angles

ANGLES

SELECTED REFERENCES The American Statistician (1947-). Amstat News (1974-). Chance (1988-), with Springer-Verlag. Journal of Business & Economic Statistics (1983-). Journal of Computational and Graphical Statistics (1992-), with the Institute of Mathematical Statistics, and the Interface Foundation of North America. Journal of Educational Statistics (1976-) with the American Educational Research Association. Journal of the American Statistical Association GASA)(1888-). Statistics Teacher Network with the National Council of Teachers of Mathematics. Stats-The Magazine for Students of Statistics (1989-). Technometrics (1959-), with the American Society for Quality Control. LILY E. CHRIST

ANGLES Defined mathematically in distinct but related ways. For example, an angle can be considered the figure formed by two rays extending from the same point. Angle also can be defined as the amount of turning necessary to bring one line or plane into coincidence with or parallel to another. The research on angles suggests a balanced approach to curriculum and teaching that includes, and more important, integrates, various conceptual frameworks for the angle concept.

MATHEMATICAL BACKGROUND An angle can be defined as the union of two rays, a and b, with the same initial point, P P is called the vertex of the angle, the rays are called the arms. The rays can be made to coincide by a rotation about P, which determines the size of the angle between a and b. That is, each arm defines a direction and the angle size is the measure of the difference of these directions. The orientation of this difference can be positive in the clockwise (as in surveying, or in "turtle math," described later) or counterclockwise (as in traditional Euclidean geometry). Methods of measuring the size of angles are based on the division of a circle. The most common are measurement by degrees and by arc length. A degree is 3~O of the circumference of a circle. The length of an arc, a, between two radii is proportional to the angle between them and to the length of the radius. The following proportion holds. Circumference (21Tr) : Arc: : 360° : Angle subtended at center. So, if the radius of a circle is known, the length of an arc on

the circumference can be used to measure the corresponding angle at the center, afro The unit, called a radian J is thus the angle at the center of a circle subtended by an arc of length equal to the radius of the circle (Gellert et al. 1977). On a unit circle, these proportions result in the measurement of the angle formed by two radii being equal to the measure of the length of the arc subtended by the angle. This notion can be generalized beyond the unit circle to a generalized arc, which can be defined as "a mapping of a directed line segment into the unit circle such that (1) it is single-valued, and (2) it is measure preserving for subsegments of length less than 'IT." Then, the generalized angle POQ can be defined as the "union of two rays OP and OQ with a common origin together with a directed generalized arc whose initial point lies on OP and whose end point lies on OQ and the signed measure of a directed generalized angle is the signed length of its directed generalized arc" (Allendoerfer 1965, 86-87). This definition is particularly useful in the study of trigonometric functions. Angles are classified according to their measure. Categories by degrees are right (90°), acute « 90°), obtuse (> 90°), straight (180°), reflex (> 180°), and full (360°). An interesting historical question remains: Why 360? The Babylonians created this measurement. They used a sexagesimal number system (base 60) rather than a decimal system (base 10). They knew that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360. With a base of 60, 6 times 60 was a natural choice. Further, this coincided with their knowledge of astronomy of the last century B.C.: The year was divided into six equal parts, each having sixty plus some fractional number of days. It is common to take degrees as a measure of rotation. One of the first sets of geometry books published in America, John Playfair's Elements of Geometry (1806), presents Euclid's Elements in a form that "renders them most useful" Gones 1944, 4). Of the three original geometry works published in America, Playfair's was the only one to introduce the notion of an angle as formed by a rotation. He does this to present an alternate proof, presenting a definition of angle that, "if while one extremity of a line remains fixed at A, the line turns about that point from AB to AG, it is said to describe the angle BAG contained by the line AB and AG" Gones 1944, 7). Angle concepts and angle measures play essential roles in analyzing and solving problems in a wide

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ANGLES

variety of geometric situations. For example, they aid in determining (a) heights that cannot be directly measured (of a redwood, for example), (b) a right angle from a hidden point, or (c) the altitude of the sun. To be able to solve such problems, students must learn about various aspects of the angle concept. To do so, students should work through the following. Awareness of turns (rotations); for example, walking paths through school halls following directions such as "forward 10 steps, right turn"; or Logo turtle geometry activities. Awareness of corners in surroundings and geometric figures; comparing angle size physically.

Thus, abstraction of shape is not a perceptual abstraction of a physical property, but is the result of a coordination of children's actions. Children "can only 'abstract' the idea of such a relation as equality on the basis of an action of equalization, the idea of a straight line from the action of following by hand or eye without changing direction, and the idea of an angle from two intersecting movements" (p. 43). Furthermore, the child constructs his representation of angle not as two intersecting lines, but rather as the "outcome of a pair of movements (of eye and hand) which conjoin" (p. 31). In fact, "Euclidean shapes . . . are at least as much abstracted from particular actions as they are from the object to which the actions relate" (p. 31).

Estimation of turns and angles with nonstandard units. Estimation of turns and angles with standard units. Connection of turn and angle as the intersection of two rays with a common endpoint (middle school). This might involve, for example, discovering that a turn (e.g., 120°) and the angle produced by that turn (60°) are complementary. Integration of regions, or the open convex part of a plane bounded by a pair of rays with a common endpoint, with rotations (beginning of secondary school). Exploration of relationships between angles and other geometric figures, such as parallel lines, leading to mathematical abstraction of these relationships. Formulation of a single, mathematically rigorous definition of angle.

THEORETICAL BACKGROUND Piaget's psychological definition contains aspects of both definitions of angle, although it emphasizes the second one. Piaget claimed that ". . . a general distinction is drawn between two major classes of shape, curvilinear or without angles, and rectilinear or with angles, though subdivisions within the two classes are hardly noticed. . . . There is no doubt that it is the analysis of the angle which marks the transition from topological relationships to the perception of Euclidean ones. It is not the straight line itself which the child contrasts with round shapes, but rather the conjunction of straight lines which go to form an angle" (Piaget and Inhelder 1967).

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RESEARCH One does not have to look far for examples of children's difficulty with the angle concept. Many students believe that an angle must have one horizontal ray, a right angle is an angle that points to the right; the angle sum of a quadrilateral is the same as its area, and two right angles in different orientations are not equal in measure (Clements and Battista 1992). This body of research indicates that students have many different ideas about what an angle is. These ideas include "a shape," a side of a figure, a tilted line, an orientation or heading, a corner, a turn, and a union of two lines (Clements and Battista 1990). Angles are not salient properties of figures to students (Clements et al. in press; Mitchelmore 1989). When copying figures, students do not always attend to the angles. Students also hold many different schemes regarding not only the angle concept, but also the size of angles. They frequently relate the size of an angle to the lengths of the line segments that form its sides, the tilt of the top line segment, the area enclosed by the triangular region defined by the drawn sides, the length between the sides (from points, sometimes but not always, equidistant from the vertex), the proximity of the two sides, or the turn at the vertex (Clements and Battista 1989). Intermediate grade students often possess one of two schemes for measuring angles. In the "45-90 schema," slanted lines are associated with 45° turns; horizontal and vertical lines with 90° turns. In the "protractor schema," inputs to turns are based on usage of a protractor in "standard" position (thus, to have a turtle at home position turn left 45°, students

ANGLES

might use an input of 135°, which corresponds to a protractor's reading when its base is horizontal) (Kieran et al. 1986). Moreover, such schemes may be resistant to change, especially through, for example, textbook definitions and examples. When they think, people do not use definitions of concepts, but rather concept images-a combination of all the mental pictures and properties that have been associated with the concept (Vinner and Hershkowitz 1980). Such images can even be adversely affected by inappropriate instruction. For example, the fact that, for many students, the concept image of an obtuse angle having a horizontal ray might result from the limited set of examples they see in texts and a "gravitational factor" (i.e., a figure is "stable" only if it has one horizontal side, with the other side ascending).

PEDAGOGICAL APPROACHES Students who know a correct verbal description of a concept but also have a specific visual image or concept image associated strongly with the concept may have difficulty applying the verbal description correctly. Instead, educators need to help them build a robust concept of angle. One approach, researched by Mitchelmore (Mitchelmore in press), uses multiple concrete analogies. To develop the concept of an angle, the teacher must provide: 1. Practical experience in various situations, leading to an understanding of angular relationships in each individual situation. 2. Angle subconcepts develop when the common features of superficially similar situations are recognized. These types of situations (e.g., turns, slopes, meetings, bends, directions, corners, opening) form different angle contexts. 3. Superficial differences between contexts initially hinder children's recognition of such common features. 4. Less obvious similarities between contexts gradually become apparent, and angle subconcepts begin to emerge. 5. A full angle concept emerges when the same common features are recognized in all angle contexts. Research on teaching activities based on these ideas revealed that most elementary-age students understood physical relations. Turns, or rotations, were a difficult concept to understand in concrete physical contexts. Other research supports the importance of

integration of all types. Some children have only understood turns and angles in a meaningful way after months of work (Clements et al. in press). Initially, they gained experience with physical rotations, especially rotations of their own bodies. During the same time, they gained limited knowledge of assigning numbers to certain turns, initially by establishing benchmarks. A synthesis of these two domains (turnas-body-motion and turn-as-number) constituted a critical juncture in learning about turns for many elementary students. An implication of the Piagetian position and the emphasis on turns is that dynamic computer environments might be useful. Turns (and angles) are critical to the view of shapes as paths, and the intrinsic geometry of paths is closely related to real world experiences such as walking. Computer games have been found to be marginally effective at promoting learning of angle estimation skills (Bright 1985). More extensive and promising are several research projects investigating the effects of Logo's turtle graphics experience on students' conceptualizations of angle, angle measure, and rotation. In one study, for example, responses of intermediate grade students in a control group were more likely to reflect little knowledge of angle or common language usage, whereas the responses of the Logo students indicated more generalized and mathematically oriented conceptualizations (including angle as rotation and as a union of two lines/segments/rays) (Clements and Battista 1989). A large group of studies has reported similar findings, although in some situations, benefits do not emerge until more than a year of Logo experience (Clements and Battista 1992). So, having these experiences over several years of elementary school is recommended. Logo experiences may foster some misconceptions of angle measure, including viewing it as the angle of rotation along the path (e.g., the exterior angle in a polygon) or the degree of rotation from the vertical (Clements and Battista 1989; Clements et aI., in press). In addition, such experiences do not replace previous misconceptualizations of angle measure (Davis 1984). For example, students' misconceptions about angle measure and difficulties coordinating the relationships between the turtle's rotation and the constructed angle have persisted for several years during their elementary schooling, especially if not properly guided by their teachers (Clements and Battista 1992). In general, however, Logo experience appears to facilitate understanding of angle measure. Logo children's conceptualizations

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APPLICATIONS FOR THE CLASSROOM, OVERVIEW

of a "larger angle" are more likely to reflect mathematically correct and coherent ideas. If activities emphasize the difference between the angle of rotation and the angle formed as the turtle traced a path, misconceptions regarding the measure of rotation and the measure of the angle may be avoided. For example, students will understand that when the Logo turtle goes forward 50, right 120, forward 50, it draws an angle with a measurement of 60°. To understand angles, students must understand the various aspects of the angle concept. They must overcome difficulties with orientation, discriminate angles as critical parts of geometric figures, and construct and represent the idea of turns, among others. Furthermore, they must construct a high level of integration between these aspects. This is a difficult task that is best begun in the elementary and middle school years, as children deal with corners of figures, comparing angle size, and turns. At the beginning of secondary school, regions should be integrated with rotations. The formulation of a single, mathematically rigorous definition of angle should follow (Mitchelmore 1989). See also Geometry Instruction; Logo; Trigonometry

SELECTED REFERENCES Allendoerfer, Carl B. "Angles, Arcs, and Archimedes." Mathematics Teacher 58(Feb. 1965):82-88. Bright, George. "What Research Says: Teaching Probability and Estimation of Length and Angle Measurements through Microcomputer Instructional Games." School Science and Mathematics 85(1985):513-522. Clements, Douglas H., and Michael T. Battista. "The Effects of Logo on Children's Conceptualizations of Angle and Polygons." Journal for Research in Mathematics Education 21 (1990):356-371. - - - . "Geometry and Spatial Reasoning." In Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). Douglas A. Grouws, ed. New York: Macmillan, 1992. - - - . "Learning of Geometric Concepts in a Logo Environment." Journal for Research in Mathematics Education 20( 1989) :450-467. Clements, Douglas H., Michael T. Battista, Julie Sarama, and Sudha Swaminathan. "Development of Turn and Turn Measurement Concepts in a Computer-based Instructional Unit." Educational Studies in Mathematics (in press). Davis, Robert B. Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex, 1984. Gellert, Walter, H. Kiistner, M. Hellwish, and H. Kastner, eds. VNR Concise Enclyclopedia of Mathematics. New York: Van Nostrand Reinhold, 1977.

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Jones, Phillip S. "Early American Geometry." Mathematics Teacher 37(1)(1944):3-11. Kieran, Carolyn, Joel Hillel, and Stanley Erlwanger. "Perceptual and Analytical Schemas in Solving Structured Turtle-Geometry Tasks." In Proceedings of the Second Logo and Mathematics Educators Conference (pp. 154-161). Celia Hoyles, Richard Noss, and Roseland Sutherland, eds. London, England: University of London, 1986. Mitchelmore, Michael C. "The Development of Children's Concepts of Angle." In Proceedings of the Thirteenth Conference of the International Group for the Psychology of Mathematics Education (pp. 304-311). Gerard Vergnaud, Janine Rogalski, and Michele Artigue, eds. Paris, France: Paris University, 1989. - - - . "The Development of Pre-angle Concepts." In New Directions in Research on Geometry and Visual Thinking. Annette R. Baturo, ed. Brisbane, Australia: Queensland University Press, in press. Piaget, Jean, and Barbel Inhelder. The Child's Conception of Space. New York: Norton, 1967. Vinner, Shlomo, and Rina Hershkowitz. "Concept Images and Common Cognitive Paths in the Development of some Simple Geometrical Concepts." In Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 177-184). Robert Karplus, ed. Berkeley, CA: Lawrence Hall of Science, University of California, 1980. DOUGLAS H. CLEMENTS MICHAEL T. BATTISTA JULIE SARAMA

APPLICATIONS FOR THE CLASSROOM, OVERVIEW Components of the curriculum consisting of real-world problems solved by mathematics. Changes in mathematics education during the last half of the twentieth century have been caused by many factors, including changes in technology, new points of view in mathematics itself, and the growth of research in mathematics education. But no factor has had a greater influence than changes in the applications of mathematics. What mathematics has significant applications? The historical position was that "applied mathematics" consists of classical analysis, such as calculus, differential equations both ordinary and partial, integral equations, and special functions. Given this position, the content of school mathematics also was defined: schools must aim for and teach all the prerequisites for calculus, and nothing else matters as much. Thus, the secondary schools emphasized plane and solid geometry, two years of algebra, trigonometry, and analytic geometry. A concentra-