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Pedagogical Implications for Problem-Centered Teaching |
| Glenda T. Lappan |
| Michigan State University |
No other decision that teachers make has a greater impact on students' opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics. Here the teacher is the architect, the designer of the curriculum:
The activity in which knowledge is developed and deployed . . . is not separate from or ancillary to learning and cognition. Nor is it neutral. Rather, it is an integral part of what is learned. Situations might be said to co-produce knowledge through activity. (Brown, Collins, & Duguid, 1989)
In order to develop productive notions about mathematics, students must have opportunities to be actually involved in doing mathematics--to explore interesting situations that can in some way be mathematicized; to look for patterns; to make conjectures; to look for evidence to support their conjectures; to make logical arguments for their conjectures; to make predictions or reach conclusions supported by evidence; to invent new ways to use their mathematical knowledge and tools to solve problems; and to abstract from experiences with solving problems the common mathematical concepts, ideas, skills, procedures, and structures that have more universal application.
In selecting a mathematical task, a teacher judges how well the task represents the embedded concepts and procedures that are the goals of instruction, how likely the students are to bump into the mathematics in the course of investigating the task, how well the task represents what is entailed in doing mathematics, and what skill development the task will or can support. An experienced teacher asks the question, "With what mathematics does the task surround the students?" Teachers also have to balance their selection of mathematical tasks to include tasks that allow and promote the usefulness of mathematics in solving authentic problems with all of their inherent messiness.
Here is an example of such a task. Consider the data in Table 13-1, taken from an advertisement in a Florida newspaper.
Table 13-1: Monthly charges for various Internet access providers |
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Suppose you are in the market for an Internet access provider. Which of the services in the table would be the best option for you to choose? The answer to this question is, "It depends!" One way to approach the problem would be to build tables and form graphical representations of each plan for different number of hours of use. Creating representations that allow comparisons is desirable. Tools such as graphing calculators or spreadsheets could be used. The solution is, of course, not a resounding endorsement for one of the services, but a more serious analysis of what ifs; an analysis that shows which plan is optimal when a desired number of hours of access is specified.
If teachers make the decision to use such a task in their classrooms, they have the responsibility to determine its mathematical potential. What mathematics can students learn from analyzing this situation? What mathematics are they most likely to use? Each of the plans can be represented by a piecewise linear relationship between cost and hours of use. This engages students in identifying variables and writing equations describing relationships that are constant for a number of hours and that change in a predictable fashion after that time. This is the essence of mathematical modeling. After modeling the situation with equations, tables of values, or graphs, students have to analyze the representation to make comparisons. They are likely to have to find points of intersection for the relationships. They may look at rates of change or slopes of lines. The important thing is that the task is rich with important, useful, connected, applicable mathematics.
Teaching through Bigger Problems
While the payoff for students can be very great, teaching through big problems increases the complexity of classroom instruction for teachers who are often imbued with the traditional view that mathematics is a well-ordered sequence of rules and procedures, mostly concerned with numbers and number operations. Many prospective and some practicing teachers do not expect mathematics to make sense, but they do expect to be able to remember a rule from which a solution can be swiftly found. They view the role of the teacher as explaining how to do the tasks and telling students when they are correct. Teachers with the traditional view can teach traditional classes confidently if they know the rules, the procedures, and when to apply them. Unless the teacher's mathematical understanding is deep and connected, however, such an approach often misses opportunities to make connections with other mathematics and with student thinking.
It takes a great deal of effort, and time, to create a new vision of what mathematics learning could be. Experience in working on "big problems" helps change how preservice teachers see themselves as learners of mathematics. "We were trained for so many years," reported Tamara, an older woman with weak mathematical background. "This is the way you do it. It becomes a way of thinking. This was the way I had always done mathematics so I've had to totally reorient myself. You have to restructure your whole way of thinking about mathematics and that alone is a big job. . . . To know mathematics means being able to say, 'this would make sense.' To know why something works and to be able to express it, you have to be able to communicate it. . . . You have to experience math, interact with it. You have to struggle to put things together and take them apart. If something doesn't work, you just try something else. . . . You need to experience it and talk about it, not just memorize it." (Schram, 1992, pp. 26, 34)
Another prospective teacher, Kim, struggled throughout the first two courses in her program. In her later courses, she began to gain confidence in her ability to think about the problems posed. "Math 201 was the hardest of the courses in the sequence because I really struggled with trying to think about math differently. . . . Now I am willing to continue working on a problem for a long time. Before, just forget it, if I didn't know the answer when I looked at a problem. I didn't even try further." (Schram, 1992, p. 32)
Preservice and inservice teachers who work on big problems become more willing to persevere with them. They develop mathematical resources, both intellectual and personal, that give them confidence in their ability to tackle real problems. They move from approaching mathematics as a technical subject to approaching mathematics as a sense-making enterprise based on careful observation, invention, making connections, gathering data, making conjectures, and seeking evidence.
Teachers face enormous complexities in attempting to create environments for learning mathematics in which students engage in making sense, individually and in groups, of big problems. Problem-centered teaching is demanding and requires of teachers an understanding of mathematics that will enable them to help students in their search to make sense of and use mathematics. Such instruction values students' thinking. Students are seen as "thinkers with emerging theories about the world" (Brooks & Brooks, 1993) rather than as passive recipients of information.
Problem-Centered Teaching
As the nature of the mathematical tasks changes, teachers must develop new classroom roles. If students are to have opportunities to explore rich problems within which mathematics will be confronted, teachers have to learn how to be effective in at least four new roles:
- Engaging students in the task;
- Pushing student thinking while the exploration is proceeding;
- Helping students to make the mathematics more explicit during whole-class and group interaction and synthesis;
- Using and responding to the diversity of the classroom to create an environment in which all students feel empowered to learn mathematics.
A reflective teacher realizes that engaging students in a task does not mean just having fun with its context. It is important, of course, for students to understand the context. But having an inclination to seek ways to mathematicize the situation they are exploring is critical. The teacher has to work with the students to help them understand which questions mathematics can help answer in the situation. This means keeping an eye on the mathematical goal in posing the task. This does not mean that the teacher structures the mathematical questions so that no thinking or work is left to the student. It means that the teacher keeps the focus on the big question embedded in the task and uses his or her judgment about whether this is a time when the students are to formulate questions for themselves or to find answers to problems in the situation that are posed by the situation or the teacher.
As the students work on the task posed--often in groups, always using tools such as calculators, computers, and physical manipulatives, as well as intellectual tools such as analogies--the teacher can assess what sense the students are making of the task and of the mathematics. By circulating among groups of students the instructor can ask students for evidence to support their conjectures and can redirect groups that are off-task or floundering. Here the teacher is a coach, a guide, an interlocutor, and an assessor of student progress and problems.
After groups have made progress on the task, the teacher and the class can come together to look at the different answers, to examine the data collected, to look at the strategies used, to examine the conjectures the groups have made and their supporting evidence, and to look at the proposed solutions and the reasoning to support the conclusions reached. It is during group sense-making that the teacher must be alert to the mathematical goals embedded in the task--to bring the mathematics alive, to help students make it more explicit and powerful, and to help students connect what they have learned to things they already know. This is where the teacher can work most effectively to set high expectations, both for students' mathematical performance and for the ways in which students engage in discussions with each other.
For teachers, it is daunting to examine what they need to know in order to help develop mathematical power for all students. Few teachers know enough to feel comfortable with this type of self-examination. However, a first step is to recognize that we all have things to learn. As students often learn most effectively in groups, so teachers will find that learning alone is unlikely to be as powerful as engaging in dialogue with other teachers. In order to get started, teachers need motivation for engaging in a daily search for tasks, materials, questions, and responses that will enable students to learn. Teachers have to focus on what students are learning rather than on simply "covering" the curriculum. Part of this has to do with the professionalism of teachers. Being professional includes managing the dilemmas of teaching in a thoughtful way, constantly trying to get smarter about the possibilities.
References
- Brooks, J. G. & Brooks, M. G. (1993).
- The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development.
- Brown, J. S., Collins, A., & Duguid, P. (1989).
- Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-42.
- Schram, P. (1992).
- Learning mathematics to teach: What students learn about mathematical content and reasoning in a conceptually oriented course. Unpublished doctoral dissertation, Michigan State University, East Lansing.
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Glenda T. Lappan is a Professor in the Department of Mathematics at Michigan State University. Her research interests are in the related areas of teaching and learning mathematics and supporting teachers to improve their practice. She is currently Vice Chair of the Mathematical Sciences Education Board and is President of the National Council of Teachers of Mathematics (NCTM), 1998-2000. She is Co-Director of the National Science Foundation funded Connected Mathematics Project and a member of the National Education Research Policy and Priorities Board. She chaired the development of the grades 5-8 part of the NCTM Curriculum and Evaluation Standards for School Mathematics and was the overall Chair for the development of the NCTM Professional Standards for Teaching Mathematics.
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