STANDARD 7 - GEOMETRY AND SPATIAL SENSE
Spatial sense is an intuitive feel for shape and space. It involves the concepts of traditional geometry, including an ability to recognize, visualize, represent, and transform geometric shapes. It also involves other, less formal ways of looking at two- and three-dimensional space, such as paper-folding, transformations, tessellations, and projections. Geometry is all around us in art, nature, and the things we make. Students of geometry can apply their spatial sense and knowledge of the properties of shapes and space to the real world.
Meaning and Importance
Geometry is the study of spatial relationships. It is connected to every strand in the mathematics curriculum and to a multitude of situations in real life. Geometric figures and relationships have played an important role in society's sense of what is aesthetically pleasing. From the Greek discovery and architectural use of the golden ratio to M. C. Escher's use of tessellations to produce some of the world's most recognizable works of art, geometry and the visual arts have had strong connections. Well-constructed diagrams allow us to apply knowledge of geometry, geometric reasoning, and intuition to arithmetic and algebra problems. The use of a rectangular array to model the multiplication of two quantities, for instance, has long been known as an effective strategy to aid in the visualization of the operation of multiplication. Other mathematical concepts which run very deeply through modern mathematics and technology, such as symmetry, are most easily introduced in a geometric context. Whether one is designing an electronic circuit board, a building, a dress, an airport, a bookshelf, or a newspaper page, an understanding of geometric principles is required.
K-12 Development and Emphases
Traditionally, elementary school geometry instruction has focused on the categorization of shapes; at the secondary level, it has been taught as the prime example of a formal deductive system. While these perspectives of the content are important, they are also limiting. In order to develop spatial sense, students should be exposed to a broader range of geometric activities at all grade levels.
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 while 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 vocabularyof 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 and to use spatial intuition to develop more generic mathematical problem-solving skills. In high school, the study of geometry expands to address coordinate, vector, and transformational viewpoints which utilize both inductive and deductive reasoning. Geometry instruction at the high school level should not be limited to formal deductive proof and simple measurement activities, but should include the study of geometric transformations, analytic geometry, topology, and connections of geometry with algebra and other areas of mathematics.
At all grade levels, 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 provide 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 constitute 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 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 understand and perform classic constructions with straight edges and compasses as well as with appropriate computer software. Formulating good mathematical definitions for geometric shapes should eventually lead to the ability to make hypotheses concerning relationships and to use deductive arguments to show that the relationships exist.
The standard geometric transformations include translation, rotation, reflection, and scaling. They are central to the study of geometry and its applications in that these movements 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 drawn 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 an important connection between geometry and algebra. Students can work informally with coordinates in the primary grades by finding locations in the room, and by studying simple maps of the school and neighborhood. 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 geometrictransformations algebraically and interpret algebraic equations geometrically.
Measurement and geometry are interrelated, and an understanding of 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 standard ones. Throughout school, they should use measurement activities to reinforce their understanding of geometric properties. All students should use these experiences to help them understand such principles as the quadratic change in area and cubic change in volume that occurs with a linear change of scale. Trigonometry and its use in making indirect measurements provides students with another view of the interrelationships between geometry and measurement.
Geometric modeling is a powerful problem-solving skill and should be used frequently by both teachers and 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 of concepts 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 use all permissible assumptions in a proof and as all 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 reason for 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 that should not be neglected.
Much of the current thinking about the development of geometric thinking in students comes from the work of a pair of Dutch researchers, Pierre van Hiele and Dina van Hiele-Geldof. Their model of geometric thinking identifies five levels of development through which students pass when assisted by appropriate instruction.
(Geddes & Fortunato, 1993)
Although the levels are not completely separate and the transitions are complex, the model is very useful for characterizing levels of students' thinking. Consistently, the research shows 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 never moves beyond identification of shapes and their properties. The discussion in this K-12 Overview draws on this van-Hiele model of geometric thinking.
In summary, students of all ages should recognize and be aware of the presence of geometry in nature, in art, and in human-built structures. They should realize 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.
Note: 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 the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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