Spatial Orientation Skill and Mathematical Problem Solving

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Journal for Research in Mathematics Education 1990, Vol. 21, No. 3, 216-229

SPATIAL ORIENTATION SKILL AND MATHEMATICAL PROBLEM SOLVING

LINDSAY ANNE TARTRE, California State University, Long Beach

The purpose of this study was to explore the role of spatial orientation skill in the solution of mathematics problems. Fifty-seven tenth-grade students who scored high or low on a spatial orientation test were asked to solve mathematics problems in individual interviews. A group of specific behaviors was identified in geometric settings, which appeared to be manifestations of spatial orientation skill. Spatial orientation skill also appeared to be involved in understanding the problem and linking new problems to previous work in nongeometric settings.

For a long time researchers have attempted to determine why some students learn mathematics or are able to solve mathematics problems better than others do. Some research has focused on the processes used to solve mathematics problems. Other researchers have tried to identify factors and skills related to doing mathematics. Among the skills found to be related to mathematics learning and achievement are spatial skills (Fennema & Sherman, 1977; McGee, 1979).

The literature contains a great deal of discussion about the possible relationship between spatial skills and mathematics. Many studies have found spatial skills to be positively correlated with measures of mathematics performance (Connor & Serbin, 1985; Fennema & Sherman, 1977). But what are spatial skills, and how and why are those skills related to mathematics?

SPATIAL SKILLS In general, spatial skills are considered to be those mental skills concerned with

understanding, manipulating, reorganizing, or interpreting relationships visually. In his review of spatial factors, McGee (1979) distinguished two major types of spatial skills:

A plethora of factor analytic studies since the 1930's have provided strong and consistent support for the existence of at least two distinct spatial abilities-visualization and orientation. (p. 909)

A factor analytic study by Connor and Serbin (1980) supported the distinction summarized by McGee. They found that

the visual-spatial tests divided into two factors.... The Cube Comparison Test, Paper Folding Test, Card Rotations Test, and DAT Space Relations Test had high loading on...[one factor]. This factor represents what ETS has referred to as the "spatial visualization" subdivision of visual-spatial ability.... [The second spatial factor] had high loadings on Hidden Patterns, Gestalt Completion, and Card Rotations. This factor represents what ETS has referred to as the "closure" subdivision of visual-spatial ability. (p. 27)

A different categorization was proposed by Linn and Petersen (1985), who identified three spatial ability categories: spatial perception, mental rotation, and spatial visualization. They defined spatial visualization as "those spatial ability tasks

This article is based on the author's dissertation at the University of Wisconsin-Madison under the direction of Elizabeth Fennema.

Lindsay Anne Tartre 217

which involve complicated multi-step manipulations of spatially presented information" (p. 1484). They distinguished it from the other two categories "by the possibility of multiple solution strategies" (p. 1484). Linn and Petersen felt that the only common characteristic of spatial visualization tasks was that the solutions required more 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 Hidden Figures Test (Ekstrom, French, & Harmon, 1976).

McGee (1979) felt that spatial visualization tasks "all involve the ability to mentally manipulate, rotate, twist, or invert a pictorially presented stimulus object" (p. 893). Kersh and Cook (1979) also suggested that tests of spatial visualization involve either the rotation or transformation of a mental object.

This study is based on the categorization expressed by McGee (1979), Connor and Serbin (1980), and Kersh and Cook (1979). Spatial visualization is distinguished from spatial orientation tasks by identifying what is to be moved; if the task suggests that all or part of a representation be mentally moved or altered, it is considered a spatial visualization task. By this definition, Form Boards and Card Rotations (Ekstrom, French, & Harmon, 1976) and the Space Relations portion of the DAT (Bennett, Seashore, & Wesman, 1973) are examples of tests of this class of spatial skills.

Spatial orientation tasks do not require mentally moving an object. Only the perceptual perspective of the person viewing the object is changed or moved. McGee (1979) stated that spatial orientation tasks "involve the comprehension of the arrangement of elements within a visual stimulus pattern and the aptitude to remain unconfused by the changing orientation in which a spatial configuration may be presented" (p. 909). This means that spatial orientation items suggest that the person understand a representation or a change between two representations.

There is not uniform agreement among researchers about either the term or the classification of spatial orientation tasks. For example, Linn and Petersen (1985) classified the Rod and Frame Test (Witkin, Dyk, & Faterson, 1962) as a spatial perception test, the Card Rotations Test (Ekstrom et al., 1976) as a spatial rotation test, and the Hidden Figures Test (Ekstrom et al., 1976) as a spatial visualization test. McGee (1979) discussed both the Rod and Frame Test and the Hidden Fig- ures Test as spatial orientation tests. ETS originally classified the Hidden Figures Test and the Gestalt Completion Test as "closure" tests and the Card Rotation Test as an orientation test (Ekstrom et al., 1976). Connor and Serbin (1980) found the Hidden Figures Test, the Gestalt Completion Test, and the Card Rotation Test to be one factor, which they labeled "closure."

In agreement with McGee (1979) and Connor and Serbin (1980) the label spatial orientation used in this paper to describe those tasks that require that the subject mentally readjust her or his perspective to become consistent with a representation of an object presented visually. Spatial orientation tasks could involve organizing, recognizing, making sense out of a visual representation, reseeing it or seeing it from a different angle, but not mentally moving the object.

218 Spatial Orientation

By this definition, the Gestalt Completion Test, the Hidden Figures Test (Ekstrom et al., 1976), the Rod and Frame Test (Witkin, Dyk, & Faterson, 1962), optical illusion, and other hidden object puzzles are examples of tests of spatial orientation skill. (See Tartre, 1990, for a more complete treatment of the classification of spatial skills.)

SPATIAL SKILLS AND MATHEMATICS

Spatial skills may be intellectually interesting in themselves, but my purpose here is to attempt to understand part of the nature of the relationship, if any, between spatial skills and mathematics. Do we use individual spatial skills in specific and identifiable ways to do mathematics? Do spatial skills serve as general indicators of a way of organizing information that may help solve many types of mathematics problems?

Although many have speculated about how spatial skills and mathematics might be related and much research lately has investigated mathematics problem-solving processes, few researchers have attempted to identify specific processes used to solve mathematics problems that might be related to spatial skills. In one longi- tudinal study by Fennema and Tartre (1985), middle school students with discrep- ant spatial visualization and verbal skills were asked to draw pictures to solve mathematics problems. 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 information in the problems. However, students with high spatial visualization skill and low verbal skill trans- lated the problem into a picture better and also had more detailed information on the picture for problems solved correctly.

The issue of whether spatial skills are general indicators of a particular way of mentally organizing information that might be helpful in many areas of mathematics has been discussed by a number of authors. One hypothesis is that mathematical reasoning and problem solving are "facilitated by a 'mental blackboard' on which the activity may be organized and the interrelatedness of components 'visu- alized'" (Anglin, Meyer, & Wheeler, 1975, p. 9). Bishop (1980) also theorized that spatial training might help students to organize the situation with mental pictures during problem solving in mathematics. He proposed that the structure of the problem might be understood through a spatial format. The frequent use of tree diagrams, Venn diagrams, charts, and other figures to organize information and show relationships among components of a problem demonstrates the plausibility of this hypothesis. Smith (1964) suggested an even more central role for spatial skills in problem solving. He stated that

the conception of spatial ability which emerges from recent research...is so all embracing that one is led to inquire whether the process of perceiving and assimilating...general patterns or configu- rations (whether spatial or non-spatial) is not in fact a process of "abstraction"....The process of perceiving and assimilating a gestalt...[is] a process of abstraction (abstracting form or structure).... It is possible that any process of abstraction may involve in some degree the perception, retention in memory, recognition and perhaps reproduction of a pattern or structure. (p. 213-214)

Brain hemispheric researchers believe that tests of spatial skill are indicators of


a mode of intellectual functioning that may be tied to global processing of problem situations or patterns (Davidson, 1979; Wheatley, 1977; Galyean, 1981). This body of research has established that there are at least two types of logical thinking processes: one type that is characterized by step-by-step, deductive, and often verbal processes and one type that suggests more structural, global, relational, intuitive, spatial, and inductive processes. Franco and Sperry (1977) identified the possible roles of these two thought processes in problem solving by stating that "nonverbal visuo-spatial apprehension seems commonly to precede and support the sequential deductive analysis involved in the solution of geometry problems" (p. 112).

Lorenz (1968) captured these ideas best when he said, I hold that Gestalt perception of this type is identical with that mysterious function which is generally called "Intuition", and which indubitably is one of the most important cognitive facul- ties of Man [sic]. When the scientist, confronted with a multitude of irregular and apparently ir- reconcilable facts, suddenly "sees" the general regularity ruling them all, when the explanation of the hitherto inexplicable all at once "jumps out" at him [or her] with the suddenness of a reve- lation, the experience of this happening is fundamentally similar to that other when the hidden Gestalt in a puzzle-picture surprisingly starts out from the confusing background of irrelevant detail. The German expression: "in die Augen springen" [to spring to the eyes], is very descrip- tive of this process. (p. 176) These descriptions of the possible relationship between spatial skills and intui-

tive structural understanding involved in problem solution sound more like the use of spatial orientation skills than spatial visualization skills. Spatial visualization tasks generally require multiple operations, often analytical, and could involve verbal mediation. Spatial orientation tasks are often difficult to analyze. Verbali- zation of the process or processes is also difficult, if not impossible. Spatial orientation tasks often require the subject to organize or make sense out of visual information. For example, in the Gestalt Completion Test, the solution is often described as appearing all at once as a whole figure (see Figure 1 for sample items). It requires that the subjects orient themselves to see the ink blots as a whole object. Mental organization of the whole structure of the spatial configuration is the important feature. Insight is the word often used to describe the solution process.

Whether this type of mental organization or apprehension is related to the organizing processes involved in problem solving and spatial orientation skill was the problem addressed in this study. The purpose of the study was to explore the role of spatial orientation skill in the solution of mathematics problems and to identify possible associated gender differences. The part of the study that investigated gender differences has been reported elsewhere (Tartre, 1990). This paper focuses on that part of the study that attempted to identify ways in which students scoring high or low in spatial orientation skill behaved differently as they solved mathematics problems.

METHOD Sample

A representative sample of 97 10th-grade students from three high schools in a midwestern city was given a spatial orientation test and a 10th-grade mathematics


This is a test of your ability to see a whole picture even though it is not completely drawn. You are to use your imagination to fill in the missing parts.

Look at each incomplete picture and try to see what it is. On the line under each picture, write a word or two to describe it.

Try the sample pictures below:

1. 2.

1. 2.

(Picture 1 is a flag and picture 2 is a hammer head.) From the Manual for Kit of Factor-Referenced Cognitive Tests (p. 27) by R. B. Ekstrom, J. W. French, &

H. H. Harmon, 1976. Princeton, NJ: Educational Testing Service. Copyright @ 1962, 1975 by Educational Test- ing Service. Reprinted by permission.

Figure 1. Gestalt Completion Test.

achievement test (Houghton Mifflin, 1971). Those students who scored in the top or bottom third of the distribution on the

Gestalt Completion Test, which was used to measure spatial orientation, were se- lected to be interviewed. This interview sample consisted of 57 students (27 high spatial orientation and 30 low spatial orientation).

Spatial Orientation Measure The Gestalt Completion Test (Ekstrom et al., 1976) includes 20 items. Part of

the representation of the object is shown, and the subject is to see or mentally organize the information in order to understand what the completed representation would be (see Figure 1). This test was chosen to measure spatial orientation skill because I concluded that it was the best test to capture the essence of pure spatial thought as described above. That is, the tasks would be solved holistically, it appeared unlikely that verbal or analytic processes would contribute to subjects' solutions, and the items directly required the structural organization of visual information in order to make sense out of the partial pictures. This test has not been used as much as other tests recently in mathematics education research, possibly because spatial visualization has been studied more and because it is so difficult to describe the process or processes used to solve it. However, in this study it is pre-


cisely that intuitive or insightful spatial organizational process that is under investigation.

Problem-Solving Interview The problem-solving interview consisted of 10 mathematics problems. Each of

the problems could be solved in more than one way, and the information presented in the problem needed to be organized in some manner to solve the problem. Seven problems concerned geometric content, and three problems concerned nongeometric content. Five geometric problems were presented visually (using a concrete or pictorial representation containing complete information for 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 nongeometric problems was presented visually and two were presented nonvisually.

Draw a square that has area equal to 2 square inches using four of the points below as vertices (corners).

I--1 inch-- I

Explain how you know that the area of your figure is 2 square inches. (Hint given verbally if appropriate: You have used horizontal and vertical

lines. Is there any other alternative?)

Figure 2. Problem 6.


For one geometric problem presented visually students were given a black and white photograph encased in glass of approximately fifty blocks stacked unevenly. They were asked to estimate and then determine how many blocks there were. They were also asked to determine how many blocks were completely hidden in the picture. Problem 3 (Figure 3) is another example of a visually presented geometric problem. The geometric nonvisually presented problem described the relationships and distances among three towns on a road and asked about the distance from one of the three towns to a fourth town. For the nongeometric visually presented problem students were given six cubes with 2, 3, 5, 6, 7, and 9 on the upper faces. They were asked to use the six numbers to make two 3-digit numbers so that when one is subtracted from the other the smallest positive difference results. One nongeometric nonvisually presented problem asked the student to determine the number of groupings of fruit that could be made using one or more of five different types of fruit.

Without calculating, what do you think is the area of the shaded figure?

I<-1 inch-- I

What is the area of the shaded figure

What is the area of the shaded figure?

Figure 3. Problem 3.


Interview Procedures The students in the sample were asked to solve the problems and explain their

solutions in individual interviews. The interviewer was not aware of the student's spatial orientation skill level during the interview or during coding. The problems were randomly ordered for each student. Students were asked not to discuss the problems with other students.

During the interview the students were encouraged to talk about what they were doing as they attempted to solve each problem. Questions were asked during and after problem solution, when appropriate, to help understand the student's thinking 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 motivation for the organization used to solve the problem. Generally, a balance was sought between the need to make the solution process as natural as possible and the interviewer's desire to obtain the maximum amount of information about the processes used by each student. Each interview was about 1 hour in length. The interviews were audiotaped, and record-keeping sheets were used during the interview for a written account of the student's overt problem-solving activities. Interviews were coded using a coding system developed for this study.

Coding Categories Some coding categories were chosen because it was hypothesized that they

manifested the use of spatial orientation skill in specific ways. A short rationale is included with the definition for the categories hypothesized to be related to spatial orientation skill. In addition, some of the categories were identified from overt behaviors observed during the interviews.

Categories applicable to all problems. The following categories were applicable to all problems:

Correct answer indicated whether or not the student found the correct answer to the problem without a hint. A hint was only possible for Problem 6 (see Figure 2).

Done like described whether or not the student indicated having previously encountered a similar problem. This statement might have occurred in response to a question asked by the interviewer, or the student could have volunteered the information in a comment about remembering or forgetting how to do it, or having just done one like it in class.

Failure to break set indicated whether or not the student demonstrated an inability to break the problem or picture apart or did not change a mind set that would provide an incomplete or inaccurate solution. For example, for Problem 6 (Figure 2), the student might have drawn shapes involving only horizontal and vertical lines and then indicated that the problem could not be solved. That student demonstrated a failure to break set. It was hypothesized that because spatial orienta-


tion skill requires reseeing or looking at something in a new way, inflexibility in breaking a mental set could indicate a lack of spatial orientation skill use.

Mental movement indicated any evidence that the student mentally moved ob- jects in the problem. For example, if a picture was present the student might have said "I picked this part up and moved it over here," indicating mental movement of part of the picture.

Misunderstood problem indicated whether or not the student demonstrated a misunderstanding or confusion about the problem. It could have been a misunderstanding that the student carried to the end of the solution or a misperception that was rectified before the end of the solution process.

Categories applicable to specific problems. The following categories only ap- plied to certain problems:

Added marks indicated the adding of marks or lines to a picture included as a part of a problem.

Drew picture indicated the drawing of a picture, diagram, chart, or any other aid that could provide visual or spatial relationships or organization for the solution process.

Drew relation indicated a marking or drawing with which the student was attempting to show a mathematical relationship visually rather than just to keep track of a counting process.

Estimate error was used for four of the problems, in which the students were asked to estimate the answer before computing it. It was thought that spatial orientation skill might be used in getting some sense of the size of the answer. The estimate error indicated the proportion of difference between the correct answer and the estimate given by the student. This category was grouped into two clusters: analytic, in which given elements needed to be analyzed and recombined, and perceptual, in which visual apprehension of the approximate magnitude of a shown figure was required. For applicable problems the estimate error was calculated using the following ratio:

student response 1- correct answer

After all the interviews were completed, they were coded on the basis of both the audio and written records. With one exception (estimate error), all categories were coded dichotomously (behavior present or absent) for each applicable prob-


lem for each student. After the coding was completed, a random sample of four interviews was recoded and intracoder agreement of at least 80% was obtained for each category.

RESULTS

A spatial orientation skill group by sex analysis of covariance (ANCOVA) was performed, using mathematics achievement as the covariate, for each behavior category by problem type. Table 1 shows the means and standard deviations for the categories for geometric and nongeometric problem types. No significant difference between spatial orientation skill groups was found for the number of correct answers for either the geometric or nongeometric problems. A significant main effect for spatial orientation skill level was obtained for three categories: failure to break set, misunderstood problem, and done like. The mean for the low spatial orientation group was greater than the mean for the high spatial orientation group for failure to break set for geometric problems, F (1, 52) = 6.48, p < .05, and for misunderstood problem for nongeometric problems, F (1, 52) = 4.78, p < .05. A significant difference favoring the high spatial orientation group was found for done like for nongeometric problems, F (1, 52) = 12.82, p < .01.

Table 1 Means (Standard Deviations) by Problem Types for Categories Applicable to All Problems

Geometric problems Nongeometric problems

Spatial orientation level Spatial orientation level

Category Low High Low High Correct answer 1.97 2.96 1.00 1.33

(1.79) (1.91) (0.83) (0.68) Done like 0.37 0.37 0.27 0.96

(0.72) (0.74) (0.52) (0.76: Failure to break set 3.37 2.07* 1.17 1.00

(1.47) (1.57) (0.91) (0.68: Mental movement 0.57 0.93 0.03 0.07

(0.86) (0.73) (0.18) (0.27: Misunderstood problem 0.87 0 74 1.60 1.04

(0.94) (0.81) (0.72) (0.85: Note. n = 30 for low spatial orientation group; n = 27 for high spatial orientation group. Means are based on 7 geometric and 3 nongeometric problems. *p < .05. **p < .01.

Table 2 shows the means and standard deviations for the categories that were only applicable to specific problems. A significant main effect for spatial orientation skill level was obtained for two categories: estimate error (perceptual) and drew relation. The mean for the low spatial orientation group was greater than the mean for the high spatial orientation group 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.


Table 2 Means (Standard Deviation) by Categories Applicable to Specific Problems

Total numiber Spatial orientation level

Category of problems Low High Added marks 5 2.63 2.48

(1.38) (1.48) Drew picture 2 1.20 1.56

(0.71) (0.58) Drew relation 2 1.23 1.81**

(0.77) (0.40) Estimate error

Analytic 2 1.15 1.32 (0.49) (0.97)

Perceptual 2 1.56 0.52* (2.43) (0.25)

Note: n = 30 for low spatial orientation group; n = 27 for high spatial orientation group. *p < .05. **p < .01.

DISCUSSION OF RESULTS

Spatial orientation skill appeared to be indicated in several ways where geometric content was involved. The low spatial orientation group had higher means for failure to break set for geometric problems. This meant that the low spatial orientation group demonstrated less flexibility in changing a formed perceptual mind set for those geometric problems. This was particularly apparent for Problem 6 (Figure 2). Approximately 10% of the low spatial orientation students were able to get the correct answer to this problem before the hint was given, whereas 41% of the high spatial orientation students found the correct answer on their own. However, 47% of the low spatial orientation and 56% of the high spatial orientation students were able to find the solution after the hint was available. The low spatial orientation students appeared to need and use the hint more than those in the high spatial orientation group to help them break the mind set of horizontal and vertical lines provided by the grid.

Failure to break set may be related to drew relation. The low spatial orientation group also had a lower mean than the high spatial orientation group for drew relation. Failure to break set for Problem 3 (Figure 3) might have been indicated by the student's inability to break the shape into parts to calculate its area. Students in the high spatial orientation group demonstrated that they could see a way to analyze the problem more often than students in the low spatial orientation group by adding marks to divide the figure into geometric shapes that they could measure (drew relation).

There was some indication that the mental movement category was related to estimate error for the perceptual problems. The difference found for estimate error indicated that the high spatial orientation group was better able to estimate the approximate magnitude of the answer for the problems in this cluster. Mental


movement for Problem 3 (Figure 3) could have been indicated by the student who, in an attempt to count completed rectangular shapes, said something like "this piece looks like it fits over here." It is possible that that statement may be describ- ing either one of two very different mental processes. It may be argued that for some students that statement indicated movement of a mental image, which is a spatial visualization skill. However, for other students it may have represented an informal mental assessment of the size and shape of a particular part of the figure, which is like the task for estimate error (perceptual), a spatial orientation skill.

It is possible that the group of behavior categories just discussed-failure to break set, drew relation, estimate error, and mental movement-may be different manifestations of the same organizational process, particularly as they apply to geometric content problems. For geometric problems, the high spatial orientation group estimated the approximate magnitude of figures more accurately than the low spatial orientation group (estimate error and perhaps mental movement), was less likely to get stuck in an unproductive mind set (failure to break set), and was more likely to add marks to show mathematical relationships (drew relation).

The two significant spatial orientation group differences found for nongeometric problems (done like and misunderstood problem) support the thesis that spatial orientation skill is manifested more broadly than just in geometric or visually presented contexts. It is an important goal of problem-solving instruction for students to be able to identify the structure of a new problem and thereby relate it to others, already solved, that are structurally like the new problem. Students in the high spatial orientation group indicated more often than students in the low spatial orientation group that they had done a problem like it before and less often that they misunderstood the problem in nongeometric settings. The combination of these two significant differences suggested that the high spatial orientation group was better able to build and link mental structures by understanding the problem and by linking the new problem to previous work.

CONCLUSIONS

The results from this study suggest that spatial orientation skill appears to be used in specific and identifiable ways in the solution of mathematics problems. These ways include accurately estimating the approximate magnitude of a figure, demonstrating the flexibility to change an unproductive mind set, adding marks to show mathematical relationships, mentally moving 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 framework was provided.

Support was also given to the idea that spatial skill may be a more general indi- cator of a particular way of organizing thought in which new information is linked to previous knowledge structures to help make sense of the new material.

Any relationship between mathematics learning and spatial skill is dependent on the specific spatial skill or test employed. The test of spatial orientation that was used for this study has not been used often for research in mathematics and spatial skills. And yet, the findings based on that test are intriguing.


Attempting to understand and discuss something like spatial orientation skill, which is by definition intuitive and nonverbal, is like trying to grab smoke: The very act of reaching out to take hold of it disperses it. It could be argued that any attempt to verbalize the processes involved in spatial thinking ceases to be spatial thinking. Spatial skill use is mental activity. Any evidence about how it is manifested must be indirect, since we cannot get into people's heads and see what they see in their mind's eye. Often, the processes involved are not even understood by the people experiencing them. The resulting indirectness of the research in this area does set limits on it but should not curtail it. If spatial skills are important to mathematics, then researchers must find ways to identify and describe the specific roles that spatial skills play in doing mathematics.

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AUTHOR LINDSAY ANNE TARTRE, Associate Professor, Department of Mathematics, California State Uni-

versity-Long Beach, 1250 Bellflower Blvd., Long Beach, CA 90840

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Spatial Orientation Skill and Mathematical Problem Solving