Differences in learning outcomes: Calculus & mathematica vs. traditional calculus

Roddick, Cheryl D

ABSTRACT: This study, conducted-at The Ohio State University, compared students from the calculus-reform course sequence Calculus & Mathematics with traditional students in two areas: conceptual and procedural understanding of calculus, and achievement in calculus– dependent courses. Task-based interviews were conducted with students from both groups after they had completed one of the two calculus sequences. These interviews were used to investigate understanding of calculus. An analysis of grades collected from the university database was conducted to investigate student achievement in calculus-dependent courses.

Results from the interviews showed that the Calculus & Mathematica students were more likely to approach problems from a conceptual viewpoint of calculus knowledge, whereas the traditional students were more likely to approach problems procedurally. The Calculus & Mathematics group also demonstrated a more general understanding of the derivative and integral than the traditional group.

Results from the analysis of grades show a significant difference in the introductory differential equations course favoring traditional students, and a significant difference in the first course of the calculus– based physics sequence favoring the Calculus & Mathematics students. Other significant differences, favoring the top third of Calculus & Mathematics students, were found in the introductory physics courses and in an introductory engineering mechanics course.

KEYWORDS: Calculus reform, technology, Calculus & Mathematica, computer algebra systems

Over a decade ago, the first university calculus reform courses began to appear across the nation. Today, while many research projects have been undertaken in this area, there are still many more questions than answers about the effectiveness of these courses. Specifically, there is a great need for research that investigates students who have completed calculus reform course sequences. Although previous studies on calculus reform have investigated student understanding [2, 3, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20], almost all of these studies have been situated within the calculus setting. Very few studies have attempted to evaluate the knowledge students take with them to other courses that require calculus as a prerequisite (Armstrong, Garner, and Wynn [1]). The focus of the study described in this article was two-fold: (1) to compare students from one of the calculus reform projects, Calculus & Mathematica, with students from a traditional sequence on their conceptual and procedural understanding of the derivative and integral after they had completed the sequence and were enrolled in a calculus-dependent course; and (2) to compare both groups on achievement, measured by a grade, in calculus-dependent courses.

Among the studies that investigated differences between Calculus & Mathematica and traditional students, results were generally positive. Porzio [20] found that Calculus & Mathematics students were better able to use and make connections between different representations. Park and Travers suggest that Mathematics facilitates student learning by encouraging an inductive, or bottom-up, approach, which enables students to explore more, and can encourage better conceptual understanding [19]. Results from a common final suggest that Calculus & Mathematics students have stronger conceptual understandings of calculus than their traditional counterparts [12], while Lefton and Steinbart found no significant differences between groups on a final exam [15]. Meel [16] uncovered some intriguing differences in understanding between honors students from third semester Calculus & Mathematics and traditional courses. On a technology-restricted written test involving limit, differentiation, and integration, he found significant differences favoring traditional students on limit tasks, conceptually– oriented items, and tasks presented without figures. According to Meel, the scoring guides for the test rewarded responses that were compatible to the formal mathematical definitions. The traditional students had been presented with the more formal presentation of calculus, while the Calculus & Mathematica students had been exposed to a more informal handling of the topics tested. On the other hand, the problem-solving interviews resulted in the finding that Calculus & Mathematica students were more successful and flexible in solving real-world problems, whereas traditional students tended to use a single strategy to solve a problem, even when the strategy produced unreasonable results. The study presented here was an effort to build on the previous studies, and to offer some perspective on the long-term effects of one of the calculus reform projects.

BACKGROUND

Calculus & Mathematica background

The four-quarter Calculus & Mathematics sequence has been offered at The Ohio State University since Fall Quarter, 1989 as an alternative to the traditional four-quarter calculus sequence. The text, which is electronic and interactive, utilizes the computer algebra system Mathematics. Class is held daily in a computer laboratory and little emphasis is placed on lectures. The main focus in this course is problem solving, which is done on the computer, and includes a variety of real-life word problems. In solving these problems, students are encouraged to discuss their ideas with each other, resulting in a lively, active class atmosphere. Tests on calculus literacy, which include paper-and-pencil computation, are given throughout the quarter, and a cumulative final exam is given on paper. Memorization of formulas and practice in hand computations are given considerably less weight than in a traditional calculus course, whereas more importance is placed on students’ writing and clear explanations.

What sets this calculus reform project apart from other calculus courses is not only the immersion in technology, but also the different approach to teaching calculus. Electronic lessons present a conceptual approach to calculus with an emphasis on problem solving. The lack of emphasis on lecture encourages more participation and responsibility on the part of the learner to construct his or her own understanding. As a result, there is a great amount of experimentation and discussion with classmates, along with teacher encouragement of student exploration. In addition, students are required to explain their homework solutions in detail, which encourages deeper understanding.

Traditional calculus background

The traditional calculus sequence offered at The Ohio State University operates under the lecture-recitation format. Students enrolled in these classes also attend class daily, but three days per week they attend a lecture where new material is presented to them, and the remaining two days are spent on solving problems in recitation class. At the time the participants of this study were enrolled in calculus, the textbook used was Calculus [6]. Although technology was not incorporated into the instruction, students were permitted to use graphing calculators on quizzes and exams. A good portion of the traditional sequence was spent on procedural proficiency, which was reflected in their homework, quizzes, and exams.

Conceptual and Procedural Knowledge

The debate over conceptual versus procedural knowledge is one that has experienced renewed vigor in light of the calculus reform movement. One major criticism of the calculus reform movement has been that students who learn calculus with the help of technology will not have the procedural skills to be successful in later calculus-dependent courses [14]. Others argue that these students have a stronger conceptual understanding and will have an advantage over students who have taken traditional calculus. Supporters of Calculus & Mathematics believe that the use of technology, together with the philosophy of learning discussed previously, can encourage student ownership of knowledge and a strong conceptual understanding [4]. This study was an effort to investigate the different types of knowledge encouraged by the two different calculus sequences.

METHOD

Sample

To investigate the first focus of the study, the researcher interviewed six students enrolled in an introductory engineering mechanics course. Three of the students had completed the Calculus & Mathematics. course sequence and the other three students had completed the traditional calculus sequence. All of these students had finished the calculus sequence one or two quarters prior to enrolling in the engineering mechanics course, and only above-average calculus students were included in the study.

The second focus of the study relates to long-term comparisons of achievement of the two calculus groups in calculus-dependent courses. This sample consists of all students who have completed either Calculus & Mathematica or the traditional calculus sequence during a two-year period, and have taken at least one calculus-dependent course during a four-year period.

Instrumentation

Task-based interviews, which followed a think-aloud protocol, were conducted and audio taped with the six students from the introductory engineering mechanics course. These interview problems were presented to the students at the beginning of the quarter for the purpose of determining student understanding of differential and integral calculus before they were exposed to it again in their current class. Problems contained in the interview were chosen to reflect the calculus used in the introductory engineering mechanics course, and were constructed to contain both procedurally- and conceptually-oriented questions. Final exams from the differential and integral courses of both groups were examined, and similar problems were used in the interview. A common interview protocol, which focused on eliciting clear explanations of solutions, was followed for all participants.

Qualitative data analysis

During the process of analyzing the student interviews, responses were classified according to whether their use of calculus involved procedural knowledge, conceptual knowledge, or a combination of both types of knowledge. The definitions of conceptual and procedural knowledge used were those discussed by Hiebert and Lefevre [11], who classify conceptual knowledge as “knowledge that is rich in relationships” [11, p. 3], and describe a network of knowledge where all pieces of information are linked to each other in a web-like fashion. Procedural knowledge is defined as a set of symbols and algorithms, where the essential features include actions or transformations that are connected and executed in a linear, or sequential fashion. Prior to classifying actual student methods, a list of possible responses to the interview questions was generated. Each response was classified according to whether the focus was procedural (through the use of algorithms) or conceptual.

In an effort to achieve some reliability on the classifications of student solution methods, two peers knowledgeable in the areas involved in the study provided their opinions on the student solutions. With one of the raters, agreement in classification was achieved in 92% of the problems. With the other rater, 85% agreement in classification was achieved. When agreement was not initially achieved, discussions with the raters resulted in a common classification.

Quantitative data analysis

Information from the university database was used to investigate the second focus of the study, which addresses the relationship between grades in calculus-dependent courses and type of calculus course taken. Information was collected for 40 Calculus & Mathematica students and 729 traditional students. Grades were recorded for each calculus course students had taken during the two-year time frame, as well as for any courses that require calculus as a prerequisite, taken during a four-year period. Since the two groups of calculus students were not randomly assigned to type of calculus course, the background variables high school rank and mathematics score on the ACT exam were used to determine the similarity of the two groups before finding whether the difference in the mean grades was significant. Students with only SAT mathematics scores had their scores converted to equivalent ACT mathematics scores. The t test and the nonparametric Wilcoxon rank-sum test (also referred to as the Mann-Whitney U-test) were then used to determine significant differences in calculus-dependent course averages. Further comparisons of course averages were made based on students’ achievement level in calculus.

RESULTS FROM INTERVIEWS

Six questions were given to the engineering mechanics students who participated in the study. These problems involve derivatives and integrals.

Calculus & Mathematics Student Responses

Pat, one of the three Calculus & Mathematics students, solved the problem correctly by hand and was able to relate the derivative to the graph of the original function. Throughout I refers to the Interviewer and S refers to the Student.

I: Why do you take the first derivative?

S: Because if you know where the first derivative is zero, you know where there is no slope, so it’s either a crest or a dip so it’s not increasing or decreasing.

I: Do you know what’s happening before this point or after this point [the max]?

S: If it’s like this (pointing at the graph), the first derivative should be positive, because it’s (the function) increasing. If it’s past this it should be negative, since it’s decreasing.

Pat demonstrated a sound understanding of the relationship between a function and its derivative. The other two Calculus & Mathematica students also made remarks that showed some conceptual understanding of the procedure they were performing. They could enunciate the reasoning involved, although not flawlessly. Rich explained his solution:

The derivative is the amount of change in the slope, so if I know that the change in the slope is 0, then I know the function isn’t increasing or decreasing at that point.

His reference to the derivative is similar to one of the traditional students’, Ed, reference to the “derivative of the slope.” Tony, the third Calculus & Mathematics student, incorporated the first and second derivatives to help him find the maximum and minimum. He made reference to the growth of the function being positive or negative.

S: If I put in 0 that means at 0 the second derivative is -6. So the derivative is decreasing there.

I: How do you know the derivative is decreasing?

S: Because its derivative is that (-6). And it has -6 growth there.

While all of the students experienced some degree of success in solving the first problem involving differentiation, there were various levels of explanation given. When asked by the researcher, all three Calculus & Mathematica students were able to discuss the concept of the derivative, whereas only one traditional student was able to offer any explanation.

Procedurally, there were mistakes in both groups, albeit fewer from the traditional. Two of the Calculus & Mathematica students had trouble factoring, and did not arrive at a correct solution without some help. One traditional student was unable to solve the problem without a graphics calculator.

Second differentiation problem

The graphical connection between a function and its derivative is the essence of the next problem. Students (see above) were given the graphs of two functions and were asked to determine which function is the derivative of the other. No equations were given, so students had to rely on their graphical knowledge of functions and their derivatives.

Figure 1 shows the plots of two functions. One is the derivative of the other. Determine which is which and explain your reasoning.

Traditional student responses

A clear dichotomy exists in the answers from the two groups. Traditional students responded with vastly different reasoning than the Calculus & Mathematics students. All three of the traditional students made a connection with these graphs to the graphs of the sine and cosine functions. Basing their answer on their knowledge that the derivative of sin x is cos x, two of the three students correctly responded that function g (solid) is the derivative of function f (dashed) – see Figure 1 above. Ed explained:

The one is a type of sine wave and the other is a type of cosine wave … The derivative of sine goes to cosine which goes to -sine which goes to -cosine and repeats the circle. I can tell that that’s a cosine and that’s a sine. If it was going from cosine to sine, it would be negative. It would be reversed. So it has to be going from sine to cosine, because they are both positive.

One of the students, Nick, responded incorrectly, only due to his belief that the derivative of cos x is sinx. Another student, Gene, attempted to discuss the concept of the tangent line.

I: Say you don’t know what these functions are at all. Can you give me another way to tell?

S: Yeah. When you have a curve, the derivative of that curve at one point is the tangent at that curve. The derivative will be tangent to that curve.

I: So how can you address that in terms of the graphs that you are looking at?

S: I would presume that the points that are on each graph are the… one of them is the function and the other one is the derivative, of that. So that’s why you have another function, instead of just a line, because it’s a function and it’s not just a point.

His comments indicate that he understands the derivative to be related to the tangent of the curve, although he has not yet recognized the graphical relationship between the function and its derivative. None of the traditional students could explain this problem without making reference to sin x and Cos x.

Calculus & Mathematica student responses

All of the Calculus & Mathematica students correctly identified the function and the derivative. Their reasoning involved making connections between the growth of the function and the sign of the derivative. Rich conveyed his thoughts the most succinctly:

[Function g] is the derivative because the slope is positive here and this function is increasing. And whenever it reaches 0 this has reached its maximum. And when the slope goes negative the function decreases. And it repeats it from there.Tony gave similar reasoning:

If on this graph there was no growth, the derivative of this function should be zero … This decreasing here, so the derivative at this point should be negative.

Pat echoed those thoughts:

If this one is increasing, this one would be positive all the way. And there’s a crest there, so it’s zero.

While all of the students were able to comment on the second differentiation problem, the two groups differed in the elements of the problems on which to focus. Traditional students related the graphs to specific functions for which they knew the derivative. Calculus & Mathematica students never suggested possible functions for the graphs; instead they focused on the more general concept of the growth of a function related to its derivative.

Traditional student responses

Two of the traditional students, Nick and Ed, responded by finding equations of each line and integrating. Ed performed his computations correctly, choosing the appropriate limits of integration for each integral. Nick arrived at the wrong answer because he chose to use -4 and 4 as his limits of integration for each integral. But his idea was correct:

Here I defined f (x), each line. And I integrated each line. I don’t know if they’re right or not, but it’s the best I could figure to do. So I integrated each individual point and I just plugged in the values.

Gene, who arrived at a correct answer using a different method, divided the graph into geometric shapes, found the area of each shape, and summed the areas to get the final answer. When asked how he got that answer, he replied,

I was taught that integration is basically the area under the function. And since I did not know how to figure out how to do the function since it’s not a congruent function, I just took the area from each part. And whenever it was on the negative side of the y-axis, I presumed it to be negative area. And then just added or subtracted the area, taking rectangles and triangle areas.

Calculus & Mathematica Student Responses

As in the previous problem, the Calculus & Mathematics students all responded in the same manner to this problem. Their method was the same as the traditional student, Gene’s: divide the function into geometric shapes, determine the area of each shape, and then sum the areas. Each student arrived at a correct answer. Rich explains:

Taking the integral over an area is the same as finding the area under the function. Actually between the function and the xaxis.

Second integration problem

What are Derivatives and Integrals?

The final question in the interview was asked to gain perspective of each student’s overall understanding of the derivative and integral concepts. Two questions were asked: What is a derivative? and What is an integral? Special attention was given to whether the responses could be classified as conceptual or procedural, and whether the understanding appeared to be general or specific. The generality or specificity of the response could be determined by the examples given and by the type of information tied to a student’s understanding of the derivative and integral.

Traditional student responses to “What is a derivative?”

In response to the question “What is a derivative?” two of the traditional students commented on its relationship to the tangent line of the function at a point. Gene wrote:

It is a tangent of a point of curve. It is the opposite of integral; it takes away a power when produced from the function; the derivative of x^sup 2^ is 2x.

Nick drew a picture of a function with a line tangent to it at (xo, Yo). He also mentioned some applications of the derivative:

At any point on a graph, the derivative of the function will give you the function tangent to it at point (x^sub 0^, y^sub 0^). The derivative of acceleration gives you velocity. You can get inflection points, local max and minimums.

Ed was the most philosophical of all the students in this study. He quickly acknowledged the difficulty of the question, then proceeded to express his ideas about derivatives.

In temporal space everything is determined by something else. For example, position is determined by your origin and velocity. Your velocity is determined by your starting speed and acceleration. The derivative tells the fundamental parts that make up an equation …. The derivative tells you where it’s been, and what its highest and lowest points were.

Calculus & Mathematica student responses to “What is a derivative?”

All of the Calculus & Mathematica students defined the derivative as the rate of change, or the measurement of the growth of the function. All of them also mentioned slope in their discussion. Tony drew a picture of a function with a tangent line at a point and referred to the slope of the tangent line.

A derivative measures the rate of change of a function.

Rich gave a similar explanation:

S: The derivative is … how the dependent variable changes with respect to the independent variable. . .

I: You’re talking about slope here. How does the derivative relate to slope?

S: It’s the same thing.

I: It’s the slope of what?

S: The original function that it’s the derivative of. Pat explained:

A derivative is a measurement of the growth of a function.

Note that students informally refer to the slope of the function and not the slope of the line tangent to the function.

Traditional student responses to “What is an integral?”

Gene’s response complements his answer about a derivative:

It’s a tool for solving volume and area; it is the opposite of derivative; it takes the power up when the function is processed; the integral of x^sup 2^ = 1/2 x^sup 3^ [sic].

Another traditional student, Nick, responded more abstractly when he wrote that an integral is the “summation of an infinite number of divisions between two limits.” He drew a curve with the area between the curve and the x-axis divided into rectangles. He also noted that

The integral from x to x^sub 0^ of f (x) dx equals the area of the surface underneath the functions and between the two limits. The integral is the antiderivative of an equation.

The third traditional student, Ed, again brought up the example of the relationships between acceleration, velocity, and position.

The integration is the summing of individual equations. What I mean is if you want to know how acceleration affects velocity, and velocity affects position, you integrate. Integration works your equation to the next level, (i.e.) to a higher order equation.

Calculus & Mathematics student responses to “What is an integral?”

All of the Calculus & Mathematics students said that the integral is the measure of the area under the curve and drew pictures to represent the summation of small areas. Two of the three students included an explanation of the limiting process.

Pat explained in detail:

S: An integral is a measurement of the area under a curve.

I: How do you find an integral?

S: The integral can measure the area under a curve. You get it by summing up all the little areas. This says add up all of the little f (x)’s times dx’s from here to here (referring to his notation and picture).

I: Okay, .. where does the limit come in?

S: To get an accurate answer, you want to let these get smaller and smaller and smaller and smaller and add up more and more and more of them.

In Rich’s response, he mentioned that the integral is a measure of area between a function and its axes, and is also the inverse of a derivative. He included a discussion of how area under a curve is related to the integral.

They used several methods; they used triangles, I think, or something like that, or trapezoids, I can’t remember. Basically they just summed up all the areas, taking into consideration that this was pretty close. (In his two pictures he drew rectangles and trapezoids.) … You make the sections smaller so the error is smaller when you look at it.

DISCUSSION

From the responses to the problems in the interviews, it is apparent that the two groups often have different ways of approaching the same problem. Results of all questions are categorized in Table 1, according to whether their approach was conceptual, procedural, or a combination. The overwhelming solution preference of the Calculus & Mathematica group was conceptual. Within the traditional group’s responses there was slightly more variety, although the majority of the responses were procedural in nature.

An important ingredient in the issue of students’ ability to apply calculus lies in the responses to the last two questions: What is a derivative? and What is an integral? Looking at the types of responses to these questions suggests a classification into two categories: general definition and specific application. See Table 2. Recall that all Calculus & Mathematica students gave a very general definition for derivative: the rate of change or growth of a function. In contrast, two of the three traditional students gave a definition relating the derivative to the tangent line of the function. Specific examples of the use of a derivative were commonly given by the traditional group.

For the integral, every Calculus & Mathematica student responded with the same two descriptions: a measurement of area under a curve and a summation of small rectangles or trapezoids. Similar consensus was not achieved among the traditional group. Two students mentioned area in their explanation; one of whom also included the idea of summing “an infinite number of divisions between two limits.” Two of the three students mentioned specific applications for which the integral can be used or specific descriptions of how the integral is found. It is interesting that, while two of the three traditional students expressed the integral as a measure of area under the curve, only one demonstrated that conceptual knowledge in a solution to a problem.

Note that many of the responses from the traditional group referred to procedures, applications of the derivative and integral, and relation to a graph. Yet the responses among the Calculus & Mathematica group were quite general, focusing mainly on the concept of the derivative and the integral. Furthermore, while both groups responded that the derivative was related to the slope of the tangent line of the function, the Calculus & Mathematica group further abstracted their explanations by describing the derivative as a rate of change, or measurement of growth. This definition shows that the concept of derivative has been detached from specific examples, and is understood in its most general form. Similarly, the Calculus & Mathematics students were able to describe the integral in a more general form.

It is the responses to these last two questions that are the most revealing in terms of the differences in student learning outcomes. These responses represent each student’s overall understanding of the derivative and integral, and will likely have a great impact on how they apply such understanding in future problems involving calculus.

RESULTS FROM STATISTICAL ANALYSIS

In this section statistical tests have been used to analyze the achievement of both groups in calculus-dependent courses. The level of achievement is measured by the grade received in each course. Since students have self-selected the type of calculus course, efforts were made to determine the degree of similarity between the two groups. High school achievement means were compared before an analysis of grades was undertaken. t-tests and the nonparametric Wilcoxon rank-sum tests were then applied to course averages to test for significant differences.

Initial Differences Between the Two Groups

With regard to background achievement, there are no significant differences between students who take the traditional calculus sequence and students who take the Calculus & Mathematica sequence. (See Appendix A.) All students have similar high school achievement levels with respect to their high school rank and ACT scores. The average high school rank of students from both sections is near the 85th percentile and the average ACT mathematics score of both groups differs by less than one point: 27.05 versus 26.29. The finding that mathematics skill level was not different for the two groups before they enrolled in a college calculus course may be considered surprising, as a common assumption is that the better students self select the Calculus & Mathematica section.

Differences in Calculus-Dependent Courses

For the statistical analysis, the two groups were compared on each individual course with a Calculus & Mathematics sample size bigger than ten. (See Appendix B.) t-tests and the nonparametric Wilcoxon rank-sum (also called the Mann-Whitney U) tests performed on the means failed to produce any significant differences at the .05 level except for two courses.

There is a significant difference at the .05 level between the two means in an introductory differential equations course. The traditional group mean is 2.7, much higher than the Calculus & Mathematics group mean of 2.06. This finding is not surprising to those who have been involved with the Calculus & Mathematics courses. Such a strong emphasis is placed on integration techniques in the differential equations courses that traditional students who have spent more time in this area have an advantage. While Calculus & Mathematics students do learn to compute integrals by hand, they do not perform the computations enough to feel as comfortable as a traditional student.

In Physics 1, the first course of the calculus-based sequence, a significant difference was found in favor of the Calculus & Mathematics group. The Calculus & Mathematics group has a mean of 2.83 while the mean for the traditional group is 2.47. Generally, this course is taken either concurrently with the first calculus course or one quarter later.

A closer look at the other courses failed to turn up any more significant differences on the groups as a whole. Although statistically insignificant, Electrical Engineering, Engineering Mechanics 1 and 2, and Physics 1, 2, and 3 all have differences favoring the Calculus & Mathematica group.

Paired Groups

Further comparisons of course averages were made on students divided into three groups: high, middle, and low achievement levels. (See Appendix C.) Based on percentile rankings in their respective calculus sequences, Calculus & Mathematica students from each level were compared to traditional students from the same level to determine whether students of similar calculus achievement perform similarly in calculus-dependent courses. The nonparametric Wilcoxon rank-sum test was used on courses with sample sizes large enough to break into three groups. Note that the samples within the three groups are not exactly the same size. Groupings were chosen so that students with the same calculus percentile ranking were not divided into two different groups.

Significant differences have been found to favor the top third of the Calculus & Mathematica group in the introductory engineering mechanics course, Physics 1, and Physics 2. With regard to the Physics 1-3 course average, Calculus & Mathematica students from the top and middle thirds significantly outperformed the traditional students. Although no other differences in course averages were significant, the top and middle thirds of the Calculus & Mathematica group had higher averages than the top and middle thirds of the traditional group, respectively. However, in Engineering Mechanics 1, Physics 1, Physics 3, and Physics 1-3, the lower third of the traditional group outperformed the lower third of the Calculus & Mathematics group.

SUMMARY

This research strengthens the findings of others who have investigated conceptual and procedural understanding between calculus groups [10, 18, 19]. Overall, findings from both qualitative and quantitative measures are encouraging for those who wish to use or are currently using calculus reform materials. Although the results from Calculus & Mathematica students are not transferable to other reform projects, one can point to certain elements of the Calculus & Mathematics project that help to shape a student’s understanding of calculus:

1. The emphasis on conceptual understanding over procedural understanding.

2. Mathematical discourse. Students spend a good deal of time communicating about mathematics, both in class with their classmates and instructor as well as in writing on their assignments.

3. Constructivist principles. Due to the deemphasis on lecturing, students must construct their own understanding of the material presented on the computer.

4. The use of technology as a tool in understanding calculus. Technology is the vehicle by which the goals of the course can be achieved.

Other calculus reform projects with these foci may encounter similar outcomes in student learning.

FUTURE RESEARCH

This interview was used as baseline information to address the issues of conceptual and procedural understanding and ability to apply calculus knowledge to engineering mechanics problems [21]. With regard to the definitions given, Calculus & Mathematica students have come closer to decontextualizing the concepts of derivative and integral than the traditional students. This difference will become more important when these same students are asked to apply their knowledge of calculus to new situations. An important question to consider is whether a general understanding is more likely to transfer than an understanding that is tied to specific examples.

In addition, statistical analysis should be done on a larger Calculus & Mathematica sample. The data collected for this study consisted of all students who had completed either calculus sequence within a two-year period. Unfortunately, the Calculus & Mathematica sample size was not large enough to allow for statistical analysis on more than a few calculus– dependent courses. These findings may indicate a slight advantage for the Calculus & Mathematica students in the applied courses such as engineering and physics, especially if they do well in their calculus sequence. Paired grouping results suggest a slight disadvantage for Calculus & Mathematica students who end up in the lower third in calculus achievement, when compared to traditional students in the lower third of calculus achievement. Yet the data should be recollected for all additional students who have completed either course sequence in order to determine whether any further significant differences can be found. Similar analyses done on other calculus reform courses would provide insightful information about lasting effects of the courses.

REFERENCES

1. Armstrong, G., L. Garner, and J. Wynn. 1994. Our Experience with Two Reformed Calculus Programs. PRIMUS. 4(4): 301-311.

2. Beckmann, C. 1988. Effect of Computer Graphics Use on Students’ Understanding of Calculus Concepts. Dissertation Abstracts International. 50(5): 1974B.

3. Crocker, D. 1991. A Qualitative Study of Interactions, Concept Development and Problem Solving in a Calculus Class Immersed in the Computer Algebra system Mathematics. (Unpublished doctoral dissertation, The Ohio State University).

4. Davis, B., H. Porta, and J. Uhl. 1994. Calculus Fl Mathematica. Reading MA: Addison Wesley.

5. Estes, K. 1990. Graphics Technologies as Instructional Tools in Applied Calculus: Impact on Instructor, Students, and Conceptual and Procedural Achievement. Dissertation Abstracts International. 51: 1147A.

6. Finney, R. and G. Thomas. 1991. Calculus. Reading MA: AddisonWesley.

7. Galindo-Morales, E. 1994. Visualization in the Calculus Class: Relationship Between Cognitive Style, Gender, and Use of Technology. Unpublished doctoral dissertation, The Ohio State University.

8. Hart, D. 1991. Building Concept Images: Supercalculators and Students’ Use of Multiple Representations in Calculus. Dissertation Abstracts International. 52(12): 4254A.

9. Hawker, C. 1987. The Effects of Replacing Some Manual Skills with Computer Algebra Manipulations on Student Performance in Business Calculus. Dissertation Abstracts International. 47: 2934A.

10. Heid, M. 1988. Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool. Journal for Research in Mathematics Education, 19(1), 3-25.

11. Hiebert, J. and P. Lefevre. 1986. Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics. Hillsdale NJ: Lawrence Erlbaum Associates. 1-27.

12. Holdener, J. 1997. Calculus & Mathematica at the United States Air Force Academy: Results of Anchored Final. PRIMUS 7(1): 62-72.

13. Judson, P. 1989. Effects of Modified Sequencing of Skills and Applications in Introductory Calculus. Dissertation Abstracts International. 49: 1397A.

14. Krantz, S. 1993. How to Teach Mathematics: A Personal Perspective. Providence III: American Mathematical Society.

15. Lefton, L. and E. Steinbart. 1995. Calculus & Mathematics: An End-User’s Point of View. PRIMUS. 5(1): 80-96.

16. Meel, D. 1998. Honors Students’ Calculus Understandings: Comparing Calculus&Mathematica and Traditional Calculus Students. Research in Collegiate Mathematics Education. III: 163- 215.

17. Melin-Conejeros, J. 1992. The Effect of Using a Computer Algebra System in a Mathematics Laboratory on the Achievement and Attitude of Calculus Students. Dissertation Abstracts International. 53(7): 2283A.

18. Palmiter, J. 1986. The impact of Computer Algebra Systems on -College Calculus. Dissertation Abstracts International. 47: 1640A.

19. Park, K. and K. Travers. 1996. A Comparative Study of a ComputerBased and a Standard College First-Year Calculus Course. In J. Kaput, A. Schoenfeld, and E. Dubinsky (Eds.) Research in Collegiate Mathematics Education. II: 155-176.

20. Porzio, D. 1994. The Effects of Differing Technological Approaches to Calculus on Students’ Use and Understanding of Multiple Representations when Solving Problems. Unpublished doctoral dissertation, The Ohio State University.

21. Roddick, C. (In preparation). Students’ Use of Calculus in an Engineering Mechanics Course.

Cheryl D. Roddick

ADDRESS: Mathematics and Computer Science Department, San Josh State University, One Washington Square, San Jose CA 95192 USA.

BIOGRAPHICAL SKETCH

This article is based upon dissertation research conducted at The Ohio State University. The author is currently an assistant professor in the math and computer science department at San Jose State University in San Jose, California. She enjoys hiking, biking, and watching Big Ten football.

Copyright PRIMUS Jun 2001

Provided by ProQuest Information and Learning Company. All rights Reserved