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http://www.jstor.org
Journalfor Researchin MathematicsEducation
1990, Vol. 21, No. 3, 216-229
SPATIALORIENTATIONSKILLAND
MATHEMATICAL
PROBLEMSOLVING
LINDSAYANNE TARTRE,CaliforniaState University,Long Beach
The purposeof this study was to explore the role of spatialorientationskill in the solution of
mathematicsproblems.Fifty-seven tenth-gradestudentswho scored high or low on a spatial
orientationtest were askedto solve mathematicsproblemsin individualinterviews.A groupof
specific behaviorswas identifiedin geometricsettings,which appearedto be manifestationsof
spatialorientationskill. Spatialorientationskill also appearedto be involved in understanding
the problemand linking new problemsto previous work in nongeometricsettings.
For a long time researchershave attemptedto determine why some students
learnmathematicsor areable to solve mathematicsproblemsbetterthanothersdo.
Some researchhas focused on the processes used to solve mathematicsproblems.
Otherresearchershave tried to identify factors and skills relatedto doing mathematics.Among the skills found to be relatedto mathematicslearningandachievement are spatialskills (Fennema& Sherman,1977; McGee, 1979).
The literaturecontainsa greatdeal of discussion aboutthe possible relationship
between spatialskills and mathematics.Many studies have found spatialskills to
be positively correlatedwith measures of mathematicsperformance(Connor&
Serbin,1985;Fennema& Sherman,1977). But whatarespatialskills, andhow and
why are those skills relatedto mathematics?
SPATIAL
SKILLS
In general,spatialskills are consideredto be those mentalskills concernedwith
understanding,manipulating,reorganizing,or interpretingrelationshipsvisually.
In his review of spatialfactors, McGee (1979) distinguishedtwo majortypes of
spatial skills:
A plethoraof factoranalyticstudiessince the 1930's have providedstrongandconsistentsupport
for the existence of at least two distinct spatialabilities-visualization and orientation.(p. 909)
A factor analytic study by Connorand Serbin (1980) supportedthe distinction
summarizedby McGee. They found that
the visual-spatialtests dividedinto two factors....The CubeComparisonTest, PaperFoldingTest,
CardRotationsTest, andDAT Space RelationsTest hadhigh loadingon...[onefactor].This factor
representswhat ETS has referredto as the "spatialvisualization"subdivision of visual-spatial
ability.... [The second spatialfactor]had high loadingson HiddenPatterns,GestaltCompletion,
and CardRotations.This factorrepresentswhatETS has referredto as the "closure"subdivision
of visual-spatialability. (p. 27)
A differentcategorizationwas proposedby LinnandPetersen(1985), who identified threespatialability categories: spatialperception,mentalrotation,and spatial visualization.They defined spatialvisualizationas "those spatialability tasks
This articleis based on the author'sdissertationat the Universityof Wisconsin-Madison
underthe directionof ElizabethFennema.
LindsayAnne Tartre
217
which involve complicatedmulti-stepmanipulationsof spatiallypresentedinformation" (p. 1484). They distinguishedit from the other two categories "by the
possibility of multiple solution strategies"(p. 1484). Linn and Petersenfelt that
the only commoncharacteristicof spatialvisualizationtaskswas thatthe solutions
requiredmore than one step. They included in this class of tasks not only Paper
Folding, Form Boards (Ekstrom,French,& Harmon,1976), and the DAT (Space
Relations)(Bennett,Seashore,& Wesman,1973) but also the HiddenFiguresTest
(Ekstrom,French,& Harmon,1976).
McGee (1979) felt that spatial visualization tasks "all involve the ability to
mentallymanipulate,rotate,twist, or inverta pictoriallypresentedstimulusobject"
(p. 893). Kershand Cook (1979) also suggested thattests of spatialvisualization
involve either the rotationor transformationof a mentalobject.
This study is based on the categorizationexpressedby McGee (1979), Connor
and Serbin (1980), and Kersh and Cook (1979). Spatial visualization is distinguished from spatialorientationtasks by identifying what is to be moved; if the
task suggests thatall or partof a representationbe mentallymoved or altered,it is
considereda spatialvisualizationtask. By this definition,Form Boards and Card
Rotations(Ekstrom,French,& Harmon,1976) and the Space Relationsportionof
the DAT (Bennett,Seashore,& Wesman, 1973) are examples of tests of this class
of spatial skills.
Spatial orientationtasks do not requirementally moving an object. Only the
perceptual perspective of the person viewing the object is changed or moved.
McGee (1979) statedthat spatialorientationtasks "involve the comprehensionof
the arrangementof elements within a visual stimulus patternand the aptitudeto
remain unconfusedby the changing orientationin which a spatial configuration
may be presented"(p. 909). This means thatspatialorientationitems suggest that
the person understanda representationor a change between two representations.
There is not uniformagreementamong researchersabouteither the term or the
classification of spatial orientationtasks. For example, Linn and Petersen(1985)
classified the Rod and FrameTest (Witkin,Dyk, & Faterson, 1962) as a spatial
perceptiontest, the CardRotationsTest (Ekstromet al., 1976) as a spatialrotation
test, and the Hidden Figures Test (Ekstromet al., 1976) as a spatialvisualization
test. McGee (1979) discussed both the Rod and FrameTest and the Hidden Figures Test as spatialorientationtests. ETS originallyclassified the Hidden Figures
Test and the GestaltCompletionTest as "closure"tests andthe CardRotationTest
as an orientationtest (Ekstromet al., 1976). Connorand Serbin(1980) found the
Hidden Figures Test, the Gestalt CompletionTest, and the CardRotationTest to
be one factor,which they labeled "closure."
In agreementwith McGee (1979) and Connorand Serbin(1980) the label spatial orientationused in this paperto describethose tasks thatrequirethatthe subject mentallyreadjusther or his perspectiveto become consistentwith a representation of an object presentedvisually. Spatial orientationtasks could involve organizing, recognizing, making sense out of a visual representation,reseeing it or
seeing it from a differentangle, but not mentallymoving the object.
Spatial Orientation
218
By this definition, the Gestalt Completion Test, the Hidden Figures Test
(Ekstromet al., 1976), the Rod and FrameTest (Witkin,Dyk, & Faterson,1962),
optical illusion, and other hidden object puzzles are examples of tests of spatial
orientationskill. (See Tartre,1990, for a more complete treatmentof the classification of spatialskills.)
SKILLSANDMATHEMATICS
SPATIAL
Spatial skills may be intellectually interestingin themselves, but my purpose
here is to attemptto understandpartof the natureof the relationship,if any, between spatialskills andmathematics.Do we use individualspatialskills in specific
and identifiableways to do mathematics? Do spatialskills serve as general indicators of a way of organizing information that may help solve many types of
mathematicsproblems?
Althoughmanyhave speculatedabouthow spatialskills andmathematicsmight
be relatedand much researchlately has investigatedmathematicsproblem-solving processes, few researchershave attemptedto identify specific processes used
to solve mathematicsproblemsthatmightbe relatedto spatialskills. In one longitudinalstudyby FennemaandTartre(1985), middle school studentswith discrepant spatial visualization and verbal skills were asked to draw pictures to solve
mathematicsproblems.When asked to tell about the problem before solving it,
students with low spatial visualization and high verbal skills tended to provide
more detailed verbal descriptions of the relevant informationin the problems.
However, studentswith high spatialvisualizationskill and low verbal skill translated the probleminto a picturebetterand also had more detailedinformationon
the picturefor problemssolved correctly.
The issue of whetherspatialskills are generalindicatorsof a particularway of
mentallyorganizinginformationthatmightbe helpfulin many areasof mathematics has been discussedby a numberof authors.One hypothesisis thatmathematical reasoning and problem solving are "facilitatedby a 'mental blackboard'on
which the activitymay be organizedandthe interrelatednessof components'visualized'"(Anglin, Meyer,& Wheeler,1975,p. 9). Bishop (1980) also theorizedthat
spatialtrainingmight help studentsto organizethe situationwith mentalpictures
during problem solving in mathematics. He proposed that the structureof the
problemmight be understoodthrougha spatial format.The frequentuse of tree
diagrams,Venn diagrams,charts,and other figures to organize informationand
show relationshipsamongcomponentsof a problemdemonstratesthe plausibility
of this hypothesis. Smith (1964) suggested an even more centralrole for spatial
skills in problemsolving. He statedthat
the conceptionof spatialabilitywhich emergesfromrecentresearch...isso all embracingthatone
is led to inquirewhetherthe processof perceivingand assimilating...generalpatternsor configurations(whetherspatialor non-spatial)is not in fact a process of "abstraction"....The
process of
perceiving and assimilating a gestalt...[is] a process of abstraction(abstractingform or structure).... It is possible thatany process of abstractionmay involve in some degree the perception,
retentionin memory,recognitionandperhapsreproductionof a patternor structure.(p. 213-214)
Brainhemisphericresearchersbelieve thattests of spatialskill are indicatorsof
LindsayAnne Tartre
219
a mode of intellectualfunctioningthat may be tied to global processing of problem situationsor patterns(Davidson, 1979;Wheatley,1977; Galyean, 1981). This
body of researchhas establishedthat there are at least two types of logical thinking processes: one type thatis characterizedby step-by-step,deductive,and often
verbal processes and one type that suggests more structural,global, relational,
intuitive,spatial,and inductiveprocesses.Francoand Sperry(1977) identifiedthe
possible roles of these two thoughtprocesses in problem solving by stating that
"nonverbalvisuo-spatial apprehensionseems commonly to precede and support
the sequentialdeductiveanalysis involved in the solution of geometryproblems"
(p. 112).
Lorenz (1968) capturedthese ideas best when he said,
I hold that Gestalt perception of this type is identical with that mysterious function which is
generallycalled "Intuition",and which indubitablyis one of the most importantcognitive faculties of Man [sic]. When the scientist, confrontedwith a multitudeof irregularand apparentlyirreconcilablefacts, suddenly"sees"the generalregularityruling them all, when the explanation
of the hithertoinexplicableall at once "jumpsout"at him [or her] with the suddennessof a revelation, the experience of this happeningis fundamentallysimilar to that other when the hidden
Gestalt in a puzzle-picturesurprisinglystartsout from the confusing backgroundof irrelevant
detail. The Germanexpression:"in die Augen springen"[to springto the eyes], is very descriptive of this process. (p. 176)
These descriptionsof the possible relationshipbetween spatialskills and intuitive structuralunderstandinginvolved in problemsolutionsoundmore like the use
of spatial orientationskills than spatial visualizationskills. Spatial visualization
tasks generally requiremultiple operations, often analytical, and could involve
verbal mediation. Spatialorientationtasks are often difficult to analyze. Verbalization of the process or processes is also difficult,if not impossible. Spatialorientationtasks often requirethe subjectto organizeor make sense out of visual information. For example, in the Gestalt Completion Test, the solution is often describedas appearingall at once as a whole figure (see Figure 1 for sample items).
It requiresthat the subjects orient themselves to see the ink blots as a whole object. Mentalorganizationof the whole structureof the spatialconfigurationis the
importantfeature.Insight is the word often used to describethe solution process.
Whetherthis type of mentalorganizationor apprehensionis relatedto the organizing processes involved in problem solving and spatialorientationskill was the
problemaddressedin this study.The purposeof the studywas to explore the role
of spatialorientationskill in the solutionof mathematicsproblemsand to identify
possible associated gender differences. The part of the study that investigated
genderdifferenceshas been reportedelsewhere (Tartre,1990). This paperfocuses
on thatpartof the study thatattemptedto identify ways in which studentsscoring
high or low in spatial orientationskill behaved differentlyas they solved mathematics problems.
METHOD
Sample
A representativesample of 97 10th-gradestudentsfrom threehigh schools in a
midwesterncity was given a spatialorientationtest and a 10th-grademathematics
220
Spatial Orientation
This is a test of yourabilityto see a whole pictureeven thoughit is not completely drawn.You are to use your imaginationto fill in the missing parts.
Look at each incompletepictureand try to see what it is. On the line under
each picture,write a word or two to describeit.
Try the sample picturesbelow:
1.
1.
2.
2.
(Picture 1 is a flag and picture2 is a hammerhead.)
Fromthe Manualfor Kit of Factor-ReferencedCognitiveTests (p. 27) by R. B. Ekstrom,J. W. French,&
H. H. Harmon,1976. Princeton,NJ:EducationalTesting Service. Copyright@ 1962, 1975 by EducationalTesting Service. Reprintedby permission.
Figure 1. GestaltCompletionTest.
achievementtest (HoughtonMifflin, 1971).
Those studentswho scored in the top or bottomthirdof the distributionon the
GestaltCompletionTest, which was used to measurespatialorientation,were selected to be interviewed.This interviewsampleconsisted of 57 students(27 high
spatialorientationand 30 low spatialorientation).
SpatialOrientationMeasure
The Gestalt CompletionTest (Ekstromet al., 1976) includes 20 items. Part of
the representationof the object is shown, and the subject is to see or mentally
organizethe informationin orderto understandwhatthe completedrepresentation
would be (see Figure 1). This test was chosen to measurespatialorientationskill
because I concludedthatit was the best test to capturethe essence of pure spatial
thoughtas describedabove. That is, the tasks would be solved holistically,it appeared unlikely that verbal or analytic processes would contributeto subjects'
solutions, and the items directlyrequiredthe structuralorganizationof visual informationin orderto make sense out of the partialpictures.This test has not been
used as much as othertests recently in mathematicseducationresearch,possibly
because spatialvisualizationhas been studiedmore andbecauseit is so difficultto
describethe process or processes used to solve it. However,in this study it is pre-
221
LindsayAnne Tartre
cisely that intuitive or insightful spatial organizational process that is under
investigation.
Problem-SolvingInterview
The problem-solvinginterviewconsisted of 10 mathematicsproblems.Each of
the problemscould be solved in more thanone way, andthe informationpresented
in the problemneededto be organizedin some mannerto solve the problem.Seven
problemsconcernedgeometriccontent, and threeproblemsconcernednongeometric content. Five geometric problemswere presentedvisually (using a concrete
or pictorialrepresentationcontainingcomplete informationfor the solution) and
one nonvisually (with written words). The seventh geometric problem was presented using a geoboard-like,dot grid framework(see Figure 2). One of the nongeometricproblemswas presentedvisually and two were presentednonvisually.
Draw a squarethathas areaequal to 2 squareinches using four of the points
below as vertices (corners).
I--1inch--I
Explain how you know thatthe areaof your figure is 2 squareinches.
(Hint given verbally if appropriate:You have used horizontaland vertical
lines. Is there any other alternative?)
Figure 2. Problem6.
222
Spatial Orientation
For one geometricproblempresentedvisually studentswere given a black and
white photographencasedin glass of approximatelyfifty blocks stackedunevenly.
They were asked to estimate and then determinehow many blocks there were.
They were also asked to determinehow many blocks were completely hidden in
the picture.Problem3 (Figure3) is anotherexample of a visually presentedgeometricproblem.The geometricnonvisuallypresentedproblemdescribedthe relationshipsand distancesamong threetowns on a road and askedaboutthe distance
from one of the three towns to a fourthtown. For the nongeometricvisually presentedproblemstudentswere given six cubes with 2, 3, 5, 6, 7, and 9 on the upper
faces. They were askedto use the six numbersto make two 3-digit numbersso that
when one is subtractedfrom the otherthe smallestpositive differenceresults.One
nongeometricnonvisually presentedproblemasked the studentto determinethe
numberof groupingsof fruitthatcould be made using one or more of five different types of fruit.
Withoutcalculating,what do you thinkis the area of the shadedfigure?
I
I<-1 inch--
shaded
ofthe
area
What
isthe
figure
What is the areaof the shadedfigure?
Figure 3. Problem3.
LindsayAnneTartre
223
InterviewProcedures
The studentsin the sample were asked to solve the problemsand explain their
solutions in individualinterviews.The interviewerwas not awareof the student's
spatialorientationskill level duringthe interviewor duringcoding. The problems
were randomlyorderedfor each student. Studentswere asked not to discuss the
problemswith other students.
Duringthe interviewthe studentswere encouragedto talk aboutwhat they were
doing as they attemptedto solve each problem.Questionswere asked duringand
afterproblem solution, when appropriate,to help understandthe student'sthinking processes more fully. In addition, on completion of each problem, students
were asked to look back at what they had done and if possible, to describe their
motivationfor the organizationused to solve the problem.Generally,a balancewas
sought between the need to make the solution process as naturalas possible and
the interviewer'sdesire to obtainthe maximumamountof informationaboutthe
processes used by each student.Each interview was about 1 hour in length. The
interviews were audiotaped,and record-keepingsheets were used duringthe interview for a written account of the student's overt problem-solving activities.
Interviewswere coded using a coding system developed for this study.
Coding Categories
Some coding categories were chosen because it was hypothesized that they
manifestedthe use of spatialorientationskill in specific ways. A shortrationaleis
includedwith the definitionfor the categorieshypothesizedto be relatedto spatial
orientationskill. In addition, some of the categories were identified from overt
behaviorsobservedduringthe interviews.
Categoriesapplicableto all problems.The following categorieswere applicable
to all problems:
Correctanswer indicatedwhetheror not the studentfoundthe correctanswerto
the problemwithouta hint.A hint was only possible for Problem6 (see Figure2).
Done like described whether or not the student indicated having previously
encountereda similarproblem.This statementmight have occurredin responseto
a question asked by the interviewer, or the student could have volunteered the
informationin a comment aboutrememberingor forgettinghow to do it, or having just done one like it in class.
Failure to breakset indicatedwhetheror not the studentdemonstratedan inability to breakthe problemor pictureapartor did not change a mind set thatwould
provide an incompleteor inaccuratesolution.For example, for Problem6 (Figure
2), the student might have drawn shapes involving only horizontal and vertical
lines and then indicatedthat the problemcould not be solved. That studentdemonstrateda failure to break set. It was hypothesizedthat because spatial orienta-
224
SpatialOrientation
tion skill requiresreseeing or looking at somethingin a new way, inflexibility in
breakinga mental set could indicatea lack of spatialorientationskill use.
Mental movementindicatedany evidence that the studentmentally moved objects in the problem.For example, if a picturewas presentthe studentmight have
said "I picked this partup and moved it over here,"indicatingmental movement
of partof the picture.
Misunderstoodproblem indicated whether or not the student demonstrateda
misunderstandingor confusion aboutthe problem.It could have been a misunderstandingthatthe studentcarriedto the end of the solution or a misperceptionthat
was rectifiedbefore the end of the solution process.
Categories applicable to specific problems.The following categories only applied to certainproblems:
Added marksindicatedthe adding of marksor lines to a pictureincluded as a
partof a problem.
Drewpictureindicatedthe drawingof a picture,diagram,chart,or any otheraid
that could provide visual or spatialrelationshipsor organizationfor the solution
process.
Drew relation indicated a markingor drawing with which the student was attemptingto show a mathematicalrelationshipvisuallyratherthanjust to keep track
of a countingprocess.
Estimate error was used for four of the problems, in which the studentswere
askedto estimatethe answerbefore computingit. It was thoughtthatspatialorientationskill might be used in getting some sense of the size of the answer.The estimate errorindicatedthe proportionof differencebetween the correctanswerand
the estimate given by the student.This category was groupedinto two clusters:
analytic,in which given elementsneededto be analyzedand recombined,andperceptual, in which visual apprehensionof the approximatemagnitudeof a shown
figure was required.For applicable problems the estimate errorwas calculated
using the following ratio:
1-
studentresponse
correctanswer
After all the interviews were completed, they were coded on the basis of both
the audio and writtenrecords.With one exception (estimateerror),all categories
were coded dichotomously(behaviorpresentor absent)for each applicableprob-
Lindsay Anne Tartre
225
lem for each student.After the coding was completed, a randomsample of four
interviewswas recodedandintracoderagreementof at least 80%was obtainedfor
each category.
RESULTS
A spatialorientationskill groupby sex analysis of covariance(ANCOVA)was
performed,using mathematicsachievement as the covariate, for each behavior
category by problemtype. Table 1 shows the means and standarddeviations for
the categoriesfor geometricand nongeometricproblemtypes. No significantdifference between spatialorientationskill groupswas found for the numberof correctanswersfor eitherthe geometricor nongeometricproblems.A significantmain
effect for spatial orientationskill level was obtainedfor three categories: failure
to breakset, misunderstoodproblem,and done like. The mean for the low spatial
orientationgroupwas greaterthanthe mean for the high spatialorientationgroup
for failure to breakset for geometric problems,F (1, 52) = 6.48, p < .05, and for
misunderstoodproblemfor nongeometricproblems,F (1, 52) = 4.78, p < .05. A
significant difference favoring the high spatial orientationgroup was found for
done like for nongeometricproblems,F (1, 52) = 12.82, p < .01.
Table 1
Means (StandardDeviations) by Problem Typesfor CategoriesApplicable to All Problems
Geometric problems
Spatialorientationlevel
Nongeometricproblems
Spatialorientationlevel
Category
Low
High
Low
High
Correctanswer
1.97
(1.79)
0.37
(0.72)
3.37
(1.47)
0.57
(0.86)
0.87
(0.94)
2.96
(1.91)
0.37
(0.74)
2.07*
(1.57)
0.93
(0.73)
0 74
(0.81)
1.00
(0.83)
0.27
(0.52)
1.17
(0.91)
0.03
(0.18)
1.60
(0.72)
1.33
(0.68)
0.96
(0.76:
1.00
(0.68:
0.07
(0.27:
1.04
(0.85:
Done like
Failureto breakset
Mentalmovement
Misunderstoodproblem
Note.n = 30 forlowspatialorientation
group.Meansarebasedon 7
group;n = 27 forhighspatialorientation
and3 nongeometric
geometric
problems.
*p < .05. **p < .01.
Table 2 shows the means and standarddeviations for the categories that were
only applicableto specific problems.A significantmain effect for spatialorientation skill level was obtainedfor two categories: estimate error(perceptual)and
drew relation.The mean for the low spatialorientationgroupwas greaterthanthe
mean for the high spatial orientationgroup for estimate error(perceptual),F (1,
52) = 4.87, p < .05. A significant difference favoring the high spatial orientation
group was found for drew relation,F (1, 52) = 9.23, p < .01.
226
Spatial Orientation
Table 2
Means (StandardDeviation) by Categories Applicableto Specific Problems
Category
Total numiber
of problems
Spatialorientationlevel
Low
High
2.63
(1.38)
1.20
2.48
(1.48)
1.56
Added marks
5
Drew picture
2
(0.71)
(0.58)
Drew relation
2
1.23
(0.77)
1.81**
(0.40)
Estimateerror
Analytic
2
Perceptual
2
1.15
(0.49)
1.56
(2.43)
1.32
(0.97)
0.52*
(0.25)
Note:n = 30 forlow spatialorientation
group.
group;n = 27 forhighspatialorientation
*p< .05.**p < .01.
OFRESULTS
DISCUSSION
Spatialorientationskill appearedto be indicatedin several ways wheregeometric contentwas involved. The low spatialorientationgrouphad highermeans for
failureto breakset for geometricproblems.This meant thatthe low spatialorientation group demonstratedless flexibility in changing a formed perceptualmind
set for those geometric problems.This was particularlyapparentfor Problem 6
(Figure 2). Approximately10%of the low spatialorientationstudentswere able
to get the correctanswerto this problembefore the hint was given, whereas41%
of the high spatial orientationstudents found the correct answer on their own.
However,47% of the low spatialorientationand 56% of the high spatialorientation studentswere able to find the solution after the hint was available.The low
spatial orientationstudentsappearedto need and use the hint more than those in
the high spatialorientationgroupto help thembreakthe mind set of horizontaland
verticallines providedby the grid.
Failureto breakset may be relatedto drew relation.The low spatialorientation
groupalso had a lower mean thanthe high spatialorientationgroupfor drew relation. Failureto break set for Problem3 (Figure 3) might have been indicatedby
the student'sinabilityto breakthe shapeinto partsto calculateits area.Studentsin
the high spatialorientationgroupdemonstratedthat they could see a way to analyze the problemmore often than studentsin the low spatialorientationgroupby
addingmarksto divide the figure into geometric shapes that they could measure
(drewrelation).
There was some indicationthat the mental movement category was related to
estimateerrorfor the perceptualproblems.The differencefoundfor estimateerror
indicated that the high spatial orientationgroup was better able to estimate the
approximatemagnitude of the answer for the problems in this cluster. Mental
LindsayAnne Tartre
227
movementfor Problem3 (Figure3) could have been indicatedby the studentwho,
in an attemptto count completed rectangularshapes, said something like "this
piece looks like it fits over here."It is possible thatthat statementmay be describing either one of two very differentmental processes. It may be arguedthat for
some studentsthat statementindicatedmovement of a mental image, which is a
spatialvisualizationskill. However,for otherstudentsit may have representedan
informalmentalassessmentof the size and shapeof a particularpartof the figure,
which is like the task for estimate error(perceptual),a spatialorientationskill.
It is possible that the group of behavior categories just discussed-failure to
breakset, drew relation,estimate error,and mentalmovement-may be different
manifestationsof the same organizationalprocess, particularlyas they apply to
geometriccontentproblems.For geometricproblems,the high spatialorientation
group estimatedthe approximatemagnitudeof figures more accuratelythan the
low spatialorientationgroup (estimateerrorand perhapsmentalmovement),was
less likely to get stuck in an unproductivemind set (failureto breakset), and was
more likely to add marksto show mathematicalrelationships(drewrelation).
The two significantspatialorientationgroupdifferencesfound for nongeometric problems(done like andmisunderstoodproblem)supportthe thesis thatspatial
orientation skill is manifested more broadly than just in geometric or visually
presentedcontexts. It is an importantgoal of problem-solvinginstructionfor students to be able to identify the structureof a new problemand therebyrelate it to
others, alreadysolved, that are structurallylike the new problem.Studentsin the
high spatialorientationgroupindicatedmore often thanstudentsin the low spatial
orientationgroup that they had done a problemlike it before and less often that
they misunderstoodthe problem in nongeometric settings. The combinationof
these two significantdifferencessuggested thatthe high spatialorientationgroup
was betterable to build and link mental structuresby understandingthe problem
and by linking the new problemto previous work.
CONCLUSIONS
The results from this study suggest that spatial orientationskill appearsto be
used in specific and identifiable ways in the solution of mathematicsproblems.
These ways include accuratelyestimatingthe approximatemagnitudeof a figure,
demonstratingthe flexibility to change an unproductivemind set, addingmarksto
show mathematicalrelationships,mentallymoving or assessing the size and shape
of part of a figure, and getting the correct answer without help to a problem in
which a visual frameworkwas provided.
Supportwas also given to the idea thatspatialskill may be a more generalindicatorof a particularway of organizingthoughtin which new informationis linked
to previousknowledge structuresto help make sense of the new material.
Any relationshipbetweenmathematicslearningand spatialskill is dependenton
the specific spatial skill or test employed. The test of spatialorientationthat was
used for this studyhas not been used often for researchin mathematicsand spatial
skills. And yet, the findings based on thattest are intriguing.
228
Spatial Orientation
Attemptingto understandand discuss something like spatial orientationskill,
which is by definition intuitive and nonverbal,is like trying to grab smoke: The
very act of reachingout to take hold of it dispersesit. It could be arguedthat any
attemptto verbalizethe processes involved in spatialthinkingceases to be spatial
thinking.Spatial skill use is mental activity.Any evidence about how it is manifested must be indirect,since we cannotget into people's heads and see whatthey
see in theirmind's eye. Often, the processes involved are not even understoodby
the people experiencingthem.The resultingindirectnessof the researchin this area
does set limits on it but shouldnot curtailit. If spatialskills areimportantto mathematics, then researchersmust find ways to identify and describethe specific roles
that spatialskills play in doing mathematics.
REFERENCES
Anglin, L., Meyer,R., & Wheeler,J. (1975). Therelationshipbetweenspatial ability and mathematics
achievementat thefourth- and sixth-gradelevels (WorkingPaperNo. 115). Madison:Universityof
Wisconsin Researchand DevelopmentCenterfor Cognitive Learning.
Bennett,G. K., Seashore,H. G., & Wesman,A. G. (1973). Differentialaptitudetests: Administrator's
handbook.New York:The Psychological Corporation.
Bishop, A. J. (1980). Spatial abilities and mathematicseducation-a review. EducationalStudies in
Mathematics,11, 257-269.
Connor,J. M., & Serbin,L. A. (1980). Mathematics,visual-spatialability, and sex roles. (Final Report).Washington,DC: National Instituteof Education(DHEW). (ERICDocument Reproduction
Services No. ED 205 305)
Connor,J. M., & Serbin,L. A. (1985). Visual-spatialskill: Is it importantfor mathematics?Can it be
taught?In S. F. Chipman,L. R. Brush,& D. M. Wilson (Eds.), Womenand mathematics:Balancing
the equation.(pp. 151-174). Hillsdale, NJ: LaurenceErlbaumAssociates.
Davidson, P. S. (1979, May). Neurologicalresearchand its implicationson mathematicseducation.
Journal of the VirginiaCouncil of Teachersof Mathematics,pp. 1-8.
Ekstrom,R. B., French,J. W., & Harmon,H. H. (1976). Manualfor kit offactor-referencedcognitive
tests. Princeton,NJ: EducationalTesting Service.
Fennema,E., & Sherman,J. (1977). Sex-relateddifferencesin mathematicsachievement,spatialvisualizationand affective factors.AmericanEducationResearchJournal, 14(1), 51-71.
Fennema,E., & Tartre,L. A. (1985). The use of spatialvisualizationin mathematicsby girls and boys.
Journalfor Research in MathematicsEducation,16(3), 184-206.
Franco,L., & Sperry,R. W. (1977). Hemispherelateralizationfor cognitive processing of geometry.
Neuropsychologia,15(1), 107-113.
Galyean,B. C. (1981). The brain,intelligence,andeducation:Implicationsfor gifted programs.Roeper
Review,4(1), 6-9.
HoughtonMifflin Company.(1971). Testsof academicprogress (FormS, mathematics,grades9-12).
Boston: Author.
Kersh,M. E., & Cook, K. H. (1979, August). Improvingmathematicsability and attitude,a manual.
Seattle:MathematicsLearningInstitute,Universityof Washington.
Linn, M. C., & Petersen,A. C. (1985). Emergenceandcharacterizationof genderdifferencesin spatial
ability:A meta-analysis.ChildDevelopment,56, 1479-1498.
Lorenz, K. Z. (1968). The role of gestalt perceptionin animal and humanbehaviour.In L. L. Whyte
(Ed.).Aspectsofform: A symposiumonform in natureand art (pp. 157-178). New York:American
Elsevier PublishingCompany.
McGee, M. G. (1979). Human spatial abilities: Psychometric studies and environmental, genetic,
hormonal,and neurologicalinfluences. Psychological Bulletin,86 (5), 889-918.
Smith, I. M. (1964). Spatial ability: Its educational and social significance. London: University of
LondonPress.
LindsayAnne Tartre
229
Tartre,L. A. (1984). The role of spatialorientationskill in the solution of mathematicsproblemsand
associated sex-related differences. (The University of Wisconsin-Madison). Dissertations Abstracts International46A: 94-95; July 1985.
Tartre,L. A. (1990). Spatial skills, gender& mathematics.In E. Fennema& G. Leder(Eds.), Mathematics and gender: Influenceson teachers and students.New York:Teachers'College Press.
Wheatley, G. H. (1977). The right hemisphere'srole in problemsolving. ArithmeticTeacher,25(2),
36-39.
Witkin, H. A., Dyk, R. B., & Faterson,H. F. (1962). Psychological differentiation.New York: Wiley.
AUTHOR
LINDSAY ANNE TARTRE,Associate Professor,Departmentof Mathematics,CaliforniaState University-Long Beach, 1250 Bellflower Blvd., Long Beach, CA 90840