Number sense-making. (Focus Issue: Planning for Instruction)

In Everybody Counts, a document from the National Research Council (1989), we are told that the major objective of elementary school mathematics should be to develop number sense. This strong statement, if taken seriously, can change the way many--but not all--teachers teach mathematics in elementary school. What is number sense? Reys and others (1991, 3-4) describe it well:

Number sense refers to an intuitive feeling for numbers and their various uses and interpretations; an appreciation for various levels of accuracy when figuring; the ability to detect arithmetical errors, and a common-sense approach to using numbers.... Above all, number sense is characterized by a desire to make sense of numerical situations.

More and more, researchers have documented examples of classrooms wherein children acquire good number sense (e.g., Fennema, Franke, Carpenter, and Carey |1993~; Howden |1989~; Lampert |1990~). Common elements about instruction in these classrooms include the following:

1. Sense-making is emphasized in all aspects of mathematical learning and instruction. This statement is particularly true of number-related aspects.

2. The classroom climate is conducive to sense-making. Open discussion about mathematics occurs both in small groups and with the class as a whole. Brown and Palincsar (1989, 395) have described why this type of climate encourages sense-making:

Environments that encourage questioning, evaluating, criticizing, and generally worrying knowledge, taking it as an object of thought, are believed to be fruitful breeding grounds for restructuring.... Change is more likely when one is required to explain, elaborate, or defend one's position to others, as well as to oneself; striving for an explanation often makes a learner integrate and elaborate knowledge in new ways.

3. Mathematics is viewed as the shared learning of an intellectual practice. Thus it is more than simply the acquisition of skills and information. Children learn how to make and defend mathematical conjectures, how to reason mathematically, and what it means to solve a problem.

4. Children learn more mathematics than they do in more traditional classroom settings. Vygotsky (1978) speaks of the zone of proximal development as a place in the learning process where a child is just ready to learn something, and interacting with peers and other people helps the child reach the next level of understanding. When children are operating at the edge of their understanding, they can learn more than when they lack this challenge.

Establishing classrooms in which such rich, connected learning takes place is not an overnight task. Several sites exist in our present curriculum wherein we as teachers can begin to think seriously about sense-making with numbers as a focus of instructional activity. The remainder of this article discusses two of these areas: first, understanding number symbols and number size; and second, alternative ways to think about computation.

Number and Symbol Meaning

The most important single element of number sense is an understanding of numbers. One cannot make sense of numbers without attaching meaning to them. …