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Constructivist Mathematics and Unicorns

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by Lee V. Stiff, NCTM President 2000-2002
NCTM News Bulletin, July/August 2001

Principles and Standards for School Mathematics is a guide to what mathematics students should learn; what teaching practices, approaches, and tools show promise; and the role that assessment plays in judging students' performance and the effectiveness of mathematics programs. Few critics of mathematics education reform challenge what Principles and Standards suggests students should learn. This may be difficult to recognize, however, because many critics of NCTM's Standards describe the mathematics content as if it were the pedagogy. Their characterizations of "constructivist math" are an example of this.

Constructivist math is a term coined by critics of Standards-based mathematics who promote confusion about the relationships among content, pedagogy, and how students learn mathematics. It is how they label classes where they see students engaged and talking with one another, where teachers allow students to question and think about the mathematics and mathematical relationships. Critics see these behaviors and infer that the basics and other important mathematics are not being taught. Because the mathematics may not be taught the way they learned it, and students may not be neatly arranged in rows as they were, and teachers do not dominate the classroom conversation as they recall, these critics intentionally equate the mathematics content to the pedagogy of reform-minded teachers.

Reform-minded teachers pose problems and encourage students to think deeply about possible solutions. They promote making connections to other ideas within mathematics and other disciplines. They ask students to furnish proof or explanations for their work. They use different representations of mathematical ideas to foster students' greater understanding. These teachers ask students to explain the mathematics.

Their students are expected to solve problems, apply mathematics to real-world situations, and expand on what they already know. Sometimes they work with other students. Sometimes they work alone. Sometimes they use calculators. Sometimes they use only paper and pencil.

Like unicorns, "constructivist math" does not exist. There are, however, several theories about learning that are categorized as "constructivism," and they can be linked to Standards-based mathematics. Constructivism addresses how students learn and what teachers can do to facilitate students' understanding. At least two definitions of constructivism exist that shed light on the teaching of school mathematics.

Radical constructivism is the philosophy that knowledge cannot be provided in some final form from parent to child or from teacher to student but must be actively assembled in the mind by each learner in his or her own way. The responsibility for expanding what one knows, or for constructing new knowledge, rests primarily on the learner and his or her efforts to achieve understanding.

Social constructivism maintains that students can better build their knowledge when it is embedded in a social context. Thus, the interaction between teacher and students is enhanced when it involves a broader community of learners--that is, students working together. Students help one another create richer meanings for new mathematical content. A type of social constructivism that applies specifically to mathematics education maintains that mathematics should be taught emphasizing problem solving; that interaction should take place (a) between teacher and students and (b) among students themselves; and that students should be encouraged to create their own strategies for solving problem situations.

Constructivist philosophies focus on what students can do to integrate new knowledge with existing knowledge to create a deeper understanding of the mathematics. Each philosophy identifies the student as an active participant in the teaching and learning process. What a teacher does to foster the integration and extension of knowledge among students can and should vary. Indeed, Principles and Standards suggests that good teachers use different strategies at different times for different purposes.

For example, there is great benefit to allowing students to construct their own algorithms for addition and subtraction. However, this does not mean that the standard algorithms for addition and subtraction cannot be taught in meaningful ways that help students integrate new knowledge or procedures with existing understandings of addition and subtraction. Nor does teaching the standard algorithm mean that standard algorithms are the first or only algorithms to which students should be exposed. Certainly, teachers can foster a greater understanding of these operations by using objects as referents for numbers and demonstrating the physical manipulations associated with each operation.

In discussions about effective mathematics teaching and learning, we must be wary of oversimplified characterizations of Standards-based mathematics. NCTM's Principles and Standards is not synonymous with constructivism or any other single teaching approach. Reducing the vision of Principles and Standards to one method of teaching or learning is a distortion of the facts and thwarts our progress toward providing a high-quality mathematics education for every student.

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