ONE POINT OF VIEW: Spatial Sense and Mathematics LearningAuthor(s): Grayson H. WheatleySource: The Arithmetic Teacher, Vol. 37, No. 6, FOCUS ISSUE: SPATIAL SENSE (FEBRUARY1990), pp. 10-11Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193836 .

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ONE POINT OF VIEW

Spatial Sense and Mathematics Learning Grayson H. Wheatley

review of United States

school mathematics reveals that rules, procedures, and an- alytic reasoning dominate the curric-

■ SPATIAL SENSE ■

ulum, whereas little attention is given to spatial visualization. What should be the roles of imagery and spatial vi- sualization in school mathematics? How does spatial visualization relate to the learning of mathematics?

Although mathematics is often con- sidered to be a collection of facts and procedures, current thinking in the field supports a view of mathematics as the activity of constructing patterns and relationships (National Council of Teachers of Mathematics, Commis- sion on Standards for School Mathe- matics 1989; National Research Coun- cil 1989). The construction of patterns and relationships requires what the authors of the Curriculum and Evalu- ation Standards for School Mathe- matics (Standards) (NTCM 1989) call spatial sense. As mathematics curric- ula shift from an emphasis on pro-

cedures to a focus on relationships, spatial sense takes on increased im- portance (Skemp 1987).

The Standards authors use the term spatial sense to refer to what has been known by a variety of other labels from spatial visualization , spatial rea- soning, spatial perception, and visual imagery to mental rotations. I suggest we think of spatial sense in terms of imagery. According to Kosslyn (1983), imagery involves the construction, re- presentation, and transformation of self-generated images. Let us examine these three components of imagery.

Construction of images. The im- ages we construct from viewing ob- jects, reading a passage, or just re- flecting are influenced by what we know. Constructing an image is not a process of taking a mental picture, but we are influenced by our knowledge of the subject. Our images may be concrete and limiting or dynamic and abstract. They are also unique.

Re-presentation. Once an image is constructed it does not remain in con- sciousness but must be "called up" when needed. We do this by re-pre- senting the images, and here again the re-presentation may not take the same form as the originally constructed im- age. For example, if you are asked to determine the number of windows in your house, you will likely re-present

several images of your house as you mentally walk around it and "look" at the windows.

Transformation. Transforming im- ages is a dynamic process. In mathe- matics, spatial sense often requires that an image be transformed in some way, for example, transforming a rhombus to a square or performing a rotational transformation. In general, spatial sense involves using mental images. We look at objects, read a passage, or just reflect and form im- ages that are often transformed as is done in mathematics. For example, in comparing two shapes we may rotate one to facilitate the comparison.

Examples of spatial sense include comparing two shapes in different ori- entations (mentally rotating one of them to make the comparison easier), recognizing the symmetry of certain shapes, relating rectangles and paral- lelograms (maintaining lengths of sides but changing the measure of the angles), drawing a diagram to assist in solving a word problem, and relating angle measure to the relative position of two rays with a common endpoint.

Spatial sense plays a major role in mathematical reasoning. Moses (1980) showed that good problem solvers use mental imagery more than poor prob- lem solvers. Turner (1982) found that the ability to transform and compare

Grayson Wheatley is a professor of mathemat- ics education at Florida State University, Tal- lahassee, FL 32306. He coordinates mathemat- ics education, develops instructional materials, and conducts research on the learning of math- ematics.

Ю ARITHMETIC TEACHER

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images mentally was highly related to success in integral calculus.

Spatial sense is useful in numerical as well as geometric settings. Just as segments can form geometric pat- terns, numbers can form numerical patterns. Number lines, coordinate axes, and graphs are ways of thinking about numerical patterns. Fractions can be thought of in terms of geomet- ric regions. Functions can be concep- tualized dynamically; a change in one variable results in a corresponding change in a second variable. More comprehensively, the formation of mathematical relationships generally involves some type of self-con- structed image.

Constructing new relationships and solving nonroutine problems are situ- ations in which imagery is particular- ly valuable. As certain mathematical ideas become more familiar, we may develop our own method that allows us to perform operations efficiently. Initially we need to understand the re- lationship, and to that end spatial sense may be quite helpful.

Reflection on mathematical activity reveals a close interplay between im- agery and analysis. Spatial and ana- lytic reasoning play complementary roles in mathematical thought, and neither should be emphasized at the expense of the other (Brumby 1982). Fischbein (1987) speaks of the dialec- tical nature of images and concepts. For example, a diagram would likely be helpful in solving the following problem:

Square tables that can seat one per- son on a side are placed next to each other, forming two rows of ta- bles. If 26 people can sit around the rectangular shape formed, how many square tables are used?

Conceiving of a diagram and then drawing it requires imagery. Solutions to many mathematics problems such as the foregoing require imagery as well as procedures and logic.

Giving all students an opportunity to develop their spatial sense is impor- tant. Using tangrams, pentominoes, and tessellations; drawing and dis- cussing geometric patterns; and con- structing and comparing three-dimen- sional figures are examples of such activities. Additional activities for

building number sense are presented in this issue. In designing school mathematics activities, efforts should be made to build spatial activities into each topic of the curriculum. The abil- ity mentally to rotate and compare im- ages of geometric shapes can be im- proved by well-designed activities in a variety of settings (Battista, Wheat- ley, and Talsma 1982; Ben-Chaim, Lappan, and Houang 1989).

While calls for change ring loud and strong, we must recognize the enor- mity of the task of building spatial sense into school mathematics curric- ula and evaluation. In most schools, memorization of facts and computa- tional proficiency are main goals, in part because standardized tests em- phasize these skills. For many years the Council has worked to increase at- tention to geometry in the elementary school curriculum. The geometry in elementary school programs tends to focus on naming and identifying sim- ple geometric shapes. The paucity of spatial activities in textbooks is clear evidence that spatial sense is not con- sidered essential in elementary school mathematics. For spatial sense to be- come a central topic of mathematics programs, its power in the learning of mathematics must be recognized and tests changed. Spatial sense is indis- pensable in giving meaning to our mathematical experience. Facilitating the needed changes will require a ded- icated and concentrated effort on the part of all of us. Let's give spatial sense a prominent place in school mathematics programs.

References

Battista, Michael, Gray son Whatley, and Gary Talsma. "The Importance of Spatial Visual- ization and Cognitive Development for Ge- ometry Learning in Preservice Elementary Teachers." Journal for Research in Mathe- matics Education 13 (March 1982):332-40.

Ben-Chaim, David, Glenda Lappan, and Rich- ard Houang. "The Role of Visualization in Middle School Mathematics Curriculum." Focus on Learning Problems in Mathematics 11 (Winter-Spring 1989):49-60.

Brumby, Mark. "Consistent Differences in Cognitive Styles Shown for Qualitative Bio- logical Problem Solving." British Journal of Educational Psychology 52 (June 1982):244- 57.

Fischbein, Efarim. Intuition in Science and Mathematics: An Educational Approach. Dordrecht, Holland: D. Reidel Publishing Co., 1987.

Kosslyn, Stephen. Ghosts in the Mind's Ma-

chine. New York: W. W. Norton & Co., 1983.

Moses, Barbara. The Relationship between Vi- sual Thinking Tasks and Problem-solving Performance. Paper presented at the Annual Meeting of the American Educational Re- search Association, Boston, April 1980.

National Council of Teachers of Mathematics, Commission on Standards for School Mathe- matics. Curriculum and Evaluation Stan- dards for School Mathematics. Reston, Va.: The Council, 1989.

National Research Council. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press, 1989.

Skemp, Richard. The Psychology of Mathemat- ics Learning. Hillsdale, N.J.: Lawrence Erl- baum Associates, 1987.

Turner, Kenneth. An Investigation of the Role of Spatial Performance , Learning Styles, and Kinetic Imagery in the Learning of Calculus. Ph.D. diss., Purdue University, 1982. W

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