My Lilly Program project was to strengthen the Engineering-&-Science Calculus II course (Math 142) by consciously integrating my research and teaching philosophies. First, an explanation of my research philosophy is in order.
My mathematical research philosophy is two-fold. One part is assimilating new knowledge into my basic foundation. This new knowledge is not learned linearly thus needs to be properly woven into existing thoughts. This weaving is a key factor not only in mathematical research but also in the general logical thinking required in most areas of life. Discussing new concepts with my colleagues clarifies the various connections between new and old knowledge. In seeing these connections, I am better able to assimilate, organize, store, and recall for later use, the new knowledge obtained. The second part is the actual problem solving. I need to ponder a problem, often for months with it continuously floating somewhere in my mind, to be able to identify the key elements and find connections with my existing knowledge, in short, to be able to see how everything interacts. Only then is it possible to combine present knowledge with an original idea to solve the problem. These two aspects of research are intimately connected. The gathering, organizing, and storing of knowledge is later used in solving a problem. A new problem adds, just by the nature of dissecting its many facets, to a better understanding of the connections between existing knowledge. With this better understanding, the existing concepts can be reorganized and more efficiently stored so that new knowledge may be added. The two aspects combine to give a solid foundation in the subject matter.
Similarly, the students need to learn how to assimilate the new material into their basic foundation and how to draw upon this knowledge when so needed. To obtain these goals, I introduced group projects into the course. Each project was their own mini-research query. On the first day of class, the students received the below philosophical explanation of Project-based Calculus.
Project-based Calculus, part of the
recent National Science Foundation
Calculus Reform Initiative, enhances the
typical calculus course with challenging
projects.
The projects, which reinforce and
expand on concepts presented in class, are designed to:
This approach to calculus is based on the premises that
one learns and retains mathematics better if one
explores it on ones own, passing from
specific examples to general principles,
discussing ideas with others along the way.
Thus the projects are multi-stepped explorations and are prepared in
groups
(of 3 to 4 students which change with each project).
You are strongly encouraged
to discuss your ideas with others. From the
instructor and teaching assistant, expect a
``minimalist'' approach to guidance. We will gladly
answer questions, but usually the reply will be in the
form of another question which will push you further but
is far from ``the answer''.
A project is yours to explore - Enjoy.
Calculus projects will not only help you gain mathematical expertise, but will also help you prepare for your upcoming career in several other aspects. Likely you will have to provide your learned expertise on a group project. This is your opportunity to develop your personal style in helping maximize the group's efforts. Likely you will have to prepare technical reports. This is your opportunity to develop such writing skills. Likely you will have to meet deadlines. This is your opportunity to develop time management skills.
Three group projects were assigned during the semester. The projects, which were to be completed in two to three weeks, were assigned so as not to interfere with the students' need for extra study time just prior to an exam. The project assignments, as given to the students, are located in Appendix A of this report. On the first day of class, the students received the following advice.
A project is a major lengthy assignment. Start immediately -- let your subconscience work for you -- immerse yourself in the project.
You should plan your first group meeting as soon as possible. But before the first meeting, you should have read the project carefully and given it some thought. At your first meeting, you should plan a method of attack keeping in mind that all members of the group are expected to understand all parts of the solution. Directly after the Tuesday-Thursday Lab period is an excellent time for regular group meetings to discuss the progress on the solution.
The project report (one per group) should clearly explain and support your conclusions. It should be a mixture of prose (written in complete sentences), equations, formulas, charts, diagrams, etc. Proper grammar, spelling, and punctuation is important. Your project solution should be written in such a way that it can be read and understood by ANY USC Calculus II student. You will be graded on your written presentation as well as the mathematical content. It it not necessary that projects be typed, but if handwritten, it must be neat and legible.
The first project explored the book's unmotivated definition of the natural logarithm function. The project was assigned on the day which we began the chapter on the logarithm function. By using ideas from Calculus I and doing computations, the students were asked to tie together their prior knowledge of the integral and the book's definition of the logarithm function as an integral.
Since it was their first project, one group member from each group met with me to present a rough draft of their project about a week prior to the due date. At this point, most groups had made substantial progress on their project but were struggling to tie it together. With a few extra hints (and several words of encouragement) most groups were able to tie the project together to gain a deeper insight into the logarithm function.
The second project were designed by my senior mentor Prof. Jim Roberts. Within the three semester calculus sequence, most students find Calculus II the most challenging due to the course's section on infinite series. To shed light on this abstract notion, Jim formulated a project, centered around the children's card game Old Maid , which introduced infinite series (without explicitly using the terminology) in a simple natural fashion. The project was due about one week before we began the chapter on infinite series so that this ``new abstract'' idea was not completely new to them when we arrived at it in the class.
To my pleasant surprise, each project arrived at the correct conclusion, furthermore, with very little help from me. The project also required that the students do a little research (e.g. read from an elementary book) into basic probability theory, which they did just fine. The last question on the project was: Give an explanation of the answer that a twelve-year-old could understand. If helpful, draw diagrams. (Don't be asking yourself -- what does she want -- be asking yourself -- what would a 12-year-old understand!) The students enjoyed this question and showed quite some originality in their answers (see Appendix C). This year, my lectures introducing infinite series draw much more classroom participation than in the prior years. This year, the students were asking questions beyond the material being presented, while in the prior years, the students' questions indicated a lack of understanding of the presented material.
Since the students liked the second project (which was designed by Jim Roberts) more than the first project (which was designed by me), I turned to Jim for help in designing the third project. The course has a short (2 lectures) introduction to complex numbers. Jim and I designed a project to help the students gain a deeper understanding of complex numbers. The project picked up where the textbook homework set left off. Combining what they had just learned about complex numbers (which, technically, was as much as they were expected to have mastered) with some basic trigonometry, they found families of integral right triangles.
The students found this project very difficult. Each group made progress but few groups were able to complete the project correctly. I will not again give a project this late into a semester when fear of finals is beginning to show its face. They probably developed a deeper understanding of (but more frustration towards) complex numbers than if we just stopped at the problems in the textbook.
Since this project is related to Fermat's Last Theorem, which is one of the most challenging open math problem to date, on the back side of third project I included the recent article Fini to Fermat's Last Theorem from the 5 July 1993 Times Magazine (see Appendix A). Over 350 years ago, Fermat wrote, about Fermat's Last Theorem, ``I have found a truly wonderful proof, which this margin is too small to contain.'' To this day, no correct proof has been found to Fermat's Last Theorem. In June 1993, Princeton professor Andrew Wiles announced a proof, which runs over 200 pages, but by the time of this project in November 1993, rumors where circulating about possible gaps in Wiles's proof. This news lead many of the students to inquire about the ``nature'' of a math proof, which was the first time, at this level of a course, that the students asked such exploratory questions.
Our goal was to strengthen and add depth to the Math 142 course through the use of group projects. Did we succeed? This is open for debate; I feel that we did succeed and I will use group projects in my further undergraduate mathematics course.
I have taught Math 142 only twice at USC: with the traditional format the Fall 1991 semester and with the project format the Fall 1993 semester. Thus there is insufficient past USC experience for a thorough comparison. The success rate (C or better) was the same both years; the class average on my (notoriously hard) hourly exams were comparable: 57% in 1991 and 49% in 1993. But my goal was added depth and not remediation; in 1993, the class average on the challenging projects was high, 80%.
For the purpose of assessment, an anonymous student questionnaire (see Appendix B) was administered during class session near the end of the semester. The students responded to the following questions on a scale of 1 (low/not much) to 5 (high/lots).
| response ---> | 1 | 2 | 3 | 4 | 5 | blank |
|---|---|---|---|---|---|---|
| question (1) | 4 | 9 | 8 | 25 | 7 | |
| question (2) | 2 | 4 | 14 | 24 | 9 | |
| question (3) | 1 | 9 | 15 | 16 | 12 | |
| question (4) | 2 | 5 | 13 | 24 | 8 | 1 |
| question (5) | 8 | 6 | 18 | 13 | 8 |
To the first four questions, more than 50% responded with a 4-or-5, while more than 75% responded with a 3-or-4-or-5. (Perhaps I was a bit too idealistic to hope for high scores on question 5). When asked on the questionnaire if they would prefer a calculus course with projects again, 64% responded YES. From these statistics, one gathers that the students feel that the Project-based Calculus course indeed did succeed. Although we did not see an improvement, between the 2 semesters, in the hourly exam scores, I feel that the questions raised, both in and out of the classroom, by the students in the Project-based Calculus course indicates that the projects did add depth to the students understanding of the material.
The student questionnaire also included the following 2 questions.
| Appendix A: | Postscript files of the three projects assigned, as given to the students: |
|---|---|
| Project # 1 | |
| Project # 2 | |
| Project # 3 | |
| Appendix B: | Postscript file of the Student Questionnaire |
| Appendix C: | Various sample project that range from the better ones to the not-so-good ones: |
| contact Prof. Girardi |
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Last modified on 14 April 1997 by Maria Girardi
URL: http://people.math.sc.edu/girardi/lilly/fr.html