| All students will develop their spatial sense through experiences which enable them to recognize, visualize, represent, and transform geometric shapes and to apply their knowledge of geometric properties, relationships, and models to other areas of mathematics and to the physical world. |
Traditionally, geometry in schools has been taught as the prime example of a formal deductive system. While this view of the content is important, its domination has led to the exclusion of other, less readily formalized topics and applications. Geometry instruction should not be limited to formal deductive proof and simple measurement activities, but should include the study of geometric transformations, analytic geometry, topology, and the connection of geometry with algebra and other areas of mathematics. Posing and solving problems in this more richly defined geometry allows students to use geometric intuition to develop more generic mathematical problem-solving skills.
Although the levels are not completely separate and the transitions are complex, the model is very useful for characterizing levels of children's thinking. One particularly pertinent finding that shows up consistently in the research is that appropriately targeted instruction is critical to children's movement through these levels. Stagnation at early levels is the frequent result of a geometry curriculum that dwells on identification of shapes and their properties.
It is not difficult to conceive of a curriculum that adheres to the van Hiele model. By virtue of living in a three-dimensional world, having dealt with space for five years, children enter school with a remarkable amount of intuitive geometric knowledge. The geometry curriculum should take advantage of this intuition in expanding and formalizing the students' knowledge. In early elementary school, a rich, qualitative, hands-on study of geometric objects helps young children develop spatial sense and a strong intuitive grasp of geometric properties and relationships. Eventually they develop a comfortable vocabulary of appropriate geometric terminology. In the middle school years, students should begin to use their knowledge in a more analytical manner to solve problems, make conjectures, and look for patterns and generalizations. Gradually they develop the ability to make inferences and logical deductions based on geometric relationships. In high school, the study of geometry expands to include coordinate geometry, trigonometry, and both inductive and deductive reasoning.
The study of geometry should make abundant use of experiences that require active student involvement. Constructing models, folding paper cutouts, using mirrors, pattern blocks, geoboards, and tangrams, and creating geometric computer graphics all allow opportunities for students to learn by doing, to reflect upon their actions, and to communicate their observations and conclusions. These activities and others of the same type should be used to achieve the goals in the seven specific areas of study that comprise this standard and which are described below.
In their study of spatial relationships, young students should make regular use of concrete materials in hands-on activities designed to develop their understanding of objects in space. The early focus should be the description of the location and orientation of objects in relation to other objects. Additionally, students can begin an exploration of symmetry, congruence, and similarity. Older students should study the two-dimensional representations of three-dimensional objects by sketching shadows, projections, and perspectives.
In the study of the properties of geometric figures, students deal explicitly with the identification and classification of standard geometric objects by the number of edges and vertices, the number and shapes of the faces, the acuteness of the angles, and so on. Cut-and-paste constructions of paper models, combining shapes to form new shapes and decomposing complex shapes into simpler ones are useful exercises to aid in exploring shapes and their properties. As their studies continue, older students should be able to perform classic constructions with straight edges and compasses as well as with appropriate computer software and to formulate good mathematical definitions for the common shapes, eventually being able to make deductions and solve problems using their properties.
The standard geometric transformations include translation, rotation, reflection, and scaling. They are central to the study of geometry and its applications in that these manipulations of figures offer the most natural approach to understanding congruence, similarity, symmetry, and other geometric relationships. Younger children should have a great deal of experience with flips, slides, and turns of concrete objects, figures made on geoboards, and paper-and-pencil figures. Older students should be able to use more formal terminology and procedures for determining the results of the standard transformations. An added benefit of experience gained with simple and composite transformations is the mathematical connection that older students can make to functions and function composition.
Coordinate geometry provides another strong connection between geometry and algebra. Students can be informally introduced to coordinates as early as kindergarten by locating entries in tables and finding points on maps and grids. In later elementary grades, they can learn to plot figures on a coordinate plane, and still later, study the effects of various transformations on the coordinates of the points of two- and three-dimensional figures. High-school students should be able to represent geometric transformations algebraically and interpret algebraic equations geometrically.
Measurement and geometry are interrelated, and understanding the geometry of measurement is necessary for the understanding of measurement. In elementary school, students should learn the meaning of such geometric measures as length, area, volume and angle measure and should be actively involved in the measurement of those attributes for all kinds of two- and three-dimensional objects, not simply the most standard, uniform ones. Throughout school, they should use measurement activities to reinforce their understanding of geometric properties. Eventually all students should understand such principles as the quadratic change in area and cubic change in volume that occurs with a linear change of scale. Another of the interdependencies between geometry and measurement is seen in high school when students learn to use trigonometry to make indirect measurements.
Geometric modeling is a powerful problem-solving skill and should be modeled for and frequently used by students. A simple diagram, such as a pie-shaped graph, a force diagram in physics, or a dot-and-line illustration of a network, can illuminate the essence of a problem and allow geometric intuition to aid in the approach to a solution. Visualization skills and understanding will both improve as students are encouraged to make such models.
The relationship between geometry and deductive reasoning originated with the ancient Greek philosophers, and remains an important part of the study of geometry. A key ingredient of deductive reasoning is being able to recognize which statements have been justified and which have been assumed without proof. This is an ability which all students should develop, in all areas, not just geometry, or even just mathematics! At first, deductive reasoning is informal, with students inferring new properties or relationships from those already established, without detailed explanations at every step. Later, deduction becomes more formal as students learn to state all permissible assumptions at the beginning of a proof and all subsequent statements are systematically justified from what has been assumed or proved before. The idea of deductive proof should not be confused with the specific two-column format of proof found in most geometry textbooks. The object of studying deductive proof is to develop reasoning skills, not to write out arguments in a particular arrangement. Note that proof by mathematical induction is another deductive method which should not be neglected.
IN SUMMARY, students of all ages should recognize and be aware of the presence of geometry in nature, in art, and in woman- and man-made structures. They should believe that geometry and geometric applications are all around them and, through study of those applications, come to better understand and appreciate the role of geometry in life. Carpenters use triangles for structural support, scientists use geometric models of molecules to provide clues to understanding their chemical and physical properties, and merchants use traffic-flow diagrams to plan the placement of their stock and special displays. These and many, many more examples should leave no doubt in students' minds as to the importance of the study of geometry.
Reference: Geddes, Dorothy, & Fortunato, Irene. "Geometry : Research and Classroom Activities" in D. T. Owens (Ed.) Research Ideas for the Classroom: Middle Grades Mathematics. New York: Macmillan, 1993.
This introduction duplicates the section of Chapter 8 that discusses this content standard. Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in Chapter 1, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences. Many of the activities provided in this chapter are intended to convey this message; you may well be using other activities which would be appropriate for this document. Please submit your suggestions of additional integrative activities for inclusion in subsequent versions of this curriculum framework; address them to Framework, P. O. Box 10867, New Brunswick, NJ 08906.