A Mathematics Class that gets right to the Core of it
Julian F. Fleron
Department of Mathematics, Westfield State College, Westfield, Massachusetts 01086
Many current efforts in first-year college mathematics courses involve simultaneous changes in content, pedagogy and methodology. However positive these efforts are, such efforts are often impractical for smaller departments or departments that do not have several people well experienced with new educational initiatives. We, my chair Catherine Lilly and I, have recently begun teaching our three credit core liberal arts mathematics class in a way that is not only entirely different, but accessible as well. The mathematical content treated in this course is similar to that treated in many liberal arts texts, but lectures have been replaced by cooperative discovery learning, writing and journalling play central roles, and the course is compatible in many ways with constructivist teaching philosophy.
Our ability to implement such changes in the course comes from our use of a problem based text. The text we use 1 leads students through a guided discovery of selected mathematical topics. Each "lesson" in the text contains a short reading motivating the mathematical topic at hand. It then contains three problem "sets" which help the students explore the mathematical topic.2 We spend the first five minutes of class discussing the reading or considering developments and questions from the previous class. For the next forty minutes the students work in groups on the problem "sets". I spend the time touring the room, answering questions, listening to group discussions, looking for misconceptions, etc. Five minutes before the end of class we share solutions and approaches to problems. The work that the students do in groups serves as rough drafts, final solutions are completed individually outside of class. The student's final solutions must be coherent, expressed in complete sentences, contain the reasoning that led to the solution, and must be intelligible to somebody who has not read the problem. The solutions are graded and extensive comments are given regarding the mathematical content and presentation as well as the written presentation. These "projects" make up the majority of the students grade. The rest is determined by class participation and student journals.
The benefits of this course are many-fold. Foremost, the students are actively engaged in mathematics for extended periods of time in a supportive environment. The role of the teacher is now that, not a lecturer. The students develop their technical writing and communication skills, their ability to precisely and coherently use and present quantitative information, and their ability to express lines of reasoning and critical elements of an argument. Since it is the text that organizes class direction, the students develop their reading skills and their ability to decipher and interpret "word problems." In a class peopled mainly by "mathematics phobes, mathematics haters, and mathematics illiterates" you have given each a chance to see mathematics in a different light. It is not a class where 50 minute exams, so destructive to such an audience, determine who passes and who fails. The students know exactly what is expected of them. They quickly see that if they come to class every day, work conscientiously, and expend the effort needed to develop final drafts of their solutions that they will pass the class with a respectable grade. Although they may not see it so clearly, they are all learning a great deal of mathematics, and in many key areas in a liberal education, their abilities are growing. The only drawback to this approach is the grading considerations. I will spend approximately 40 hours grading this semester for each of my classes of 35 students. This is a tremendous amount of work, but I think it is important to assess their work with a level of respect commensurate with the effort they have expended.
I believe this is a wonderful environment to learn in. It is supportive and it most efficiently utilizes the time and efforts of both the students and teacher. I have found this approach remarkably successful, and the environment it creates a model for my other classes. That alone is reason enough to share such an experience. Beyond this, the fact that such an approach is accessible in many settings and can be adapted to include a variety of educational initiatives and philosophies makes it both timely and significant.
1. We use Harold Jacobs' Mathematics: A Human Endeavor, published by W.H. Freeman, ISBN number 0-7167-2426-X. The importance of the organization of the text cannot be understated, nor can the fact that we are extremely happy with the book. Return to text.
2. The structure and organization of this book is radically different from any other than I have seen. Return to text.
J_Fleron@FOMA.WSC.Mass.Edu
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