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Tom Carpenter

School mathematics curricula in the U.S. have traditionally separated arithmetic and algebra. This historic separation has deprived students of powerful schemes for thinking about mathematics in the early grades, says UW-Madison education professor Thomas Carpenter. Carpenter directs the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA).

Separating arithmetic and algebra makes it more difficult for students to learn algebra in the later grades. But simply pushing the current high school algebra curriculum down into the elementary school won’t work. A broader conception of algebra emphasizes the development of algebraic thinking, rather than just skilled use of algebraic procedures. Students in the elementary grades can begin to engage in meaningful discussion about mathematical proof and make significant progress in understanding its nature and importance. Development of their algebraic reasoning is reflected in their ability to generate, represent, and justify generalizations about fundamental properties of arithmetic.

In their research, NCISLA researchers Tom Carpenter, Linda Levi, Patricia Berman, Jae-Meen Baek, and Margaret Pligge have found that when students working with mathematics make generalizations and represent them for their classmates, they articulate unifying ideas that make important mathematical relationships explicit. Underlying this pedagogical approach is a conception of mathematical understanding as constructing mathematical relationships and reflecting on and articulating those relationships.

For the last 5 years, Carpenter and his colleagues have worked intensively with a group of teachers to study the development of students’ algebraic reasoning in the elementary grades and to construct instructional approaches that support that development. Their work with 100 elementary school teachers and their students in Grades 1 through 6, including in-depth studies of three classes, has provided the following insights.

A window on student thinking

When students make generalizations about properties of numbers or operations, they make explicit their mathematical thinking. Generalizations provide the class with fundamental mathematical propositions for examination, while also opening up students’ thinking for analysis and discussion. Although students often have a great deal of implicit knowledge of properties of arithmetic operations, they typically have not explicitly examined generalizations about properties of numbers and operations or thought systematically about them. The trick for educators, says Carpenter, is to find an instructional context in which students’ implicit knowledge can be made explicit. Discussion of appropriately selected true and false number sentences provides such a context.

As an example: In one class exercise, children were asked whether it is true that 0 + 5,869 = 5,869. After some discussion, the group came up with the generalization: “Zero added with another number equals that number.” They also came up with the following generalizations: “Zero subtracted from another number equals that number,” and “Any number minus the same number equals zero.” In addition, one student came up with several related generalizations about multiplication.

Students engaged in discussions about whether their generalizations were accurately stated and demonstrated a good conception of the appropriate use of counter examples to challenge other students’ claims or generalizations. For example, one student proposed a generalization: “When you put zero with one other number, just one zero, with the other number, it equals the other number.” Another student challenged the generalization with the counterexample 100 + 100 = 200.This led the students to revise the generalization so that it was more accurate.

True, false, and open number sentences provided a context in which these students could begin to convert their implicit understandings into explicit generalizations. Number sentences generated by the teacher provided the initial basis for drawing out generalizations. But once the classes started to talk about generalizations, making generalizations became a class norm, and students would propose generalizations on their own.

In most of the classes studied, students would write generalizations on sheets of paper and post them in some location in the room. When generalizations were difficult to state clearly in natural language, the students would use symbols to express the generalizations precisely. For example, students represented a conjecture about changing the order of numbers in addition as follows: For all numbers a and b, a + b = b + a.

Conclusions

Elementary school students can learn to adapt their thinking about arithmetic so that it is more algebraic in nature. They can learn that the equal sign represents a relation, not a sign to carry out a calculation. They can learn to generalize and to express their generalizations accurately using natural language and symbols. Although not all students in the elementary grades will master mathematical proof, they can begin to engage in meaningful discussion about proof and make significant progress in understanding its nature and importance.

Understanding justification and proof takes years to develop. Although many sixth-grade students in a NCISLA case study were not yet able to generate proofs by themselves, most of them learned to recognize the limits of examples and the value of general arguments. They engaged in discussions of the nature of proof that made explicit important issues that most students never encounter at any point in their education. These experiences could provide a foundation for deepening these students’ understanding of proof in the future.

“One of the things that was striking about the classes we worked in was that the students were engaged in sense making,” says Linda Levi. “They thought that mathematics should make sense and that they could make sense of it. Students persisted for extended periods of time working on a problem, because they thought they should be able to figure it out.”

All students benefit by engaging in the kinds of interactions that are required to make generalizations explicit, represent them accurately with natural language and symbols, and demonstrate that they are valid for all numbers. Learning to use precise language and communicate about mathematical ideas addresses not only an important goal of the mathematics curriculum, but also important issues of equity. The best students have always figured out how to derive generalizations and thereby make mathematics easier to learn and apply. Helping students make generalizations explicit gives all students access to powerful ideas of mathematics.

A new book is available, “Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School” by Carpenter, Levi, and Megan Loef Franke (Heinemann, 2003).

For more information about research into mathematics education, see the NCISLA Web site at http://www.wcer.wisc.edu/NCISLA/.

Funding for NCISLA projects is provided by the National Science Foundation and the U.S. Department of Education’s Institute of Education Sciences (formerly the Office of Educational Research and Improvement). Carpenter’s study of the development of algebraic reasoning in the elementary grades is funded by the National Science Foundation.