Are We Introducing Mathematical Symbols Too Soon?Author(s): Kathy B. HamrickSource: The Arithmetic Teacher, Vol. 28, No. 3 (November 1980), pp. 14-15Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191821 .

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Are We Introducing Mathematical Symbols Too Soon? By Kathy B. Hamrick

Some of the errors children make in elementary mathematics are a result of confusion associated with the written symbolization of mathematics. Try an experiment with your first, second, or even third grade class. First, cover four counters with a piece of paper and place three counters next to the paper. Say, "I have seven counters in all; you can see three. How many do I have hidden?" Later, give the class the prob- lem 3 + D = 7. How many correctly guessed that four counters were hidden

Kathy Hamrick is an assistant professor of mathe- matics and computer science at Augusta College in Augusta, Georgia. She teaches algebra and statis- tics courses as well as mathematics for elementary teachers. This article is based on her doctoral dis- sertation. A companion article containing much of the research can be found in the May 1979 issue of the Journal for Research in Mathematics Educa- tion.

but answered "10" for the problem 3 + D = 7? This is not unusual. Many children make mistakes when working with the written form of problems they could correctly solve if the problems were done orally. It is possible that the meanings associated with written sym- bolization are not the meanings that are used by a child who works ver- bally.

The learning of the written symbol- ization of mathematics is similar in many respects to the learning of the written symbolization of language, that is, learning to read. However, the error contrasts between oral and written work often found when children work with mathematical symbols are not as evident when children work with lan- guage symbols. Perhaps this is because a child is not considered ready to read until he has an adequate speaking and hearing knowledge of the words and

sentences he is expected to read, while there is little consideration of the read- iness factor of a spoken vocabulary in relation to the mathematical symbols a child is expected to read.

The similarities between learning to read mathematical symbols and learn- ing to read language symbols were summarized by Hickerson (1959, p. 241) who stated,

Since arithmetic is a system of symbolism just as language is a system of symbolism, why shouldn't the accepted principles underlying the understanding and use of language symbols ap- ply to the understanding and use of arithmetic symbols?

What are the "accepted principles" un- derlying the understanding and use of language symbols that we should apply to the teaching of arithmetic symbols? One such principle, generally accepted by language educators is the impor- tance of an adequate spoken vocabu-

Arithmetic Teacher

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lary as a readiness factor for learning to read. Tinker and McCullough (1975, p. 7) described beginning reading as follows:

For the beginner, learning to read entails learning that printed symbols stand for speech. The child reads when he says the correct printed words and recognizes their meaning because of his previous experience in comprehending speech in meaningful sequence. He discovers that printed words "talk" sense.

It is possible that the printed symbols of mathematics as well as of language nlay lack meaning to many children because the children lack the adequate verbal facility necessary to make the printed mathematical symbols "talk" sense.

Other experts in reading or language education, Harris (1970), Lee and Al- len (1963), and Bond (1954) agree that successful reading is dependent on the child's ability to relate the written sym- bols to spoken symbols. The spoken symbols provide a link between written symbols and the meaning to be as- signed to the written symbols. When this link is broken - that is, the child does not have adequate verbal facil- ity - the child cannot assign meaning tò the written symbols. For this reason, verbal facility is an important read- iness factor for learning to read.

There are important similarities be- tween language symbols and arith- metic symbols and the learning of each. It is also true that arithmetic symbols are more compact than lan- gUage symbols and the word attack skills used with language symbols often dknnot be used with arithmetic sym- bols. However, both are printed sym- bols and our main goal in teaching ei- ther type of symbol is that the child assigns meaning to the symbols. When learning language symbols, the child assigns meaning to the written symbols by associating the written symbols to already known sound symbols. The al- ready known sound symbols arouse meaning in the mind of the learner. In mathematics, we often ask the child to assign meaning directly to the symbol or symbols. For example, we ask the child to relate the symbols "2 + 3 = 5" to an act of joining two disjoint sets and counting the new set. If we applied language principles to mathematics, we would first teach the child to relate the spoken number sentence "two plus

three equals five" to the action of join- ing the sets. Then the written symbols could be associated with an already known sound symbol.

In essence, if we apply the principles of learning language symbols to learn- ing arithmetic symbols, we are propos- ing a new readiness factor for learning the written symbolization of arith- metic.

A dissertation study in 1976 (Ham- rick 1976) supports the importance of verbal facility with the mathematical symbols of addition and subtraction number sentences. In this study, 38 first-grade students were classified as ready or not ready according to scores on a readiness test that was based on verbal facility with addition and sub- traction. The two groups were each then split into an immediate and de- layed symbolization group. All stu- dents received 12 weeks of instruction. The group of interest was the one com- posed of "not ready" students who had written symbolization of addition and subtraction delayed until they were comfortable with addition and sub- traction in a verbal mode. Both the sta- tistical evidence and the opinions of the teachers supported this method of delaying symbolization for not ready children. A similar study was done by McKillip (1976) with the topic of place value. The results were the same.

This readiness factor for learning arithmetic symbols can be explicitly stated as follows: Children are ready for the introduction of written symbol- ization of a topic when they have mas- tered the objectives of the topic orally, perhaps with the aid of pictures or ma- nipulatives.

This definition of readiness can be applied topic by topic to assess a child's readiness before the in- troduction of written symbolization of the topic. Essentially, a given child is ready for the written symbolization of, say, addition when that child can add orally using objects if necessary. The definition does not seem strong when one considers that a child is not ex- pected to read a word or sentence until he can meaningfully say the word or sentence. If a child does not have ade- quate language facility, experiences are provided for the child, telling stories, and so on, that are designed to bring

the child to readiness before the writ- ten language symbols are introduced. Arithmetic experiences in which the child says number sentences or sym- bols in response to, say, the joining of sets can also be employed to bring the not ready children to readiness for mathematical symbols of addition. For example, one activity that worked well in the dissertation study involved the use of a cardboard barn. The teacher put three toy horses in the barn then put two more in the barn. The children took turns telling a story about what had happened. The teacher then told them how to tell a "math story" about what happened. The number sentence 3 + 2 = 5 (using the "plus" and "equals") was the "math story."

Other activities involved the chil- dren telling the "math story" for the physical union of disjoint sets, or pic- tures of the union of disjoint sets. Later, the children were told number sentences and they moved objects or drew a picture to "show" the number sentence. When the children were com- fortable with activities of these types, and only then, the written number sen- tences of addition were introduced.

Are we introducing mathematical symbols too soon? If we accept the pro- posal that the principles of languages be applied to the learning of arithmetic symbols, we are.

References

Bond, Guy L., and others. Pre-Primers, Three of Us, Play With Us, Fun With Us, with Teacher's Guide. Chicago: Lyons and Carna- han, 1954.

H am rick, Kathy B. "An Investigation of Oral Language Factors in Readiness for the Writ- ten Symbolization of Addition and Sub- traction." Doctoral dissertation, The Univer- sity of Georgia, 1974.

Harris, Albert J. How to Increase Reading Abil- ity. New York: Davis McKay Company, Inc., 1970.

Hickerson, J. Allen. "Similarities Between Lan- guage and Arithmetic." Arithmetic Teacher, 6 (November, 1959):241-44.

Lee, Doms M., and Richard V. Allen. Learning to Read Through Experience. New York: Ap- pleton-Century-Crofts, 1963.

McKillip, William D. "First Grade Verbal and Manipulative Mode Study, 1974-1975." Proj- ect for the Mathematical Development of Children, Athens, Georgia. Report of Princi- pal Investigator, November, 1975. Mimeo- graphed.

Tinker, Miles A., and Constance McCullough. Teaching Elementary Reading. New Jersey: Prentice Hall, Inc., 1975. W

November 1980 15

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