One Point of View: Helping Children Understand Rational NumbersAuthor(s): Thomas E. KierenSource: The Arithmetic Teacher, Vol. 31, No. 6 (February 1984), p. 3Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190998 .

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One Point OF X7ÍGCD

Helping Children Understand

Rational Numbers By Thomas E. Kieren

University of Alberta, Calgary, AB T5M 0Z8

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1984 The natural number system is the

first one that a child studies in school. Instruction builds on the child's knowledge of this system. The me- chanics of counting are nurtured by stressing such ideas as unit identifica- tion (what is to be counted), succes- sor, counting on, and counting back. Images of number such as "stair- cases" and sequences of objects are fostered. Attention is paid to the child's conservation of number and ability to establish a one-to-one corre- spondence. A bridge is then made from these concepts to operations and such thinking tools as grouping and ungrouping.

A second number system that chil- dren encounter is that of the rational numbers, starting first with the posi- tive "fractional" numbers. Although we have used the children's knowl- edge of natural numbers to begin in- struction ("count the number of parts, count the number shaded"), this stat- ic part-whole approach is inadequate to develop the concepts related to rational numbers.

What are the mechanisms, images, and language needed for the develop- ment of rational-number ideas? Math- ematically, the rational numbers arise as solutions to equations like

ax = b, where a and b are integers and а ф 0.

The Editorial Panel encourages readers to send their reactions to the author with copies to NCTM (1906 Association Drive, Reston, VA 2209 1) for consideration in '* Readers' Dia- logue."

Such equations represent problems like "There are 2 pizzas and 3 chil- dren. How much is each child's share?" Thus, a key tool, analogous to counting natural numbers, is divid- ing up equally, or partitioning. The curriculum should provide many par- titioning activities in which the ob- jects themselves are not naturally di- visible (seventeen cars for three car- trailers) and activities in which they might be divisible (eight cookies for three children). As with counting numbers, children come to school with a background in rational num- bers as well. Even young schoolchil- dren can grasp halving and apply this action to generate halves, quarters, and eighths.

Two considerations are important in developing these activities. The first is that rational numbers are multi- plicative and additive. Thus, division into ten parts comes from "halving of fifths," a multiplicative notion. In di- viding seven among four, a child may say "one and a half and half of a half," or symbolically, 7-^4=1 + 1/2 + 1/4. These examples also reflect the need for an informal language of fractions that precedes formal sym- bolic development. These examples show rational numbers to be both measurement numbers, "how much," and operator numbers, "one half of one fifth is one tenth."

Today, decimal representation of rationals is stressed. Care must be taken so that the knowledge building and informal language use discussed earlier are not lost. Otherwise the child may become computationally

facile with decimals but may have a limited understanding of them. We should not assume that the grouping- ungrouping action that allows one to go from ones to tens to hundreds will also allow one to go from ones to tenths to hundredths: dividing up is simply different from ungrouping. A tenth is a derived idea for the child, whereas a hundredth is, of necessity, an abstraction based on other divid- ing-up processes.

My theme is that rational-number ideas are sophisticated and différent from natural-number ideas. We need to help children develop the knowl- edge-building tools, the images, and the informal language needed to un- derstand this number system. Chil- dren come to class with a background that can foster their ability to learn rational numbers. We need to develop this background for formal work with decimals, fractions, and rational alge- braic forms. Such a program will make rational numbers a tool for problem solving rather than useless, partly remembered knowledge, w

Professional Dates

NCTM 62d Annual Meeting 25-28 April 1984, San Francisco, Calif.

NCTM 63d Annual Meeting 17-20 April 1985, San Antonio, Tex.

NCTM 64th Annual Meeting 2-5 April 1986, Washington, D.C.

For a printed listing of local and regional meetings, contact NCTM, Dept. PD, 1906 Association Dr., Reston, VA 22091, (703)620-9840.

February 1984 3

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