Investigating Students’ Conceptual Understanding of Decimal Fractions Using Multiple Representations.
Copyright © 2003 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.
Martinie, Sherri L., and Jennifer M. Bay-Williams. “Investigating Students’ Conceptual Understanding of Decimal Fractions Using Multiple Representations.” Mathematics Teaching in the Middle School 8 (January 2003): 244–47.
Discussion
WE ADMINISTERED THIS INSTRUMENT TO FORTY- three sixth graders. The students worked individually and had as much time as they needed to complete the ques- tions. An item was scored correct if the student had accu- rately completed the given representation in a way that correctly identified the size of the two decimal numbers. For example, in the first question, students had to label the number line with a 0 and a 1 and correctly place 0.06 close to 0 and 0.6 slightly to the right of 1/2.
Even though each of these tasks required some concep- tual knowledge to represent the answer correctly, stu- dents’ success with the decimal tasks varied for each rep- resentation. Many students could accurately show 0.6 and 0.06 in one or two representations but not the others. The number of students scoring all correct (4) to none correct (0) are shown in table 1. Only six students (14%) of those tested were able to represent the decimal numbers in all four situations. Note that 77 percent of the students showed some conceptual understanding of decimals by providing correct responses to one, two, or three of the tasks, but they were not able to represent the numbers correctly for all the models.
Students’ success with the different models varied greatly (see table 2). Students were correct most often when explaining decimal numbers using the 10 × 10 grid and using money. Although 58 percent of students an- swered the place-value question correctly, most compared the tenths place of each decimal. Only six students (14%) stated that six-tenths is more than six-hundredths or made any quantitative comparison of the two decimals.
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Conceptual Understanding Using Multiple Representations
TABLE 1 Correct Responses on Decimal Questionnaire
NUMBER OF PERCENT OF NUMBER OF STUDENTS STUDENTS
CORRECT RESPONDING RESPONDING RESPONSES CORRECTLY CORRECTLY
4 6 14% 3 14 33% 2 12 28% 1 7 16% 0 4 9%
Total 43 100%
TABLE 2 Correct Responses for Each Item on the Decimal Questionnaire
PERCENT OF NUMBER OF STUDENTS STUDENTS RESPONDING
ITEMS ON RESPONDING CORRECTLY QUESTIONNAIRE CORRECTLY (OUT OF 43 STUDENTS) Number line 11 26% 10 × 10 grid 28 65% Money 28 65% Place value 25 58%
The number line was the most difficult of the four models. In fact, of the fourteen students who missed only one representation, eleven missed the number line. The most common error (made by sixteen of the thirty-two students who missed this question) was to label 0 and 1 on the number line, place 0.6 accurately, then incorrectly place 0.06 or leave it off entirely (see fig. 2). Notice that the student whose work is shown in figure 2 considered 0.06 to be halfway between 0 and 0.6, confusing one-tenth the size of 0.6 with one- half the size of 0.6. Another common error was to label 0.06 on the number line to the left of 0.6 but to place both decimal numbers inaccurately between 0 and 1 (see fig. 3). Students seemed to understand that 0.06 was smaller than 0.6 but did not indicate the sizes of the decimals in relation to 0 and 1. Figure 4 shows an- other common error, which was to identify 0.06 as larger than 0.6, perhaps with the idea that the longer decimal is larger, as is true with whole numbers. In two of the student samples, students used 0.5 and 0.05 as benchmarks to try to identify the correct placement of 0.6 and 0.06. This approach illustrates students’ at- tempts to apply what they know about the sizes of these decimals, specifically, that 0.5 is one-half and 0.6 is slightly larger than one-half.
Follow-up Assessment on Linear Representation
IN THE NUMBER-LINE MODEL, STUDENTS HAD DIFFI- culty labeling endpoints of 0 and 1 and relating the values 0.6 and 0.06 to the endpoints. Because we could not deter- mine whether students were struggling with the number line or with the relative values of the decimals, we de- signed another assessment that included four number-line tasks of increasing complexity:
1. Draw a number line that shows the numbers 1 through 5. 2. Draw a number line that shows 2.5. 3. Draw a number line that shows 0.4. 4. Draw a number line that shows 0.4 and 0.04.
What percentage of your students would successfully plot the numbers for each of these four tasks? Figure 5 shows the results for the forty-three sixth graders that we tested.
Most students understood the number line in relation to whole numbers, but many could not place the decimals, espe- cially those less than 1. Only one in five students was able to place 0.04 and 0.4 accurately! Recall that on the first test, 26 percent of students were able to label and place 0.6 and 0.06 correctly. This additional task revealed that students’ diffi- culty with a number-line representation was specific to those decimals less than 1, in particular, those less than 1/10.
Implications for Teaching and Learning
TO MAKE SENSE OF DECIMALS, STUDENTS NEED multiple experiences and contexts in which to explore them. Our assessment instrument using four representa- tions indicates that students may appear to understand decimals using some models, but they may lack a pro- found overall understanding of decimal concepts. In in- struction, therefore, teachers must include many represen-
246 M A T H E M A T I C S T E A C H I N G I N T H E M I D D L E S C H O O L
Fig. 2 A student places 0.6 correctly but is unable to place 0.06 correctly.
Fig. 3 This student’s solution recognizes that 0.6 is greater than 0.06, but the student does not indicate the relative sizes of the deci- mal numbers compared with 0 and 1.
Fig. 4 A student places 0.06 to the right of 0.6, apparently based on the misconception that the longer decimal is greater in value.
Fig. 5 Results of follow-up assessment using four number-line tasks of increasing complexity
0
20
40
60
80
100
Label number line 1–5
Label number line with 2.5
Label number line with 0.4
Label number line with 0.4
and 0.04
P er
ce nt
C or
re ct
Follow-up Tasks
93% 83%
57%
20%
Percent of Accuracy
V O L . 8 , N O . 5 . J A N U A R Y 2 0 0 3 247
tations of decimal concepts to broaden and deepen stu- dents’ understanding.
Teachers should also use multiple representations to assess students’ understanding. Without the number-line question in our assessment instruction, we might have concluded that our students had a sound understanding of decimals and their relative magnitude. Mistakes can reveal student misconceptions or overgeneralizations and pro- vide opportunities for learning, both for the teacher and students. An instrument that asks students to provide dif- ferent representations and explanations for a particular concept can be an eye-opener for teachers and can guide instructional decisions to enable students to deepen their understanding of concepts. The purpose of our decimal questionnaire was to identify student misconceptions and use that information to guide instructional planning.
Collecting data from students often results in more ques- tions. In our classrooms, the surprising difficulty of the num- ber line led to a follow-up inquiry to find out more about what students could and could not do. The follow-up number-line questions revealed that students’ number-line difficulties were specifically related to the size of the numbers, in particular, to decimals less than 1/10. We might offer several possible ex- planations for the students’ difficulty with locating numbers less than 1/10 on a number line. One explanation is that stu- dents were asked to draw and label all parts of the model, in- cluding the 0 and the 1 without any visual organizers already marked for them. This task was also the only one that called for approximation; students might not have been able to estimate approxi- mate positions for the two values, even though they could illustrate exact representations (such as the shading required in item 2, fig. 1). Students might also have been inexperienced with number lines. Using the number line to discuss the approximate magnitude of decimal numbers (as well as fractions and percents) is an effective tool for developing students’ num- ber sense (Bay 2001). Given that students struggle with the number-line model and knowing that decimals often appear in linear models in real-life situations, such as on a ther- mometer or metric ruler, we must recognize the importance of including linear models in our teaching of decimal concepts.
Summary
PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) states, “Students must learn mathematics with understanding, actively building new knowledge from expe- rience and prior knowledge” (p. 11). With decimals, prior knowledge of whole numbers may cause misunderstandings. For students to fully understand the similarities and differ-
ences of decimals and whole numbers, instruction must emphasize conceptual development, including the use of a variety of decimal representations.
References
Bay, Jennifer M. “Developing Number Sense on the Number Line.” Mathematics Teaching in the Middle School 6 (April 2001): 448–51.
National Council of Teachers of Mathematics (NCTM). Princi- ples and Standards for School Mathematics. Reston, Va.: 2000.
Resnick, L. B., P. Nesher, F. Leonard, M. Magone, S. Omanson, and I. Peled. “Conceptual Bases of Arithmetic Errors: The Case of Decimal Fractions.” Journal for Research in Mathemat- ics Education 20 (January 1989): 8–27.
Sackur-Grisvard, C., and F. Leonard. “Intermediate Cognitive Or- ganizations in the Process of Learning a Mathematical Con- cept: The Order of Positive Decimal Numbers.” Cognition and Instruction 2 (2) (1985): 157–74.
Sowder, Judith. “Place Value as the Key to Teaching Decimal Op- erations.” Teaching Children Mathematics 3 (April 1997): 448–53.
Wearne, D., and J. Hiebert. “Constructing and Using Meaning for Mathematical Symbols: The Case of Decimal Fractions.” In Number Concepts and Operations in the Middle School, edited by James Hiebert and Merlyn Behr, pp. 220–35. Reston, Va.: National Council of Teachers of Mathematics, 1988. �
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