# Rational Numbers: An Integration of Research

By Thomas A. Romberg; Elizabeth Fennema et al. | Go to book overview

partition-reducer. Transforming the size of the units in the operand to which a rational number is applied is conceptually very different from changing the number of units. Rather than duplicating or reducing the number of units, the size of the units is stretched or shrunk, and a transformation of the measure of the units of the operand quantity takes place, thus the term stretcher/shrinker.

These different conceptualizations demand different cognitive structures for their understanding and require special considerations for modeling problem situations and answering questions about the situation. The duplicator/partition-reducer requires skill in partitioning quantity and in the understanding of and skill with partitive division. The stretcher/ shrinker interpretation also requires partitioning skill but at the same time requires extensive understanding of measurement concepts and understanding of and skill with quotitive division.

In modeling problem situations, attention must be given to whether the problem situation can be interpreted in terms of a change in size of units of quantity or in terms of a change in the number of units of quantity. Some problems allow for either interpretation, others for just one or the other, and we suspect that there are some problems for which the operator construct is inappropriate. These situations, we hypothesize, will require a part-whole, quotient, or ratio interpretation of rational number.

We suspect that one of the difficulties students have in understanding rational number is due to the fact that the symbolic representation of a rational number involves two numbers. Yet a rational number represents a single quantity, or in the case of rational number as operator, a single exchange function. It has been hypothesized that many errors that children make with rational number operations and relations is due to the fact that they perceive a rational number as two entities. The analysis of rational number as operator that we have presented gives some insights into the two-entity versus one-entity phenomenon. As an exchange function, a rational number x/y can be seen as a composite of an x-for-1 exchange and a 1-for-y exchange, or as a direct x-for-y exchange; giving children experience at the manipulative level and at the symbolic level in representing rational numbers in both forms may facilitate their development of a concept of rational number as a single entity.

REFERENCES

Behr, M., Harel, G., Post, T., & Lesh, R. ( 1990, April). On the concept of rational numbers: Towards a semantic analysis. Paper presented at the annual meeting of the American Educational Research Association, Boston.

Behr, M., Harel, G., Post, T., & Lesh, R. ( 1992). "Rational number, ratio, and proportion". In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 308-310). New York: Macmillan.

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