Numbers and operations

Glasgow, Robert, Gay Ragan, Wanda M. Fields, Robert Reys, and Deanna Wasman. “The Decimal Dilemma.” Teaching Children Mathematics 7 (October 2000): 89–93.

Copyright © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

89OCTOBER 2000

I f you are aware of the results given in the

media reports about the Third International

Mathematics and Science Study (TIMSS), you

probably know that fourth graders from the United

States (U.S.) scored above the international average

in mathematics and that eighth and twelfth graders

scored below average (Mullis et al. 1997). As an edu-

cator, you are aware of the dangers of looking only at

averages of test scores. Rich information can be

gleaned from the TIMSS data that will help us learn

more about what our students know and are able to

do. The data from a large-scale study, such as the

TIMSS, often raise questions about what the numbers

really mean. This article addresses one such question

that arose from examining part of the third- and

fourth-grade TIMSS data. The process that we used

may be as valuable as the information that we found.

Perhaps this process will help you answer questions

that arise as you reflect on the TIMSS results.

The Dilemma The TIMSS data are reported in mathematical con- tent categories (see timss.bc.edu). In the category

of fractions and proportionality, U.S. third graders performed at the international average, whereas U.S. fourth graders performed above the interna- tional average. However, a close look at the released test items reveals an interesting phenome- non. U.S. students did not do as well on questions involving decimals as they did on questions involving fractions (see fig. 1).

Questions M-5 and J-7 differ primarily in repre- sentation of the possible responses (decimal and fractional), yet the performance levels of U.S. stu- dents were dramatically different for these two questions. Thirty-two percent of U.S. fourth graders answered the decimal ques- tion (M-5) correctly, whereas 80 percent answered the fraction question (J-7) cor- rectly. Although this difference between understanding levels can be seen in the inter- national scores, the difference in U.S. stu- dents is more extreme. The results of ques- tion I-2 seem to further highlight a deficiency in understanding decimal notation.

The data in figure 1 generate many questions concerning decimals in third- and fourth-grade classrooms. Figure 1 suggests growth in under- standing decimals and fractions from third to fourth grade, indicating that students are probably receiving instruction on both topics. The dilemma

Bob Glasgow, Gay Ragan, Wanda Fields, and Deanna Wasman were doctoral students at the University of Missouri when this study was done. Glasgow, [email protected], teaches and works with preservice teachers at Southwest Baptist University in Bolivar, MO 65613. Ragan, [email protected], is currently an assistant professor at Southwest Missouri State University, Springfield, MO 65804. Fields, [email protected], has taught at both the middle school and high school levels. She has also taught college algebra courses at a com- munity college and at the University of Missouri—Columbia. Reys, [email protected], is on the faculty at the University of Missouri. Wasman, [email protected], is teaching in the Department of Mathematical Sciences at Appalachian State University, Boone, NC 28608.

The authors gratefully acknowledge Linda Coutts, mathematics coordinator for Columbia Public Schools, and Vicki Robb, principal of Russell Boulevard Elementary School, for their help in collecting data from teachers and students that are reported.

Robert Glasgow,

Gay Ragan,

Wanda M. Fields,

Robert Reys, and

Deanna Wasman

Decimal Dilemma The

90 TEACHING CHILDREN MATHEMATICS

is this: Why does decimal understanding lag behind fractional understanding? Is it a result of when decimals are taught or more a result of how they are taught in the U.S.? Reflect on the items and results in figure 1. Would your students per- form at the U.S. levels? Why is decimal under- standing weaker than fractional understanding in U.S. third and fourth graders? How might you col- lect data to answer these questions?

The Details We examined the details surrounding the decimal dilemma by looking at national and state informa- tion concerning the coverage of decimals in U.S. third- and fourth-grade classrooms.

The national level The TIMSS addressed national curricular differ- ences by asking an agency of each participating country to report which test items were included in its country’s intended curriculum by the fourth grade. For the U.S., the National Research Council selected persons who were familiar with mathe- matics curricula across the country to perform a Test-Curriculum Matching Analysis. The U.S. was the only country to identify 100 percent of the questions as being included in the fourth-grade curriculum. Other countries, such as Japan (identi- fying 89% of the questions as being covered), Sin- gapore (74%), and the Netherlands (52%), were less optimistic about what their curricula included, even though the fourth graders in all three coun- tries outperformed the U.S. fourth graders. Even at the third-grade level, the U.S. curriculum review- ers found 100 percent of the questions to be

included in the intended curriculum. Again, Japan (75%), Singapore (51%), and the Netherlands (23%) were more cautious. The reviewers were therefore confident that the intended curriculum for U.S. third and fourth graders would prepare them to perform well with decimals, but the actual results on the TIMSS did not support this opti- mism. Was the intended curriculum misreported, or is it not being taught?

The state level We next investigated curriculum expectations at the state level, since most states have curricular frameworks that serve as guidelines for local school districts. An examination of five state cur- ricular frameworks (Alabama, California, Min- nesota, Missouri, and New Jersey) found that dec- imal notation is studied by the time that students are in the fourth grade in all five states. Each framework expects that students should correctly answer question I-2 in figure 1. All but one frame- work mentioned using shaded regions to represent decimal notation as found in question M-5. All five frameworks promote using money to introduce decimals, as well as looking at decimal equiva- lences for common fractions. Are these frame- works representative of elementary school educa- tion in the U.S.? What does your state or district framework suggest for decimal instruction in these grades?

The Discoveries We then considered the decimal dilemma by exam- ining a single school district. We examined the cur- riculum of one local school district and found that

F IG

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E 1 The TIMSS questions with average percent of students answering correctly

Grade U.S. International

Question I-2 0.4 is the same as . . . 3rd 21% 21% a) four c) four hundredths b) four tenths d) one-fourth 4th 40% 39%

Question M-5 Which number represents the shaded part of the figure? 3rd 18% 33% a) 2.8 b) 0.5 c) 0.2 d) 0.02 4th 32% 40%

Question J-7 Part of the figure is shaded. What fraction of the figure is shaded? 3rd 63% 42% a) 5/4 b) 4/5 c) 6/9 d) 5/9 4th 80% 61%

in the third grade, decimals were developed within the context of money; in the fourth grade, students were expected to be able to read, write, compare, add, and subtract decimals through the hundredths place. To see whether the classroom practice matched the intended curriculum, we surveyed third- and fourth-grade teachers and interviewed several students at each grade level.

The district level—teachers Teachers were asked when they first introduce dec- imals (see fig. 2) as well as what contexts they use to teach decimals (see fig. 3). Twenty-eight third- grade and thirty fourth-grade teachers of mixed- ability classrooms responded. Teachers who taught multiple-grade classrooms were excluded.

The intended curriculum of the district appears to be the curriculum implemented, since all teach- ers gave a particular time when decimals are first introduced at their grade level. Knowing that deci- mals are introduced predominantly toward the end of the school year, after the time when the TIMSS was administered, provides insight as to why the U.S. student performance on decimals was low. It is interesting to note the varying responses in fig- ure 2. If the instructional schedule varies within a school district, it is certainly expected to vary across the country. When do you introduce deci- mals in your classroom? When do other teachers in your district first introduce decimals?

The contexts in which these teachers use deci- mals seem consistent with the conclusions reached by examining state curricular frameworks (see fig. 3). The predominant way to teach decimals is to use money, but the use of pictorial representations and fraction equivalences seems consistent with the level of preparation required for the TIMSS questions discussed in this article. How do you pre- sent decimals to your students? Do you think that other teachers in your district teach decimals in ways significantly different from yours?

Teachers predicted what percent of their stu- dents would answer each of the three TIMSS items correctly (see fig. 4). Not surprisingly, teachers thought that their students would do best on question J-7, which dealt with fractions. Teach- ers’ predictions for the items did not differ greatly from the overall U.S. results shown in figure 1. More surprising was the wide range of responses. For example, for question M-5, fourth-grade teachers’ predictions ranged from 0 percent to 95 percent. Out of thirty teachers, fourteen predicted 20 percent or less and four predicted 80 percent or higher. Similar ranges were found at both grades for all questions. What percent of your students would answer each question correctly? Would the predictions of teachers in your district have such a

wide range?

The building level—students Finally, in an effort to gain some insight into what students might have been thinking when they answered the TIMSS questions, we talked with stu- dents. We conducted interviews with fifteen third graders and twenty fourth graders from the same school district in which the teacher survey was conducted. In the United States, the TIMSS was administered in April 1995. Accordingly, we con- ducted our interviews in late March.

We initially presented the students with the three TIMSS questions (I-2, M-5, and J-7). Ques- tion I-2 was presented first without the multiple- choice responses. Students were asked to name “0.4” and were encouraged to say it in more than one way. If they did not say “four tenths” (zero of fifteen third graders and one of twenty fourth graders did), they were given the multiple choices. Responses indicated a very low level of under- standing of the decimal-notation terminology. Nearly all the students of both grades initially said “zero point four,” and only one of fifteen third

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F IG

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E 2 When teachers introduced decimals

F IG

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E 3 Contexts that teachers used to teach decimals

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1st quarter

2nd quarter

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Pictorial Metric system

Equivalence to fractions

Base 10 blocks

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92 TEACHING CHILDREN MATHEMATICS

graders and three of the remaining nineteen fourth graders chose “four tenths” when given choices. When we asked about their experiences with these kinds of numbers, most of the third graders claimed little exposure to decimals. Most did not even recall having heard the term decimal before

the interview. Some students said that they had seen these types of numbers when studying money. The only third grader who was able to choose the correct response claimed knowledge from outside the school setting. The student said, “I’ve seen on basketball games there’s like a zero point four seconds left on the clock, zero point four, and they say there’s four tenths of a second left on the clock.”

On question M-5, students were given the multiple choices immedi-

ately. Three of fifteen third graders and eight of twenty fourth graders chose the correct answer. These numbers are somewhat deceiving, because when asked to explain their answers, it was clear that of the eleven students who answered correctly, only five (one third grader and four fourth graders) were able to explain why 0.2 was the correct answer. For example, one fourth grader answered M-5 as follows:

(c) 0.2 because if there was a number here [in place of the 0], it would probably be a 9. And I know it is not (b), 0.5, because that is half of ten.

The answer, even though correct, indicates that the student’s understanding of decimals is limited. Students had great difficulty relating the decimal notation to the picture. The predominant incorrect answer was “2.8” (given by twelve of fifteen third graders and six of twenty fourth graders). This response is consistent with the most common

incorrect answer chosen when the TIMSS was administered. When students were asked to explain their reasoning, they said things like, “Maybe it is two point eight because there are two shaded and there are eight that are not.” Other students tried to use their knowledge about fractions to help them. One student responded in this way:

Well, we did this before so our teacher had us do this and I was real good at it and she explained it real well, so there was only two shaded parts and the rest was eight. I put two eighths and then two slash eight. Two goes with the shaded part and eight with the unshaded part.

Fourth graders did claim to have studied deci- mals more than third graders. Some of them reported to have studied decimals in terms of money and explored the relationship of fractions to decimals on a number line. Very few students reported using pictures, such as the one in question M-5, to study decimals.

Question J-7, which deals with fractional nota- tion, yielded results very different from those gen- erated by the decimal questions. Four of fifteen third graders and seventeen of twenty fourth graders were able to answer correctly without being given the multiple choices. When given the multiple choices, three more third graders and one more fourth grader answered correctly. Most stu- dents, even if answering incorrectly, showed some understanding of fractional notation. Nearly all the students at both grade levels reported studying fractions in school. Their performance levels at both grade levels indicate clear differences between students’ fractional and decimal thinking.

The Discussion Our interviews with students document that most students are not as familiar with decimals as with fractions. Our survey of teachers supports this

F IG

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E 4 Teachers predicted what percent of their students would answer each question correctly.

Question Grade Average Teacher Prediction

I-2 3 22%

4 34%

M-5 3 25%

4 33%

J-7 3 54%

4 71%

What percent of your

students would

answer each question

correctly?

observation by indicating that teachers have lower expectations of their students for responses on dec- imal questions than on fractional questions. Although district curricula and state frameworks address decimal notation, decimals take a back seat to fractional notation in third and fourth grades. We also found that decimals may not be addressed until late in the year in many classrooms. When decimals are studied, students may rarely see fig- ures like the one used in question M-5. Perhaps teachers should use pictorial representations more frequently when discussing decimals and should attempt to teach decimals in conjunction with teaching fractions. What do you think?

Have we gained new insight into the decimal dilemma? We obviously still have a great deal to think about. The data from the TIMSS only begin to tell the story of what students know and understand. Our examination shows the complexity of trying to expand on a few of the TIMSS items. This examina- tion included looking at the expectations for students at national, state, local, and classroom levels. We hope that our questions throughout this article, as well as the structure of our explorations, will invigo- rate your thinking about what your students are learning. Whether it concerns the decimal dilemma or other questions that emerge from the TIMSS, we challenge you to investigate the situation in your classroom, make a plan, and attack the problem. We encourage you to examine the TIMSS data in detail and then explore related information at the national, state, district, and building levels. Find out what other teachers in your district are doing. Most impor- tant, try the TIMSS questions with your own stu- dents. The results may surprise you.

Bibliography “California Department of Education Academic Performance

Index.” www.cde.ca.gov/board/mcs–intro.html. World Wide Web.

“ENC Standards and Frameworks.” www.enc.org/reform/ fworks/states.htm. World Wide Web.

Minnesota Department of Children, Families, and Learning. Minnesota K–12 Mathematics Frameworks. Saint Paul, Minn.: SciMath, 1998.

Missouri Department of Elementary and Secondary Education. Missouri’s Framework for Curriculum Development in Math- ematics K–12. Jefferson City, Mo.: Missouri DESE, 1996.

Mullis, Ina V.S., Michael O. Martin, Albert E. Beaton, Eugenio J. Gonzalez, Dana L. Kelly, and Teresa A. Smith. Mathe- matics Achievement in the Primary School Years: IEA’s Third International Mathematics and Science Study. Chest- nut Hill, Mass.: International Association for the Evaluation of Educational Achievement (IEA), 1997.

Rosenstein, Joseph G., Janet H. Caldwell, and Warren D. Crown. New Jersey Curriculum Framework. Trenton, N.J.: New Jersey Mathematics Coalition and New Jersey Depart- ment of Education, 1996.

“Third International Mathematics and Science Study.” timss.bc.edu. World Wide Web. ▲

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