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This large study examines data over a long period of time regarding a calculus "reform" course. Grades and attitude are studied, with advice for novices at assessment.
Background and Purpose
For many years now the mathematical community has struggled to reform and rejuvenate the teaching of calculus. My experience with one of these approaches is in the context of a small comprehensive college or university of 2400 students. Our university attracts students of modest academic preparation and achievement. Most freshmen come from between the 30th and 60th percentiles of their high school class. The central half of our students have mathematics SAT scores between 380 and 530 (prior to the re-norming of the exams). We offer support courses in elementary algebra, intermediate algebra, trigonometry, and precalculus for any student whose placement examinations show they lack the requisite skills for the first mathematics course required by their major.
In view of the characteristics of our student body, the nature of the improvements seen in calculus performance are extremely important. One or two sections of the six normally offered for first semester calculus were taught using a reform calculus approach and text developed by Dubinsky and Schwingendorf at Purdue University. The remaining sections were taught using a traditional approach and text supplemented by a weekly computer lab.
The goals we established for this course were to improve the level of understanding of fundamental concepts in calculus such as function, limit, continuity, and rate of change, and to improve student abilities in applying these concepts to the analysis of problems in the natural sciences and engineering. We hoped to reduce the number of students who withdrew from the course, who failed the course, or who completed the course without mastering the material at a level necessary to succeed in further mathematics courses or to successfully apply what they had learned in courses in science and engineering. Performance in these courses has frequently been sharply bimodal, and it was hoped that we could find an approach which would narrow the gap between our better students and our weaker students. Our past experience had been that between 20 and 45% of beginning calculus students failed to perform at a level of C- or better. Approximately half of that number withdrew from the course before completion, the remainder receiving grades of D or F. Furthermore, students passing with a C- or D in Calculus I seldom achieved a passing grade in Calculus II.
In this reform approach, the way in which material is presented is determined by extensive research into how mathematics is learned. The very order in which the topics appear is a reflection not of the finished mathematical structure, but of the steps students go through in developing their understanding from where they start to where we wish them to end. Mathematics is treated as a human and social activity, and mathematical notation is treated as a vehicle which people use for expressing and discussing mathematical ideas. A cooperative group learning environment is created and nurtured throughout the course. Students work in teams of 3-4 students in and out of class and stay with the same group for the entire semester. Group pride and peer pressure seem to play a significant motivational role in encouraging students to engage with the material in a nontrivial way.
The computer is used not for practice or for drill, but rather for creating mathematical objects and processes. The objects and processes which the students create are carefully chosen to guide the students through a series of experiences and investigations which will enable them to create their own mental models of the various phenomena under investigation. Basic concepts are developed in depth and at leisure, looking at them from many viewpoints and in many representations. ISETL and Maple are used to describe actions in mathematical language and to generate algorithmic processes which gradually come to be viewed as objects in their own right upon which actions can be performed. Objects under study are dissected to reveal underlying processes and combined with other processes to form new objects to study. The time devoted to what the student views as "real calculus" — the mechanics of finding limits, determining continuity, taking derivatives, solving applications problems, etc. is perhaps only one-half that in a traditional course. We hoped that the learning of these skills would go far more easily, accurately, and rapidly, because the underlying ideas and concepts were better understood, and in fact this did seem to be the case.
Method
Two assessment models were used, each contributing a different viewpoint and each useful in a different way. One was a quantitative assessment of student performance and understanding based upon performance on the final examination for the course. A parallel qualitative assessment was obtained through careful observation and recording of student activities, discussions, insights, misconceptions, thought patterns, and problem solving strategies. Some data came from student interviews during and after the classes. Other data came from the frequent instructor/student dialogues generated during classroom time, laboratory and office hours.
The quantitative assessment provided a somewhat objective measure, unclouded by the instructor's hopes and aspirations, of the general success or failure of the effort. Although rich in numerical and statistical detail, and valuable in outlining the effectiveness (or lack thereof) for various sub-populations of students, it offered few clues, however, as to directions for change, modification, and improvement.
The goal of the qualitative assessment was to gain a more complete idea of what the student is actually thinking when he or she struggles with learning the material in the course. Without a clear understanding of the student's mental processes, the instructor will instinctively fill in gaps, resolve ambiguities, and make inferences from context that may yield a completely inappropriate impression of what the student understands. If we assume that the student reads the same text, sees the same graph, ponders the same expression as we do, we will be unable to effectively communicate with that student, and are likely to fail to lead them to a higher level of understanding. The qualitative assessment, although somewhat subjective, proved a rich and fertile source for ideas for improvement, which were eventually reflected through both assessment vehicles.
The two methods, used in tandem, were quite effective, over time, in fine tuning the course delivery to provide an enhanced learning environment for the vast majority of our students. By clarifying what it is that was essential for a student to succeed in the course, and by constantly assessing the degree to which the essential understandings were being developed (or not) and at what rate, and in response to which activities, the delivery of the course was gradually adjusted.
Findings
The instructors of the traditional sections prepared a final examination to be taken by all calculus sections. The exam was designed to cover the skills emphasized in the traditional sections rather that the types of problems emphasized in the reform sections. In an attempt to minimize any variance in the ways individual instructors graded their examinations, each faculty member involved in teaching calculus graded certain questions for all students in all sections.
At the start of this experiment, the performance of students in the reform calculus sections, as measured by this exam, lagged significantly behind students in traditionally taught sections. During the third semester of the experiment the students in the reform sections were scoring higher than their cohorts in the traditional sections. Furthermore, the segment of our population who most needed support, those in the lower half or lower two-thirds of their calculus section (as measured by performance on the final examination) proved to be those most helped by these changes in pedagogy. Although every class section by definition will have a lower half, it is quite a different story if the average grade of those in the lower half is a B than it is if it is a D. By the third semester of the experiment, the bottom quartile score of reform sections was higher than the median score of the traditional sections (see the graphs on the next page).
Student interviews, discussions, and dialog quickly revealed that what the student sees when looking at a graph is not what the teacher sees. What students hear is not what the instructor thinks they hear. Almost nothing can be taken for granted. Students must be taught to read and interpret the text, a graph, an expression, a function definition, a function application. They must be taught to be sensitive to context, to the order of operations, to implicit parentheses, to ambiguities in mathematical notation, and to differences between mathematical vocabulary and English vocabulary when the same words are used in both. Interviews revealed that the frequent use of pronouns often masks an ignorance of, or even an indifference to, the nouns to which they refer. The weaker student has learned from his past experience, that an instructor will figure out what "it" refers to and assume he means the same thing.
| Final Exam Grades/ Traditional Sections | ||||
| Percentile Range: | 0-24 | 25-49 | 50-74 | 75-100 |
| semester 1 | C | C | B | A |
| semester 2 | F | C | C+ | A- |
| semester 3 | D | C | B | B |
| Final Exam Grades /Reform Sections | ||||
| Percentile Range: | 0-24 | 25-49 | 50-74 | 75-100 |
| semester 1 | F | D | C | B- |
| semester 2 | F | C- | C | B |
| semester 3 | C | B | B | B+ |
| Summary of Final Examination Scores (out of 160 points) | ||||||
| Term 1 | Term 2 | Term 3 | ||||
| Reform | Trad | Reform | Trad | Reform | Trad | |
| Maximum: | 146 | 159 | 103 | 114 | 154 | 158 |
| 3rd Quartile | 84 | 115 | 83.5 | 91 | 123 | 118.5 |
| Median | 86 | 91.5 | 66.5 | 75 | 105 | 93 |
| 1st Quartile | 43.5 | 71 | 44.5 | 54.5 | 100.5 | 74 |
| Minimum | 16 | 43 | 31 | 2 | 55 | 9 |
Use of Findings
When first implemented, students resisted the changes, and their final exam grades lagged behind those of the other sections. Some of the students found the method confusing. On the other hand, others found this the first satisfying experience with mathematics since fourth grade, and reported great increases in confidence. Withdrawals in these courses dropped to zero. Analyzing the program, we found that among the students with a negative reaction, the responsibility and independence required of them was a factor. As a result, in future offerings I hired lab assistants; morale improved and student reactions became highly positive; scores meanwhile rose modestly. The challenge of the following semester was to convert student enthusiasm within the classroom setting to longer periods of concentration required outside the classroom. Here I myself became more active in guiding students. Student explorations were replaced by teacher-guided explorations, but simultaneously, I pushed students harder to develop a mastery of skills and techniques which the problems explored. Each group was required to demonstrate their problem solving skills in front of the other groups. This led to a drop in morale to more normal levels but also to sharp increases in student performance as demonstrated throughout the semester and also on the common final examination.
The dramatic changes in student performance were not observed the first or even the second time the course was offered. It required an ongoing process of evaluation, modification, revision, before achieving a form which appears to work for our students with their particular backgrounds. A break-in period of 1-1/2 to two years may be necessary for experimental approaches to be adapted to local conditions before the full effectiveness of the changes made can be determined and measured. The "curricular material" was essentially the same through all deliveries of the course. What was refined and addressed over time was the pedagogy and classroom approach, as well as student attitudes and morale issues.
Success Factors
The approach to teaching and assessment outlined in this paper requires that the professor take the time to figure out what is going on inside his or her student's minds. The professor should make no assumptions and let the students explain in their own words what it is that they see, read, hear, and do. The professor must then take the time to design activities that will help students replace their naive models with more appropriate ones. The syllabus and use of class time must be adapted to allow this to happen. Some students may be resistant to thinking about the consequences of their mathematical action and will be resistant to reasoning about what the mathematical symbols communicate. Some faculty may have difficulty understanding what you are doing, and prefer that you "train" or "drill" them to perform the required manipulations. But if approached with sensitivity, perseverance, and helpfulness most students can be encouraged to expand their horizons and indulge in some critical thinking. Some advice:
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