space geometry, instruction

SPACE GEOMETRY, INSTRUCTION

Education Development Center. MathFINDER Mac1toshVersion 1.1. Armonk, NY: LearningTeam, 1993. Edwards, Lois A. Data Insights. Pleasantville, NY: :unburst, 1991. Electronic Portfolio. Jefferson City, MO: Scholastic, 995. Fey, James T., and M. Kathleen Heid. Math Exploation Tholkit. College Park, MD: University of Maryland, 985. Hoffer, Alan. Algebra Xpresser. Acton, MA: Bradford, 991. Jackiw, Nicholas. The Geometer's Sketchpad. Berkeley, :A: Key Curriculum, 1991. Laborde, Jean-Marie. Cabri Geometry. France: Unirersite de Grenoble 1, 1990. Learning in Motion. Measurement in Motion, 1994. Learning Technology Center. Jasper Software. 'l"ashville, TN: Vanderbilt University, 1992. Mathematics TestBuilder. Acton, MA: Bradford, 1993. Morgan Media. Learner Profile. Pleasantville, NY: iunburst, 1994. O'Brien, Thomas C. The King's Rule. Pleasantville, 'N: Sunburst, 1985. - - - . Safari Search. Pleasantville, NY: Sunburst, L991. Olive, John, and Leslie P. Steffe. CandyBar. (Also :alled TIMA: Bars). Acton, MA: Bradford, 1994. Probability Theory. Hanover, NH: True Basic, 1988. Professional Systems International. StatExplorer. Glenview, IL: Scott, Foresman, 1993. Rosenberg, Jon. Math Connections. Pleasantville, NY: SunburstlWings for Learning, 1993. Schwartz, Judah, L., and MichalYerushalmy. The Geometric Supposers. Pleasantville, NY: Sunburst, 1985. Thompson, Patrick. Blocks. San Diego, CA: San Diego State University, 1993. Wolfram, Stephen. Mathematica. Champaign, IL: Wolfram, 1991. ZHONGHONGJlANG

SPACE GEOMETRY, INSTRUCTION Traditionally, the investigation of solid geometric shapes such as prisms, pyramids, cones, cylinders, and spheres, as well as the mathematical space in which these shapes are considered. The major curricular goals for three-dimensional (3D) geometry can be grouped into three clusters: (a) how objects, groups of objects, and space itself are organized or structured; (b) how objects in 3D space are measured; and (c) how objects are described. Moreover, consistent with the recommendations of the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics 1989),

another important, but often-neglected, general goal is the development of spatial visualization skills. As students progress through the grades, their structuring, measuring, describing, and visualizing must become increasingly sophisticated, especially for those who must deal with the demands of subjects such as 3D analytic geometry, calculus, and many topics in science and engineering. The study of space geometry, like other areas of geometry, can beneficially utilize the van Hiele theory (Clements and Battista 1992) as a framework for choosing appropriate instructional tasks and gauging the development of students' thinking. This theory suggests that, in grades K-14, students' thinking about 3D geometry should progress from reasoning holistically about solids based on their appearance (level 1), to identifying the components of solids and informally describing solids in terms of their properties (level 2), to logically relating properties of solids and thereby hierarchically classifying them (level 3), and finally to reasoning formally and constructing proofs within an axiomatic system for 3D geometry (level 4).

STRUCTURE The structure of a 3D object refers to how its components are put together, how they are spatially organized and interrelated. The van Hiele theory suggests that structural knowledge can be developed by having students first become familiar with common solids as wholes, next by examining their components and investigating how these components are spatially related, and finally by describing these relationships as properties of solids. Although reaching van Hiele level 2 in 3D geometry is a reasonable goal for the upper elementary grades, and perhaps level 3 by eighth grade, these levels are rarely reached in traditional curricula. For instance, Gutierrez, Jaime, and Fortuny (1991) found that although 38 out of 41 preservice teachers and lout of 9 eighth graders had completely acquired van Hiele levell, only 11 of the teachers and none of the eighth graders had completely acquired level 2, and 3 of the teachers and none of the eighth graders had completely acquired level 3. A commonly employed instructional activity is to have students learn about the components of 3D shapes by identifying their faces, vertices, and edges, as well as tracing and comparing their faces. The goal of these activities should be for students to start mentally constructing the structure of solids as they decompose them into their constituent parts. When

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Bourgeois (1986) utilized such an approach with third graders, the students' ability to select solids that were "pictured" by nets improved. However, although students who employed the analytic strategy of counting faces of the solids or nets were quite successful in associating solids to nets, very few students employed such an analytic strategy and thus could have been considered to be moving into van Hiele level 2. A similar, but mentally more demanding, activity is to have students make nets for solids. This requires students not only to examine the parts of solids but how they are related. Jean Piaget's research indicates that children have difficulty imagining nets for solids before the ages of 8 or 9 (Bourgeois 1986). Another way to have students investigate the structure of 3D shapes is for them to examine the cross sections of the shapes, another level 2 activity. Davis (1973) implemented a 25-minute training session in which sixth- , eighth- , and tenth-grade students cut styrofoam solids and were asked questions about the cross sections. He then had the students select the correct drawing of cross sections of the solids. The number of students (out of 30) who scored 87.5% or higher was 2 for sixth grade, 15 for eighth grade, and 21 for tenth grade. The author suggested age 14 or higher as the mastery level for the concept of cross-sectioning geometric solids. Battista and Clements' (1996) work with third graders gives us further insight into how students think about the components of solids. As these students enumerated edges of polyhedra that they had built out of sticks and connectors, they frequently miscounted the edges (which they noticed because their counts were inconsistent with previous counts or the counts of other students). It was not until they organized their counts (for example, they might count the edges on the top, then the bottom, then the lateral sides of a prism) that they were able to correctly enumerate the edges. These students imposed a different spatial structure on the parts of the solid as a result of attempting to count them. In fact, the counting task forced students to reflect on, and many times alter, their structuring of the objects. That reflection on counting acts should help students spatially structure a situation is consistent with Piaget's theory that the mental construction of shapes consists of the coordination and abstraction of the child's mental actions in dealing with the shapes (Piaget and Inhelder 1956). Thus, the key to helping students structure 3D shapes and move to higher van Hiele levels is the presentation of problems in which students concretely investigate parts of shapes and reflect on how these parts are related.

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MEASUREMENT Measurement, particularly of volume and sur· face area, has traditionally been a major curricula] emphasis in 3D geometry. The research indicates: however, that students are not learning this material effectively. As we shall see later, a major reason for this seems to be students' meager understanding of the structure of 3D objects. A typical volume problem presents students with a diagram of a 3D cube array and asks them to determine its volume or the number of cubes in it. BenChaim, Lappan, and Houang (1985) found that less than 50% of middle grade students could solve such problems. The results of the second National Assessment of Educational Progress (NAEP) showed that less than 40% of 17-year-olds could solve problems of this type (Hirstein 1981). Typical errors that students in grades 5-8 made on these problems include counting the cube faces shown in the diagram, and perhaps doubling that number, and counting the number of cubes showing in the diagram, and perhaps doubling that number (Ben-Chaim, Lappan, and Houang 1985). BenChaim, Lappan, and Houang suggested that (a) the counting-the-faces error was due to students dealing with the picture strictly as a two-dimensional object; (b) students who counted cubes showed an awareness of the three-dimensionality of the depicted object; (c) students who did not double their counts did not seem to visualize the hidden portions of the objects; and (d) many students had difficulty relating isometriclike drawings to the rectangular solids they represent. They concluded that these errors were related to students' visualization ability. Hirstein (1981), on the other hand, attributed many of the errors students make on this type of problem to confusion between volume and surface area. In fact, Enochs and Gabel (1984) found that preservice elementary teachers' understanding of these two concepts was poor and that these students did not really understand either concept. Indeed, whereas 77% of these students were certain that the volume of a rectangular solid could be found by multiplying the length, width, and height, only 44% were certain that the volume was equal to the area of the base times the height, and only 58% were certain that the volume could be found by counting the number of unit cubes that filled the solid. The situation was worse for surface area, with only 34% knowing that it could be determined by counting the number of unit squares on the outside of the solid. Apparently, many of these students had memorized formulas for obtaining quantities that they did not

SPACE GEOMETRY, INSTRUCTION

understand. Because of students' fuzzy notions about both concepts, and because the concepts have traditionally been taught one after the other, it is easy to see why students' ideas about them are confused and intermingled. The authors suggest that these concepts be taught with a "hands-on" approach and that formulas be mentioned only at the end of exploration. Battista and Clements (1996) provide a more elaborate description of students' solution strategies and errors in dealing with 3D cube arrays. Their data suggest that many students are unable to enumerate the cubes in a 3D array because they cannot coordinate the separate views of the array and integrate them to construct one coherent mental model of the array. Up to 64% of third graders and 33% of fifth graders showed evidence of this type of thinking. Ben-Chaim, Lappan, and Houang (1985) reported data suggesting that about 39% of fifth-eighth graders might be using this type of thinking. Eventually, as students become capable of coordinating views, they restructure the arrays. Those who complete a global restructuring utilize layering strategies. Those in transition utilize strategies that indicate a local, piece-by-piece, structuring.

were more easily interpreted than layer plans, orthogonal views, and coordinates. Grade 9 and grade 11 students were more successful than grade 7 students in interpreting layer plans, orthogonal views, and coordinate descriptions. Although other research suggests that middle school students often have difficulty with perspective drawings, 85% of this study's grade 7 students correctly interpreted such drawings. Ben-Chaim, Lappan, and Houang (1989) gave students three weeks of instruction on representing solids and constructing solids from drawings. To assess their ability to communicate spatial information, students were given 3D models made up of 10 small cubes taped together and were to devise a description of what their building looked like. Before instruction, only 26% of students gave correct building descriptions; after instruction, 83% gave correct descriptions. Again, thinking about the van Hiele levels can be helpful. Students' work with describing 3D shapes must be coordinated with their conceptual development concerning those objects. For example, to expect students who are at van Hiele level 1 to give verbal descriptions that describe the properties of shapes, rather than the vague, holistic descriptions characteristic of this level, simply encourages students to rotely memorize such descriptions.

DESCRIPTION One of the major goals of 3D instructional units should be to promote the development of the basic concepts and language needed to reflect on and communicate about spatial relationships in 3D environments. Such knowledge, along with visualization skills, is not only critical in mathematics but in science and engineering. Students must explore various ways to pictorially represent solids and learn to correctly visualize 3D configurations described in diagrams and different types of building instructions. Mitchelmore (1980) found four stages of sophistication in children's drawing of 3D solids. In the first stage, the figure is drawn as a single, orthogonal face or as a general holistic outline. In the second stage, several visible faces are drawn but not properly depicted or connected, or hidden faces are included. In stage 3, drawings attempt to represent a single point of view and to depict depth. In stage 4, parallel edges on the object are represented by parallel lines in the drawing. The sophistication of students' drawings increased with grade level. Cooper and Sweller (1989) examined students' ability to interpret various representations of simple three-dimensional objects. For all ages studied, objects, perspective drawings, and verbal descriptions

CURRICULAR RECOMMENDATIONS At the primary level, students need experiences building with 3D objects. For instance, they can build a house or castle out of wooden geometric solids. Or one student might make a simple figure out of connecting cubes and another student could try to duplicate it. As students progress, they can also start thinking about the parts of shapes. For instance, students can use ink to stamp the faces of solids on cards; other students can try to identify which solids made the cards. At the intermediate level, students can build polyhedra from sticks and connectors. They can identify the shapes of shadows made by geometric solids and explore notions of perspective by examining how various shapes look from different angles. They can make boxes that contain a given number of cubes. They can predict how many cubes fit in boxes, then check their answers with cubes. Given various kinds of diagrams of cube configurations, including pictures of several orthogonal views, they can build the configurations with connecting cubes. Toward the end of this period, they can be introduced to the

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formal concept of volume (initially, students should focus on enumerating cube arrays). As students move into junior high, they can continue many ofthe activities started in the intermediate grades. But they also can deal with more abstract situations. They need to make sense of volume in wider contexts. For instance, they need to understand the relationship between linear and volume measurements and to make sense out of situations that involve rational, not just whole number, measurements. At the high school level, students should study 3D coordinate systems, 3D transformations, cross sections of solids, and more complex volumes, such as those for nonpolygonal solids and shapes that must be decomposed. Problems that require interpretation of increasingly complex drawings should continue to be given to students. As students move through the grade levels, they should be presented with tasks that encourage more and more visualization. However, because visualization ability differs widely among individuals, at all grade levels some students will require the presence of actual physical materials. It is essential to recognize that visual ability can be improved by giving students many opportunities to manipulate physical objects and to reflect on these manipulations.

lected Solid Figures." Journal for Research in Mathematics Education 4(1973): 132-140. Enochs, Larry G., and Dorothy L. Gabel. "Preservice ElementaryTeachers' Conceptions ofVolume." School Science and Mathematics 84(1984):670-680. Gutierrez, Angel, Adela Jaime, and Jose M. Fortuny. "An Alternative Paradigm to Evaluate the Acquisition of the van Hiele Levels." Journal for Research in Mathematics Education 22(1991):237-251. Hirstein, James J. "The Second National Assessment in Mathematics: Area and Volume." Mathematics Teacher 74(1981):704-708. Mitchelmore, Michael C. "Prediction of Developmental Stages in the Representation of Regular Space Figures." Journal for Research in Mathematics Education 11 (1980):83-93. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston,VA:The Council, 1989. - - - . Principles and Standards for School Mathematics. Reston, VA: The Council, 2000. Parzysz, Bernard. "Knowing vs Seeing. Problems of the Plane Representation of Space Geometry Figures." Educational Studies in Mathematics 19(1988):79-92. Piaget, Jean, and Barbel Inhelder. The Child's Conception of Space. London, England: Routledge and Kegan Paul, 1956. MICHAEL T. BATTISTA MICHAEL MIKUSA DOUGLAS H. CLEMENTS

See also van Hiele Levels

SELECTED REFERENCES Battista, Michael T., and Douglas H. Clements. "Students' Understanding of Three-dimensional Rectangular Arrays of Cubes." Journal for Research in Mathematics Education 27(3)(1996):258-292. Ben-Chaim, David, Glenda Lappan, and Richard T. Houang. "Adolescents' Ability to Communicate Spatial Information: Analyzing and Effecting Students' Performance." Educational Studies in Mathematics 20(1989): 121-146. - - - . "Visualizing Rectangular Solids Made of Small Cubes: Analyzing and Effecting Students' Performance." Educational Studies in Mathematics 16(1985): 389-409. Bourgeois, Roger D. "Third Graders' Ability to Associate Foldout Shapes with Polyhedra." Journal for Research in Mathematics Education 17(1986):222-230. Clements, Douglas H., and Michael T. Battista. "Geometry and Spatial Reasoning." In Handbook of Research on Mathematics Teaching. Edited by Douglas A. Grouws. New York: Macmillan, 1992, pp. 420-464. Cooper, Martin, and John Sweller. "Secondary School Students' Representations of Solids." Journal for Research in Mathematics Education 20(1989):202-212. Davis, Edward J. "A Study of the Ability of School Pupils to Perceive and Identify the Plane Sections of Se-

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SPACE MATHEMATICS The achievements of the space program owe much to mathematics. In the study of the solar system, large numbers are rounded and scientific notation is used. Calculations of center of mass, center of pressure, and trajectories are used in studying rocketry. Geometry is used in calculating the orbital paths of satellites and planets and in the measurement of the earth. One of the first measurements of the Earth was made by a Greek mathematician, Eratosthenes, over two thousand years ago. He noted that at noon on the first day of summer the sun appeared to be directly overhead at the city of Syene in Egypt. At the same time, in the city ofAlexandria a pole would cast a shadow such that the angle of the Sun's rays to the pole measured about 7.2°. By setting up a proportion using the given angle and the known distance between the two cities, 5,000 stadia 7.2°

circumference of the Earth 360°