A. PROJECT TITLE: THE MADISON PROJECT OF SYRACUSE UNIVERSITY AND WEBSTER COLLEGE

B. PROJECT DIRECTOR: Professor Robert B. Davis, Mathematics Department, Smith Hall, Syracuse University, Syracuse, New York 13210 U.S.A. (315)476-3768 or (315)476-5541, ext 23

C. PROJECT HEADQUARTERS:

1. Contact: The Madison Project, 918 Irving Avenue, Syracuse, New York 13210, U.S.A. (315)476‑3768.

2. Special facilities or activities available for visitor viewing: By special arrangement it is possible to visit classrooms in various schools, including culturally deprived situations, non-graded schools using various forms of flexible programming and team teaching, etc. In addition, it is possible to view project films (which also show actual classroom lessons and to talk with project personnel about specific problems of various sorts. Some project classrooms center around "mathematics laboratories."

D. PRINCIPAL PROFESSIONAL STAFF: Leon Henkin, Co-Director for Berkeley, California Implementation Program; Donald Cohen, Resident Coordinator for New York City; Diane Resek, Resident Coordinator for Berkeley, California; Beryl S. Cochran and Leah Horwitz, Film Preparation; William McConnell, Co‑Director for Implementation Programs; Edith Biggs, Robert Wirtz, Marion Walter, Katherine Vaughn, Katie Reynolds Hannibal, and William Betts, Visiting Specialist Teachers; Herbert Ginsburg, Co-Director for Piagetian Studies; Joyce Statz, Co-Director for LOGO Computer Studies, Lucian Hall, Resident Coordinator for Richmond, Virginia; George Grossman, New York City Bureau of Curriculum.

E. PROJECT SUPPORT: 1. Funding agencies: National Science Foundation; United States Office of Education; Alfred P. Sloan Foundation; Marcel Hoizer Foundation; and a group of industries and trade unions in the St. Louis area, plus contributions from participating schools and colleges. 2. Associated agencies: Syracuse University; University of California (Berkeley); Webster College; the Public School System of: Berkeley, California; Richmond, Virginia; New York City; Syracuse, New York; and the State Departments of Education in Delaware and in California.

F. PROJECT HISTORY: 1. Principal originators: Robert B. Davis, Beryl S. Cochran, Donald E. Kibbey, Sister M. Francetta Barberis, S.L., and Jacqueline Grennan Wexler. 2. Date and place of initiation: 1957; Syracuse, New York and Weston, Connecticut.

3. Evolution and development of the project: The Project was originally started in order to provide University faculty members who taught teachers with up-to-date first-hand experience in directly teaching children, on the principle that he who teaches experimental physics should himself be engaged in experimental physics and if the principle holds in physics, why not also in education. The original target-population were low achieving children in grade seven who were significantly below grade level in mathematics. These children were found to possess considerably more mathematical ability than had been suspected. (The Project's work in mathematics thus closely parallels the work of Hughes Mearnes and of Herbert Kohl in the area of creative writing.) For the next 6 years the Project sought to explore this unsuspected mathematical ability of children, working also with high ses children at various grade levels; in the process of doing this, it became necessary to develop a new mathematics curriculum, with selection of topics, sequencing of topics, new notations, new definitions, etc., that were more suitable for creative work by young children. (Thus, in this phase of its activities, the Project generally resembled Seymour Papert and Wallace Feurzeig's work on LOGO, and William Johntz's work on SEED.) The success of this program led to the creation of large‑scale teacher education programs. Joint ventures with the Elementary School Science (ESS) Project of ES1/EDC, and with the British Nuffield Mathematics Project (and other British educators) led to a greater use of manipulatable physical materials, small group work, math labs, and student projects. In its most recent stage, the Project is focussing on two matters: first, the re‑orientation of methods of teaching, of testing, and of curriculum planning that now seems necessary in the light of the discoveries made by Jean Piaget and the Geneva group of cognitive psychologists; and, second, with the new possibilities for school mathematics that have been created by the Papert-Feurzeig development of the LOGO computer programming language, and the various LOGO developments in hardware, in software, in pedagogical techniques, and in curriculum design.

G. PROJECT OBJECTIVES: 1. Overall project purpose: In broad terms, the purpose of the Project can be stated as follows: The Project has seen educational settings where children explore significant aspects of our present culture, and from such exploration learn what this culture is, and how they themselves can function creatively within it. (This general theme is well described in Casey and Liza Murrow, Children Come First American Heritage Press, New York: 1971, and in Edith Biggs and James MacLean, Freedom to Learn An Active Learning Approach to Mathematics Addison-Wesley Publishing Company, Inc. Canada: 1969.) The Project is concerned with helping schools and teachers set up such learning environments which implies a need to study them more closely in order to see more clearly what really is involved, and to prepare appropriate study materials that can be used successfully in such environments. The Project has focussed on mathematics as its main content area partly because Project personnel happen to be mathematicians, and partly because mathematics plays a central role in schools and in education.

2. Specific objectives:

(a) To study the process of cognitive growth in children.

(b) To study effective learning environments.

(C) To produce and test learning materials for use by children and by teachers that will reflect what can be found out from activities (a) and (b). The content area is mainly mathematics, plus some science‑ but provision is made for the inclusion of other content areas.

H. UNIQUE CHARACTERISTICS OF THE PROJECT: The project is different primarily in two ways: first, its concern for the great ability of most children which is untapped by most traditional forms of schooling; and, second, its concern for the laws of human cognitive growth.

I. SPECIFIC SUBJECTS, GRADE, AGE AND ABILITY LEVELS: Subjects: Primarily mathematics, plus some science‑ but opportunities are provided to move into nearly any area of study, by pursuing appropriate themes and appropriate methods. Grade: Project practice is to attempt to avoid grade‑level segregation of children, and consequently Project materials are un-graded. Most materials have been developed for use by children between the ages of 8 years old, up to beginning college work; however, some materials for use by younger children have been developed. Ability levels: Project methods allow for adaptation to any ability level, although the same curriculum is not used for children of widely different abilities or interests.

J. MAIN METHODS OF INSTRUCTION USED IN THE PROJECT: Independent study, laboratory investigations, seminars, discussion sessions and small‑group work.

K. PRESENT COMMERCIAL AFFILIATIONS: Books are presently available from Addison‑Wesley Publishing Company, Inc., San Hill Road, Menlo Park, California 94025, U.S.A. Twelve films are scheduled for release in the near future by Houghton Mifflin Company, 110 Tremont Street, Boston, Massachusetts 02107. Subtraction and Division Using Beans and Beansticks; Experience with Fractions: Suppose It Comes Out Even; Experience with Fractions: Suppose It Doesn't Come Out Even; Fractions and the Meaning of Division; Fractions on the Number Line, Using String The Number Line, Using the Overhead Projector; Area, Using Geoboards. In addition, some "shoebox" kits for math lab experiments are available from Math Media, Inc., P. 0. Box 345, Danbury, Connecticut 06810. These kits are entitled: Discs; Geoboards; Peg Game; Tower Puzzle; Centimeter Blocks, Weights and Springs.

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DESCRIPTION OF MATERIALS ALREADY PRODUCED: 1. Discovery in Mathematics (Publishers, Addison-Wesley Publishing Co., Inc.) Student discussion guide, plus text for teachers. This book provides a supplementary program in coordinate geometry, axiomatic algebra, and applications to science, suitable especially for grades 4‑8. It is concerned particularly with creative learning experiences of a non-routine nature. 2. Explorations in Mathematics Student discussion guide, plus text for teachers. This book is concerned with introductory ideas in algebra, statistics, mathematical logic, matrix algebra, and some applications to physics. Special emphasis is placed upon historical background, and the study of this book can be closely related to various units in social studies (such as the life and times of Rene Descartes). It is suitable for grades 6 through 9, inclusive. 3. A Modern Mathematics Program as it Pertains to the Interrelationship of Mathematical Content, Teaching Methods, and Classroom Atmosphere. (The Madison Project). 1963. Report submitted to the Commissioner of Education, U.S. Office of Education, Fall, 1963. The provides a general view of Madison Project activities. 4. A Modern Mathematics Program as it Pertains to the Interrelationship of Mathematical Content, Teaching Methods, and Classroom Atmosphere. (The Madison Project). 1965. Report submitted to the Commissioner of Education, U.S. Office of Education, Fall, 1965. Note that this is distinguishable from item 3 above only by the date. The 1965 report is the most comprehensive description presently available of Madison Project materials and activities. 5. The Madison Project A Brief Introduction to Materials and Activities (1965). 6. Notes on the Film: First Lesson. (This pamphlet accompanies the film of the same name.) 7. Robert B. Davis, Some Remarks on "Learning By Discovery". 8. Robert B. Davis, The Next Few Years. 9. Robert B. Davis, Experimental Course Report/Grade Nine.

10. Doris Machtinger, Experimental Course Report/Kindergarten.

11. Donald Cohen, Inquiry in Mathematics Via the Geoboard.

12. Supplementary Modern Mathematics for Grades 1 through 9. In-Service Course #1 for Teachers. This is a complete "packaged" in‑service course, including films, written materials, and laboratory equipment.

13. Supplementary Modern Mathematics for Grades 2 through 9. In-Service Course #2 for Teachers. This is a sequel to item 12 above.

14. The Journal of Children's Mathematical Behavior, (Vol. 1, No. 1 Winter 1971-72) presently available. This informal journal discusses the development of mathematical thought in children and how to study it.

15. A Concrete Approach to Introductory Ideas in Mathematics (booklet accompanying film series listed in item 16).

16. Film series (16mm, sound, black and white) A Concrete Approach to Introductory Ideas in Mathematics. Individual titles: Readiness for Place Value Numerals; A Sixth Grade Lesson on Place Value Numerals; Subtraction Using Beans; Addition and Multiplication Using Plastic Washers; Addition and Division Using Beans and Beansticks; Subtraction and Division

Using Beans and Beansticks; Experience with Fractions: Suppose It Comes Out Even; Experience with Fractions: Suppose It Doesn't Come Out Even; Fractions and the Meaning of Division; Fractions on the Number Line, Using String; The Number Line, Using the Overhead Projector; Area, Using Geoboards.

17. Audio tape recording #D‑1: + = 2 x . This is a recording of an actual classroom lesson with fifth grade children, proving algebraic theorems from a set of axioms selected by themselves.

18. Film (16mm., sound, black and white) A Lesson with Second Graders. This film shows an actual classroom lesson involving signed numbers, the number line, and Cartesian co‑ordinates.

Viewing this film is one of the best introductions to project activities.

19. Film (16mm., sound, black and white) Complex Numbers via Matrices. This film shows an actual classroom lesson. Seventh grade students use the isomorphism between rational numbers an a subset of the set 2 by 2 matrices to facilitate an extension into complex numbers.

20. Film (16mm., sound, black and white) Matrices. An actual classroom lesson. Fifth and sixth graders explore the algebra of 2‑by‑2 matrices.

21. Film (16 mm., sound, black and white) Solving Equations

With Matrices. An actual classroom lesson, similar to item 1 above, but less sophisticated. Sixth grade students.

22. Film (16mm., sound, black and white) Average and Variance An actual classroom lesson, with 6th grade children.

23. Film (16mm., sound, black and white) Graphing an Ellipse. An actual classroom lesson, with 7th grade students.

24. Film (16mm., sound, black and white) Circles and Parabola An actual classroom lesson, with 6th grade children.

25. Film (16mm., sound, black and white) First Lesson. An actual classroom lesson, with a mixed class of children from grades 3 to 7.

26. Film (16mm., sound, black and white) Second Lesson. This lesson occurred on the day following that shown in item 25 above, with the same students.

27. Film (16mm., sound, black and white) Weights and Springs. A "laboratory" lesson, with 6th grade children.

28. Film (16mm., sound, black and white) Graphing a Parabola. This is a portion of the film listed in item 20.

29. Film (16mm., sound, black and white) Guessing Functions. A seventh grade class of culturally deprived urban children.

30. "shoebox" packages for physical experiments related to the mathematics program, or for physical and tactile experiences related to the learning of mathematics. Titles: Discs, Geoboards, Peg Game, Tower Puzzle, Centimeter Blocks, Weights and Springs.

31. (The Project also makes use of physical materials prepared by Z. P. Dienes, by ESS, by the Nuffield Project, and by others as well as desk calculators of various sorts.

M. MATERIALS AVAILABLE FREE: This varies according to the availability of reprints of various articles. Contact the Madison Project, 918 Irving Avenue, Syracuse, New York 13210.

N. MATERIALS PURCHASABLE: Items 1 and 2 from Section L. Item 1: Student text $3.21; Teacher text $8.40. Item 2: Student text $3.40; Teacher text $9.24. (Order from Regional Office, Addison‑Wesley Publishing Company, Inc.) Item 11 available from Walker Educational Book Corporation, 720 Fifth Avenue, New York, New York 10019. Item 14, The Journal of Children's Mathematical Behavior available from The Madison Project, 918 Irving Avenue, Syracuse, New York 13210. Price (subject to change): $1.00 Items 15 and 16: Available soon from Houghton Mifflin Co., 110 Tremont Street, Boston, Massachusetts 02160. Item 29: "shoebox" kits available from Math Media, Inc., P. 0. Box 345, Danbury, Connecticut 06810. Price for complete set of 6: $19.00. Four or more sets: $17.25 each.

0. ADDITIONAL MATERIALS BEING DEVELOPED: Please refer to present and future issues of The Journal of Children's Mathematical Behavior.

P. LANGUAGE OF MATERIALS: 1. As originally written: English. 2. Have been or will be translated: Explorations in Mathematics; Parts Two and Five have been translated into Japanese. Japanese translation available from: Addison‑Wesley Publishing Company, Inc., International Division, Reading Massachusetts.

Q. COUNTRIES IN WHICH MATERIALS ARE USED: U.S.A., Canada, Great Britain, Japan, Korea, India, Vietnam, Ghana, Nigeria, Uganda, Israel, Australia, New Zealand.

R. PROJECT IMPLEMENTATION. The project materials are widely used. Exact figures are unknown. Exemplary classrooms may be located by contacting The Madison Project coordinators listed in D. See, especially, classrooms in Berkeley, California and in New York City.

S. TEACHER PREPARATION: 1. Consultant service available for teachers using the materials: In New York City; in Berkeley, California; in St. Louis, Missouri; in Washington, D. C., and elsewhere. Contact the Project at its Syracuse address for information. Consultants

are prepared to assist teachers, administrators, or interested parents in a variety of ways, including program planning and practical classroom implementation. 2. Activities conducted for pre‑service and in‑service teacher training: Extensive in‑service programs are available in New York City, in Delaware, and in California. Details available from the Project. 3. Available pre‑service and/or in‑service teaching materials for science educators to use in preparing teachers: In‑Service Course I (dealing with ways of combining arithmetic, algebra, and analytic geometry to provide a broad elementary school program), consists of printed materials and ten film excerpts available at a cost of $30.00 for rental of the films and $100.00 for 30 copies of the printed materials (or, single copies $3.50). A more diverse in‑service or pre‑service teacher education package which includes consideration of mathematics in open classrooms is now undergoing trials. Preliminary versions are available at a cost of $3.00.

T. PROJECT EVALUATION: 1. Has the effectiveness of the materials been evaluated: Yes, internally and by the California State Department of Education, Far West Laboratory for Research and Development. 2. Pertinent published research studies: "Research Report of the Specialized Teacher Project 1968‑69." California State Department of Education, Sacramento, California, February 1970; Kathleen Devaney, "An ALERT Report on The Madison Project," 1972. Available from the Educational Information Products Division, Far West Laboratory for Educational Research and Development, 1 Garden Circle, Hotel Claremont, Berkeley, Calif. 94705.; Alan Barson, Beryl Cochran and Robert Davis, "Child‑Created Mathematics," The Arithmetic Teacher, March 1970.; J. Robert Cleary, "A Study of Test Performance in Two Madison Project Schools and One Control School," Webster College, St. Louis, Missouri. 3. Brief abstract of in‑house or unpublished research: By far the best evidence concerning the mathematical behavior of children in Project classes is provided by actual videotape and film records showing what children do: conjecturing theorems proving them, analyzing new problem situations, etc. These films and videotapes show not merely that the children do do all of this‑ they show precisely how the children do it. These films are available for loan or rental. For details write to the Project. 4. Evaluative data available to interested individuals: Please see the preceding remark (question T‑3).

U. PROJECT PUBLICITY: 1. Davis, Robert B. "Report of the Syracuse University Webster College Madison Project." American Mathematical Monthly Vol. 71, No. 3 (March, 1964) pp. 306‑308.

2. _. "The Madison Project's Approach to a Theory of Instruction," Journal of Research in Science Teaching Vol. 2 (1964), pp. 146‑162. 3. , "The Next Few Years," The Arithmetic Teacher Vol. 13, No. 5 (May, 1966), pp. 355‑362. 4. . The Changing Curriculum: Mathematics. Association for Supervision and Curriculum Development, NEA, 1967. 5. Yeomans, Edward. Education for Imitative and Responsibility Comments on a visit to the Schools of Leicestershire County. National Association of Independent Schools, Boston, Mass. 2nd edition, February 1968. 6. Pine, Patricia. "New Math Road Show," American Education Vol. 4, No. 7, July‑August 1968. 7. M. Vere De Vault and Thomas Kriewall, Perspectives in Elementary School Mathematics, Charles Merrill Company, 1969. 8. "Nuffield Mathematics Project: Teacher's Guides," Mathematics Teaching No. 53, Winter, 1970, pp. 53‑56. 9. Ginsburg, Herbert, The Myth of the Deprived Child, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1972.

10. Davis, Robert B., "Observing Children's Mathematical Behavior as a Foundation for Curriculum Planning," The Journal of Children's Mathematical Behavior Vol. 1, No. 1 (Winter1971‑72).

V. BRIEF SUMMARY OF PROJECT ACTIVITIES SINCE 1970 REPORT: The Project has a new major focus: the careful observation of the mathematical behavior of children. This may provide a sounder foundation for curriculum design, and a more effective point of intervention for changing school mathematics, than any that have been used previously. The "careful observation of children" involves depth studies in the sense of Piaget, and NOT a main focus on superficial verbal behavior.

W. PLANS FOR THE FUTURE: 1. The process of preparing films and videotapes for release by Houghton Mifflin, or otherwise, will continue with a growing list of films becoming available. 2. The program of observing and describing the mathematical behavior of children will become a main emphasis in Project activity. (Cf., Robert B. Davis, "The Problems of Relating Mathematics to the Possibilities and Needs of Schools and Children," in H. Freudenthal, ed., Educational Studies in Mathematics, D. Reidel Publishing Company (Holland), June 1971. 3. The establishment of consultants and in‑service study opportunities in various geographical areas will be expanded especially in California, New York City, Delaware, Baltimore, and the area around Washington, D.C.). 4. Methodological and even philosophical matters, in specific cases which the Project judges to be of immediate practical relevance, will be pursued. (Cf. Robert B. Davis, Mathematics Teaching‑ With Special Reference to Epistemological Problems. Monograph No. 1 (Fall, 1967), of the Journal of Research and Development in Education, College of Education, University of Georgia, Athens, Georgia 30601. 5. More attention will be given to mathematics in relation to open education. (Casey and Liza Murrow, Children Come First, American Heritage Press, New York, 1971). 6. An elementary school mathematics program, for presentation to children via the PLATO computer system, will be developed jointly with UICSM. 7. Children will be studied as they learn the BBN/MIT LOGO computer programming language. 8. The Project will participate in an international comparison of how curriculum development and innovation efforts are undertaken in various nations of Europe and Asia.