preservice elementary teachers' conceptions of volume

670

Preservice Elementary Teachers^Conceptions of Volume

Larry G. EnochsDorothy L. Gabel

The metric system of measurementis frequently studied in elementarymathematics and science methodscourses for preservice teachers. In-cluded in this "typical" course isusually a unit consisting of measur-ing volumes and areas. Althoughstudents should have a good graspof these concepts before enteringcollege, many have difficulty inmeasuring and calculating volumesand surface areas of objects evenwhen they are rectangular solids. Ifwe are to prepare these preserviceelementary teachers to teach mathe-matics and science to children, theabilities to calculate, operationallydefine, and understand the meaningof volume is necessary.

Little research has been done onpreservice elementary teachers’ con-ception of volume. Bilbo and Mil-kent (1978) compared two differentapproaches for teaching volumeunits of the metric system to non-

science majors. They found that by teaching volume before length,students understood volume better. In the treatment group, area was not

School Science and MathematicsVolume 84 (8) December 1984

Conceptions of Volume 671

taught because the researchers felt that it might confound the students’conception of volume. It is unclear from this research whether this wasthe reason why students did better using the volume first approach orwhether it was because mathematical formulas were not introduced.

"The major difficulty that students appear to have in deter-mining volume and surface area is that they rely on the use offormulas to solve the problems rather than on the conceptualmeaning of volume and surface area.^

In working with volume and surface area, students must use spatialvisualization skills. Siemonkowski and MacKnight(197I) found that ahigh correlation exists between three dimensional conceptualization andsuccessful grades in college courses. Bishop (1978) asserted that manyprofessions require a considerable amount of spatial ability. These pro-fessions included artists, engineers, and scientists. While preserviceelementary teachers will not likely be taking advanced science courses,their success would probably be enhanced in their required sciencecourses if they possessed a higher ability in three dimensional conceptua-lization. Further, the degree of success in their required science coursesshould strengthen their ability to teach mathematics and science at theelementary level.Frandson and Holder (1969) reported that deficiency in spatial-

visualization-aptitude can be compensated for by instruction. Bishop(1978) added that spatial representations, over time, are built up throughactions performed on objects in space. It follows, according to Bishop,therefore, that topological, projective, and Euclidian abilities can im-prove and develop as the result of experiences. With more precise analy-sis of the specific tasks involved in the learning of the concept of volumethe student’s performance should be improved.

Procedure

In order to improve instruction on teaching the meaning of volume, aneffort was made to identify students’ misconceptions of volume and sur-face area through written observation and interviews. Following a classon surface area and volume, eight students were interviewed. Surfacearea was included in instruction because many students do not dis-tinguish between volume and surface area. A thorough understanding ofvolume would include this distinction. A protocol was used for each in-terview including probes and interviewer assistance where appropriate.


672 Conceptions of Volume

The written observation consisted of each respondent completing twotypes of questionnaires, one dealing with volume and one with surfacearea prior to instruction.

Subjects

One hundred and twenty-five preservice elementary teachers participatedin this study at a large midwestern university in the fall of 1981. Theparticipants were enrolled in a freshman basic skills science course forelementary education majors. Ninety-eight percent of the population wasfemale.

^Ninety-eight percent of the population was female.^

Instruments

In order to probe the misconceptions of preservice elementary teacherswith respect to surface area and volume, the study-specific SurfaceArea/Volume Misconception Inventory (SAVMI) was developed. Sixforms of the SAVMI were developed. Each form had 13 identical multi-ple choice items that listed possible ways to calculate surface area andvolume (See Figure 1). Three forms were used to ask students to answerthese items in relation to the volume of a regular solid, a cylinder or an ir-regular solid (a rectangular solid with a small rectangular solid removedfrom it). The other three forms were used to ask students to answer itemsin relation to the surface area of these same geometric figures. Each re-spondent was given two forms of the SAVMI, one volume form and onesurface area form. Each form contained a different geometric figure.The figure types were not intended to represent equal difficulty. Possiblescores ranged from 0 to 13. Figure 1 shows one form of this instrument.

Prior to administering the SAVMI, content validity was established bysurveying a panel of five judges deemed to be expert in the field of prob-lem solving. These judges responded to four statements by rating the de-gree the SAVMI assessed misconceptions concerning surface area andvolume. The data resulting from this assessment were subjected to theKendalFs Coefficient of Concordance Test (Siegel, 1956) and a coeffi-cient of 0.62 was found considering ties. The Spearman-Brown reliabilitycoefficients for the various forms of the instrument were 0.81, 0.67,0.82, 0.83, 0.92, 0.61. The SAVMI was administered to 125 subjects atthe beginning of the semester.



Name

Date.

Section.

Directions: On this test you will be asked about possible ways to find the volume orthe surface areas of certain objects. If the method given describes a waythe answer can be obtained mark "yes" by blackening "a." If themethod cannot be used mark "no" by blackening "b." If you are un-certain if the method can be used mark ’*c."

How could the volume (capacity) of the following hollow object be deter-mined?

yes no uncertain Methodsa b c 1. Multiply the area of the base x heighta b c 2. Use the formula 6 (L x W x H)a b c 3. Measure the volume of water that would fill the object to

the brima b c 4. Count the number of cubes that are 1 inch on each side

that would fill the objecta b c 5. Use the formula 2(L x W) + 2(L x H) + 2(W x H)a b c 6. Multiply the length x width x heighta b c 7. Count the number of marbles that are 1 cubic inch in

volume that you can pour into the object to fill ita b c 8. Use the formula 2[(L x W) + (L x H) + (W x H)]a b c 9. Count the number of squares 1 inch on a side that could

be placed on the outside of the objecta b c 10. Count the number of cubes that are 1 inch on a side that

would fit in the base and multiply this number by thenumber of layers that would fill the object

a b c 11. Use the formula 6L + 6W + 6H12. The volume (capacity) of the object in cubic inches

is: a. 40 b. 66 c. 76 d. 240 e. Some other answera b c 13. Did you use one of the 11 methods listed here to deter-

mine the volume of the object?14. Which method did you use _________?15. If your answer to question twelve was "Some other

answer" please describe how you obtained your answerbelow.

FIGURE 1Sample SAVMI Test



Interviews

In addition to the administration of the SAVMI, eight preservice elemen-tary students, all females, were interviewed. Early in the course, but afterthe instructional unit on the metric system, a laboratory practical testwas administered to all (125) students. The test consisted of determiningthe volume and the surface area of an irregularly shaped geometric figuremade from 5 to 8 cubograms. Two points were awarded for the correctvolume, two for the correct surface area, and one point for the correctunits. The scores ranged from 0 to 5. (Cubograms are cubes one centi-meter on a side that weigh one gram.) The eight interview subjects wereselected from a pool of students making zero on this test. Only volun-teers were used.

Interviews were used to gather the data which were based on a struc-tured protocol. The protocol consisted of a series of problems and alter-nate problems on volume and surface area using three solid three dimen-sional models, hollow models and drawings of each. Alternate modelswere available for use when a respondent had to be prompted or assisted.Figures used during the interviews are listed in Table 1.The interviews were conducted by the two investigators conducting the

study. Interviews were conducted in an informal setting in a non-threat-ening atmosphere. The respondents were assured that these interviewswere not to be connected in any way to their course evaluations butrather would assist them in identifying their problem-solving errors. Eachrespondent was instructed to "think-aloud" as they solved the problemand to write down their calculations. The interviews were audio-taped.

Following the interviews three tapes were randomly selected andlistened to by the two investigators. Each investigator listed the errorsand/or misconceptions made by the respondent. These lists werecompared and synthesized into seven categories of errors. These cate-gories, with typical examples, were used to design a coding form for usein coding the eight tapes (TABLE 2). Each tape was coded using the re-sultant form by both investigators. All coding sheets were then comparedfor agreement which was found to be 96%.

Findings

Table 3 presents data from all forms of the SAVMI instrument. Table 4gives details from the two forms of the instrument concerned with rec-tangular solids, and Table 5 gives the results of an analysis of varianceaccording to the type of figure (regular, rectangular, cylinder, irregular)and the kind of measurement (volume, surface area).



Table 1

Interview Protocol

FigureSurface Area Volume

Cubogram (plus alternate)

Hollow Box (plus alternate)

Solid Cube (plus alternate)

Drawing of Regular Box

Irregular Box (plus alternate)

Drawing of Irregular Box

Irregular Cubogram (plus alternate)

Hoi low Cylinder (plus alternate)

Solid Cylinder (plus alternate)

Diagram of Cylinder

SA

SA

SA

SA

SA

SA

SA

SA

V

V

V

V

V

V

V

V

V

V

Although there was no significant main effect for the type of figure forwhich students answered the questions on volume, the means were pro-gressively lower as the type of geometric shape became more complexand volume was less easy to determine by using a formula. For bothvolume and surface area, the means were quite low (7.74 and 6.70 re-spectively) indicating that although these preservice elementary educa-tion teachers have a statistically significant better understanding ofvolume than surface area, they really do not understand either of theseconcepts well.

This is illustrated in Table 4 which lists the percentages of studentswho selected correct and incorrect methods for determining the volumeand surface area of the simplest geometric shape, the regular rectangularsolid. Of the five correct ways to calculate the volume of a rectangularsolid, 77% of the students were certain that volume could be obtained bymultiplying the length X width X height. Only 44% were certain thatvolume was equal to the area of the base X height. Students were evenless certain on how to calculate the surface area of a regular rectangularsolid even though in the problem itself they were told that it was the sum



Table 2

Interview Coding Form and Error Frequencies

ErrorExampleSAVolume

Concept/definition ofvolume or surface area

Formula memorizing mode

Confusion between length,area and volume

Unit memorizing mode

Conversion of cm^ to ml

Multiplication of unitsnot correct or unitsincorrect

Arithmetic wrong

Only gives formula or 7doesn’t know formulacan’t use formula

Uses formula sponta- 6neously, incorrectly,or in cases whereinapplicable

Uses units interchange- 4ably

Unable to transfer to 1in3, in2, etc.

Answer to question in- NA*correct

Answer 1

Answer

*Not applicable

of all the areas on the surface. While 67% could correctly determine thevolume only 26% could determine the surface area.

These results led the researchers to interview eight students in order to

determine more precisely the misconceptions of volume and surface area

held by elementary education majors who had difficulty with the con-cepts of volume and surface area as described previously.

Table 2 presents the error frequencies encountered by way of the inter-

views. Numbers are given merely to indicate some of the common mis-conceptions and problems that students encounter rather than serve as a

method of comparison between volume and surface area. The major dif-ficulty that students appear to have in determining volume and surfacearea is that they rely on the use of formulas to solve the problems rather



than on the conceptual meaning of volume and surface area. This wasmade apparent by their giving a formula spontaneously in many in-stances where the formula was incorrect or inapplicable or giving aformula and not knowing how to apply it. It was evident that thesestudents did not understand the concepts of volume and surface area(after two hours of instruction) because they used units interchangeablyfor length, volume, and area (cm, cm3, cm2). Frequently, even whenstudents were able to get a correct answer using the metric units used inclass, they were unable to do a similar problem when the units werechanged to the more familiar English system. Compounding students’difficulties in determining volume and surface area were the mistakesthey made in arithmetic and in algebra (not knowing for instance cm xcm2 = cm3).

Table 3

Means for Volume and Surface Area of DifferentShaped Figures

Figure

MeasurementRegularCylinderIrregularCombined

Volume 7-8.447»7.587-7.137-7.74n - 43n - 43n -39n - 125

Surface 7-6.667-6.747-6.707-6.70Area

n - 38n » 43n - 44n - 125

Combined 7-7.617-7.167-6.907-7.22n - 81n ° 86n - 83n - 250

Discussion

If elementary teachers are expected to teach volume in the schools theymust first understand the concept themselves. If they do not, volume willbe taught as a formula to be memorized and applied, rather than as anentity in itself. This exploratory study examined the procedures that


Conceptions of Volume

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TABLE 5Analysis of Variance for Surface Area Volume Data (SAVMI)

Source Sum ofdfMean F Sig.SquaresSquare

Main Effects 41.54313.85 1.61 NSFigure 6.2723.14 .3 NSMeasurement 34.32134.32 3.98 P .05

2-Way InteractionsFigure x Type 48.80224.40 2.83 NS

Explained 90.73518.15 2.10 NSResidual 2105.152448.63Total 2195.882498.82

<(. . . they must teach the concept. . /

students use to solve volume and surface area problems as well as mis-conceptions. Findings from this study indicate that a large percentage ofelementary education majors do not understand the concepts of volumeand are unable to distinguish volume from surface area. Students’concept of volume is bemuddled. They were found to solve problems us-ing a "memorizing mode" rather than basing their answers on the con-cept itself. In order to remedy this situation, science and science methodsinstructors need to be aware that many preservice elementary teachers donot understand the concept of volume, and therefore, they must teachthe concept instead of assuming that it is already understood and self-evi-dent to all students. This research does not indicate how volume and sur-face area should be taught. Since many students associate the meaning ofvolume exclusively with formulas, however, it seems reasonable to inferthat using an exclusive formula approach is not beneficial. Perhaps usinga hands-on approach without mention of formulas until the end of theinstruction would be more effective in teaching the concept. Instructorsalso need to be made aware of other errors that students make (SeeTables 2 and 4) and devise methods to eliminate them.

This study does not indicate whether students are unable to determinethe volume and surface area of objects because they have low spatial-visualization aptitude or not. However, additional practice in workingwith geometric solids in determining their volume and surface area mightnot only enable students to understand the concepts of volume and sur-face area but also improve their spatial-visual aptitudes.



References

Bilbo, T. E. and M. Milkent. "A Comparison of Two Different Approaches for TeachingVolume Units of the Metric System." Journal of Research in Science Teaching, 1978, 75,53-57.Bishop J. "Developing Students’ Spatial Abilities." Science Teacher, November, 1978, 2-23.Frandson, A. and R. Holder. "Spatial Visualization in Problem Solving Complex VerbalProblems." Journal of Psychology, 1969, 73,229-233.Siemonkowski, F. and F. MacKnight. "Spatial Cognition: Success Prognosticator in Col-lege Science Courses." Journal of College Science Teaching, October, 1971, 56-69.Siegel, S. Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill,1956.

Larry G. Enochs Dorothy L. GabelCollege of Education Science and EnvironmentalKansas State University EducationManhattan, Kansas 66506 Indiana University

Blooming ton, Indiana 47405

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preservice elementary teachers' conceptions of volume

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