Changes in Methodology: Environments for Learning

"Telling is not teaching, and listening is not learning" (Sally Berenson, North Carolina State University). Every reformer either knows already or quickly discovers the need to focus on learning and on creating an environment in which learning can take place. For some, the physical location is the traditional classroom; for some, a laboratory of the "lab science" type; for some, both. In most projects, "lecture" is being partially or totally replaced by multiple classroom and/or lab activities. The operative word here is "activity" -- active involvement of the student learners replacing passive reception of "the word" from an all-knowing lecturer. These classroom and lab activities often involve experimentation, discovery, and open-ended problems.

Multiple activities go hand-in-hand with multiple representations of the subject matter. Well-designed projects challenge students to use all the tools at hand -- pencil-and-paper, calculator (with or without graphics), computer (definitely including graphics), and, most importantly, prior experience with mathematical concepts and techniques -- to attack problems not previously solved, either by the students themselves or (in examples) by the instructor or textbook author. These activities are also the vehicles for written and oral communication. Students talk to each other and to the instructor about mathematics, formally (as in presentations at the blackboard) and informally (as in small group activities in or out of class or in help sessions). Students are expected to read their textbooks -- often a new and unfamiliar expectation. And many of the activities lead much more naturally to written responses (reports or journals, for example) than to tests.

In most cases, student activities involve teamwork. Students can learn a lot from and with each other that they would never learn working in isolation or listening to a lecture. However, cooperative learning environments also raise issues on which we do not yet see consensus emerging: issues of individual vs. corporate responsibility. Our students have been programmed to compete with each other, to see other students (rather than ignorance) as the enemy. And we are conditioned to using that sense of competition as a motivator and to seeing cooperation as "cheating." So how do we get students to share their learning experiences? And how do we evaluate their efforts when they do?

These are difficult questions to answer, but they are not impossible. Neil Davidson (University of Maryland) has been using cooperative learning environments in calculus and precalculus instruction for over 20 years, and he has assembled practical advice on how to do it from a number of practitioners [4]. Bill Medigovich observes that just saying "groups are good" is not enough -- in particular, it doesn't work to order students to work together outside of class and continue the traditional student-to-teacher focus in the classroom. The behavior we want students to engage in has to be modeled and practiced in the classroom -- under supervision -- or it will never be learned. Janet Ray (Seattle Central Community College) adds that the hardest part of teaching a reformed course is to learn to structure and manage the classroom for effective learning.

Ray Marshall (former Secretary of Labor, now in economics at the University of Texas-Austin), in an address to an NSF invitational conference on systemic reform in education, made a distinction between incentives that are positive, negative, and perverse. The ones that don't work well, in economics or in education, are negative ("You do what I want, or I'll punish you") or perverse ("You do what I want, and I'll punish you anyway"). Superior achievement, both in school and in the workplace, has been demonstrated with positive incentive systems: You do what we all want, and we'll all be rewarded. In the calculus reform movement, a positive incentive system that rewards cooperative achievement has a catchy description coined by Robert White, President of the National Academy of Engineering, in an address to the 1987 NSF conference on calculus reform [8]: In the science/mathematics/engineering pipeline, calculus should become a pump, not a filter.

The filtering issue was a major stimulus to the reform of calculus instruction, because over half of the students taking calculus across the country were failing to achieve satisfactory grades. (Another major stimulus was the observation that, even with satisfactory grades, most students were not able to use calculus in any other course.) Consider this: Our salaries are paid by our students (or their families) through tuition and taxes. We are not paid by the medical, engineering, and business schools who want us to screen their applicants. Thus, we must assign the highest priority to learning (our students' best interests, whether they know it or not) and relegate certification to a distinctly inferior role. An important precursor to certification is assessment of student knowledge, but assessment and feedback are also important tools for education, and we should see them primarily in that light. We are responsible, first and foremost, to our students.

What we know about learning and what we are learning about learning tell us that teaching to tests is counterproductive. And now that we have a wide range of other student activities to observe -- lab reports, project reports, worksheets, group interactions -- we are free to de-emphasize tests as the primary focus of the course. While we may never see a consensus on just how important tests are, everyone understands that we can and should construct and use other forms of assessment as well.

Here is an example of how de-emphasis of tests can be carried out: The calculus committee at Duke, which sets policy for the laboratory course now being taken by all students who start with Calculus I, has allocated 35% of the grade to the laboratory, 20% to the final exam, and the rest to the discretion of the instructor (within broad guidelines); the instructor may use in-class tests, group or individual project reports, quizzes, homework, end-of-term essays, portfolios, or whatever he or she considers appropriate. In most sections, the split between grades given to individuals and grades given to groups is about 50-50, plus or minus 5%.

The dominant paradigm in the educational literature is constructivism, which may be described briefly as the belief that all learning is constructed by the learner in response to challenges to refine or revise what is already "known" in order to cope with new situations. This paradigm specifically contradicts the dominant belief among college-level teachers of mathematics (and probably other subjects) that knowledge is "transmitted" from knower to learner. The theory and practice of constructivism, along with the data supporting its effectiveness, are largely unknown to the mathematics faculty, but every reformer who has focused on what students learn, rather than on what teachers teach, has independently discovered at least the basic ideas of constructivism. There is a simple, highly convincing proof that constructivism better explains what's in students' heads than does "transmissionism": Ask your students to write down what they are thinking as they work through a problem. You will find in their writing ideas or thought patterns that they didn't learn from any lecture or any textbook -- ideas that you wouldn't want to take credit for. Since those ideas didn't come from the external sources, their owners must have made them up.

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