New classroom rules to promote preservice elementary teachers' mathematics learning

This article describes instruction in a university mathematics course for preservice elementary teachers modeled from progressive instructional strategies used in elementary mathematics classrooms. The instruction is designed to encourage students to solve problems creatively and independently, to communicate their thinking in small group and whole-class discussion, and to construct deep understanding of mathematical ideas. The classroom rules governing this instruction are reviewed in detail.

Do university mathematics courses required for preservice elementary teachers prepare them for the elementary classrooms envisioned by the National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (NCTM, 2000)? These mathematics courses usually fall under the jurisdiction of the mathematics department and are taught by mathematics faculty or graduate students using some form of a traditional, lecture-recitation format. Although mathematics reform has inspired active instructional revision in some university mathematics departments, there remains a strong, recalcitrant adherence to a tradition of direct instruction among mathematicians, many of whom, the brightest and most capable students in their courses, recall with nostalgic fondness and appreciation the rigor, discipline, and intensity of theorem-proof graduate courses, mirroring the deductive approach of the mathematics textbook, beginning with foundational axioms and building an edifice of powerful and elegant theory. One very successful university mathematician summarized mathematics instruction this way: the professor's role, as the master artist, is to paint a beautiful picture of mathematics for the students, who, as apprentices, copy the master's work to improve their skill; students expect and want, he continued, to see the professor exquisitely portray and carefully delineate elegant mathematics in university mathematics courses.

RELATED RESEARCH

Research suggests that mathematics instruction the majority of preservice elementary teachers have received is preoccupied with procedures and based upon lectures (Battista, 1999; Manouchehri, 1997; O'Brien 1999). "Traditional mathematics teaching ... is still the norm in our nation's schools. For most students, school mathematics is an endless sequence of memorizing and forgetting facts and procedures that make little sense to them." (Battista 1999, p. 426) In mathematics classrooms of many colleges and universities, instructional approaches such as multiple representations, open explorations, and meaningful investigations of ideas have not replaced the lecture (Manouchehri, 1997). Ball (1990, 1996) concluded that preservice elementary teachers have a weak, fragmented knowledge of mathematics, mostly acquired facts and memorized rules; she further asserted that they have rarely seen or experienced a kind of teaching that, focusing on conceptual understanding, engages students in complex reasoning in authentic contexts. In a meta-analysis of 151 research studies Hembree (1990) discovered that preservice elementary teachers have the highest level of mathematics anxiety of any major on university campuses. Emenaker (1996) found that preservice elementary teachers, with improvement in conceptual understanding, relied less on the memorization of facts, algorithms, and formulas.

There is a strong interconnectedness with regard to conceptual understanding, mathematics anxiety, and memorization. If students possess a deep conceptual understanding of mathematics, they will need to rely less on arbitrary, memorized facts and rules and will exhibit less mathematics anxiety.

From 13 years of teaching mathematics courses for preservice elementary teachers, reading their mathematics autobiographies, conducting interviews, and conversing with them, the author believes most preservice elementary teachers are anxious, if not terrified, about passing courses like college algebra, which they often consider the main roadblock to their career teaching elementary school. Because of fear and anxiety, they have developed survival skills to navigate these classes. They work hard to master algorithms presented by the professor on the board, especially those that also appear throughout the homework, and concentrate on getting the correct answer because they know that correct answers will be aptly rewarded on the next test. They have learned there is one correct answer to a problem and one method to get that answer, the professor's, and a one-to-one correspondence between word problems and formulas; for every word problem there is a formula, and vice versa. They use textbook examples as templates for homework problems and memorize vast quantities of information: facts, formulas, rules, algorithms, sequences of buttons to push on the graphing calculator, etc.

From the author's experience teaching mathematics in high school and supervising high school student teachers in mathematics, he has observed that preservice elementary teachers often have experienced a cyclic routine in traditional high school mathematics courses: the teacher introduces the new material, quite often as a series of algorithmic examples, students work on homework problems individually while the teacher circulates throughout the classroom helping them, then the teacher answers questions on the homework. Preservice elementary teachers, as students in traditionally taught high school and university mathematics courses, often have experienced teacher-centered mathematics instruction that focuses on rules, formulas, and answers. If they experience a mathematics course that is student-centered, emphasizes active-learning, communication, and reasoning, they most likely will be able to teach children mathematics effectively at the elementary level in a manner consistent with the vision presented in the National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (NCTM, 2000).

Progressive reformers in elementary mathematics education have inspired a radical overhaul of mathematics instruction the author uses in Math for the Elementary Teacher, a required university mathematics course for preservice elementary teachers. Cognitively Guided Inquiry (Carpenter, Fennema, and Franke, 1996; Carpenter, Fennema, Franke, Levi, and Empson, 1999), the Purdue Problem-Centered Mathematics Project (Cobb et al., 1991), and the work of Constance Kamii (Kamii, 1985a, 1990b; Kamii and Dominick, 1998; Kamii and Warrington, 1999) espouse a constructivist view of mathematics learning, that the teacher cannot transmit mathematical knowledge directly to students, but students construct it by resolving situations that they find problematic. All three projects accentuate student thinking and active learning, are problem-centered and intensely interactive, and highlight communication, reasoning, and conceptual understanding. Driscoll (2000), Davis, Mayer, and Noddings (1990), and Fosnot (1996) provide a more detailed explication of constructivism and its implications for classroom practice. In every elementary classroom there are rules for students to follow, usually on poster board in bright colors, visible to all; in Math for the Elementary Teacher there are also rules for students to follow, but rules quite different from those governing a traditional university mathematics classroom. Preservice elementary teachers can profit greatly from participating in a university mathematics course using instruction consistent with the NCTM Principles and Standards for School Mathematics, compatible with a constructivist view of learning, and modeled from progressive practice in elementary mathematics education.

NEW CLASSROOM RULES

In Math for the Elementary Teacher students and instructor constitute a learning community, which has rules to encourage students' mathematical thinking, to promote active learning, communication, and reflection, and to enhance students' confidence and independent thought. At times students, working in groups of four, are responsible for teaching the class; at other times the instructor is responsible. The instructor typically introduces new content by the following approaches: (a) one or two problems, which involve the fundamental concepts of the lesson, may be counterintuitive, and may challenge students' preconceived ideas; (b) a game such as "What's my rule?" (Kamii, 1985) to draw students energetically into the learning process; or (c) a short (10 -20 minutes) lecture of key concepts, conventions, and notation.

Problem solving is an essential component of Math for the Elementary Teacher. There is a distinction in the class between exercises and problems. Finding the sum of 1/2 and 1/3 is an exercise; a problem is a whole different breed of cat, leading to Classroom Rule 1: A "problem" emphasizes fundamental mathematical concepts, challenges students (is not trivial or overwhelming), elicits discussion and debate, and has no immediately apparent solution. Students must be able to understand a "problem", be able to follow more than one path to its solution, and have the tools (cognitive and technological) to solve it. A problem often is, but need not be, a word problem embedded in a real-life context. For example, the following is a good "problem" for students of Math for the Elementary Teacher:

You have two identical containers, one holding grape juice and the other the same amount of water. You fill a measuring cup with grape juice, pour it into the water, stir the solution thoroughly, then, after filling the measuring cup with solution, pour it into the water container. Is there more grape juice in the water container or water in the grape juice container?

Homework problems are assigned almost every night and come from many sources, sometimes created by students or the instructor, at other times arising from class discussions or student ideas. Problems originate from web sites on the Internet, publications, elementary classrooms in which students are doing their field experience, real-life situations, and from the textbook (Bennett and Nelson, 2001). Student groups teaching the class may assign homework problems, the instructor may ask student groups to create a problem or choose a problem from the textbook, or the instructor may assign homework problems.

An "open" problem in mathematics research is one on whose solution the mathematics research community has not reached agreement. Classroom Rule 2: Every problem, no matter who submitted it for class consideration, is an open problem until the learning community of students and instructor agrees that the problem has been "solved". The word "solution" is another redefined term for students of Math for the Elementary Teacher, and is not synonymous with the answer. At the beginning of the course the instructor informs the class of the next rule. Classroom Rule 3: The "solution" to a problem is both the answer and the reasoning used to obtain the answer. Students' reasoning may include explanation, justification, and clarification of their method of solving a problem. It usually involves pictures, diagrams, tables, graphs, and other visual representations, and may require the use of manipulatives. A problem is "solved" only when a group of students, or individual student, presents a solution on which there is consensus (general agreement) among the members of the learning community; there can be no dissenting skeptic with a sound argument against either the reasoning or the answer. The instructor insists that reasoning is crucial for a problem to be "solved", so students must convince their peers and the instructor both of the compelling logic of their reasoning and the correctness of their answer.

During the homework review time students first compare their solutions with members of their group; often they are quite surprised how other students thought about the problems and there is usually a great deal of animated conversation. Students then add to a "work list" problems that have occasioned perplexity or disagreement. Classroom Rule 4: A "work list" is a catalog of open problems not yet solved by the learning community. A member of the learning community can add a problem to the "work list" at any time. After the class, working in groups, has discussed work list problems, the instructor asks, "Can any group help us solve any problem on the work list?" Usually one or two groups volunteer, but, if that does not occur, the instructor uses a random number generator to choose a group to present the first problem on the work list. As students solve problems, they are eliminated from the work list.

Classroom Rule 5: Clarification, interpretation, focus, representation, and analysis represent progress and accomplishment in solving problems. Students must prepare a "progress report" for problems they have not solved. According to Polya (1957), "understanding the problem" is the critical first phase of problem solving. If students in a group cannot solve a problem, they must prepare a progress report, in which they help the learning community understand the problem under consideration. This may include a description, interpretation, and/or elaboration of what the problem is asking; it may also include definitions of terms and ideas in student' own words, pictures or diagrams used to depict the information, solution avenues attempted, and discussion of various viewpoints on the problem. Students acclimatize slowly to the idea that their own thinking has value and that, although they have not found the correct answer to a problem, they have learned a great deal from examining the problem from different angles, investigating various representations of the problem, communicating their ideas to one another, scrutinizing their approaches, and summarizing the status of their knowledge about the problem for presentation to the whole class.

Classroom Rule 6: The instructor is a facilitator of student construction of knowledge not a transmitter of knowledge. From a constructivist perspective the instructor cannot pour knowledge into students as if they were empty vessels waiting to be filled. The instructor's role as a facilitator is a complex one, including, among other responsibilities, assisting students to develop productive small-group relationships, encouraging mathematical dialogue, accenting conflicts between alternative solutions, and describing students' ideas in more sophisticated terms and more conventional notation (Cobb et al., 1991). Accompanying this rule is Classroom Rule 7: The instructor does not solve open problems for students. The instructor may request students to reflect upon inconsistencies or ramifications of their thinking and may answer factual or information questions about open problems, for example, providing a definition and numerical example of the union of two sets; the instructor may also suggest problem-solving strategies (Polya, 1957), such as making a simpler problem or drawing a picture, when it does not interfere with students' own thinking about the problem.

At first teacher-dependent students greet these rules with some suspicion and anxiety because they are accustomed to hard-working mathematics teachers conscientiously bending over backwards to help them whenever they get stuck and do not get the answer in the back of the book. During the semester, since the instructor attempts to be pleasant in not solving problems and informative about how this rule advances independent thought, students begin to smile, saying, "You're not going to tell us the answer, are you?" By the end of the semester they stop asking.

At times the instructor mediates directly in students' whole-class attempts to reach consensus either by offering a suggestion, challenging an assumption, or supplying information not recognized by students. Occasionally, all student groups, adamantly convinced of a solution, are ready to continue to the next problem, although they all have the wrong answer from an incorrect interpretation of the problem. The instructor, a member of the learning community not convinced by the solution, suggests that they reconsider certain aspects of the problem.

DISCUSSION

Mathematics learning is a rich, deep process, emphasizing conceptual understanding, reasoning, communication, problem solving and real-life applications. If a student "solves" a problem in Math for the Elementary Teacher, she or he must convince group members of the solution, defend that solution indisputably under the scrutiny of the whole class, and write coherently and persuasively about that solution on exams. She or he often utilizes a great wealth of informal, intuitive, and creative ideas when confronted with challenging problems whose solutions demand in-depth conceptual understanding.

The instructor of Math for the Elementary Teacher has established a problem-solving curriculum that encourages students to construct deep understanding of mathematical ideas and to become autonomous, active learners confident of their own thinking. The successes of this enterprise is most evident when students are vigorously debating various solutions to problems they have created, examining one another's reasoning, assumptions, and interpretations, and have the board peppered with pictures and diagrams to support conflicting arguments. Achieving the goal of constructivist mathematics learning, students are constructing their own mathematics in resolving situations that they find problematic (Cobb et al., 1991). As a result, they are better able to teach in classrooms envisioned by the NCTM and actualized by progressive reformers.

REFERENCES

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Kamii, C., & Warrington, M. A. (1999). Teaching fractions: Fostering children's own reasoning." In National Council of Teachers of Mathematics Yearbook, pp. 82-91. Reston, Va.: National Council of Teachers of Mathematics.

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DR. JOHN K. ALSUP

Education

Black Hills State University

Spearfish, SD, 57799

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