initial decimal concepts: are they really so easy?

Initial Decimal Concepts: Are They Really So Easy?Author(s): Judith ZawojewskiSource: The Arithmetic Teacher, Vol. 30, No. 7 (March 1983), pp. 52-56Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190654 .

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Initial Decimal Concepts: Are They Really So Easy?

By Judith Zawojewski

Fig. 1 Fig. 2

(Model)

2-thirds ^ ^ 3

(Word name) (Symbol)

3-tenths 10

(Model) (Word name) (Fraction symbol)

^ _ ; y _

0.3 (Decimal notation)

Decimals have long been considered an important mathematical topic and are even more important today. The increasing availability of calcula- tors and the increasing use of the metric system are giving decimals a more prominent place in the mathematics curriculum than previously.

The achievement data available indicates that children have difficulties with the concepts about decimals. An item on the Michigan Educational As- sessment Program Mathematics Inter- pretive Report of 1974 dealt with the identification of the place value of a two-place decimal. Only 26 percent of the students in seventh grade throughout Michigan were able to perform at a mastery level for four out of the five parallel items (Coburn et al. 1974). It was felt that these items indicated that

Judith Zawojewski is on the faculty of the National College of Education. She has taught middle-school children in the Baker Demon- stration School as well as preservice teachers in the Undergraduate School.

pupils do not understand decimals well. Reports on the results of the National Assessment indicated that performance by 13-year-olds was lower on all exercises involving zeros (e.g., 0.04, 0.0034). The students tended to ignore the decimal and treat the two choices as whole numbers. The authors suggest that "one source of difficulty appears to be a lack of a firm understanding of the place-value interpretation of decimals" (Carpen- ter et al. 1981).

The authors also noted that only 55 percent of the 13-year-olds could change a common fraction expressed in tenths and hundredths to decimals. About 50 percent of the same students could change decimals expressed in tenths and hundredths to fractions. These results made it clear that we should be spending more time having children become familiar with decimals, their meanings and uses, before rushing directly to decimal computations. (Carpenter et al. 1981)

These data reflected what this writer had seen in the classroom. In an attempt to find ways to deal with

these difficulties, a classroom study was developed to investigate the initial learning of decimal concepts.

The Study The study was designed to look at the initial learning of decimal concepts at three grade levels, namely grades 4, 5, and 6. The accompanying instructional unit that was developed incorporat- ed research done by Galloway (1976). The content of the unit focused on four areas of decimal concepts: tenths, hundredths, place value to the hundredths place, and applications in measurement and money. The instructional unit was developed and piloted at the Baker Demonstration School of the National College of Education in Evanston, Illinois. After revision, the unit was taught in a fourth-, a fifth-, and a sixth-grade class. The three groups were hetero- geneous and they were taught in self- contained classrooms by the regular teachers at a suburban school in the Chicago area. The curriculum of the particular school did not provide for

52 Arithmetic Teacher



instruction in decimals until late in the fifth-grade year. The fourth and fifth graders had not received any instruction in decimals that year. The unit was a second introduction for some of the sixth-grade students.

Description of the unit Section I: Fractions (lessons 1, 2, 3)

The major emphasis of the first section was on reviewing key fraction ideas essential to the understanding of decimal concepts. Connections were made between the model, the word name, and the symbolic name for fractions (fig. 1). The initial fraction work was based on an article by Ellerbruch and Payne in the 1978 NCTM Year- book.

Section II: Tenths (lessons 4, 5, 6, 7)

In this section identifying and using decimal notation for tenths was taught. The students had already worked with the connections between the model, the word name, and the fraction symbol, so the decimal notation for tenths was the only new idea presented (fig. 2). This section also included decimal notation for mixed numbers in tenths. The models and oral names for the numbers were em- phasized: for example, 1.3 was read as "1 and 3 tenths."

Language development through oral work was very important to the development of the concepts. Oral counting exercises were used to rein- force the language. In one activity the teacher began counting by tenths and each student was to say the next con- secutive tenth. When a number such as 2 and 9 tenths was reached, the children were to then follow with "3 whole units." Students who made er- rors were to go to the end of the sequence of students. In that way they were the ones to receive the most turns. After such activities, the children were asked to write number sequences counting by tenths.

Use of the number line to present fractions of a unit of distance was carefully covered. Long rectangles were used initially as a transition from a region model to a distance model.

Fig. 3

One whole unit

. - Turn the strips over U . - u > H D

0.20 0.2

The teachers had to keep two major ideas in front of students. The first was to focus on the whole unit of distance, and the second was to label the distances at the ends of the inter- vals rather than in the middle of the interval or unit. The ordering and comparing of decimals was presented on both the number line and with the region models.

Section III: Hundredths (lessons 8, 9, 10)

The focus of this section was on decimal notation for hundredths. Connec- tions between the model, the word name, the fraction symbol, and the decimal notation paralleled those in Section II. Notation of hundredths greater than one and counting by hundredths were also included. Initially the counting was approached orally, with students counting in a manner similar to the counting by tenths. Dis- cussion was encouraged to find the similarities and differences in counting by tenths and hundredths. For example, in tenths one should "turn the corner to the next whole unit" after reaching 9 tenths. When counting by hundredths, a whole unit is built after reaching 99 hundredths.

The instruction continued to stress the connections between the model, the oral names, and the symbols.

Section IV: Place value (lessons 11, 12, 13, 14, 15, 16)

Individual models for each student were used to help develop place- value ideas. First, students cut out regions to show tenths and hundredths of the same whole unit. The hundredths were cut in individual pieces as well as in strips of ten hundredths. Using the ten-hundredths strips, the students could turn them over to see that the unmarked side of the strip was identical to the tenth strips. Using this physical representation, students were able to show that 20 hundredths is equivalent to 2 tenths. The decimal notation used to accompany this notation was 0.20 = 0.2 (fig. 3).

The idea of place value was further developed using examples such as "26 hundredths." Two strips of ten hundredths and six individual hundredths were used to show 26 hundredths. The two strips of ten hundredths were then turned over to demonstrate that the region shaded was identical to 2 tenths and 6 hundredths (fig. 4).

Fig. 4

One whole unit

- Turn the strips over U u

111 O i 26 hundredths 2 tenths 6 hundredths

March 1983 53



This section also introduced comparing and ordering one-place and two-place decimals. The models were used to help students see that in determining the order of two decimals of differing place value, the digit to con- sider is the digit in the tenths place. A number-line model was used as an alternate method in determining order. Students first numbered a line in tenths; and then using the idea that 0.1 = 0.10, they renumbered the line in tens of hundredths. This notion helped students to locate numbers such as 32 hundredths and 40 hundredths. Then by locating an approxi- mate placement of 32 hundredths and another decimal numeral, they could make a comparison by deciding which was greater on the basis of which number was to the right of the other.

Throughout this section, money was used as another intepretation of place value. Writing 0.26 as 26 hundredths or 2 tenths 6 hundredths was compared to writing $0.26 as 26 cents or as 2 dimes 6 pennies.

Section V: Application (lessons 17, 18) The last two lessons dealt with the use of decimals in metric measure. First, the centimeter was used as a whole unit and the students measured line segments to the nearest tenth of a centimeter. The meter was then used as the whole unit and the students practiced writing a number of meters and centimeters as a decimal numeral. For example: 1 meter 13 centimeters can be written as 1.13 meters. The students were also expected to be able to reverse the process.

Results of the Study Posttest/retention test

An in-depth achievement test was administered at the conclusion of the unit. The test was also administered 3 weeks later as a retention test. The test was subdivided into content areas: tenths, hundredths, and mixed tenths and hundreds, place value, and applications. Table 1 shows the means

Table 1 Means for the Subtests of the Posttests and Retention Tests

o 4. Grade 4 Grade 5 Grade 6 Section o 4. (No. of items) Post Reten. Post Reten. Post Reten.

Te(n1t6h)s 13.9 14.4 14.4 14.2 15.0 15.6

Hundredths < A < л A ¿ t /> t л^ -, % л л % л ¿ /|^ч 14.1 < A < л 14.6 A ¿ 12.1 t /> t 12.7 л^ -, 14.1 % л л % 14.6 л ¿

Mixed T/H & place value 8.5 7.8 8.0 7.8 9.1 9.4 (12)

Application 54 42 44 43 63 63

Table 2 Percent of Students Scoring at Least 80% Correct and Less Than 60% Correct for the Subsections for the Posttests and Retention Tests

Section - r- Grade 4 Grade 5 Grade 6 Achieve. level Post Reten. Post Reten. Post Reten.

-.. 2*80% 76% 88% 82% 8Î% КЮ% 100% lenms <60% 4% 4% 5% 5% 0% 0%

Hunareatns < 6Q% %% 8% 2?% {Q% Q% 4% Mixed T/H & ^ 80% 44% 32% 23% 33% 52% 67%

place value < 60% 32% 48% 36% 41% 19% 22% a™i,w;™ ^80% 20% 12% 14% 10^ 33% 26% Appucauon a™i,w;™

< 60% 48% 68% 68% 62% 41% 37%

for each subtest at each grade level. Strong achievement was evident in the content areas of tenths and hundredths.

In order to look at mastery aspects of the work, "mastery" was arbitrari- ly defined to be 80 percent correctly answered on each subsection of the test. Table 2 indicates the percent of students scoring at least 80 percent correct and less than 60 percent correct for each subtest. Examination of the percents of students at mastery level on both the post test and the retention test indicates a great stabil- ity in the knowledge of tenths. The percent of students at mastery level actually increased on the retention test. Situations requiring the use of place value posed specific problems as did work with measurement and money.

Analysis of selected test items

Specific types of items on the test indicated some interesting aspects of learning decimals. Although the sam- ple was small and taken from a limited population, the findings have some bearing on the approaches to the initial teaching of decimal concepts.

Items in which students were to continue counting sequences were quite well mastered, though two trou- ble spots were apparent. In counting by hundredths, many students com- pleted a sequence in this way: 12.08, 12.09, 13.00. Although students appeared to be comfortable with the idea that 100 hundredths makes one whole unit, the momentum of the pattern set by the previously learned tenths seemed to interfere. The other difficult type of sequence for students was




found in examples such as this: 20.97, 20.98, 20.99, 30.00. This is interesting in light of the fact that this particular type of error was expected and the participating teachers gave this type of sequence special attention in the unit.

The use of the number line to show tenths was considered to be very difficult by the teachers teaching the unit. The items dealing with the number lines, however, were answered correctly by at least 80 percent of the students at all three grade levels. This seemed to indicate that careful teaching focusing on the whole unit and the fractional distance from zero can pro- duce mastery results.

Place value was previously known to this writer to be difficult for children. For this reason, very careful treatment of the idea of place value was developed for the unit of study. Students first learned the tenths and hundredths notations separately. Then equivalence between a number of hundredths and tenths was established through a great deal of work with region models, as described previously. The idea that 2 tenths is equivalent to 20 hundredths was to be firmly established as the basis for further place- value work. The region models were then again used to show the physical equivalence of 24 hundredths to 2 tenths 4 hundredths. The covering of the "20 hundredths" with "2 tenths" helped the students to see the meaning of place value.

Even after very careful development, some items stood out as being very difficult for students. Figures 5, 6, and 7 show three test items and the percent of students scoring correctly on the items by grade level. The results on those three items indicate that even more careful development of the notion of place value than was provided in this study is needed.

The use of money as an application in the initial learning of decimals caused some confusion. Students had difficulty understanding the use of the decimal notation as dollars, and tenths and hundredths of dollars. Stu- dents apparently do not interpret $1.26 as 1 and 26 hundredths dollars. They think about it as two separate units: dollars and cents. Thinking of

dimes and pennies as fractional parts of a dollar appeared difficult for children.

Measurement was surprisingly difficult for the children involved in the study. Mastery was not obtained. This may have been due to the lack of previous measurement work experi- enced by the students. Another factor may have been that the instructional unit used measurement as a short follow-up to complete the unit rather than incorporating the ideas of measurement throughout. It might have

been more appropriate to include measurement directly after the work with the number line. The results were in conflict with those of Gallo- way and others who found that children as young as third grade were able to master measurement to the nearest tenth of a unit using decimals.

Examination of teacher observa- tions indicated that, overall > the sixth- grade teacher felt that the majority of the sixth-grade students were able to grasp the materials presented. The results of the tests on mixed tenths

Fig. 5

Item #35 Post Ret.

4th (30%) (56%) - « 5th (52%) (52%) -

6th (63%) (70%) -~

Write the decimal that - tells the amount shaded 1111

Fig. 6 ^^^.^^.^^.^^-^

Item #36 Post Ret

4th (76%) (72%)

5th (67%) (52%)

6th (52%) (56%) ZZZZZZZZZZ

Shade 0.3 of this square. I - I - I - I - I - I - I - I - I - I -

Fig. 7

Item #37 Post Ret.

4th (84%) (56%) Which one does not mean 0.52?

5th (76%) (76%) Make an "X"

ßth 6th (Avo/ (67 /o) ' ia-7o/' (67 /o) 52 hundredths ßth 6th (Avo/ (67 /o) ' ia-7o/' (67 /o) 5 hundredths 2 hundredths 5 tenths 2 hundredths

March 1983 55



and hundredths and place value, as well as the results on applications were low, however. Both the fourth- grade and fifth-grade teachers felt that the work on fractions and tenths was within the reach of the majority of the children in those grades. The same teachers felt less comfortable with the hundredths, but still thought the work was within the reach of most of their students. They both reported more difficulty with place value and measurement than with tenths and hundredths.

More than 50 percent of all students reported that they felt all parts of the unit were "easy." Generally, they thought they had mastered more than the achievement test indicated. The students indicated less comfort with place value ahd applications when compared to their level of confidence with tenths and hundredths. This pattern paralleled the achievement data.

Implications for Curriculum Work This investigation of class learning with decimals has implications for curriculum development and suggests the need for further studies of this kind.

Tenths and hundredths, as taught in this unit, appear to be appropriate for mastery learning with the fourth-, fifth-, and sixth-grade students in the study. Fourth- and fifth-grade students may need more time on the work with hundredths, but the ap- proach used was effective. Emphasis should be placed on models and oral names for decimal numerals. The work with hundredths should focus carefully on developing the concepts for 0.01 to 0.10 and also developing concepts for 0.10 to 0.99.

Major attention was given to developing place- value ideas in this unit, but some difficulties were still en- countered, especially in the fourth and fifth grades. Some possible sug- gestions would be to separate the initial learning of tenths and hundredths from the work with place value. It may be that children need some time to internalize the concepts learned in the simple notation of tenths and hun-

Table 3 Mean Scores for Seventh-grade Pretest and Remedial Group Pretest, Posttest, and Retention Test

Mixed T/H & Test Tenths Hundredths place value Applications

No. of items Ï6 16 12 Ш

^N^mf 13-6 12'5 8'6 58 Remedial pretest t < й Q ̂ A i ii

N = i3 11'° y-J *v Jl

Remedia^ posttest 15 5 n 8 96 75 Remedial

retention test 15.3 13.7 9.6 6.5 N = 13

dredths. There may also be a factor of readiness involved. A larger percent- age of the sixth graders than of the fourth or fifth graders were able to master place- value ideas, and this dif- ference may be due to the ' 'review" aspect or a maturing of the students. Perhaps place value should be introduced and explored more slowly in the fifth grade, with expectations for mastery of place-value concepts for tenths and hundredths delayed until sixth grade.

Follow-up Follow-up work was conducted at a local junior high school where the achievement test was administered to the total seventh-grade population. The unit was then taught to a group of 13 students who scored lowest on the pretest. After completion of the instructional unit, this group was administered a posttest and retention test, as had been done in the original study. Table 3 shows the mean scores on the various test situations.

The total seventh-grade pretest results were lower than the sixth-grade posttest means for each subtest. Inter- estingly, the posttest and retention test scores of the remedial seventh- grade group who studied the unit were similar to the posttest and retention test means of the sixth grade in the original study. This seems to imply that remedial seventh-grade students may need the same careful building of concepts as was done initially with the fourth, fifth, and sixth grades in the

study. Even more powerful is the abil- ity of remedial junior high students to show significant achievement when taught in a careful and meaningful manner.

Conclusion Helping children understand the concepts of decimals prior to their use in computation should be a major goal of elementary school mathematics education. We can reduce later problems in decimal computations if a strong foundation of understanding has been built first. This study provided some insights and information into ways that may be applied to the initial teaching of decimal concepts.

References

Carpenter, Thomas P., Mary Kay Corbitt, Hen- ry S. Kepner, Jr., Mary Montgomery Lind- quist, and Robert E. Reys. "Decimals: Re- sults and Implications from National Assessment." Arithmetic Teacher 28 (April 1981):34-37.

Coburn, Terrerice G., Leah M. Beardsley, Alan A. Edwards, and Joseph N. Payne. Michigan Educational Assessment Program Mathematics Interpretive Report, 1974 Grade 4 and 7 Tests. Monograph No. 8. Pontiac: The Michigan Council of Teachers of Mathematics, 1975.

Ellerbruch, Larry W. and Joseph N. Payne. "A Teaching Sequence from Initial Fraction Concepts Through the Addition of Unlike Fractions." In Developing Computational Skills, 1978 Yearbook of the National Council of Teachers of Mathematics. Reston, Va.: The Council 1978.

Galloway, Patricia Winner. "Achievement and Attitude of Pupils Toward Initial Fractional Number Concepts at Various Ages from Six to Ten Years and of Decimals at Ages Eight, Nine and Ten." Unpublished doctoral disser- tation, University of Michigan, 1976. Щ




initial decimal concepts: are they really so easy?

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