A Core Mathematics Class that Appears to Serve its Purpose
Julian F. Fleron
Department of Mathematics, Westfield State College, Westfield, Massachusetts 01086
"The goal of teaching is learning, not teaching." -The mathematician Hugo Rossi
Introduction
Mathematical Explorations, MA0150, is a three credit mathematics course at Westfield State College.1 This course can be used to fulfill part of students' analytical science core requirement, which consists of three courses; at least one in mathematics and one in a laboratory science. The course description for the course reads:
An exploration of topics from elementary theoretical mathematics, including some of, but not limited to, the following topics: sets, logic, number systems, geometry, elementary number theory, and counting. Designed to give the liberal arts major an appreciation of mathematical reasoning and content.
Within this description the choice of content and philosophical approach to this course varies broadly by teacher and semester. Overall this course, and the others that make up the core, have not been entirely successful. The drop/failure rate over the last few years is unacceptably high, but probably at a level that is common to many mathematics departments at similar institutions with comparable core requirements.
I am currently (spring 1995) teaching two sections of this course for the first time. It has been a wonderful, and I think, extremely successful experience. I would like to briefly provide you an idea of the nature of these classes. For me it was an entirely new way of teaching. I would not have had the courage to attempt such "radical"2 changes without the guidance and support of my department chair, Professor Catherine Lilly. It is my hope that sharing the story of this experience will help allow others to embark on similar undertakings in their classrooms.3
The Setup
In this class many of the current "reforms" in mathematics education, and education in general, are employed. This includes: cooperative learning, constructivist cognitive theory, writing in mathematics, journalling, etc. The crucial ingredient that allows one to introduce so many "reforms" simultaneously is the text that is used: Mathematics, a Human Endeavor: a book for those who think they don't like the subject by Harold R. Jacobs.4 It is an enlightening and enthusiastic book; one of the best texts I have ever used. The content is broad and interesting. This semester we are treating: mathematical ways of thinking, large numbers and logarithms, an introduction to statistics, and topics in topology. But what is most important about the book, the way I use it, is the way it is organized. Each chapter is made up of approximately 6 lessons, each organized around a central theme. Each lesson has a reading which, unlike the norm, is a brief two or three page introduction to the theme. It treats historical and philosophical relevance, or applications, or some other form of intellectual motivation of the theme, but not content in the traditional sense. The remainder of the lesson, usually between 4 and 6 pages, is devoted to exercises. These are not your run of the mill exercises, the kind that have managed to beat most of the population into a state of indifferent submission. Instead, they are designed to lead the students through a process of guided mathematical discovery.5 Learning takes place in the process of reading, interpreting, considering, discussing, and exploring the exercises. In a very real sense the book acts as a map that helps the student chart the discovery of a personal mathematical exploration.
Having described the book, let me describe how I run the class. Class meets three times a week, fifty minutes a period, for 15 weeks. The class meets in a room where there are eight large tables. Students sit in groups, of their own choosing, but fixed for the semester, 4 or 5 to a table. Before class begins I write a quote on the side board, something related to mathematics, learning or education.6 Each lesson begins with me asking students, by name, questions about the reading for the lesson, something they have done before class. The students then begin to work on the exercises in the lesson. They work in groups and discussion is encouraged. I tour the room, helping students when they have questions and otherwise just observing students discovering. I ask questions and point out inconsistencies when I see them, and usually once a period we stop for five minutes so we can discuss interesting thoughts or difficulties. Again, I call on students by name to answer questions or to provide explanations of what their group had done. In a fifty minute class, on average, 45 minutes are spent with the students working on the exercises in their groups.
The students use their work from class as rough drafts for solutions to the "exercises." At the end of each chapter the students hand in final drafts of their solutions. I believe that in a class like this it is important for students to develop their technical writing and communication skills, be able to precisely and coherently present quantitative information, as well as to coherently express lines of reasoning and critical elements of an argument. Accordingly, the final solutions are to be expressed in complete sentences which coherently express their solution and the reasoning that lead to this solution in such a way that the solution is intelligible to someone who has neither read nor previously considered the exercise.
Results and Observations
Of my 70 students, not a single student was enrolled in this class for a reason other than to fulfill their core requirement. Not a single student said they enjoyed mathematics. And a full two-thirds admitted on the first day that they were frightened of this class. By the end of the third day their entire outlook had changed. When I arrive five minutes before class begins, half of the class is usually seated in their groups working already. Many days I feel the class would continue without interruption if I didn't show up at all. Unlike most other classes at this level, there is not a chronic attendance problem even though attendance does not count in the determination of their course grade.7
The work that the students do is incredible. It's not mathematically perfect, but it is much better than I had ever imagined it would be. The final solutions that the students hand in are usually about 20 pages per chapter. Usually 10% of the students type their solutions. (All 20 pages worth!) They really work extremely hard on these projects. In addition to these projects, the students keep journals which they write in once a week or so and hand in at various times during the semester. In a course which is so atypical of other mathematics courses, for me as well as them, these journals serve a very important role for all parties. They make the students reflect on their experience in this class. And they help me because in them issues are raised that I might not have recognized otherwise. The one issue that stands out involves the central role of writing. Some of the students argue that their is no place for such emphasis on writing in a mathematics class.
The only truly problematic issue is grading their work. I believe what they are doing is extremely important, otherwise I would not expect them to work so hard on it. In return, I think it is important to assess their work with a level of respect commensurate with the work they have put into it. This means grading their projects like one might upper level papers, not simply "freshman math homework." With 70 papers of 20 pages this is quite a challenge. I grade the papers in the following manner. In each of the categories: mathematical correctness, depth of understanding, completeness, coherence and clarity, and neatness, organization, grammar, spelling and effort, I award 1 to 5 points. I provide comments liberally, be it on mathematical issues, writing issues, or reasoning issues. Because I provide so much feedback it is necessary to indicate the severity of the issues I comment on. I use + whenever the solution is "correct" and the comments are merely suggestions or stylistic improvements. I use a check to indicate that the difficulty is more serious and has resulted in deductions. And I use a cross over parts of solutions that are "badly incorrect." But no matter how serious the difficulty I try to make my comments constructive and engaging. Simply telling the students they are wrong, or even providing them with "correct answers," thwarts, and essentially ends, their journeys. On the other hand, giving their efforts legitimacy by pointing out inconsistencies and misconceptions in a way that will help them reflect on the difficulties enables them to continue their exploration.
That is my grading philosophy. I think it is commensurable with my philosophy for the nature of the class as a whole, and I think this is an important issue. It is however, about enough to kill me. I grade about 40% of the solutions from each lesson. To grade one students solutions to one lesson, in the manner described above, takes about two and one-half minutes on average. This is about 3 hours per lesson. At six lessons per chapter, plus the further exploration problems that are occasionally assigned, this is about 20 straight hours of grading per chapter. Since we will cover about 4 chapters, this is 80 total hours of grading for the semester for two classes of 35 students! That is a tremendous amount of grading, concentrated over four short time periods during the semester. And is not easy grading. To minimize the handling of so many long papers I have found it essential to grade a lessons worth of problems at a time, that's about 20 problems at once, in addition to keeping track of all the grading schemes so the papers will be graded uniformly.
Conclusions
For all the work the grading requires, I still believe it is well worth it. I started with 72 students, most of whom where mathematics phobes and, in some sense, mathematics illiterates. I still have 70 students. I'm not sure there are any mathematics phobes left. There are no mathematics illiterates either. There are two classes of mostly hardworking students. Some of whom still don't want to be writing sentences in a mathematics class and aren't having a great time. But I think it is safe to say, most of them are enjoying this class more than they thought they ever could enjoy a mathematics class. Less than 5% of the students are in danger of failing. Every student has learned some mathematics in this class. In fact, I would venture a guess that the average student has spent more time actively engaged in doing mathematics than they have over any other two year period in their lives. Accordingly they have grown in their understanding of mathematics, in their reasoning and critical thinking abilities, in their quantitative communication skills, etc. I believe these things are exactly what a course like this should be about. Additionally, I believe that it is these types of things that are the intended benefits of requiring mathematics to be included as part of the core requirements.
As a final note I would like to include what I, and perhaps the students, find most remarkable about this class. Namely, that the students are doing it themselves. That they are the key. The success of this class, which I find remarkable, isn't because there is a master teacher. I certainly am not. Nor is it because there is a perfect textbook; although it really is an awesome book. This class is a success because finally the students have been given an opportunity to explore mathematics as well as a rich, supportive environment in which to do so. They finally have an opportunity to succeed in a mathematics class, and essentially every one of them has accepted that opportunity.
Footnotes
1. Westfield State College is liberal arts college of 3500 full-time students and 1500 part-time students, located in the western part of Massachusetts. It is one of the oldest coeducational teachers colleges in the United States. Return to text.
2. I cannot use this word recently without being reminded of the following quote by Dan Kennedy: "When I was young all of my radical friends were in reform school. Today all of my radical friends are in school reform." From "A Tale of Two CD's" in The American Mathematical Monthly, Aug.-Sept. 1994. Return to text.
3. I would happily share more detailed information about this course with anybody who is interested. This includes the course syllabus, format and grading instructions, grading sheets, students reflections, etc. Return to text.
4. First published in 1970, the book is now in its third edition and is published by W.H. Freeman. The ISBN is 0-7167-2426-X. Return to text.
5. This methodology fits well with constructivist cognitive theory, an educational philosophy that is rapidly, and radically, reshaping much of mathematics education. For a brief discussion of constructivism in mathematics education see "Constructivism in Mathematics Education: A View of How People Learn" by A. Selden and J. Selden in Undergraduate Mathematics Education (UME) Trends, March 1990. For much more in depth treatment see Constructivist Views on the Teaching and Learning of Mathematics, Journal for Research in Mathematics Education Monograph #4, edited by R.B. Davis, C.A. Maher, and N. Noddings, 1990. Return to text.
6. To give you an idea of the type of quotes, three recent examples are:
"Imagination is more important than knowledge." - Albert Einstein.
"Education is learning more than it is being taught. It is the chemistry of curiosity exposed to information. In that sense all of life is potentially school. And even I can pass that." - Bob Guiccone, jr.
"Hilbert once had a students in mathematics who stopped coming to his lectures, and Hilbert was finally told that the young man had gone off to be a poet. Hilbert is reported to have remarked, 'I never thought he had enough imagination to be mathematician.'" - George Polya Return to text.
7. Actually, that is not entirely true. Part of their grade is based on class participation. A large part of this grade is determined by their responses to my questions, which of course they cannot answer if they are not in class. Return to text.
J_Fleron@FOMA.WSC.Mass.Edu
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