Computers, Concrete Materials and Teaching Fractions

470

Computers, Concrete Materials and TeachingFractionsStanley BallDepartment of Educational

Leadership and CounselingThe University of TexasEl Paso, Texas 79968

(<. . . the software provides a transition from concreteto symbolic level for students^ skill development infraction problem-solving.^

The teaching of fractions continues to be a topic of concern to mathematics educators,presumably because of the difficulty in developing meaningful concepts of fractions andbinary operations on fractions. This article describes the development and use of a uniton fractions containing concrete materials and computer software (modeled after afraction strip chart). Designed to simulate concrete materials, the software provides atransition from concrete to symbolic level for students’ skill development in fractionproblem-solving.

Fraction Unit Development

Recent suggestions for mathematics curriculum development include the use of concretematerials (McBride and Lamb, 1986), but only if students have the prerequisite skillsand are developmentally ready to learn the concepts taught. Also, it is necessary that themanipulative model is at an appropriate level of abstraction and is consistent with therequired symbolic representation (Wiebe, 1983). These criteria can be satisfied withfraction strip chart materials beginning as early as second grade and continuing throughelementary school. For example, Silvia (1986) discusses the effectiveness of a fractionstrip chart in providing concrete experiences with fractions for nine-to-eleven-year-olddeaf students.Microcomputers are a valuable tool for teaching concepts in mathematics, particularly

when the computer activities are integrated meaningfully into the curriculum (Ball,1986). Berlin and White (1986), for example, use computer simulations to promote thetransition from concrete manipulation to abstract thinking. This article examines theeffectiveness of a software package designed to be consistent with a fraction strip chartmodel.

The software package was designed to:

1) maximize interaction with the microcomputer to provide a transition from concrete

manipulative activities to an understanding of fraction concepts expressed using symbols,2) integrate the microcomputer into the curriculum, and3) use graphics to simulate students’ use of concrete materials.

School Science and MathematicsVolume 88 (6) October 1988

Teaching Fractions 471

The Materials

The entire unit used in this study consisted of the software package plus class materials,all of which focused on the fraction strip chart concept. First, a poster board chart withappropriately sized strips was prepared for classroom demonstration use. Individualcharts were laminated for students’ use. In addition, scissors, transparent tape, anenvelope for the fraction pieces as well as a large supply of strips which matched theindividual charts were available during each lesson. New concepts were alwaysintroduced with the physical strip materials and not with the software.Twelve computer lessons were written which simulated the concrete processes involved

in 1) defining the concept of fraction, 2) deciding more than or less than for fractions,3) finding equivalent fractions, 4) adding and subtracting fractions, and 5) multiplyingfractions. The graphics displayed the same fraction strip chart students used at theirseats. Colored shading represented the various lengths of fraction strips.The graphics were also consistent with the process of finding fraction names for

strips. This process involved lining up the strip in question even with the left edge of thechart, starting at the top, and sliding the strip down until the right edge of the stripmatched a vertical line in one of the rows. The fraction name for the strip was recordedwith the denominator being the total number of equivalent parts in that row and thenumerator being the number of parts covered up by the strip. The answer was to be inlowest terms as students were encouraged to select the row closest to the top of thechart.

Figure 1 displays a graphic seen on the monitor screen at the end of a fractionaddition lesson. When the graphic first appears, the following figures are present:

1) the fraction strip chart (with no shading) in the upper lefthand corner,2) the face,3) two strips (with no shading) in the lower lefthand corner, and,4) the directions below the face.

As the fraction addition problem appears below the strip chart, portions of the twostrips at the bottom are shaded representing the two fractions to be added together.Then the shaded strip just above the two bottom strips appears as a result of making a"train" (taping the two strips together end-to-end) from the two original strips 1/2 and2/3.

Students notice the strip is longer than "one whole." Subsequently, with prompts, ifnecessary, they decide that a portion equal to one whole needs to be removed. To keeptrack of the shortening of the original strip, the whole number "1" is written to theright of the equal sign beneath the chart where the original two fractions were written.Finally, the shaded part in the strip chart in the upper screen indicates the correctposition of the remaining strip once the whole part is removed. The fraction name forthis strip is written next to the "1" to complete the problem.The program was interactive in two important ways. First, the face to the right of the

chart is a partial reinforcement to students. With a correct response the face smiles.Otherwise, it frowns. Second, the process of measuring lengths, cutting and tapingtogether (which were originally done at the concrete level) were simulated at thecomputer. For example, when a student selected a row in the table, the strip wascolored in that row to simulate the actual placing of a strip there. It was possible thenfor the student to decide if the selection was correct.


472Teaching Fractions

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1/21/3i/()1/51/61/71/81/91/10

.-"i HERE IS THE PORTIONOF THE MONITORSCREEN WHEREINSTRUCTIONS AREGIVEN.

PRESS RETURNWHEN READYTO CONTINUE

FIGURE 1. Sample monitor screen display

Procedure

To investigate the effectiveness of the instructional unit, five fourth-grade classes in twoschools were involved in the study. Three classes were selected as treatment (one at oneschool and two at the other) and two as control (one at each school). The treatmentteachers were selected because of their involvement in a summer computer workshopand expressed interest in the study. The control teachers were the other fourth gradeteachers at the two schools. Both schools were located in middle class neighborhoods.There were 44 male and 47 female students in the five classes.

Pretests on fractions were given to all classes during the same week in the fall.Paper-pencil pretests and post-tests had the same format and were slight modificationsof chapter tests in the textbooks. Some questions were slightly modified to use fractionstrips. Some items simply required students to calculate the answer to a fraction



problem involving a binary operation or to respond with greater than, less than, orequal. Other items asked students to draw a fraction strip to show their answers. Thefollowing two problems serve as examples:

1) Add:3+

!

112) Draw a picture of strips showing the sum of - + -

The test required about twenty minutes to complete. Cronbach alpha (SSPS, 1986), ameasure of internal consistency reliability, was .78 for the pretest and .84 for thepost-test. The post-test was administered approximately three months after the pretest,at the end of the instruction on fractions. The total time spent on teaching fractions wassix to eight weeks depending on the individual class.

In treatment classes, each new fraction concept was introduced with strip chartmaterials and followed by additional strip chart activities. Then students worked on thecomputer lessons. Additionally, teachers were encouraged to supplement these activitieswith off-computer fraction work sheets. One microcomputer was available in each classwhich each student used for 20-30 minutes per session per day. New topics werepresented once all students had completed the computer lesson on previous concepts.The treatment included the emphasis on the concrete experiences with the fraction stripchart followed by simulation interaction with the computer.

Control classes were assigned activities from the fraction workbooks selected by thedistrict. Teachers had virtually no contact with treatment group teachers due to differentteaching schedules. However, control group teachers were encouraged to use thefraction strips as they felt appropriate. The regular instructional materials suggested theuse of the fraction chart and other pictorial representations of fractions, which wereused on a limited basis depending on the individual teacher. No computers were used inthe control classes.

Statistical Results

Scores of students who took both the pretest and the post-test were analyzed. Table 1

TABLE 1

Mean Pretest and Post-test Scores

Group

Test Treatment Control t-value

Pre

Post

Adj. Post

M =

SD =

M =

SD =M =

5.163.4214.392.7614.37

M =

SD =

M ==SD =

M =

4.892.9311.264.6011.31

.39

4.07**

n = 56 n = 35

**p < .001


474 Teaching Fractions

shows the pretest, post-test and adjusted (due to pretest) post-test means for treatmentand control groups. The t-values shown in this table indicate that there was nosignificant difference between pretest means, but there was a significant differencebetween post-test means (treatment versus control).

Table 2 shows the one way covariance analysis using pretest scores as the covariateand post-test scores as the dependent variable. The post-test scores adjusted fordifferences in pretest scores had a significant treatment effect.

Source

Covariate(Pretest)

TreatmentEffect

ExplainedResidualTotal

TABLE 2

Analysis of Covari

SS

90

20129110571348

iance

df

1

12

8890

MS

90

201145.51215

F

7.5*

16.8**12

* p <.01

Discussion

These results validate the effectiveness of the treatment as administered in this study.The computer was used in the treatment group classes but not in the control groupclasses. However, the extent to which concrete fraction strip materials were used in thecontrol classes was subject to individual teacher preference. Therefore, it is difficult to

assess the relative influence of the extensive use of concrete materials versus the use ofthe computer. Nevertheless, the combination improved students* skills in fractionproblem-solving. The purpose of the project was to investigate the feasibility ofcomputer software which provided a transition from the concrete to the symbolic level.There was no interest in attempting to determine the relative influence of the computeritself.

Informal student observations and interviews provided additional validation of theeffectiveness of this unit on fractions. Treatment group students seemed confident withanswers using actual strip measurements. This was true whether finding equivalentfractions with a given strip or finding the fraction name for a train of two strips tapedtogether. This same confidence seemed present in their interactions with the computer.

Previous work with students using the fraction strip chart suggested that cutting andtaping strips together became tedious before students were ready to develop symbolicalgorithms for finding answers (Ball, 1986). The computer performed an importantfunction of supplying additional practice at this important skill level. Students in thetreatment classes had previously used the computer in the class. Thus the results are notlikely due to any computer novelty effect.



Important research remains to be done investigating the relative timing andapplicability of the use of concrete manipulatives and subsequent computer simulationinteraction. This project, however, suggests that there are advantages to using computeractivities to provide transition from the concrete to the symbolic level in mathematicsproblem-solving.

References

Ball, S. (1986). Cognitively integrating computers into curriculum-need and examples[Summary]. Conference Proceedings of the Sixth Annual Texas Computer EducationAssociation, 188-109.

Ball. S. (1986). From story to algorithm. School Science and Mathematics, 86(5),386-394.

Berlin, D., and A. White. (1986). Computer simulation and the transition from concretemanipulation of objects to abstract thinking in elementary school mathematics.School Science and Mathematics, 86(6), 468-479.

McBride, J. W., and C. E. Lamb. (1986). Using concrete materials to teach basicfraction concepts. School Science and Mathematics, 86(6), 480-488.

Silvia, E. (1986). A fractions curriculum for deaf children. School Science andMathematics, 86(2), 126-136.

SPSS Inc. (1986). SPSSX User’s Guide (2nd ed.). Chicago: Author.Wiebe, J. W. (1983). Physical models for symbolic representations in arithmetic. School

Science and Mathematics, 83(6), 492-502.

# # #

GALAXY IS LARGEST KNOWN;SHOWN TO HAVE TIDAL EFFECT

A distant and heretofore not very spectacular galaxy known to astronomers forthe past 20 years by its number�Markarian 348�now tentatively enjoys thedistinction of being the largest known galaxy in the universe.

Dr. Susan M. Simkin and colleagues report in the March 13 issue of Science thatMarkarian 348 is about 1.3 million light years in diameter. For comparison, our owngalaxy, the Milky Way, is about 100,000 light years across.

What’s more, says the Michigan State University professor of astronomy, thegreat cloud of hydrogen surrounding the galaxy moves in tidal fashion.

Just as the moon causes tides in the Earth’s oceans, Simkin explains, a smaller,nearby galaxy exerts sufficient gravity to make the hydrogen cloud undulate aroundthe core of Markarian 348. However, the movement would be noticeable to Earthobservers only over a period of hundreds of millions of years.Astronomers have long speculated on the existence of tidal effects as the cause of

spiraling movement in galaxies but it has never been proved in detail.


Computers, Concrete Materials and Teaching Fractions

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