ONE POINT OF VIEW: Sense and Nonsense about Fractions and Decimals

ONE POINT OF VIEW: Sense and Nonsense about Fractions and DecimalsAuthor(s): Joseph N. PayneSource: The Arithmetic Teacher, Vol. 27, No. 5 (January 1980), pp. 4-7Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191663 .Accessed: 17/06/2014 16:04

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ONE POINT OF VIEW

Sense and Nonsense about

Fractions and Decimals By Joseph N. Payne,

university of Michigan, Ann Arbor, Michigan

With our certain, albeit slow, movement to the metric system and with the widespread use of calculators, there is general agreement that decimals will be introduced earlier in our elementary school mathematics curriculum. Deci- mals for tenths, for example, have been taught successfully in grade three. Nevertheless, there are major questions, substantial disagreements, and some sheer nonsensical statements being made about fraction concepts, fraction computation, and decimal computation.

The height of nonsense is evidenced by a recent suggestion given to a fifth grade teacher by her principal: "Skip all the work on fractions. Do only decimals. We no longer need fractions because of calculators and the metric system." This principal is ignorant of both the essential practical uses of fractions and the need for fractions in subsequent courses.

Fraction concepts will remain an in- tegral part of our lives irrespective of calculators or metrication. Can one

imagine a waiter or waitress not knowing a half-liter container for wine? Or a child having no name for one part of a candy bar shared equally with three friends? Or a sibling not knowing what part of an estate each of seven children will get in an inheritance? Fractions are needed in everyday affairs to de- scribe parts of wholes, parts of amounts of money, and parts of numbers, as well as in measurement.

Even a cursory examination of mathematics and science courses re- veals the need for fractions. Knowing only decimals, how could an algebra student solve ax = b, or handle even the simplest formulas in physics? Al- though an elementary school curriculum should not be determined solely by needs in subsequent courses, ignor- ing future needs is not very sensible. Fraction Computation We are beginning to see a shift in the levels at which we teach computation with fractions. One reason for the change is the evidence about the level

January 1980 5



at which computation can be taught successfully. Another reason is the de- creased need for certain computations as our adoption of the metric system comes about.

The curriculum changes in the 1960s advanced computational topics with fractions, introducing them much earlier in the curriculum. For example, equivalent fractions and addition and subtraction of unlike fractions were moved into grade four and division of fractions into grade five. At these earlier levels, however, the computations proved impossible for a large number of pupils. Other factors compounded the learning problems. Inadequate and confusing concrete models were used. Insufficient time was allotted for developmental work. Not enough attention was given to the necessity for master- ing essential prerequisites for the computational algorithms. Even without metrication, the realities of instruction would force the movement of fraction computation into higher grades.

For the present time, it makes sense to delay formalization of rules with equivalent fractions until grade five and to do more concrete and meaningful developmental work in all grades, five through eight. Such a move seems especially wise in view of the fre- quency with which the misuse of equivalent fractions is identified as a major source of error.

The size of denominators in addition and subtraction, a well recognized cri- terion of difficulty for fractions and mixed numbers, can be held to numbers most often encountered. Eighths, sixteenths, and thirty-seconds will be used less. Fifths and tenths will be used more.

By simplifying the denominators and by using better instructional se- quences appropriate for given grade levels, we may be able to teach so that kids learn what we want to teach them about fraction computation. I am still appalled by the number of children in grade seven who add 2/5 and 1/5 and get 3/10. Our research shows clearly that adequate developmental work almost completely eliminates this and similar computational difficulties.

Nowhere is there more nonsense than the point of view held by many on multiplication and division of frac-

tions. In a recent article in a leading educational journal (not an NCTM publication), a writer offered some weak help on teaching multiplication of fractions. Then the writer com- mented that multiplication is not im- portant anymore but teachers should continue to teach it because it is on tests. Horrors! Just because of tests? This is hardly a defensible reason for long range curriculum planning. How does one solve ('/2)x = b without multiplication of fractions? Or find 2/3 of a dozen? Or find 1/2 of a recipe (admit- tedly one from an "old" but excellent cookbook) that calls for 3/4 cup of butter? Multiplication is essential and it is not difficult to teach. I am im- pressed with the reasons often stated for delaying division of fractions a bit, even until sixth or seventh grades after multiplication of fractions is well learned. Both multiplication and division are needed by the end of grade eight, however.

In summary, I see some delay in the grade levels at which fraction computation is introduced and mastery ex- pected. Further, the computation will involve fewer complications with large and unwieldy denominators. With better development and about the same amount of time in the curriculum, there is hope for substantial improve- ment in our work with fraction computation.

Fractions, Then Decimals, or Vice- Versa?

I have participated in many arguments on the question of whether fractions or decimals should come first. Recently, I have quit arguing, not because I lost but because I think this is not the major question. The question is not which should come first but what meaning we want children to have for fractions and decimals.

I know of no way to teach the meaning of 0.7 without taking a strip of paper as a unit, splitting it into ten equal parts, and holding up seven of them. This idea is identical to the one for

fractions. So the question is merely one of the nature of the symbolism introduced. Our research on fraction and decimal concepts convinces me that the notation is not the difficulty. The difficulty rests with the adequacy of the concrete or spatial representation. We must give careful attention to the unit, to equal partitions of the unit, and to the use of verbal names before symbols in an effort to enhance the quality of the quantitative ideas we help children build.

The claim that decimals can be introduced solely by extending place value to the right fosters meaningless dependence on symbols. How does a student know that 3.7 is between 3 and 4 just by naming the places? I have difficulty explaining this if I cannot rely on concrete interpretations and say "three whole things and seven-tenths of another whole thing. The seven- tenths is a part of the fourth whole thing so 3.7 is between 3 and 4." Such concrete interpretations are essential for almost all estimation work with decimals and for meaningful computation. Again, fraction ideas are evident. Certainly we must exploit a student's knowledge of whole numbers when we teach decimals. We must recognize, however, the insufficiency of depend- ing on whole number knowledge ex- clusively.

There are claims that decimal computation is easier. Computational rules for decimals may be easier to state than computational rules for fractions. "Line up the decimal points and places" seems simpler than "Find a common denominator, add numer- ators, and write the common denominator." The ease with which rules are stated, however, often masks the difficulties in understanding the rules.

Difficulties with decimals, in fact, parallel the well-known difficulties with fractions. Understanding equivalent decimals - for example, that 0.6 = 0.60 = 0.60 - corresponds to the difficulties with equivalent fractions. We can give rules about appending or an- nexing zeros and ignore the reasons. We do so, however, with the great risk of students having extensive sets of memorized rules, some correct and some incorrect. We are not likely to de- velop the critical skills of estimation,

6 Arithmetic Teacher



thoughtful mental computation, and reasonable quantitative thinking.

All of us have seen errors such as 0.2 X 0.3 = 0.6. We can restate the rule about placing the decimal point. But why does it work? The best way I know to explain it to a child is to draw a square, cut it into 100 equal parts, and mark offa region two-tenths by three- tenths. Then it's obvious that there are 6 parts in the region and that each part is a hundredth. This is precisely the same development that I would use for multiplication with fractions. I think that it is neither more difficult nor less difficult to do the computation with decimals meaningfully than it is with fractions.

With the level of difficulty for development about the same, I see operations in decimals being done at about the same grade level as operations on fractions.

Questions

Recent NCTM publications have pre- sented the varied components of the complex we call place value for whole

numbers. We need similar analyses for decimals and more research on instructional strategies. Certainly we want pupils to view a decimal such as 0.34 not only as 3 tenths 4 hundredths but also as 34 hundredths. How do we teach this most effectively? What are other components of place value for decimals? We need similar analyses for all the major computational algorithms with decimals, more evidence on effec- tiveness of postulated instructional se- quences, and better notions of how fraction and decimal computation in- terrelate.

In the upper grades, there is a natu- ral interference as multiplication with fractions is taught. If the multiplier is less than one, the product is less than the other factor. In finding 2/3 of a number, you multiply the number by 2/3; but you can think of it as dividing by 3 and multiplying by 2. We cannot hope to eliminate all these apparent contradictions but we should be working on ways to minimize the difficulties. Perhaps some of the recent work on viewing fractions in multiplication as operators will help.

We need better instructional se- quences for percent and more careful analysis of the connections among per- cents, fractions, and decimals. How can we teach the necessary algorithms and at the same time teach sensible es- timations and approximations with percent? The highly practical use of percent should give these problems high priority.

Summary Nonsensical claims about deletion of fractions ignore both the practical and mathematical objectives in our curriculum. Suggestions on decimals often ignore the need for fraction ideas in teaching decimals.

It seems sensible for fractions to remain, for decimal concepts and notation to come earlier, and for fraction computation to |>e reduced in com- plexity and moved upward in the curriculum.

Many questions remain. With the help of researchers and good teachers, we are now in a position to tackle some of the difficult questions remaining. D

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January 1980 1



Article Contentsp. 4p. 5p. 6p. 7

Issue Table of ContentsThe Arithmetic Teacher, Vol. 27, No. 5 (January 1980), pp. 1-60Front MatterONE POINT OF VIEW: Sense and Nonsense about Fractions and Decimals [pp. 4-7]Let's Do ItWithout Pencil and Paper [pp. 8-12]

Strategies for Teaching Children Gifted in Elementary Mathematics [pp. 14-17]When You Use a Calculator You Have to Think! [pp. 18-21]Errors That Are Common in Multiplication [pp. 22-25]From the File [pp. 25-25]Leap Years and Such [pp. 26-27]Ideas [pp. 28-32]What's Going On [pp. 33-33]A Look at the Past [pp. 34-37]An Experience in Everyday Economics [pp. 38-41]A "Friend" in Need [pp. 42-43]Speed and Accuracy in Mathematics [pp. 44-45]ADAM Comes to Nueva [pp. 46-47]An American Teaches in Shanghai Middle School #2 [pp. 48-49]Is It Necessary to Invert? [pp. 50-52]Books and Materials [pp. 52-52]From the File [pp. 53-53]Reviewing and ViewingNew Books for PupilsReview: untitled [pp. 54-54]Review: untitled [pp. 54-54]Review: untitled [pp. 54-54]Review: untitled [pp. 54-54]Review: untitled [pp. 54, 58]

Readers' Dialogue [pp. 56-56]Reviewing and ViewingNew Books for TeachersReview: untitled [pp. 58-58]Review: untitled [pp. 58-58]

NCTM Nominations for the 1980 Election [pp. 59-59]Nominees for the 1981 Election [pp. 59-59]Professional Dates [pp. 60-60]Back Matter

Documents

ONE POINT OF VIEW: Sense and Nonsense about Fractions and Decimals