ONE POINT OF VIEW: Fractions and Other Rational NumbersAuthor(s): Thomas R. PostSource: The Arithmetic Teacher, Vol. 37, No. 1 (SEPTEMBER 1989), pp. 3, 28Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193717 .

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ONE POINT OF VIEW

Fractions and Other Rational Numbers

Thomas R. Post

results of na- tional and inter-

national assess- ments indicate that students have signif- icant difficulties in learning about ratio-

nal numbers. In 1979 only 24 percent of the nation's thirteen-year-olds could estimate the sum of 12/13 and 7/8 given the following possibilities: 1 , 2, 19, 21, and I don't know. Fifty-five percent selected either 19 or 21 as the estimated sum!

Why do students encounter such difficulties? Surely part of the answer resides in the fact that instruction in rational numbers is poorly conceptu- alized and implemented by the vari- ous textbook series. Most approaches do not deal significantly with any type of manipulative material, even though many excellent ones exist (fraction circles, Cuisenaire rods, chips, and number lines are examples). Students are expected to formulate fraction-re- lated concepts primarily on the basis

Thomas Post is professor of mathematics edu- cation at the University of Minnesota, Minne- apolis, M N 55455. He has been coprincipal in- vestigator, with M. Behr and R. Lesh, on four NSF-sponsored projects since 1979 and has been conducting research on the learning of ra- tional-number and proportional-reasoning concepts.

of their interaction with the printed page, which, by definition, can only include pictorial models (iconic repre- sentations) and symbols. This limita- tion reflects a very severe misunder- standing of the nature of human learning. It is further exacerbated by the fact that most texts focus on op- erations with fractions rather than on such fundamental concepts as parti- tioning and order and equivalence. Thus, very early in their instruction on rational numbers, students can be found generating long lists of equiva- lent fractions, adding, subtracting, multiplying, and dividing, all without having the foggiest idea of what they are doing.

Even well-conceived instruction on fractions involves more complex forms of numerical relationships than previously encountered. I am speak- ing here of the transition from additive to multiplicative structures, a truly major transition in mathematical do- mains. Prior to the introduction of ra- tional numbers, virtually every type of numerical problem situation can be viewed as a variation on the counting theme. With the beginning of serious attention to multiplicative relation- ships, students find that the additive baggage that has served them so well in the past no longer applies, for ex- ample, 1/2 + 1/3 ф 2/5. Students for the first time must employ conditional and relativistic kinds of thought pro-

cesses and begin to identify new types of relationships between numbers. For example, 1/3 is'less than 1/2 even though 3 is greater than 2, but 3/7 is greater than 2/7 because 3 is greater than 2. Multiplicative patterns work, but not additive ones. For example, 2/3 = 4/6 but 2/3 ф 3/4. New rules must be learned for addition and sub- traction, but the old rules for multipli- cation seemingly continue to work.

Rational numbers are susceptible to various interpretations other than the part- whole model. Decimals, ratios, operators, number lines, and indi- cated divisions are all considered sub- constructs of rational numbers. Ide- ally as students progress through the curricula, these interpretations are sorted out and understood. This task is difficult for many. Major changes must be made in the ways students encounter these concepts individually and in the amount of attention paid to their collective interrelationships. The latter is almost totally absent in the school curricula.

Important questions about chil- drens' conceptual development in the rational-number domain are being ad- dressed by the research community. We already know much about the way in which students are able to under- stand these concepts and the accom- panying difficulties in teaching them. It is time for such results to be seri- ously reflected in the school curricula.

The ubiquitous nature of rational- number concepts in all of mathemat- ics surely makes them one of the most important conceptual domains to be studied by students. Decimals, frac- tions, ratios, and other multiplica- tively based relationships literally dominate the junior high school math- ematics curricula. These rational- number ideas eventually play a major role in the development of proportion- al-reasoning abilities, which, in turn, become the intellectual and mathe- matical cornerstone of much of what is to come in the secondary school years.

Bibliography Cramer, Kathleen, Thomas R. Post, and Mer-

lyn J. Behr. "Cognitive Restructuring Abil-

(Continued on page 28)

SEPTEMBER 1989 3

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Objective

Using the display of names from Greek and Roman mythology, stu- dents will apply classifications by vowels and consonants, vertical and horizontal symmetries, and fractional relation to classify the component let- ters.

Directions 1. Reproduce a copy of the work- sheet for each student. 2. Review the properties, vowels and consonants, and horizontal and verti- cal symmetry. Also review the defini- tion of a fraction as a part of the whole. 3. Give the students time to read and complete the worksheet. 4. Allow time for small groups of stu- dents to discuss their answers and to agree on solutions.

Answers

See table 3.

^^^^n Number of Vowel Consonant Vertical Horizontal

Name letters part part symmetry symmetry

ZEUS 4 2/4 2/4 U E PEGASUS 7 3/7 4/7 AU E PANDORA 7 3/7 4/7 АО DO DIANA 5 3/5 2/5 1 A D 1 HERCULES 8 3/8 5/8 H U ECH ULYSSES 7 3/7* 4/7 U Y E APOLLO 6 3/6 3/6 АО О JASON 5 2/5 3/5 АО О CIRCE 5 2/5 3/5 1 CEI CUPID 5 2/5 3/5 U 1 CID PLUTO 5 2/5 3/5 U О Т О PERSEUS 7 3/7 4/7 U E ATALANTA 8 4/8 4/8 A T ORPHEUS 7 3/7 4/7 О H U О H E URANUS 6 3/6 3/6 U A

*The у is a vowel in this example.

Extensions

Encourage students to find other names from mythology to analyze. Consider rotational symmetries for the letters N, O, S, and Z. Make state-

ments using the language of inequali- ties, such as, 'The consonant part of Jason is greater than the vowel part"; "the vowel part of Apollo is equal to the consonant part"; and "half the letters in Uranus are vowels."

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ity, Teacher Guidance, and Perceptual Dis- tractor Tasks: An Aptitude-Treatment Interaction Study." Journal for Research in Mathematics Education 20 (January 1989): 103-10.

Behr, Merlyn J., Richard Lesh. Thomas R. Post, and Edward A. Silver. "Rational Num- ber Concepts." In The Acquisition of Math- ematics Concepts and Processes, edited by Richard Lesh and Marsha Landau, 91-125. New York: Academic Press, 1983.

Kieren, Thomas. "On the Mathematical. Cog- nitive, and Instructional Foundations of Ra- tional Numbers." In Number and Measure- ment, edited by Richard Lesh. 101-44. Columbus. Ohio: ER1C/SMEAC. 1976.

Lesh. Richard. Merlyn J. Behr, and Thomas R. Post. "Rational Number Relations and Pro- portions." In Problems of Representation in the Teaching and Learning of Mathematics, edited by Claude Janvier, 41-58. Hillside. N.J.: Lawrence Erlbaum Associates, 1987.

Post, Thomas, and Kathleen Cramer. "Re- search into Practice: Children's Strategies in Ordering Rational Numbers." Arithmetic- Teacher 35 (October 1987): 33-35. Щ

28 ARITHMETIC TEACHER

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