was changing rapidly where the graph had a steep slope, and was changing slowly where the graph had a more shallow slope. Eight of fifteen calculus students and four of nine post-calculus students gave this response. Again, we see a difference between students who have taken calculus, half of whom used slope in their responses, and students who have not, only a quarter of whom used slope.

In part (c), the subjects were asked how rapidly the yeast population was growing at 4 am. Most students used the average change over an interval to estimate the rate of change at the point of interest. Of these, a few used the interval from midnight (the start of the graph) to 4 am, which produces a less accurate estimate.⁴ Most subjects used the interval from 3 am to 4 am, and several used intervals from 3 am to 5 am. A few used shorter intervals near 4 am. In any case, the rates of change over these various intervals correspond to the slopes of secant lines. Since the derivative is usually constructed as the limit of the slopes of secant lines, these students were demonstrating knowledge that is useful in the study of the derivative. Some calculus and post-calculus students drew a line tangent to the curve at the point of interest and used the slope of that line to judge the rate of growth. This corresponds to the derivative, and is taught in first semester calculus.⁵ None of the pre-calculus students used a tangent line.

In part (d), the students were then given a table of values corresponding to points on the graph, and again asked about the rate of change at 4 am. Students who had used an interval in part (c) generally used the same interval in part (d), but were able to obtain more precise results because they did not have to estimate values from the graph. Of course, students who had used tangent lines in part (c) couldn’t draw tangent lines on the table, but one student did explain how the results of his calculations from the table related to the slope of a tangent line.

In part (e) the students were given a new graph showing the area around 4 am greatly enlarged, and a table of values giving the number of cells every 15 minutes rather than every hour. Most took advantage of this increased precision to obtain estimates of the rate of change at 4 am using intervals from 3:45 am to 4 am or from 3:45 am to 4:15 am. Nearly all said that the new estimates were better than the ones made from the less detailed information.

Finally, in part (f) the students were asked to describe a procedure for getting a better estimate or a precisely correct answer. All post-calculus, two-thirds of calculus, and half of pre-calculus students said that using smaller intervals would give a better approximation. Seven of the ten post-calculus students and three of the fifteen calculus students said that if they had a formula for this function, they would find the derivative symbolically and evaluate it at t = 4 to get an exact answer. Seven post-calculus students and four calculus students said that they could get an exact answer by finding the limit of the average rate of change over intervals of decreasing width around t = 4. About half of the pre-calculus and calculus students said that if they had the exact value of the slope of the tangent line, then that would be the exact answer. Other pre-calculus students talked about taking shorter intervals, but did not give any sign that they thought of this as a limiting process leading to an exact answer. For them, it was just a procedure for finding somewhat better approximations.

Since finding the instantaneous rate of change as a limit of average rates of change over intervals of decreasing width is one of the big ideas of calculus, it is not surprising that students who have taken calculus would be more likely to come up with this answer than students who have not. What is perhaps more surprising is that there would be such a difference between second-semester calculus students and upper division mathematics majors. After the first semester of calculus, this idea does not get much attention in the curriculum, so one might expect not to see any further improvement in understanding this concept once first semester calculus is completed. Contrary to this expectation, 70% of post-calculus students, but only 27% of second-semester calculus students said that an exact answer could be obtained by taking a limit of average rates of change over decreasing intervals. Hauger takes this as an indication that students continue to building their understanding of calculus concepts after they leave the course, and suggests a couple of ways in which this might occur.

Hauger does not mention what seems to me to be the most obvious explanation, which is that not every calculus student goes on to be an upper-division mathematics major. Generally speaking, only the more successful ones do. Perhaps these post-calculus students do not have a significantly better understanding of the limit process then they had when they were calculus students, but as calculus students they may have had a better understanding than their classmates. In essence, the post-calculus students may represent the likely future knowledge state not of all calculus students, but only of the few calculus students who gave the most sophisticated answers.

Hauger: Speeding up and Slowing Down
Hauger’s second paper (Hauger, 1997) is a qualitative analysis of responses of four of the subjects interviewed in the larger study. He writes:

One of the goals of the larger study from which this report comes was to examine knowledge that precalculus students have and which could be used in their construction of knowledge of rate of change in calculus. Part of that examination included looking at what these students did correctly and thinking about how that knowledge might support their learning of calculus. But I was also interested in seeing what errors these students made and, in the case where they discovered and corrected their errors, how those discoveries came about and led to refining their knowledge of rate of change. That is the main story of this paper. (page 4)

This paper focuses on the first interview task, with some attention to the second task and brief mentions of the third. Task 1 started with this description (page 42):

“Two people start at opposite corners of a room and walk toward each other. As they walk, they both slow down as they get closer to each other, pass, and then they both speed up as they get further apart. This takes a total of eight seconds. The opposite corners of the room are 20 feet apart.

Tasks 2 and 3 were similar to Task 1. In Task 2, the two people walked at a constant speed. Task 3 involved the people speeding up as the approached one another, and slowing down after they passed. All three tasks asked the student to do the following four things (page 42):

a. On the graph paper supplied, draw a graph showing the distance between the two people at each moment in time. Describe your graph.
b. How does your graph show the two people slowing down? speeding up?
c. Now complete the table of values for your graph.
d. How does your table of values show the two people slowing down? Speeding up?

A correct graph for the first task resembles a parabola, with vertex around (4, 0) and sides reaching (0, 20) and (8, 20). A correct graph for task 2 passes through the three points, but forms a straight-sided V shape rather than a curved, parabola-like shape.

Hauger opens with some information on how the 37 subjects responded to the first interview task. Seventeen drew incorrect graphs. Six subjects realized their error as soon as they drew the graph. Eight realized their error when asked for explanations or for tables of values. One realized her error when presented with task 2. Two of the 17 students who drew incorrect graphs never realized their error. Eight of the seventeen students who drew incorrect graphs for task 1 drew V-shaped graphs that were appropriate for task 2, which they had not yet seen.

From the 37 subjects, Hauger selected four pre-calculus students who all made the same error, drawing, in response to task 1, the V-shaped graph that was appropriate for task 2, and who all noticed and corrected their errors. In an effort to discover what knowledge students bring to the study of calculus and how they use this knowledge to solve problems, Hauger studied the ways in which these four students reasoned about rate of change, and the factors that led them to recognize their errors and construct correct graphs.⁶

Noting that several students realized their “error” as soon as they drew the graph, before any other questions were asked, Hauger suggests that the act of drawing the graph may be important to the students’ thinking. All 37 subjects immediately realized that the graph should go through the points (0, 20), (8,20), and some point near (4, 0). Several started by drawing straight line segments connecting those points, the simplest possible graph through the points. Perhaps this was not so much an indication that they really thought the graph was composed of straight segments as it was just their way of starting to think about what the graph might look like. Some students began criticizing this graph and altering it without any prompting, which suggests that it was never intended as a final product, but was more a matter of thinking on paper.

One of the four subjects whose interviews are examined in detail fit this pattern. This student said that straight lines had constant slope and thus represented constant speed, which was not desired in task 1. Changing speed had to be represented by curved lines, with steeper slopes for greater speeds and shallower slopes for slower speeds. She continued to discuss her graphs in terms of slope throughout the first three tasks.

Other students drew the straight-sided, V-shaped graph and appeared satisfied with it, until they were asked to explain how their graph showed slowing down and speeding up. One of the four subjects of this paper, when asked how his straight-sided graph showed slowing down, responded: “It doesn’t show . . . the speed is the same because it doesn’t go in a curve.” At that point, Hauger took the interview on to task 2, returning later to task 1. It is not clear how the subject might have adjusted his task 1 graph if asked to do so at this point.

In his work on task 2, this subject said that his graph showed steady pace “’Cause they take same time going a certain distance.” Hauger at first interpreted this to mean that the same distance of 5 feet was covered in each second, but later, due to other things the subject said, wondered whether the subject had meant that each half of the graph covered 20 feet in 4 seconds. If the subject meant the latter, then the same reasoning would lead him to conclude (incorrectly) that both the V-shaped graph and the parabolic graph showed constant speed. Hauger consistently asked about distance covered in one-second intervals, steering the subject to thinking in terms of distance per unit time, rather than total time versus total distance or time per unit distance. By the time they returned to task 1, the subject apparently had cleared up whatever misunderstanding had been present, and was able to draw a correct graph immediately.⁷

The third subject continued to be satisfied with her graph until she was asked to create the table of values and explain how slowing down and speeding up were reflected in the values. Considering the average rate of change over one-second intervals, she realized that slow motion should involve less distance in a fixed amount of time, while fast motion should involve more distance. Prior to that point, she had maintained that the downward slope of her V-shaped graph showed slowing down, and the upward slope showed speeding up. It appears that she was confusing increasing distance with increasing speed, and similarly for decreasing distance and speed.⁸ Later she makes comments that are ambiguous, and could indicate that she thinks steep slopes correspond to high speeds or to increasing speeds.⁹ By the third task, this subject was clearly describing the situation correctly.

The fourth subject did not realize that her V-shaped graph was incorrect for task 1 until she was working working with task 2, which involves motion at a constant speed. She was at a loss for how to proceed until Hauger drew her attention back to the graph she had drawn for task 1. At that point, she saw that her graph and table from task 1 involved the same amount of change over each interval, and thus were correct for task 2 but not task 1.

When this subject finally drew a parabola-like graph for task 1, she put the “vertex” at (5, 0) rather than (4, 0), noting that slowing down would make the initial 20 feet take longer, while speeding up would make the final 20 feet take less time. She also drew her graph with the appropriate concavity. It is not clear that she ever understood that the location of the “vertex” was unimportant to the task as written. In a similar way, she put the lowest point of her graph for task 3 at (2, 0) rather than (4, 0), noting that the first part involved speeding up, and so would take less time than the second, which involved slowing down.¹⁰ Since she drew this graph with appropriate concavity, and since the location of the lowest point is irrelevant, this was counted as a correct graph, although it appears likely that some misconceptions remained.

Hauger notes that ideas about slope and average change over an interval were very powerful tools for these subjects in their efforts to represent slowing down and speeding up. He recommends that teachers “provide more opportunities for students to use their knowledge of shape of graph and rate of change over intervals to explore rate of change.” By showing how these students used knowledge of average rate of change to reason about instantaneous rate of change, Hauger’s work supports the idea that some understanding of average rate of change over an interval is a necessary precursor to developing a concept of instantaneous rate of change.

Continue with this paper.

Footnotes

¹ This theory comes out of a course I took two years ago, C&I 407, “Research Issues in Mathematics Education: Historical and Contemporary Issues in Equity,” with Prof. Rochelle Gutiérrez. I have emailed her for references, but with no success. She does remember presenting the idea to our class, but doesn’t remember the specific papers from which she got the information. She gave me the names of a couple of researchers whom she remembers as working in that area, but an ERIC search on their names turned up nothing relevant. If either of us comes up with anything, I’ll revise this paper accordingly.