I. Number and Title of Course:

MED 604 Teaching of Geometric Concepts

Fundamental to the high school curriculum are the concepts of geometry. This course was developed to give our graduate students the background of learning theory, pedagogy, mathematical models and new developments specific to the teaching of geometric concepts in the high school curriculum. No such course exists at present.

A. Students will understand the latest research on how young people learn mathematics and the implications for classroom instruction.

B. Students will have a better understanding of transformational geometry and its role in the new curriculum.

C. Students will have an understanding of the of how students learn geometric concepts, and will implement a variety of techniques for teaching these topics by the development of exemplary curriculum.

A. Understanding How Students Learn Geometry

1) Pierre van Hiele levels of learning geometry and the developmental theories of Jean Piaget.
2) Students will examine developmental levels of Piaget and their implications for curriculum design. Student will also examine the developmental levels put forth in the theory of learning by Pierre van Hiele an appropriate curriculum and activity for each level. While other psychologist will be examined, the remainder of the course will be organized by van Hiele levels to exemplify his theory.

B. van Hiele's Level 1 Geometry - Recognize properties of space & shapes in space.

Students will engage in learning about the models for rich environments appropriate for developing activities at this level. Some of the models are:

1) Geoboard activities

2) Mirror & Mira activities

3) Tessellations & Symmetry

4) Informal solid geometry

5) Introductory Logo

6) Geometry of paper folding

7) Computer geometry activities

8) Congruency

C. van Hiele's Level 2 Geometry - Logically ordering shapes and making inferences - "Informal Proofs"

In this part of the course the students will learn how to use the materials introduced in the previous part of the course as a basis for forming arguments for key ideas in mathematics. What follows is a listing of some of the key ideas that will be developed for informal proofs.
 

1) Parallelism

a) Congruent angles formed by parallel lines cut by a transversal

b) Hierarchy of polygons based upon parallelism

c) Sum of angles of a triangle

d) Sum of angles of polygons

e) Central, interior, exterior angles of polygons

f) Sum of exterior angles of polygons

2) Pythagorean Theorem

a) Informal proofs

b) Irrationality of lengths

c) Applications

d) Generalizable Pythagorean Theorem

3) Constructions:

a) with a compass and straightedge

b) with Mira

c) with angle templates

d) with paper folding

4) Proofs of theorems pertaining to the measurement of area and volume

5) Proofs that pertain to circles and arcs.

D. van Hiele's Level 3 - Geometry formally within a mathematical system.

Students will study the formal geometric system of transformational geometry to exemplify the type of activity in this level. Transformational geometry is the basis for the geometry of the New York State Curriculum and other integrated curricula. Topics to be covered are listed below:

1.) The Geometry of Transformations

a) Maps and Mappings

b) One-to-One Mappings; Transformations

c) Mappings in Algebra; Functions

d) Isometries

e) Problems Solved by Reflections

f) Properties of Isometries

g) Rotations

h) Translations and Glide Reflections

i) Symmetry

j) The Fundamental Theorems of Isometries

2.) The Algebra of Transformations

a) The Composite (Product) of Mappings

b) The Algebra of Translations

c) The Algebra of Half Turns

d) The Algebra of Rotations

e) Groups

f) Transformation Groups

g) Symmetry Groups

A. Books and Articles

Battista, Michael T. "MATH STUFF Logo Procedures: Bridging the Gap between Logo and School Geometry." Arithmetic Teacher V35 p7-11 September 1987:.

Battista, Michael T. and Clements, Douglas " Research into Practice: Using Spatial Imagery in Geometric Reasoning. " Arithmetic Teacher v39 p 18-21 November 1991.

Brown, Richard G. Transformational Geometry. Lexington, MA: Ginn and Company, 1973.

Burger, William F. "Geometry." Arithmetic Teacher V32 p52-56 February 1985.

Burger, William F., and Michael Shaughnessy. "Characterizing the van Hiele Levels of Development in Geometry." Journal for Research in Mathematics Education V17 p31-48. January 1986

Crowley, Mary L. "The van Hiele Model of the Development of Geometric Thought." found in Learning and Teaching Geometry, K-12, Yearbook 1987. National Council of Teachers of Mathematics, Reston, VA, 1987.

Fey, James T., William F. Atchison, Richard A. Good, M. Kathleen Heid, Jerry Johnson, Mary G. Kantowski, and Linda P. Rosen. Computing and Mathematics: The Impact on Secondary School Curricula. College Park, Md.: University of Maryland, 1984.

Fuys, David "Van Heile Levels of Think in Geometry" Education and Urban Society v17 n4 p447-62 August 1985

Gordon, Myles "What is the Geometric Supposer a Case of?" Report n 90-5. Educational Development Center, Newton, MA 1990

Gagatsis, A. and Patronis, T. Using Geometrical Models in a Process of Reflective Thinking in Learning and Teaching Mathematics." Educational Studies in Mathematics v 21 n 1 February 1990.

Geddes, Dorothy, David Fuys, C. James Lovett, and Rosamond Tischler. "An Investigation of the van Hiele Model of Thinking in Geometry among Adolescents." Paper presented at the annual meeting of the American Educational Research Association, New York, March 1982.

Geddes, Dorothy, David Fuys, and Rosamond Tischler. "An Investigation of the van Hiele Model of Thinking in Geometry among Adolescents." Final report, Research in Science Education (RISE) Program of the National Science Foundation, Grant No. SED 7920640. Washington, D.C.:NSF, 1985.

Hoffer, Alan. "Geometry Is More than Proof." Mathematics Teacher v74 p11-18. January 1981

Hoffer, Alan. "Van Hiele-based Research." In Acquisition of Mathematical Concepts and Processes, edited by R. Lesh and M. Landau. New York: Academic Press, 1983.

Lindquist, Mary Montgomery and Shulte, Albert P., eds. Learning and Teaching Geometry, K-12, Yearbook 1987. National Council of Teachers of Mathematics, Reston, VA, 1987.

O' Daffer P. and Clemens S. Geometry - An Investigative Approach. 2nd Addison Wesley Publishing Co. Reading MA 1992

Papert, Seymour. Mindstorms. New York: Basic Books, 1980.

Prevost, Fernand J. "Geometry in the Junior High School." Mathematics Teacher v78 p411-18 September 1985.

Shaughnessy, J. Michael, and William F. Burger. :Spadework Prior to Deduction in Geometry." Mathematics Teacher v78 p419-28 September 1985.

Senk, Sharon L. "How Well Do Students Write Geometry Proofs?" Mathematics Teacher v78 p448-56 September 1985.

Suydam, Marilyn N. "The Shape of Instruction in Geometry: Some Highlights from Research." Mathematics Teacher v78 p481-86 September 1985.

Usiskin, Zalman. "Van Hiele Levels and Achievement in Secondary School Geometry." Final report, Cognitive Development and Achievement in Secondary School Geometry Project. Chicago: University of Chicago, 1982.

van Hiele, Pierre M. Begrip en Inzicht [Understanding and insight]. Netherlands: Muussess Purmerend, 1973.

van Hiele, Pierre M. "A Child's Thought and Geometry.' In English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele, edited by Dorothy Geddes, David Fuys, and Rosamond Tischler as part of the research project "An Investigation of the van Hiele Model of Thinking in Geometry among Adolescents," Research in Science Education (RISE) Program of the National Science Foundation, Grant No. SED 7920640. Washington, D.C.:NSF, 1984a. (Original work published in 1959).

Wallace, Edward and West, Stephen Roads to Geometry New Jersey Prentise Hall 1992

Wilson, M. "Measuring a van Hiele geometry sequence: A reanalysis" , Journal for Research in Mathematics Education V3, p260-271 1990

Yusuf, Mian "Logo Based Instruction" a paper presented at the Annual Meeting of the Mid-Western Educational Research Association. Chicago IL. Eric Document #ED348218 1991

B. Periodicals

Arithmetic Teacher

Computers in the Teaching of Mathematics and Science

For the Learning of Mathematics

Journal of Research in Mathematics Education

Mathematics Teaching

Mathematics Teacher

This class will be presented in a class discussion/lecture format. Student will present projects, write reaction papers to synthesize the readings and develop exemplary curriculum materials. The evaluation will be based upon the quality of this work and classroom tests.

Admission into the Masters program.

3: 0 semester hours

IX. Departmental approval.

This course was examined with the recommended procedures and was approved by the graduate faculty of the Mathematics Department on May 13, 1993.

__________________________Department Chair

MED 604 Teaching of Geometric Concepts

This course will focus on the major approaches to teaching geometry in the secondary schools. Students will examine the traditional Euclidean approach as well as the transformational, computer based and integrated approaches to the teaching and learning of geometric concepts in the high school. The major psychological issues surrounding the learning of geometry will be discussed.

XI. Statement of Qualifications

A doctorate degree in mathematics education is the minimum formal education required.

Several members of the department who could teach this course are: Tom Giambrone (EdD, SUNY Buffalo 1983), Eileen Schoaff ( Ph.D. SUNY Buffalo 1988) Luella Johnson (PhD SUNY Buffalo 1990), and Betty Krist (Ed. D SUNY Buffalo 1980).

XII. Support Services Required.

none.