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Auxiliary Files for PCTM Magazine Subscribers
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Geometry for All Students(Adopted 3/18/99) As the education community wrestles with the challenge of providing future leaders with education appropriate for the demands of an uncertain future, decisions must be made about the content included in the mathematics curriculum. Various organizations, e.g., The National Council of Teachers of Mathematics (NCTM), Mathematics Association of America (MAA), and U. S. Department of Education through the National Assessment of Educational Progress (NAEP) and National Voluntary Mathematics Test (NVMT), have identified content strands which form a basis for our Kindergarten through grade 12 mathematics curricula. The content strands suggest a foundation of mathematics which all students should learn. The following topics are consistently evident: Number Sense, Properties, and Operations; Measurement; Geometry and Spatial Sense; Data Analysis, Statistics, and Probability; and Algebra and Function. The strands do not define separate courses but create unifying themes across grade-level experiences. The mathematics curriculum is reconceptualized to model conceptual growth from informal/intuitive understanding to generalized/formal knowledge. The focus of these strands throughout the grade-levels is to develop the learners thinking abilities, use the structure and nature of the geometric concepts, to analyze and solve problems that arise in their everyday activities. This paper, GEOMETRY for ALL STUDENTS, addresses the structure of the geometry strand and its role in the total mathematics experience of the basic education for all students. The mathematics curriculum, in particular the geometry component presented in this document, should be view as an evolving body of knowledge always building on the learners prior experiences, to make more formal and complex levels of geometric knowledge accessible. Geometry throughout the years has been a long-standing challenge in the K-12 curriculum. A textbook author and mathematics leader of the sputnik era, Carl Allendoerfer (1969) wrote: The mathematical curriculum in our elementary and secondary schools face a serious dilemma when it comes to geometry(sic). It is easy to find fault with the traditional course in geometry, but sound advice on how to remedy these difficulties is hard to come by . . . . curricular reform groups at home and abroad have tackled the problem, but with singular lack of success or agreement . . . . We are, therefore, under pressure to do something about geometry; but what shall we do?(p. 165) A mathematician and curriculum developer, Zalman Usiskin (1987) indicated that geometry is faced with performance and curriculum problems. There is no geometry curriculum at the elementary school level. As a result, students enter high school not knowing enough geometry to succeed. There is a geometry curriculum at the secondary level, but only about half of the students encounter it, and only about a third of these students understand it. To improve performance requires more geometry study, which requires increased numbers of more knowledgeable teachers, which requires that more people want to study geometry, a desire usually associated with better performance. (p. 29) The 1992 National Assessment reported that 77% of high school students in the United States have taken a course in geometry, an improvement from 53% reported 14 years before. Usiskin (1997) indicated that we already have quite a bit of geometry for all students who stay in school and this has occurred with little fanfare over the past 10 years. More students are exposed to geometry using both synthetic and algebraic approaches, with and without coordinates and transformations. However, there is little emphasis on the important issues of connection and coordination across grade structures, so many students experience duplication or discontinuity going from elementary, through middle and to the secondary levels. This has created little uniformity and continuity in the geometry curriculum with each division (grade-level cluster) in the school structure tending to start the curricular experiences from scratch. There are also challenges about the level of formality which have not received the attention they deserve. When a learning experience is required by ALL students, rationale for studying this area must be stronger than when the experience is completed by a subgroup who are focused on study of the content. Any change in rationale must be accompanied by a change in content. Analysis of geometry concepts suggest that:
The results of the Third International Mathematics and Science Study (TIMSS, 1998) given to students at the time they complete secondary education provides clear evidence that the challenges identified by Allendoerfer and others have not been met. The general knowledge assessment items were chosen based on their likelihood of arising in real-life situations and not on their connection to a particular curriculum. These items contained concepts from geometry such as finding the length of a ribbon required to bind a rectangular package(drawing provided). The US average on this item was 32% correct while the International average was 45% correct. Overall, the US score was among the lowest of the twenty-one TIMSS nations. During the TIMSS study, additional assessments were created to compare students taking advanced mathematics courses. In the US, these were students who take a full year of a high school course that included calculus in its title. This assessment shows that among the content areas (Numbers, Equations and Functions; Calculus; Validation and Structure; Probability and Statistics; Geometry) the United States students performance was relatively weakest in geometry, scoring 15th of 16 nations participating. A sample item involves establishing and justifying that a triangle was isosceles based upon knowledge of various angle measures created by a given altitude. The results show a 19% average for U.S. students versus 48% average for the International students. Given the fact that these U. S. students should be among our best, this begs the question, Has our geometry curriculum focused upon small insignificant ideas at the expense of developing conceptual understanding of big ideas and their applications? During the last three decades, the developmental model of geometric thought presented in the research of Dina van Hiele-Geldorf and Pierre van Hiele has gained international recognition. Currently, the van Hiele model is probably the most helpful paradigm for planning the geometry strand in K-12 instruction. This model identifies the geometric thinking process in five levels: 1) visualization [recognizing and naming the figures]; 2) analysis [describing the attributes]; 3) informal deduction [classifying and generalizing by attributes]; 4) deduction [developing proofs using axioms and definitions]; and 5) rigor [working in various geometrical systems]. This philosophy asserts that learners move sequentially from visualization toward the rigor level. In most school geometry programs, the van Hiele model is contradicted by the traditional reality in that geometry is viewed as a high school course where most students are exposed to a formal, abstract level with little to no regard for their appropriate conceptual readiness. Students are expected to function in a deductive, proof-saturated environment with little prior exposure. (NCTM, 1988) A curriculum with a K -12 geometry strand based on the van Hiele philosophy would recognize the critical role of concrete manipulatives and technology for modeling ideas. Physical materials permit students to interact and experiment with objects and shapes as total entities rather than relationships among the component parts of these objects and shapes. As students progress through the geometry strand, this experimentation and observation leads to analysis and classification of objects and shapes as well as realization of interrelationships of properties within figures and shape. While students are maturing relative to their geometric knowledge, the program must provide opportunities to explore interrelationships among figures and with concepts in other strands of the curriculum. These connections are made more evident for a wider range of students by the existence of dynamic geometry capabilities on calculators and on computer systems. Several examples of resources currently available are: Microworlds (Logo), Cabri, TI-92, Geometers Sketchpad, Geometry Inventor, Geometric superSupposer, and Tesselmania. These systems make a what if ... model of instruction a truly functional part of the curriculum and extend the options far beyond the classical tools of compass and rulers. These systems also provide effective alternative to the flat page, 2-D images of the 3-D objects that students confront daily. Leaders and teachers realize that geometry is important enough to warrant a place at all levels of K - 12 instruction, but have no agreement regarding the content, sequence or timing of geometry to be taught. Even though inroads have been made, failure to take advantage of the geometry done at previous levels creates a situation where there is not sufficient time to develop the geometry that is recommended, except possibly for the very best students. This weakness in curricular organization contributes to secondary students who have only rudimentary knowledge of the scope and power of geometry. A variety of efforts has been made to modify the geometry in the school program, but an additional challenge is to give current classroom teachers the vital opportunity to experience geometry from perspectives beyond their own high school experiences. Contemporary geometry must be more than a series of two-column proofs founded upon axioms in Euclidean space. Addressing these challenges will require modification in the conceptual and mathematical knowledge of large numbers of preservice and inservice teachers. School geometry deals with the investigation of shapes and objects in two and three dimensions, their location, relationships, and properties. The tools(processes) used in these investigations include classification, analysis, representation and visualization. Application of these tools in school geometry must extend beyond the geometry that Euclid knew. It must include the geometry of concepts that are very important today maps, networks and flexible forms. As shapes and objects are studied, discoveries about similarities and differences among them are made. Recognition of shapes and objects in a variety of forms, and as components of complex forms, becomes possible. Projective geometry experiences provide students with observation about geometric forms as they exist in the world around them and contribute to the development of ones spatial sense. Two and three dimension spatial sense is a fundamental component of the early study and assessment of geometry. Once students understand spatial relationships, they can use the dynamic nature of geometry to connect mathematics to their world. The geometry content strand extends well beyond merely identifying geometric shapes or using procedures to apply spatial visualization skills in order to understand relationships. It includes both informal and formal construction of geometric figures, focusing on the geometric principles behind those constructions. Geometry provides a rich context for the development of reasoning skills, including making conjectures and validating them. Geometry is increasingly used to model and solve real-world problems, often connecting to algebraic representations through a coordinate system. Students are expected to have had much experience with basic identification of shapes such as triangles, circles, and rectangles, and with identifying and measuring line segments and angles. The application of transformational geometry techniques provides students with new insights into geometric relationships. Graduates should also understand similarity and congruence relationships. They must learn how to use a protractor and compass, plot points on a coordinate plane, apply algebraic reasoning, and dynamic technologies (handheld and desktop) in the study of relationships among geometric shapes. While formal proof must have a place in the geometry curriculum, its place as the basis of geometry for ALL students must be reconsidered. Investigation in the form of student projects can add excitement and insight into the students and teachers joint learning experiences. Utilizing the model of geometric thinking presented in the van Hiele philosophy and the knowledge-base students establish within the data collection, organization, analysis, and interpretation strand, the geometry curriculum should reflect a flow of observation, conjecture, validation, plausible argument and for some students formal proof. This curriculum must facilitate identification of given factual information and its utilization to arrive at a valid conclusion. Thus, clarifying the distinction between conjectures build upon the information and valid conclusions based upon facts. All this can be accomplished without requiring structured formal proof of ALL students. This points to a rich, developmentally appropriate strand of geometric ideas distributed throughout the K - 12 curriculum, without submerging the students in a heavily saturated deductive- proof system. The K -12 geometry curriculum built around six major themes will provide students with a knowledge-base for the challenges of the future. These themes are: 1) Shape; 2) Measurement (metric and non-metric); 3) Geometric Relations (Similarity and Congruency); 4) Dimension; 5) Reasoning (Induction and Deduction); and 6) Spatial Visualization (one, two and three dimensions). The school mathematics curriculum should be designed so that the students knowledge of a topic evolves through the different levels of understanding indicated in the model cited above. With careful structure throughout the school program, all students will experience the processes of geometry observation, formulation of hypotheses, testing of hypotheses, proof, and applications as well as make connections between these processes and those in other mathematics disciplines. The Geometry for All Students program is designed to meet the needs of ALL students as they systematically study geometric shape and structure and increasingly use formal reasoning and proof in their study.(NCTM, 1998) These recommendations are supported by the discussion draft of NCTMs Standards 2000 Principles and Standards for School Mathematics. Attention should be given so that ALL students: a) analyze characteristics and properties of two- and three-dimensional geometric objects; b) select and use different representational systems, including coordinate geometry, Euclidean and non-Euclidean models, and graph theory; c) recognize the usefulness of transformations and symmetry in analyzing mathematical situations; d) recognize the usefulness geometric representations in making sense of other areas of mathematics; and e) use visualization and spatial reasoning to solve problems both within and outside of mathematics.
Structuring the Geometry Program It is recommended that ALL students who complete the geometry program available in basic education should demonstrate functional knowledge of the following standards:
Primary Level Recognize and name geometric shapes in two and three dimensions (circle, sphere, square, cube, triangle, rectangle, pyramid, prism, cylinder, cone) and recognize models of shapes in real life Demonstrate an understanding of 2-D shapes by a) building models of shapes with concrete objects and manipulatives b) drawing models c) constructing models on geoboards and/or graph paper Demonstrate an understanding of transformations of figures by a) performing reflections, rotations and translations of shapes using models and drawings of shapes b) explore and explain lines of symmetry in shapes and in nature Demonstrate the capability of creating complex shapes by combining basic shapes Demonstrate the capability of decomposing complex shapes into a combination of basic shapes Describe, interpret, and apply ideas of position, direction and distance to locate objects on a plane and in space Demonstrate a basic awareness of spatial relationships
Intermediate Level Classify and compare triangles and quadrilaterals according to sides and/or angles Create and analyze geometric designs by using a) branching/recursive patterns b) tessellations (copy and original creations) Understand the concepts of set, line, point, and plane Give descriptions of geometric figures based upon relationships and properties of sides and angles observed by using a variety processes, e.g., sorting, building, modeling, tracing, measuring, and constructing Analyze and describe (written and verbal) planar figures (polygons and circles) a) using properties and measures of component parts of the figure b) using basic shapes of lesser area c) using tables of data representing linear and region measures of figures Demonstrate knowledge of 2-D and 3-D shapes a) describing in words how geometric shapes were constructed b) constructing using manipulatives, geoboards and computer software c) drawing figures using plane and isometric paper d) describing distance between points along horizontal and vertical lines of a coordinate system e) finding familiar solids in the environment and describe them f) identify a 3-D shape from a set of 2-D views, e.g., match a solid with its front, side, and top views Show relationships between and among figures using translations, reflections and rotations Describe location and movement, making and using coordinate maps to represent positions Define the basic properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms, cones and cylinders. Develop a working definition of the concepts of area and volume. Analyze simple transformations of geometric figures and rotations of line segments.
Middle Level Demonstrate knowledge of properties lines, segments, rays and angles by a) creating and describing models representing these ideas b) identifying these concept as components of a draw or natural object c) using intersecting lines, segments, and rays to explore properties of angles d) creating perpendicularity and bisections utilizing a variety of construction tools Use patterning to create complex geometric figures Draw and label perpendicular and parallel lines Draw, label and measure complementary, supplementary, and vertical angles and list their properties Classify, describe, and compare familiar polygons according to their distinguishing geometric properties as well as being regular or irregular Identify, name, draw and list Euclidean properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms, cones and cylinders. Develop a formal understanding of the concepts of area and volume. Identify, name and draw prisms, cylinders, cones, pyramids and spheres Distinguish between similar and congruent polygons Analyze plane figures using coordinated plane a) parallel and perpendicular lines b) right triangles and rate of change Explain the surface area of cubes, prisms, pyramids, and cylinders using nets of the three dimensional shapes. Explore translations, reflections and rotations in the Euclidean plane Approximate the value of _ (pi) through experimentation Create and describe cross-sectional views of 3-D shapes (Slice clay/play dough models) Describe 3-D images created by rotating a 2-D figure around a given axis Generate transformations using dynamic computer software. Analyze and describe geometric patterns, such as tessellations and sequences of shapes. Analyze objects using tessellations, symmetry, congruence, similarity, scale, and angles and identify their applications in practical situations. Distinguish between conjecturing and generalizing
High School Level Describe and carry out procedures for constructing perpendicular lines, parallel lines and bisectors (segment and angles), using a variety of tools such as compass and straightedge, paper folding and dynamic geometry software Explore geometry in coordinate environments modeling navigational, polar, and spherical situations Solve problems involving a) proportionality and similarity of polygons b) inscribed and circumscribed polygons c) 2-D figures in the coordinate plane Construct the image of figures using transformations a) translations b) rotations c) reflections d) dilations Model situations geometrically to formulate and solve problems a) discrete/finite systems (networks) b) trigonometric relationships Analyze figures and shapes including their symmetries, cross-sections, truncations. Visualize 3-D shapes from different perspectives Apply geometric principles to describe physical phenomena, e.g., surface tension, cracking, resultant forces, etc. Identify and describe geometric patterns in natural phenomena, designed art and engineered objects Use established properties to develop a logical conclusion regarding given criteria as well as judge, construct and communicate proofs
In addition, the college intending students should have the opportunity to gain foundational knowledge in the following areas:
Prove two polygons are congruent or similar using deductive proofs based upon algebraic and coordinate structures as well as geometric systems. Analyze and contrast deductive and analytic proofs of theorems. Select method of proof best suited to a particular theorem and support choice of proof methodology as well as structure of proof. Identify and prove the properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals using deductive proofs Identify corresponding parts in congruent or similar polygons to solve problems. Explore geometric concepts utilizing a variety of representations(Euclidean and non-Euclidean based) a) coordinates b) trigonometric relationships c) networks d) transformations e) vectors d) matrices Use the properties of angles, arcs, chords, tangents, and secants to solve problems involving circles. Use analytic geometry procedures to prove geometric properties of 2 and 3-D figures. Explore non-Euclidean spaces and introductory topics of topology
Bibliography
The National Council of Teachers of Mathematics publications:
The committee is appreciative of the many individuals who gave of their time to read the numerous drafts of the position paper. Their comments were carefully reviewed so that the committee could create a position which is representative of these recommendation and provide guidance to districts as they for focus their geometry curricula. The committee members are:
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The Pennsylvania Council of Teachers of Mathematics is a non-profit organization affiliated with the National Council of Teachers of Mathematics.
© 1996-2007, Pennsylvania Council of Teachers of Mathematics. All rights reserved.