GCMP
PROJECT
TITLE: EDUCATIONAL RESEARCH COUNCIL OF AMERICA MATHEMATICS
PROGRAM
(ERCMP)
B. PROJECT DIRECTOR:
Mr. John F. Mehegan, Director, Educational
Research
Council Mathematics Program, 614 West Superior,
Cleveland,
Ohio 44113, U.S.A. (216)
696‑8222, ext. 241.
C. PROJECT
HEADQUARTERS: 1. Contact: Project director. 2. Special facilities or activities
available for visitor viewing: Classes in participating schools and ERC
facilities.
D.
PRINCIPAL PROFESSIONAL STAFF: John F. Mehegan, Director; William T. Hale,
Assistant Director; S. Edwin Humison, K‑8 Coordinator; Mrs. Rae Marie Creps,
Research Associate; Miss Lucille McCraith, Research Associate; Miss Mary
Muesegaes, Research Assistant; Miss Margaret Russell, Research Associate.
E. PROJECT
SUPPORT: 1. Funding agencies: The Cleveland Foundation, The Good Works
Foundation, The Grant Foundation, The George Gund Foundation, The Louise H.
& David S. Ingalls Foundation, Inc., The Martha Holden Jennings Foundation,
The Kettering Foundation, The Laurel Foundation, The Lubrizol Foundation,
various industrial firms and individual contributions. 2. Associated agencies:
Not answered.
F. PROJECT
HISTORY: 1. Principal originator: Dr. George H. Baird, President, Educational
Research Council of America. 2. Date and place of initiation: March, 1959;
Cleveland, Ohio. 3. Evolution and development of the project: In 1961 K‑3 materials were published. In 1965 K‑6 materials were published. In 1968 junior high materials
were published, K‑6 were revised, and Units for high
school students not in academic mathematics. In 1972 individualized K‑8 materials are being processed.
G. PROJECT
OBJECTIVES: 1. Overall project purpose: To develop, for all children in grades
K‑12, an individualized comprehensive, sequential mathematics program that is both mathematically correct and pedagogically sound. 2. Specific objectives: To develop skillful computation,
measurement skills, and geometric concepts and skills.
UNIQUE
CHARACTERISTICS OF THE PROJECT: A learning continuum with associated behavioral
objectives to assist the teacher to individualize the curriculum. Diversified
learning materials, i.e., games, investigations, and manipulative materials,
are an integral part of the program. The program has a unit format, each unit
of work consisting of: 1. A list of entering behaviors.
2. A
pretest to measure entering behaviors and the behaviors to be induced by study
of the unit. 3. Write‑in pupil pages. 4. Related
diversified learning materials. 5. A diagnostic test to be administered before
practicing the induced behaviors. 6. A teacher guide with suggestions for pre‑page activities, use of the pages, and an answer key. 7. A
list of terminal objectives. 8. A post‑test.
I. SPECIFIC
SUBJECTS, GRADE, AGE AND ABILITY LEVELS: Mathematics for all students in K‑8, Basic Math for 9‑12 students not in academic mathematics,
and use of the computer in mathematics in 7‑12.
J. MAIN
METHODS OF INSTRUCTION USED IN THE PROJECT: Independent study, laboratory
investigations, lectures, seminars, discussion sessions, computer assisted
instruction, and films and filmstrips.
K. PRESENT
COMMERCIAL AFFILIATIONS: Science Research Associates, Inc., 259 East Erie
Street, Chicago, Illinois 60611, U.S.A.
L.
DESCRIPTION OF MATERIALS ALREADY PRODUCED: 1. K‑6 Pupil
Texts and Teacher Guides. 2. K‑6 Tests. 3. Addition, Subtraction,
Multiplication, and Division Units (available only to Council Schools). 4.
Games, Investigations, and Manipulative Materials. 5. Filmstrips. 6. Junior
High Student Texts and Teacher Guides. 7. Informal Geometry Booklets. 8. A
BASIC Primer. 9. Elementary Functions with BASIC
10. Key
Topics for the Primary Teacher.
11. Key
Topics for the Intermediate Teacher.
12. ERC
Catalog of Materials and Services.
M.
MATERIALS AVAILABLE FREE: Item 12.
N. MATERIALS
PURCHASABLE: Items 1, 2, 10, 11. Write to Science Research Associates for price
list. Items 4, 5, 6, 7, 8, 9. Write to the Educational Research Council for
price list.
0.
ADDITIONAL MATERIALS BEING DEVELOPED: Geometry Kits and Measure ment Kits for K‑8, and units for Fractional Numbers.
P. LANGUAGE
IN WHICH MATERIALS: 1. Were originally written: English. 2. Have been or will
be translated: Not answered.
COUNTRIES
IN WHICH MATERIALS ARE USED: United States.
PROJECT IMPLEMENTATION:
(In Council Schools) Information on adoption of the entire program is unknown.
1. Total number of teachers using any of the materials: 2,000. 2. Total number
of students using any of the materials: 70,000. 3. Total number of schools
using any of the materials: 175. 4. The totals stated in 1, 2 and 3 are
estimated. 5. Name and location of selected schools where the program is in
use: State of Ohio: Aurora, Avon Lake, Bay Village, Berea, Brooklyn, Cleveland
Catholic Schools, Chardon, Cuyahoga Heights, Fairview Park, Independence, some
Cleveland Lutheran Schools, Hayfield, North Olmsted, Olmsted Falls. Out of
State: Bradford, Pennsylvania; Brockton, Massachusetts; Greenville, Michigan;
Imlay City, Michigan; Muskegon, Michigan; Niles, Michigan; Owatonna, Minnesota.
3. TEACHER
PREPARATION: 1. Consultant services available for teachers using the materials:
The ERCMP staff (all teachers) is available at all times to conduct workshops,
teach demonstration lessons, and assist the teacher to implement ERCMP or other
programs. 2. Activities conducted for pre‑service and in‑service teacher training: In‑service
workshops at Council schools; one, two, or three day workshops in Council
schools: $5 per day for Council teachers, $25 per day for Non‑Council teachers. 3. Available pre‑service and/or in‑service teaching materials for
science educators to use in preparing teachers: None.
E. PROJECT
EVALUATION: 1. Has the effectiveness of the materials been evaluated: Yes,
internally. 2. Published research studies: (a) Comparative Data for the Greater
Cleveland Mathematics Tests, Grades 1‑3, Test 1, Form A (A Teacher's Guide
to the Interpretation of Scores), 1964. (b) Comparative Data for the Greater
Cleveland Mathematics Tests, Grades 1‑3, Tests 2‑4, Form A (A Teacher's Guide to the Interpretation of
Scores), 1964. (C) Comparative Data for the Greater Cleveland Mathematics
Tests, Grade 4, Tests 1‑2, Form A, Including a Supplement of
Data for Tests 3‑4, Form A, 1964. (d) The Performance
of GCMP Students on the Stanford Arithmetic Tests, Grades Four through
Six, 1964‑65, 1965. (Edward) A Descriptive
Analysis of the use of the Greater Cleveland Mathematics Program in Grades One,
Two, and Three, 1966. (f) A Technical Report on the Analysis of Test Results
from an Investigation of Two Teaching Methods in the Multiplication with Whole Numbers, 1966. 3. Brief abstract of in‑house or unpublished research. (a) Comparative Study of
Arithmetic Test Scores Before and
After the
Introduction of GCMP.
A comparison
of the mean grade equivalent scores on the
Arithmetic
Reasoning and Computation subtests of the Stan: Achievement Tests indicates
that the performance of pupil (Grade 3 and 4) seems to have improved somewhat
on both the Arithmetic Reasoning and Computation subtests after introduction of GCMP materials.
(b) An
Interim Report on the Results of GCMT, Intermediate Series.
A
descriptive analysis of the four parts of each of seven tests, corresponding to
seven student booklets of the GCMP was made in order to help establish
guidelines to evaluate pupils' achievement on the concepts and skills stressed
in the program.
(C) Grades
2, 3 and 4 Math Project.
Preparation
of the final reports is in process on the comparative
studies of the teaching methods of computational skills to students in grades 2‑4. The programs in the studies
include experimental units to be published in 1974, the corresponding materials
in the current GCMP and the most
widely used
non‑GCMP mathematics programs.
The
following analyses were made in the studies:
(i)
Attainment of each objective by different program groups by various IQ levels.
(in)
Comparison of the overall achievement of various program groups after
differences in IQ and initial
ability in
computation are adjusted (i.e., analysis of covariance using IQ and the initial
computational score as the covariates).
(iii)
Analysis of errors made by students of varying I levels in the experimental
group. Its purpose is to help teachers understand students' misconception and
help ERC's Mathematics Department develop remedial
materials
for these students.
(d) Grade 1
Project.
A
comparative study is underway to compare the experiment program to be published
in 1974 and the current GCMP.
(Edward)
Pre‑Algebra Projects.
(i) Development
of the Diagnostic and Placement Inven tory to be given to students at the end
of grade 6 or the beginning of grade 7 to identify students' strength and
weaknesses in seven areas: Numeration, whole numbers fractional numbers,
decimals, integers, rational numb and geometry.
(in)
Evaluation of the pre‑algebra programs and the
accompanying achievement tests.
(f) Basic
Mathematics Project.
Report to
the Teachers on the Results of the "Circles" Test and the "Boxes
and Cylinders" Test.
An
investigation of the students' performance on the pretest and post‑test on the booklets Circles and
Boxes and
Cylinders in the Basic Mathematics Program,
After the
Introduction of GCMP.
A
comparison of the mean grade equivalent scores on the
Arithmetic
Reasoning and Computation subtests of the St
Achievement
Tests indicates that the performance of pupil
(Grade 3
and 4) seems to have improved somewhat on both
Arithmetic
Reasoning and Computation subtests after intro‑,
duction of
GCMP materials.
(b) An
Interim Report on the Results of GCMT, Intermediate
Series.
A
descriptive analysis of the four parts of each of seven
tests,
corresponding to seven student booklets of the GCMP
was made
in order to help establish guidelines to evaluate
pupils'
achievement on the concepts and skills stressed b
the program.
(C) Grades
2, 3 and 4 Math Project.
Preparation
of the final reports is in process on the corn‑i
parative
studies of the teaching methods of computational
skills to
students in grades 2‑4. The programs in the
ies include
experimental units to be published in 1974, the
corresponding
materials in the current GCMP and the most
widely used
non‑GCMP mathematics programs.
The following
analyses were made in the studies:
(10)
Attainment of each objective by different program
groups
by various IQ levels.
(in)
Comparison of the overall achievement of various program groups after
differences in IQ and initial ability in computation are adjusted (i.e.,
analysis of covariance using IQ and the initial computational score as the
covariates).
(iii)
Analysis of errors made by students of varying IQ levels in the experimental
group. Its purpose is to help teachers understand students' misconception and
help ERC's Mathematics Department develop remedial materials for these
students. (d) Grade 1 Project. A comparative study is underway to compare the
experimental program to be published in 1974 and the current GCMP. (Edward) Pre‑Algebra Projects. (i) Development of the Diagnostic and
Placement Inventory to be given to students at the
end of grade 6 or the beginning of grade 7 to identify students' strength and
weaknesses in seven areas: Numeration, whole numbers fractional numbers,
decimals, integers, rational numbers and geometry.
(in)
Evaluation of the pre‑algebra programs and the
accompanying
achievement tests.
(f) Basic
Mathematics Project.
Report to
the Teachers on the Results of the "Circles" Tea
and the
"Boxes and Cylinders" Test.
An
investigation of the students' performance on the
pretest
and post‑test on the booklets Circles and
Boxes
and Cylinders in the Basic Mathematics Program,
After the
Introduction of GCMP.
A
comparison of the mean grade equivalent scores on the
Arithmetic
Reasoning and Computation subtests of the Stan: Achievement Tests indicates
that the performance of pupil (Grade 3 and 4) seems to have improved somewhat
on both the Arithmetic Reasoning and Computation subtests after intro‑ ¥ duction of GCMP materials.
(b) An
Interim Report on the Results of GCMT, Intermediate Series.
A
descriptive analysis of the four parts of each of seven tests, corresponding to
seven student booklets of the GCNJ was made in order to help establish
guidelines to evaluate pupils' achievement on the concepts and skills stressed 11
the program.
(C) Grades
2, 3 and 4 Math Project.
Preparation
of the final reports is in process on the comparative
studies of the teaching methods of computational skills to students in grades 2‑4. The programs in the studies include
experimental units to be published in 1974, the corresponding materials in the
current GCMP and the most
widely used
non‑GCMP mathematics programs.
The
following analyses were made in the studies:
(i) Attainment
of each objective by different program groups by various IQ levels.
(in)
Comparison of the overall achievement of various program groups after
differences in IQ and initial
ability in
computation are adjusted (i.e., analysis of covariance using IQ and the initial
computational score as the covariates).
(iii)
Analysis of errors made by students of varying L levels in the experimental
group. Its purpose is to help teachers understand students' misconception and
help ERC's Mathematics Department develop remedial
materials
for these students.
(d) Grade 1
Project.
A
comparative study is underway to compare the experiment program to be published
in 1974 and the current GCMP.
(Edward)
Pre‑Algebra Projects.
(i)
Development of the Diagnostic and Placement Inventory to be
given to students at the end of grade 6 or the beginning of grade 7 to identify
students' strength and weaknesses in seven areas: Numeration, whole numbers
fractional numbers, decimals, integers, rational numb and geometry.
(in)
Evaluation of the pre‑algebra programs and the
accompanying achievement tests.
(f) Basic
Mathematics Project.
Report to
the Teachers on the Results of the "Circles" Test and the "Boxes
and Cylinders" Test.
An
investigation of the students' performance on the pretest and post‑test on the booklets Circles and
Boxes and
Cylinders in the Basic Mathematics Program,
546
to
determine the gain and attainment of major objectives. The Basic Mathematics
Program is intended for non‑college bound students of the tenth
or eleventh grade. Pretest and post‑test are also available for the
tests accompanying the other books in the Basic
Mathematics Program. 4. Evaluative data available to interested individuals:
Additional information is available from the Director of
the Evaluation and Testing Department of the
Educational Research Council of America, Rockefeller Building, Cleveland, Ohio
44113.
J. PROJECT
PUBLICITY: The Grade Teacher February, 1971.
T¥ SUMMARY
OF PROJECT ACTIVITIES SINCE 1970 REPORT: Development of units including pupil
pages, behavioral objectives, tests, teacher guide, and diversified materials.
PLANS FOR
THE FUTURE: Publication of a complete K‑4 program for the school year 1974‑75 and a 5‑8 program for the school year 1975‑76.
A. PROJECT
TITLE: THE MADISON PROJECT OF SYRACUSE UNIVERSITY AND WEBSTER COLLEGE
B. PROJECT
DIRECTOR: Professor
Robert B. Davis, Mathematics Depart‑
ment,
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 class‑
rooms in
various schools, including culturally deprived situa‑
tions, non‑graded schools using various forms of flexible pro‑
gramming
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 "mathe‑
matics
laboratories."
D.
PRINCIPAL PROFESSIONAL STAFF: Leon Henkin, Co‑Director
for Berkeley, California Implementation Program; Donald Cohen, Resi dent
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, CoDirector 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 mem‑
bers 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 ex‑
perimental
physics ‑‑ and if the principle holds in
physics,
why not
also in education. The original target‑population were
low ses
children in grade seven who were significantly below
grade level
in mathematics. These children were found to po‑
ssess
considerably more mathematical ability than had been sus‑
pected.
(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 pro‑
gram led to
the creation of large‑scale teacher education pro‑
grams.
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 techni‑
ques, 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 AddisonWesley 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.
668
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 nonroutine 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 child
ren,
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. Seveni grade students use the isomophism between
rational numbers an a sub‑set 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 Nuf field 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
theorms 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 UniversityWebster 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 Iniative 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,
PrenticeHall, Inc., Englewood Cliffs, New
Jersey, 1972.
10. Davis,
Robert B., "Observing Children's Mathematical Be‑
havior 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.
A. PROJECT
TITLE: MINNESOTA MATHEMATICS AND SCIENCE TEACHING PROJECT (MINNEMAST)
B. PROJECT
DIRECTOR: James H. Werntz, Jr., Professor of Physics, Director, Center for Educational
Development, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
(612)373‑4537.
C. PROJECT
HEADQUARTERS: 1. Contact: Minnemath Center, University of Minnesota, 720
Washington Avenue, S.E., Minneapolis, Minnesota 55455, U.S.A. 2.
Special
facilities or activities available for visitor viewing: None.
D.
PRINCIPAL PROFESSIONAL STAFF: M. R. Boudrye, Research Associate, Administrator.
E. PROJECT
SUPPORT: 1. Funding agencies: University of Minnesota. 2. Associated agencies: Formerly
funded by the National Science Foundation.
F. PROJECT
HISTORY: 1. Principal originator: Paul C. Rosenbloom, Professor of Mathematics.
2. Date and place of initiation: 1961; University of Minnesota. 3. Project
terminated: September, 1970.
G. PROJECT
OBJECTIVES: 1. Overall project purpose: To produce coordinated mathematics and science curriculum for grades K‑6. 2.
Specific objectives: To develop process acquisition, attitudinal changes and scientific literacy.
H. UNIQUE
CHARACTERISTICS OF THE PROJECT: Broad spectrum of elementary school children of varying capacities and backgrounds.
I. SPECIFIC
SUBJECTS, GRADE, AGE AND ABILITY LEVELS: Coordinated mathematics and science,
grades K‑3; college level, teacher
preparatory.
J. MAIN
METHODS OF INSTRUCTION USED IN THE PROJECT: Independent study, laboratory
investigations, seminars, and discussion sessions.
K. PRESENT
COMMERCIAL AFFILIATIONS: Textbook in mathematics: "Ideas in
Mathematics," published by W.B. Saunders Co., Philadelphia, Pa. 1970.
L.
DESCRIPTION OF MATERIALS ALREADY PRODUCED: 1. Minnemath Reports (terminated
1969).
2. Coordinated
units: 1‑29 for grades K‑3.
3. Overview.
4. Living
Things in Field and Classroom.
5. Extending
Man's Senses.
6. Ideas
in Mathematics.
7. Questions
and Answers about MINNEMAST.
M.
MATERIALS AVAILABLE FREE: Item 7 only. Please address the project headquarters.
N.
MATERIALS PURCHASABLE: Item 2 ‑ 5, information available by writing
project headquarters. Item 6 from W. B. Saunders Co., Philadelphia,
Pennsylvania, U.S.A.
0.
ADDITIONAL MATERIALS BEING DEVELOPED: Pre‑service and in‑service teacher aids.
P. LANGUAGE
IN WHICH MATERIALS: 1. Were originally written: English. 2. Have been or will
be translated: None.
Q. COUNTRIES
IN WHICH MATERIALS ARE USED: United States and Canada.
R. PROJECT
IMPLEMENTATION: 1. Total number of teachers using any of the materials: 200. 2.
Total number of students using any of the materials: 50,000+. 3. Total number
of schools using any of the materials: 125. 4. Number of teachers have adopted
the entire program: 20 school systems. 5. Number of students involved: 6,000.
6. Number of schools involved: 125. 7. The totals stated in 1,2,3,5, and 6 are
estimated. 8. Name and location of selected schools where this program is in
use: St. Paul, Minnesota; West St. Paul, Minnesota; South Hadley,
Massachusetts; Pittsfield, Massachusetts; Newton, New Jersey; Quincy,
Massachusetts; Redlands, California.
S. TEACHER
PREPARATION: 1. Consultant services available for teachers using the materials:
Write to the Center at address in C‑l. 2. Activities conducted for pre‑service and in‑service teacher training: Summer
workshops, supported by National Science Foundation, in various areas.
Information available from NSF, Student and Curriculum Improvement Section,
Washington, D. C. 20550. 3. Available pre‑service
and/or in‑service teaching materials for
science educators to use in preparing teachers: Pre‑servic and in‑service materials were planned, and
some early development completed. No funds have been
available for completing these materials and early phases are no longer
available for distribution.
T. PROJECT
EVALUATION:
1. Has
the effectiveness of the materials been evaluated: Yes,
internally.
2. Published
research studies:
(a)
Hively, W., H. Patterson, and Sara H. Page, "A Uni‑
verse‑Defined System of Arithmetic Achievement Tests",
Journal
of Educational Measurement Vol. 5, No. 4, Winter
1968,
p. 275.
(b)
Johnson, P. E., "On the Communication of Concepts in
Science",
Journal of Educational Psychology, 1969, Vol. 60,
p.
32‑40.
(C)
Murray, F., "Reversibility Training in the Acquisition
of
Length Conservation", Journal of Educational Psychology
Vol.
59, No. 2, 1968, p. 82‑87.
(d)
Murray, F., "Operational Conservation of Illusion‑
Distorted
Length", British Journal of Educational Psych‑
ology
Vol. 38, Part 2, June 1968, p. 189‑193.
3. Brief
abstract of in‑house or unpublished research:
(a)
Reports on the direct evaluation of most of the K‑3
units
have been completed. These reports contain a detailed
description
of the test domain for the unit and a summary
of
the results obtained during the field test of the unit.
(b)
The MINNEMAST Experiment with Domain Referenced
Achievement
Testing Systems A detailed description of the
evaluation
procedures used in the MINNEMAST evaluation
project.
(C)
A Curriculum Evaluation and Revision Based on Domain
Referenced
Achievement Test System This paper describes
how
an individual unit (Unit 2) was evaluated and revised
based
upon the results of that evaluation. It also pre‑
sents
the results of a subsequent evaluation.
(d)
The Use of Sample Test Items As Objectives for
Instruction
‑ The Effects Upon the Teacher and Upon the
Learner.
This paper describes a study in which six kinder‑
garten
teachers were provided with sample test items for a
MINNEMAST
unit while a matched sample of six other teachers
were
provided with the unit only. The study sheds some
light
on the role of objectives in instruction.
(Edward) Future Uses
of Domain Referenced Achievement Testing
Systems This
paper outlines some of the potential uses
for Domain
Referenced Achievement Testing Systems. It also
points to some
of the pitfalls in those applications.
(f) An
Introduction to Domain Referenced Achievement
Testing An
overview of the psychological basis for the
evaluation
model utilized in the MINNEMAST Project. The
paper also
includes a glossary of terms as utilized in the
testing
model
(GCMP) The
Experimental Analysis of Educational Objectives
Ph.D.Thesis,
University of Minnesota, George Rabehl. The
paper
represents the philosophical and scientific rationale
for casting
the formulation of educational objectives into
an
experimental context. It shows that the specification
691
of relevant
and irrelevant conditions is not a once and for
all
activity but requires instead a continuing self‑correc‑
tive
process involving the steps of hypothesis, application,
analysis,
and a reformulation of educational intent. What
is achieved
is a framework for analyzing, describing, and
comparing
curricular materials; for making inferences from
student
performances beyond a finite set of items; and
finally a
basis for proposing and interpreting psycholog‑
ical
studies in terms of the actual characteristics of
educational
requirements.
(h) A Comparison
of Two Conceptual Frameworks for Teaching
the Basic
Concepts of Rational Numbers Ph.D. Thesis, Uni‑
versity of
Minnesota, Donald Sension. This paper compares
the effects
of two physical models for teaching fraction
concepts on
student performance. It utilizes a Domain
Reference
Achievement Testing System.
(i) An
Investigation of the Effectiveness of Independent
Study of
Novel Mathematics Material in the Elementary
School
Ph.D.Thesis, University of Minnesota, Lester
Becklund.
The paper presents the results of a study which
examined
the role of the teacher in presenting some novel
mathematics.
A MINNEMAST game unit on vectors and trans‑
formations
was used.
(j)
Arithmetic Achievement Test Performance of MINNEMAST
Mathematics
Pupils in the Third and Fourth Grades This
paper
presents a summary of the results of a two year
study of
the performance of pupils in the MINNEMAST mathe‑
matics program
on selected arithmetic achievement tests.
(K) The
Relationship Between Concepts of Conservation of
and
Number. The purpose of this study was to
describe
the relationship between attainment of concepts
of
conservation of length and number. The concepts were
embodied in
a compound number‑length task. Two aspects
of
performance were investigated: (10) the comparative
performance
of solvers and non‑solvers on conservation of
number and
conservation of length, and (2) the stability
of
performance characteristics across the age span sampled.
Fifty‑five children, 21 females and 34 males, ages 6
through 9 years,
participated in the study.
U. PROJECT
PUBLICITY: 1. Ahrens, R. B., "MINNEMAST ‑‑
The Coordinated Science and Mathematics Program", Science and Children Vol
65, December 1965, p. 811‑814. 2. Bray, Edmund C.,
"MINNEMAST, An Elementary Math‑Science Program", School
Science and Mathematics June 1969. 3. Bray, Edmund C., "The MINNEMAST
Elementary MathematicsScience Program", The Physics
Teacher May 1968. 4. Maxwell, Graham, "Some Notes and Comments on the
Minnesota Mathematics and Science Teaching Project", The Australian
Mathematics Teacher March 1969.
5. Rising,
Gerald R., "Research and Development in Mathematics and Science Education
at the Minnesota School Mathematics and Science Center and the Minnesota
National Laboratory", School Science and Mathematics Vol. 65, December
1965, p. 811‑814. 6. Rosenbloom, P. C., Journal
of Research in Science Teaching (1963), p. 276‑280. 7.
Subarsky, Zachariah, "Curriculum Construction for K‑6 Science and Math ‑‑ a Strategy", Science and
Children November 1968. 8. Subarsky, Zachariah, "The Systems Concept in
Science", The Instructor January 1968. 9. Victor, Laurence, "Systems:
An Organizing Principle for Science Curricula", Science and Children
January/February 1968, p. 17‑20. 10. Werntz, James H., "A
Style of Understanding", Nature and Science Vol. 4, No. 12, March 13,
1967.
V. BRIEF
SUMMARY OF PROJECT ACTIVITIES SINCE 1970 REPORT: Preparation for sale of printed materials and laboratory kits.
W. PLANS
FOR THE FUTURE: Continued availability of materials. Write to the Center.
693
A. PROJECT
TITLE: SCHOOL MATHEMATICS STUDY GROUP (SMSG)
B. PROJECT
DIRECTOR: Dr. E. G. Begle, SMSG ‑ School of Education,
Stanford
University, Stanford, California 94305, U.S.A.
(415)321‑2300 X2681.
C. PROJECT
HEADQUARTERS: 1. Contact: Project director, SMSG. 2. Special facilities or
activities available for visitor viewing: There are no facilities for viewers.
D. PRINCIPAL
PROFESSIONAL STAFF: None.
E. PROJECT SUPPORT:
1. Funding agencies: National Science Foundation. 2. Organizational agencies:
Stanford University and the Conference Board of the Mathematical Sciences.
F. PROJECT
HISTORY: 1. Principal originators: Ad hoc Conference of Mathematicians. 2. Date
and place of initiation: March 1958; Yale University.
G. PROJECT
OBJECTIVES: 1. Overall project purpose: To bring together classroom teachers
and research mathematicians in a joint effort to improve the pre‑college mathematics curriculum. 2.
Specific objectives: The primary purpose of the SMSG was to foster research and
development in the teaching of school mathematics. The work of SMSG consisted
primarily in the development of courses, teaching materials and teaching
methods. It was a part of SMSG's task, in cooperation with other mathematical organizations, to encourage exploration of the hypotheses
underlying mathematics education.
H. UNIQUE
CHARACTERISTICS OF THE PROJECT: Not answered.
I. SPECIFIC
SUBJECTS; GRADE; AGE AND ABILITY LEVELS: Mathematics; kindergarten through
grade 12; teacher training materials.
J. METHODS
OF INSTRUCTION USED IN THE PROJECT: Normal classroom procedures.
K. PRESENT
COMMERCIAL AFFILIATIONS: The monograph series "New
Mathematical
Library" is published by Random House, Inc., 457 Madison Avenue, New York,
New York 10022, U.S.A. The filmed course for elementary school teachers is
distributed by Modern Learning Aids, 1212 Avenue of the Americas, New York, New
York, 10036, U.S.A.
L.
MATERIALS PRODUCED:
First
Course in Algebra:
1. Student's
Text, Parts I and II.
2. Teacher's
Commentary, Parts I and II.
Programed
First Course in Algebra (Revised Form h):
3. Student's
Text, Parts I and II.
4. Student's
Response Booklet.
5. Teacher's
Commentary.
Geometry;
6. Student's
Text, Parts I and II.
7. Teacher's
Commentary, Parts I and II.
Geometry
with Coordinates:
8. Student's
Text, Parts I and II.
9. Teacher's
Commentary, Parts I and II.
Intermediate
Mathematics:
10. Student's
Text, Parts I and II.
11. Teacher's
Commentary, Parts I and II.
Elementary
Functions:
12. Student's
Text.
13. Teacher's
Commentary.
Introduction
to Matrix Algebra:
14. Student's
Text.
15. Teacher's
Commentary.
Analytic
Geometry:
16. Student's
Text.
17. Teacher's
Commentary.
Algorithms,
Computation and Mathematics:
18. Student's
Text.
19. Teacher's
Commentary.
20. FORTRAN,
Student's Text.
21. FORTRAN,
Teacher's Commentary.
22. ALGOL,
Student's Text.
23. ALGOL,
Teacher's Commentary.
Calculus:
24. Student's
Text, Parts I and II.
25. Teacher's
Commentary, Parts I and II.
26. Student's
Text, Part III.
27. Teacher's
Commentary, Part III.
Calculus of
Elementary Functions:
28. Student's
Text (2 parts).
29. Teacher's
Commentary (2 parts).
777
Mathematics
for Junior High School:
30. Volume
1, Student's Text, Parts I and II.
31. Volume
1, Teacher's Commentary, Parts I and II.
32. Volume
2, Student's Text, Parts I and II.
33. Volume
2, Teacher's Commentary, Parts I and II.
Introduction
to Secondary School Mathematics:
34. Volume
1, Student's Text, Parts I and II.
35. Volume
1, Teacher's Commentary.
36. Volume
2, Student's Text, Parts I and II.
37. Volume
2, Teacher's Commentary.
Introduction
to Algebra:
38. Student's
Text, Parts I and II.
39. Teacher's
Commentary, Parts I and II.
Mathematics
for the Elementary School:
40. Book
K, Teacher's Commentary.
41. Book
1, Student's Text.
42. Book
1, Teacher's Commentary.
43. Book
2, Student's Text.
44. Book
2, Teacher's Commentary.
45. Book
3, Student's Text, Parts I and II.
46. Book
3, Teacher's Commentary, Parts I and II.
47. Grade
4, Student's Text, Parts I and II.
48. Grade
4, Teacher's Commentary, Parts I and II.
49. Grade
5, Student's Text, Parts I and II.
50. Grade
5, Teacher's Commentary, Parts I and II.
51. Grade
6, Student's Text, Parts I and II.
52. Grade
6, Teacher's Commentary, Parts I and II.
Mathematics
for the Elementary School ‑ Special Editions:
53. Book
K, Teacher's Commentary.
54. Book
1, Student's Text, Parts I and II.
55. Book
1, Teacher's Commentary, Parts I and II.
56. Developing
Mathematics Readiness in Pre‑School Programs.
Mathematics
Through Science:
59. Graphing,
Equations and Linear Functions, Students Text,
Part
58. Measurement
and Graphing, Teacher's Commentary, Part I.
59. Graphing,
Equations and Linear Functions, Student's Text,
Part II.
60. Graphing,
Equations and Linear Functions, Teacher's
Commentary,
Part II.
61. An
Experimental Approach to Functions, Student's Text,
Part III.
62. An
Experimental Approach to Functions, Teacher's
Commentary,
Part III.
Supplementary
Units:
65. Junior
High School, Student's Text.
66. Junior
High School, Teacher's Commentary.
778
67. Essays
on Number Theory I.
68. Essays
on Number Theory II.
69. Development
of the Real Number System.
Probability:
70. Primary
Grades, Student's Text.
71. Primary
Grades, Teacher's Commentary.
72. Intermediate
Grades, Student's Text.
73. Intermediate
Grades, Teacher's Commentary.
74. Classroom
set of Spinners for Primary Grades.
75. Classroom
set of Spinners for Intermediate Grades.
76. Introduction
to Probability, Basic Concepts, Student's
Text, Part
I.
77. Introduction
to Probability, Special Topics, Student's
Text, Part
II.
Supplementary
and Enrichment Series:
78. SP‑16
Algebraic Structures.
79. SP‑23
Radioactive Decay.
80. SP‑26
Mathematical Theory of the Struggle for Life.
81. SP‑27
1 + 1 = ?.
82. SP‑28
Order and the Real Numbers: A Guided Tour.
83. SP‑29
The Mathematics of Trees and Other Graphics.
Reprint
Series:
84. RS‑1
The Structure of Algebra.
85. RS‑2
Prime Numbers and Perfect Numbers.
86. RS‑3
What is Contemporary Mathematics?.
87. RS‑4
Mascheroni Constructions.
88. RS‑5
Space, Intuition and Geometry.
89. RS‑6
Nature and History of
90. RS‑7
Computation of ir
91. RS‑8
Mathematics and Music.
92. RS‑9
The Golden Measure.
93. RS‑10
Geometric Constructions.
94. RS‑11
Memorable Personalities in Mathematics: Nineteenth
Century.
95. RS‑l2
Memorable Personalities in Mathematics: Twentieth
Century.
96. RS‑l3
Finite Geometry.
97. RS‑l4
Infinity.
98. RS‑15
Geometry, Measurement and Experience.
Spanish
Translations:
99.
Matematicas Para El Primer Ciclo, Secundario, 2 parts (JI‑RS) 100.
Matematicas Para El Primer Ciclo Secundario, 2 parts, Comentario (CJI‑RS).
101. Matematicas Para El Primer Ciclo Secundario, 2 parts (JII‑RS) 102.
Matematicas Para El Primer Ciclo Secundario 2 parts, Comentario (CJII‑RS).
103. Matematicas Para La Escuela Secundaria, Primer Curso de Algebra, 2 parts
(F‑RS), set.
104. Matematicas
Para La Escuela Secundaria, Primer Curso de
Algebra, 2
parts, Comentario (CF‑RS).
105. Matematicas
Para La Escuela Secundaria, Geometria, 2 parts
(G‑RS),
set. $3.00.
106. Matematicas
Para La Escuela Secundaria, Geometria, 2 parts
Comentario
(CG‑RS). $3.00.
107. Matematicas
Para La Escuela Secundaria, Matematica
Intermediate,
2 parts (I‑RS). $3.00.
108. Matematicas
Para La Escuela Secundaria, Funciones
Elementales
(E‑RS). $2.00.
109. Matematicas
Para La Escuela Secundaria, Introduccion Al
Algebra De
Las Matrices (A‑RS). $2.00.
110. Geometria
Analitica (GA). $2.00.
111. Matematicas
Para La Escuela Primaria, Grado 4, 2 parts,
Comentario
(SE‑4). $4.00.
112. Matematicas
Para La Escuela Primaria, Grado 5, 2 parts,
Comentario
(SE‑5). $4.00.
113. Matematicas
Para La Escuela Primaria, Grado 6, 2 parts,
Comentario
(SE‑6). $4.00.
114. Estudios
De Matematicas, Conceptos de Geometria
Intuitiva
(SM‑5). $2.00.
115. El
Curso Conciso En Matematicas Para Los Profesores De
Escuela
Primaria (SM‑9). $2.50.
116. Introduccion
A Sistemas Numericos (SM‑14). $2.50.
Soviet
Studies in the Psychology of Learning and Teaching
Mathematics:
117. Volume
1 ‑ The Learning of Mathematical Concepts.
118. Volume
II ‑ The Structure of Mathematical Abilities.
119. Volume
III ‑ Problem Solving in Arithmetic and Algebra.
120. Volume
IV ‑ Problem Solving in Geometry.
121. Volume
V ‑ The Development of Spatial Abilities.
122. Volume
VI ‑ Instruction in Problem Solving.
Investigations
in Mathematics Education:
123. A
Journal of Abstracts and Annotations, Volume 1.
124. A
Journal of Abstracts and Annotations, Volume 2.
125. A
Journal of Abstracts and Annotations, Volume 3.
126. A
Journal of Abstracts and Annotations, Volume 4.
Studies in
Mathematics:
127. Euclidean
Geometry Based on Ruler and Progractor
Axioms (SM‑2).
128. Structure
of Elementary Algebra (SM‑3).
129. Geometry
(SM‑4).
130. Concepts
of Informal Geometry (SM‑5).
131. Number
Systems (SM‑6).
132. Intuitive
Geometry (SM‑7).
133. Concepts
of Algebra (SM‑8).
134. Brief
Course in Mathematics for Elementary School Teachers
(SM‑9).
135. Applied
Mathematics in the High School (SM‑b).
780
104. Matematicas
Para La Escuela Secundaria, Primer Curso de
Algebra, 2
parts, Comentario (CF‑RS).
105. Matematicas
Para La Escuela Secundaria, Geometria, 2 parts
(G‑RS),
set. $3.00.
106. Matematicas
Para La Escuela Secundaria, Geometria, 2 parts
Comentarlo
(CG‑RS). $3.00.
107. Matematicas
Para La Escuela Secundaria, Matematica
Intermediate,
2 parts (I‑RS). $3.00.
108. Matematicas
Para La Escuela Secundaria, Funciones Elementales (E‑RS).
$2.00.
109. Matematicas
Para La Escuela Secundaria, Introduccion Al
Algebra De
Las Matrices (A‑RS). $2.00.
110. Geometria
Analitica (GA). $2.00.
111. Matematicas
Para La Escuela Primaria, Grado 4, 2 parts,
Comentario
(SE‑4). $4.00.
112. Matematicas
Para La Escuela Primaria, Grado 5, 2 parts,
Comentario
(SE‑5). $4.00.
113. Matematicas
Para La Escuela Primaria, Grado 6, 2 parts,
Comentario
(SE‑6). $4.00.
114. Estudios
De Matematicas, Conceptos de Geometria
Intuitiva
(SM‑5). $2.00.
115. El
Curso Conciso En Matematicas Para Los Profesores De
Escuela
Primaria (SM‑.9). $2.50.
116. Introduccion
A Sistemas Numericos (SM‑l4). $2.50.
Soviet Studies
in the Psychology of Learning and Teaching
Mathematics:
117. Volume
1 ‑ The Learning of Mathematical Concepts.
118. Volume
II ‑ The Structure of Mathematical Abilities.
119. Volume
III ‑ Problem Solving in Arithmetic and Algebra.
120. Volume
IV ‑ Problem Solving in Geometry.
121. Volume
V ‑ The Development of Spatial Abilities.
122. Volume
VI ‑ Instruction in Problem Solving.
Investigations
in Mathematics Education:
123. A
Journal of Abstracts and Annotations, Volume 1.
124. A
Journal of Abstracts and Annotations, Volume 2.
125. A
Journal of Abstracts and Annotations, Volume 3.
126. A
Journal of Abstracts and Annotations, Volume 4.
Studies in
Mathematics:
127. Euclidean
Geometry Based on Ruler and Progractor
Axioms (SM‑2).
128. Structure
of Elementary Algebra (SM‑3).
129. Geometry
(SM‑4).
130. Concepts
of Informal Geometry (SM‑5).
131. Number
Systems (SM‑6).
132. Intuitive
Geometry (SM‑7).
133. Concepts
of Algebra (SM‑8).
134. Brief
Course in Mathematics for Elementary School Teachers
(SM‑9).
135. Applied
Mathematics in the High School (SM‑10).
136. Mathematical
Methods in Science (SM‑11).
137. A
Brief Course in Mathematics for Junior High School
Teachers
(SM‑12).
138. Inservice
Course for Primary School Teachers (SM‑13).
139. Introduction
to Number Systems (SM‑14).
140. Calculus
and Science (SM‑l5).
141. Some
Uses of Mathematics (SM‑16).
142. Mathematical
Concepts of Elementary Measurement (SM‑17).
143. Puzzle
Problems and Games Project (SM‑18).
144. Reviews
of Recent Research in Mathematics Education (SM‑l9).
Conference
Reports:
145. Conference
on Elementary School Mathematics (CR‑1).
146. Orientation
Conference for SMSG Experimental Centers (CR‑2).
147. Orientation
Conference for SMSG Elementary School Experi‑
mental Centers
(CR‑3).
148. Orientation
Conference for Geometry with Coordinates (CR‑4).
149. Conference
on Future Responsibilities for School Mathe‑
matics (CR‑5).
150. Mathematics
Education for Below Average Achievers (CR‑6).
151. A
Conference on Mathematics for Gifted Students (CR‑7).
152. A
Conference on Mathematics Education in the Inner City
Schools (CR‑8).
153. A
Conference on Responsibilities for School Mathematics in
the 70's
(CR‑9).
NLSMA Reports:
154. No.
1 X‑Population Test Batteries.
155. No.
2 Y‑Population Test Batteries.
156. No.
3 Z‑Population Test Batteries
157. No.
4 Description and Statistical Properties of
X‑Population
Scales.
158. No.
5 Description and Statistical Properties of
Y‑Population
Scales.
159. No.
6 Description and Statistical Properties of
Z‑Population
Scales.
160. No.
7 The Development of Tests.
161. No.
9 Non‑Test Data.
162. No.
10 Patterns of Mathematics Achievement in Grades 4,5,
and 6: X‑Population.
163. No.
11 Patterns of Mathematics Achievement in Grades 7 and 8
X‑Population.
164. No.
12 Patterns of Mathematics Achievement in Grades 7 and 8
Y‑Population.
165. No.
13 Patterns of Mathematics Achievement in Grade 9:
Y‑Population.
166. No.
14 Patterns of Mathematics Achievement in Grade 10:
Y‑Population.
167. No.
15 Patterns of Mathematics Achievement in Grade 11:
Y‑Population.
168. No.
16 Patterns of Mathematics Achievement in Grade 10:
Z‑Population.
169. No.
19 The Non‑Intellective Correlates of Over‑and
Under‑
achievement
in Grades 4 and 6.
170. No.
21 Correlates of Mathematics Achievement: Attitude and
Role
Variables.
171. No.
22 Correlates of Mathematics Achievement: Cognitive
Variables.
172. No.
23 Correlates of Mathematics Achievement: Teacher Back‑
ground and
Opinion Variables.
173. No.
24 Correlates of Mathematics Achievement: School‑
Community
and Demographic Variables.
174. No.
25 Correlates of Mathematics Achievement: Teacher
Assigned
Grades.
175. No.
26 Correlates of Mathematics Achievement: Summary.
176. No.
28 Teacher Effectiveness in Mathematics Instruction.
ELMA
Technical Reports:
177. No.
1 Kindergarten Test Batteries, Description and
Statistical
Properties of Scales.
178. No.
2 Grade 1 Test Batteries, Description and
Statistical
Properties of Scales.
179. No.
3 Grade 2 Test Batteries, Description and Statistical
Properties
of Scales.
180. No.
4 Grade 3 Test Batteries, Description and Statistical
Properties
of Scales.
Geometry
Units for Elementary School:
181. Sets
of Points, Student's Text.
182. Unit
I ‑ Teacher's Commentary.
183. Congruence,
Student's Text.
184. Unit
II ‑ Teacher's Commentary.
185. Congruence
and Familiar Geometric Figures, Student's Text.
186. Unit
III ‑ Teacher's Commentary.
187. Measurement
of Curves (length), Students Text.
188. Unit
IV ‑ Teacher's Commentary.
189. Measurement
of Plane Regions (Area), Student's Text.
190. Unit
V ‑ Teacher's Commentary.
191. Measurement
of Space Regions, (Volume), Student's Text.
192. Unit
VI ‑ Teacher's Commentary.
193. Measurement
of Angles, Student's Text.
194. Unit
VII ‑ Teacher's Commentary.
195. Side
and Angle Relationships for Triangles, Student's Text.
196. Unit
VIII ‑ Teacher's Commentary.
197. Circles
and Constructions, Student's Text.
198. Unit
IX ‑ Teacher's Commentary.
199. Whole
Numbers as Coordinates of Points, Student's Text.
200. Unit
x ‑ Teacher's Commentary.
201. Integers
as Coordinates of Points, Student's Text.
202. Unit
XI ‑ Teacher's Commentary.
Secondary
School Mathematics:
203. Chapters
1 and 2, Student's Text.
204. Teacher's
Commentary.
205. Chapters
3 and 4, Student's Text.
206. Teacher's
Commentary.
207. Chapters
5 and 6, Student's Text.
208. Teacher's
Commentary.
209. Chapters
7 and 8, Student's Text.
210. Teacher's
Commentary.
211. Chapters
9 and 10, Student's Text.
212. Teacher's
Commentary.
213. Chapters
11 and 12, Student's Text.
214. Teacher's
Commentary.
215. Chapters
13 and 14, Student's Text.
216. Teacher's
Commentary.
217. Chapters
15 and 16, Student's Text.
218. Teacher's
Commentary.
219. Chapters
17 and 18, Student's Text.
220. Teacher's
Commentary.
221. Chapters
19 and 20, Student's Text.
222. Teacher's
Commentary.
223. Chapters
21 and 22, Student's Text.
224. Thacher's
Commentary.
225. Chapters
23 and 24, Student's Text.
226. Teacher's
Commentary.
227. Chapters
25 and 26, Student's Text.
228. Teacher's
Commentary.
229. Chapters
27 and 28, Student's Text.
230. Teacher's
Commentary.
Secondary
School Advanced Mathematics:
231. Chapters
1 and 2, Student's Text.
232. Teacher's
Commentary.
233. Chapter
3, Student's Text.
234. Teacher's
Commentary.
235. Chapters
4 and 5, Student's Text.
236. Teacher's
Commentary.
237. Chapters
6 and 7, Student's Text.
238. Teacher's
Commentary.
239. Chapter
8, Student's Text.
240. Teacher's
Commentary.
Secondary
School Mathematics ‑ Special Editions:
241. Chapters
1, 2, and 3, Student's Text.
242. Chapters
4 and 5, Student's Text.
243. Chapters
6 and 7, Student's Text.
244. Chapters
8 and 9, Student's Text.
245. Teacher's
Commentary for Chapters 1‑9.
246. Chapters
10 and 11, Student's Text.
247. Chapters
12, 13, and 14, Student's Text.
248. Chapters
15 and 16, Student's Text.
249. Chapters
17 and 18, Student's Text.
250. Teacher's
Commentary for Chapters 10‑18.
Miscellaneous
Publications:
251. Very
Short Course in Mathematics for Parents.
252. Philosophies
and Procedures of SMSG Writing Teams.
253. SMSG:
The Making of a Curriculum.
New
Mathematical Library:
254. Niven
‑ Numbers: Rational and Irrational (NML‑l).
255. Sawyer
‑ What is Calculus About? (NML‑2).
256. Beckenback
and Bellman ‑ An Introduction to Inequalities
(NHL‑3).
257. Kazarinoff
‑ Geometric Inequalities (NNL‑4).
258. Davis
‑ The Lore of Large Numbers (NML‑6).
259. Zippin
‑ Uses of Infinity (NML‑7).
260. Yaglom
‑ Geometric Transformations I (NHL‑8).
261. Olds
‑ Continued Fractions (NML‑9).
262. Ore
‑ Graphs and Their Uses (NML‑10).
263. Hungarian
Problem Book I (NNL‑11).
264. Hungarian
Problem Book II (NML‑12).
265. Aaboe
‑ Episodes from the Early History of Mathematics
(NML‑l3).
266. Grossman
and Magnus ‑ Groups and their Graphs (NML‑14).
267. Niven
‑ Mathematics of Choice (NML‑15).
268. Friedrichs
‑ From Pythagoras to Einstein (NML‑l6).
269. The
Contest Problem Book II (NML‑l7).
270. Chinn
and Steenrod ‑ First Concepts of Topology (NML‑l6).
271. Coxeter
and Greitzer ‑ Geometry Revisited (NNL‑19).
272. Ore
‑ Invitation to Number Theory (NML‑20).
273. Yaglom
‑ Geometric Transformations II (N}IL‑21).
274. Sinkov
‑ Elementary Cryptanalysis (NML‑22).
275.
Honsberger ‑ Ingenuity in Mathematics (NML‑23). NEWSLETTERS
Information concerning SMSG was disseminated through its Newsletters which
appear at irregular intervals. The following Newsletters are still available
until the supply is exhausted: 276. No. 15. Reports on various SMSG Projects. 277. No. 17. Lists Supplementary
Publications and selected list of inexpensive books for supplementary use. 278. No. 19. Report of a survey of in‑service
programs for mathematics teachers. 279. No. 21.
Reference guide to the New Mathematical Library Description, topical
classification and index with suggested grade levels. 280. No. 23. Panel on Supplementary Publications. 281. No. 24. Reports on various
SMSG projects. 282. No. 25. Articulation of
Content of SMSG Texts, grades 7-10. 283. No. 28.
Articulation of Content of SMSG Texts, grades 1‑3 and 4. 284. No. 30. Status Reports, Recent Publications. 285. No. 33. Mathematics for
Disadvantaged and Low Achieving Students. 286. No. 35. Status
Reports ‑ Recent Publications.
287. No.
36. Final Report on a New Curriculum Project. 288. No. 37. SMSG
Publications.
MATERIALS
AVAILABLE FREE: Newsletters Nos. 15, 17, 19, 21, 23, 24, 25, 28, 30, 33, 35, 36
and 37; Report No. 1 ‑ The Programmed Learning Project; Report No. 2
‑ The Special Curriculum Project; Pilot Program on Mathematics learning
of Culturally Disadvantaged Primary School Children; Report No. 3 ‑ A
FilmFilm Text Study; Report No. 4 ‑ The Special Curriculum Project:
1965‑66; Report No. 5 ‑ The Slow Learner Project: The Secondary
School "Slow‑Learner" in Mathematics; Report No. 7 ‑ Preliminary
Report on an Experiment with Junior High School Very Low Achievers in
Mathematics: Report No. 7 ‑ Final Report on an Experiment with Junior
High School Very Low Achievers in Mathematics; Report No. 8 ‑ The
Mathematics Through Science Study: Attitude Changes in a Mathematics
Laboratory; Report No. 9 ‑ Teacher Knowledge and Student Achievement in
Algebra. ELMA Reports No. 1 ‑ A Longitudinal Study of Mathematical
Achievement in the Primary School Years: Description of Design, Sample,
and Factor Analyses of Tests; No. 2 ‑ A Longitudinal Study of
Mathematical Achievement in the Primary School Years: Curriculum and Socio‑Economic
Comparisons and Predictions from Previous Achievement. Available on request
from SMSG Headquarters.
MATERIALS
PURCHASABLE: Prices subject to change: See Newsletter No. 37, March 1972. For
New Mathematical Library series, a special school edition is available to
students and teachers from Random House, Inc./School Division, P. 0. Box 457,
Westminster, Maryland 21157, U.S.A. Attn: Order Department for $1.95 per copy.
A hard‑bound library edition priced at $2.95 is available from the same
address.
ADDITIONAL
MATERIALS BEING DEVELOPED: Other NLSMA reports in preparation.
LANGUAGE IN
WHICH MATERIALS: 1. Were originally written: English. 2. Have been or will be
translated; Arabic, Bengali, Chinese, Cypriot, Dutch, French, Greek,
Indonesian, Italian, Japanese, Korean, Portugese, Swedish, Turkish, Vietnamese.
COUNTRIES
IN WHICH MATERIALS ARE USED: Not answered.
PROJECT
IMPLEMENTATION: Not answered.
CHER
PREPARATION:
¥ Consultant
services available for teachers using the
materials:
None.
Activities
conducted for pre‑service and in‑service teacher training: None.
3. Available
pre‑service and/or in‑service teaching materials
for science
educators to use in preparing teachers: Studies
in
Mathematics, Volumes 1‑19.
T. PROJECT
EVALUATION: 1. Has the effectiveness of the materials been evaluated: Yes, by
project staff. 2. Pertinent published research studies: NLSMA Reports. 3. Brief
abstract of in‑house or unpublished research: Other NLSMA Reports in
preparation. 4. Additional evaluative data available to interested individuals:
Arrangements for this are being planned.
U. PROJECT
PUBLICITY: Begle, E.G. "SMSG: The First Decade" The Mathematics
Teacher Volume LXI, No. 3, March 1968, p.239‑245.
V. BRIEF
SUMMARY OF PROJECT ACTIVITIES SINCE 1968 REPORT: Continuation of new
junior high school curriculum project. Additional analyses of NLSMA data.
W. PLANS
FOR THE FUTURE: Phase out of all current activities, except publication program
for existing materials and maintainance of NLSMA data bank, on August 31,
1972.
786
A. PROJECT
TITLE: SECONDARY SCHOOL MATHEMATICS CURRICULUM IMPROVEment STUDY
B. PROJECT
DIRECTOR: Howard F. Fehr, Teachers College, Box 120, Columbia University, New
York, New York 10027, U.S.A.
C. PROJECT
HEADQUARTERS: 1. Contact: Project director. 2. Special facilities or activities
available for visitor viewing: Experimental classes ‑ Courses I‑VI ‑
around metropolitan New York; or reference to some persons in charge of SSMCIS
programs in schools in other parts of the country.
D.
PRINCIPAL PROFESSIONAL STAFF: Director: Howard F. Fehr; Research Associates:
Jeremy Kilpatrick, James Lovett; Research Assistants: John Camp, David
Fuys, Howard Kellogg.
E. PROJECT
SUPPORT: 1. Funding agencies: U.S. Office of Education and National Science
Foundation. 2. Associated agencies: Teachers College, Columbia University.
F. PROJECT
HISTORY: 1. Principal originator: Howard F. Fehr. 2. Date and place of
initiation: November, 1965; Teachers College, Columbia University. 3. Evolution
and development of the project: Based on OECD seminars in Paris (1959),
Dubrovnik (1960), Athens (1965), and the Cambridge Report (1963), a group of
twenty mathematicians and educators met in June 1966 to formulate a position
paper stating the aims and procedures of the study, to construct a blueprint
for the proposed 7‑12 mathematics courses, and to make detailed
recommendations for the mathematical content of Course I. Using these
recommendations as a guide, a team of eight mathematics educators wrote a
textbook for Course I. In each subsequent year (1967‑72) a pre‑planning
conference of SSMCIS staff and advisors and a June working conference including
writers met to review and revise previous texts and to make specific
recommendations for the next text to be written.
G. PROJECT
OBJECTIVES: 1. Overall project purpose: To formulate, construct, and test a
unified secondary school mathematics program (7‑12) for University‑bound
students. 2. Specific objectives: To write, evaluate, and revise text materials
and teachers commentaries for Courses 1‑VI; to assist experimental
schools in the implementation of the SSMCIS program.
H. UNIQUE
CHARACTERISTICS OF THE PROJECT: 1. SSMCIS materials are geared to the college‑bound
student in the upper 20% of academic ability.
818
2. Traditional
division of school mathematics into arithmetic, algebra, geometry, advanced
algebra, calculus is replaced by a unified study of mathematics in which all
branches of mathematics are unified through fundamental concepts (set,
relation, mapping, and operation) and structures (group, ring, field, and
vector space). 3. Probability, statistics, computer programming, and linear
algebra are integrated into the program. 4. Course VI materials are composed of
a text suitable for a half year of study and five booklets (Algebra and
Morphisms; Applications of Probability; Statistics; Matrices, Determinants, and
Eigen Values; Differential Equations) which provide the teacher and students
with options in choosing topics of study.
SPECIFIC
SUBJECTS, GRADE, AGE AND ABILITY LEVELS: Mathematics: Grades 7‑12; ages
11‑18; upper 20% of academic ability.
MAIN
METHODS OF INSTRUCTION USED IN THE PROJECT: Independent study, lectures,
discussion sessions, and the computer, used as a tool in problem solving.
PRESENT COMMERCIAL
AFFILIATIONS: None.
DESCRIPTION
OF MATERIALS ALREADY PRODUCED: 1. Text and Teachers Commentary. 2. Course I,
Book 1, 2 Final Version. 3. Course II, Book 1, 2 Final Version. 4. Course III,
Book 1, 2 Final Version. 5. Course IV, Book 1, 2 Final Version. 6. Course V,
Book 10, 2 Revised. 7. Course VI, Book 1 Experimental, not available to public.
8. Booklets Algebra and Morphisms; Matrices, Determinants, and Eigen Values;
Statistics. 9. Bulletins.
MATERIALS
AVAILABLE FREE: Bulletin 7 (Spring 1972). Send name and address to be included
on our bulletin mailing list.
MATERIALS PURCHASABLE:
Teachers College Press, 1234 Amsterdam
Avenue, New
York, New York 10027, U.S.A.
Textbooks:
Courses I‑IV Part I and Part II $3.25 each part;
Courses I‑IV
Teachers Commentary $5.25.
ADDITIONAL
MATERIALS BEING DEVELOPED: Course V (grade 11) text; Course VI (grade 12) text;
and additional booklets.
LANGUAGE IN
WHICH MATERIALS: 1. Were originally written: English. 2. Have been or will be translated:
Hebrew, French, and perhaps Spanish.
COUNTRIES
IN WHICH MATERIALS ARE USED: United States of America;Israel; Quebec, Canada;
Belgium; Brazil.
R. PROJECT
IMPLEMENTATION: 1. Total number of teachers using any of the materials: 300. 2.
Total number of students using any of the materials: 12,000. 3. Total number of
schools using any of the materials: 200. 4. The totals stated in 10, 2 and 3
are estimated. 5. Name and location of selected schools where the program is in
use: The materials are used in various cities and districts throughout the
U.S.A. (In particular: California, Connecticut, Maryland, Michigan, Missouri,
New Jersey, New York).
S. TEACHER
PREPARATION: 1. Consultant services available for teachers using the materials:
SSMCIS director and staff research assistants can be contacted by telephone or
by mail for assistance regarding program implementation. 2. Activities
conducted for pre‑service and in‑service teacher training: Summer
institutes aimed at preparing teachers to teach the SSMCIS materials are
sponsored by N.S.F. ‑‑ See N.S.F. announcement. 3. Available pre‑service
and/or in‑service teaching materials for science educators to use in
preparing teachers: None.
T. PROJECT
EVALUATION: 1. Has the effectiveness of your materials been evaluated: Yes,
internally. 2. Published research studies: Fey, James T.,PATTERNS OF VERBAL
COMMUNICATION IN MATHEMATICS CLASSES, Unpublished Ph.D. dissertation, Teachers
College, Columbia University, 1968. Hoban, Brother Michael, C.S.C.,
TRANSFORMATION GEOMETRY IN THE JUNIOR HIGH SCHOOL, Unpublished Ph.D.
dissertation, Teachers College, Columbia University. 3. Brief abstract of
in‑house or unpublished research: SSMCIS evaluation efforts have been
primarily formative in nature and include: Staff visits to experimental
classes, written teachers' comments about text materials, bi‑yearly
staff‑teachers meetings, and end‑of‑course exams. Other
evaluation being conducted at the present time includes: Student Attitudes
Survey and Comprehensive Course III Examination. Reports concerning them
are available as of May, 1972. 4. Evaluative data available to interested
individuals: Technical Reports regarding Attitude Survey and Course III
Evaluation are available as of May, 1972.
U. PROJECT
PUBLICITY: Not answered.
V. BRIEF
SUMMARY OF PROJECT ACTIVITIES SINCE 1970 REPORT: Producing, experimenting and
writing Courses IV, V, VI for grades 10, 11, 12. Training teachers in N.S.F.
institutes.
W. PLANS
FOR THE FUTURE: Completion of the total program by June 30, 1973.
A. PROJECT TITLE:
UNIVERSITY OF ILLINOIS ARITHMETIC PROJECT AT
EDUCATION
DEVELOPMENT CENTER THE ARITHMETIC PROJECT
B. PROJECT
DIRECTORS: Prof. David A. Page, Department of Mathematics, University of
Illinois at Chicago Circle, Box 4348, Chicago, Illinois 60680, U.S.A. Mr. Jack
Churchill, Education Development Center, 55 Chapel Street, Newton,
Massachusetts 02160, U.S.A.
C. PROJECT
HEADQUARTERS:
1.
Contact: Mr. Jack Churchill (617)969‑7100.
2.
Special facilities or activities available for visitor
viewing:
Visitors are welcome to inspect written materials of
the
course and to view one or more of the course films by
appointment.
D.
PRINCIPAL PROFESSIONAL STAFF: Jack Churchill, Associate Director and Editor.
E. PROJECT
SUPPORT: 1. Funding agencies: National Science Foundation; Education
Development Center; Ford Foundation; Carnegie Corporation. 2. Associated
agencies: University of Illinois; Education Development Center.
F. PROJECT
HISTORY: 1. Principal originators: David A. Page and Jack Churchill. 2. Date
and place of initiation: 1958; University of Illinois, Urbana, Illinois. 3.
Evolution and development of the project: The Project was formed with a grant
from the Carnegie Corporation of New York. David Page had previously worked
with the late Max Beberman at the University of Illinois Committee on School
Mathematics from its early years, and had worked with the Physical Science
Study Committee. After five years of developing topics and testing them in
elementary classrooms, the Project moved to Education Development Center (then
Educational Services Inc.) in Massachusetts, to prepare materials which would
help teachers learn how to introduce the ideas in their classrooms. Some five
years of further development and refining went into these materials, including
use in 17 institutes for teachers in schools in the Boston area and Illinois.
The result was a 19‑week packaged, self‑contained course which was
released in 1969 for both in‑service and pre‑service use. The
completed course has been used with over 2500 teachers in approximately 75
school systems and colleges, and was revised in 1972 on the basis of this
experience. The program is now available in two parts, each containing ten
sessions, which may be used separately.
G. PROJECT
OBJECTIVES: The central theme of the project is that the study of mathematics
should be an adventure, requiring and deserving hard work. Children who grasp
some of the inherent
846
fascination
of real mathematics while they are in elementary school are well on the way to
success in further study of mathematics and science. Students who are not to
continue a formal study of mathematics deserve a taste of the subject that is
at least as appealing. The project is not attempting to develop a systematic
curriculum for any grade level, in the view that determining an adequate
curriculum is not possible until more alternatives exist to choose among. What
is needed are frameworks that provide day‑to‑‑day, "here‑is‑something‑totry"
ideas for the classroom. The emphasis is on things that the teacher can begin
working with soon. The term "new mathematics" is avoided by the
project. More properly, the project seeks novel ways of doing old
mathematics‑‑new structure or schemes within which can be found
large numbers of interrelated problems revealing significant mathematical
ideas. Teachers participating in an institute work a number of sequences
of such problems each week to become acquainted with the mathematics, and then
begin to make up and try out their own sequences. Throughout its work, the
project has found that improved computational skills usually follow work with
its materials. Children will do impressive amounts of computation in order to
solve problems that interest them.
UNIQUE CHARACTERISTICS
OF THE PROJECT: The Project is designed to convey both mathematics and pedagogy
in an indirect way; to free teachers from the limitations inherent in any
particular text or program; to enable teachers to capitalize on interesting
ideas wherever and whenever they appear (as often from students as from texts);
to encourage teachers to uncover and follow their own best instincts about what
is interesting in mathematics; in short, to teach the creative teaching of
mathematics. No specific teaching style is prescribed; the emphasis
on the creativity of the teacher is in the spirit of open education, although
open classroom styles are not shown in the films. The Project's target
population is all teachers of young children. Course materials are more
valuable in grades 2 through 6 than in K‑l. Within this range, however,
the Project's topics can be adapted and applied extensively either in
themselves or in connection with other programs.
SPECIFIC
SUBJECTS, GRADE, AGE AND ABILITY LEVELS: Mathematics, grades kindergarten
through six; in‑service and pre‑service elementary teachers.
MAIN
METHODS OF INSTRUCTION USED IN THE PROJECT: Independent study, seminars,
discussion sessions and classroom teaching with children. The course for
teachers is based on written lessons, films of classes, discussions, careful
correcting of written work with attention to sources of errors, and the inventing
and adapting by participating teachers of new materials for classes.
K. PRESENT
COMMERCIAL AFFILIATIONS: None.
L. DESCRIPTION
OF MATERIALS ALREADY PRODUCED:
1. General
Information.
2. Ways
to Find How Many.
3. Maneuvers
on Lattices.
4. Well‑Adjusted
Trapezoids.
5. Number
Lines for the Orbiting Atomic Teacher.
6. Do
Something About Estimation.
7. Teaching
Creativity in Mathematics.
8. Arithmetic
With Frames.
9. Functions.
10. A
Sample and Description of Course I.
11. Book:
Number Lines Functions and Fundamental Topics
12. Written
Lessons:
(a)
Introduction to Frames and Number Line Jumping Rules.
(b)
Consecutive Jumps. Distances Jumped. Competing
Number
Line Jumping Rules.
(c)
Parentheses and "Multiplying Before You Add". Stand‑
still
Points.
(d)
Effects of Using Rules in Different Orders.
(Edward)
Introduction to Maneuvers on Lattices.
(f)
Frame Equations. Midpoints. Rules Moving Two Points.
(GCMP)
Rules Moving Two Points, Continued. Composition of
Number
Line Rules.
(h)
Composition, Continued.
(i)
Some Wrong Answers. Composing Number Line Rules to
Move
Two Points to Two Points.
(j)
Artificial Operations.
(k)
More Work With Artificial Operations.
(1)
Maneuvers on Lattices, Continued.
(m)
More Work With Competing Rules. Lower Brackets.
(n)
Lower Brackets and Upper Brackets.
(0)
Graphing Equations With Lower and Upper Brackets.
(p)
Simultaneous Equations. Points and Lines in a Plane.
(q)
Number Plane Jumping Rules.
(r)
Number Plane Rules, Continued.
13. Films:
(a) A
First Class With Number Line Rules and Lower
Brackets
(Lee Osburn, Grade 5).
(b) Which
Rule Wins (Phyllis
R. Klein, Grade 3).
Cc) Standstill
Points (David A. Page, Grade 5).
(d) Three
A's Three B's and One C (David A. Page, Grade 5).
(Edward) A Seven‑Fold
Lattice (Francis X. Corcoran, Grade 5).
(f) Frames
and Number Line Jumping Rules (Lee Osburn,
Grade 5).
(GCMP) Rules
Moving Two Points (David A. Page, Grade 5).
(h) Introduction
to Composition (Marie L. Hermann, GCMP
Grade 5).
(i) Surface
Area With Blocks CPhyllis R. Klein, Grade 1).
(i) Some
Artificial Operations (Phyllis R. Klein, Grade 4).
(K) Counting
With Dots (David A. Page, Grade 2).
848
(1) A
Periodic Lattice (Phyllis R. Klein, Grade 5). (m) Lower and Upper Brackets
(Carol Daniel, Grade 4). (n) Inequalities With Lower Brackets (Francis X.
Corcoran, Grade 5). (of) Graphing With Square Brackets (David A. Page, Grade
5). (p) Graphing Absolute Value Equations (Marie L. Hermann, Grade 2). (q)
Jumping Rules in the Plane Part I (Lee Osburn, Grade 6). (r) Jumping Rules in
the Plane Part II (Lee Osburn, Grade 6). (s) Rotations in the Plane (David A.
Page, Grade 5). 14. Supplements: (a) Answers to Common Questions About the
Institute. (b) Computing With Positive and Negative Numbers. (C) Answers to
Questions About the Film "Standstill Points" (d) Dividing By Zero.
(e) Maneuvers on Lattices. (f) Arithmetic With Frames. (GCMP) Functions. (h)
Using Blocks to Introduce Other Bases of Numeration to a Fourth Grade. (i)
"Surrounding" With Centimeter Blocks. (j) Well‑Adjusted
Trapezoids. (K) Ways to Find How Many. (1) More Suggestions for Lattices. (K) Using
Centimeter Blocks to Introduce Prime Numbers to a Third Grade. (1) Graphing
Number Line Jumping Rules. (m) More Problems With Composition of Number Line
Rules. (n) Graphing Simultaneous Equations. (of) Examples of Questions Dealing
With DDxD. (p) More Work With Number Plane Rules. (q) Hybrid Rules: Jumping
Rules From the Line to the Plane and From the Plane to the Line. (r)
Bibliography. 15. Discussion Notes (for each written lesson and film). 16.
Corrector's Guides (for each written lesson).
M. MATERIALS
AVAILABLE FREE: Items 1‑10 are free in small quantities
from
Education Development Center.
N.
MATERIALS PURCHASABLE: Items 2 and 3, $0.20 each in quantities larger than 2,
available from Education Development Center. Item 11, $3.80 (deduct 25% discount
on orders from schools), available from the Macmillan Company, 866 Third
Avenue, New York, New York 10022. Information on the cost of course materials
is available from EDC.
0.
ADDITIONAL MATERIALS BEING DEVELOPED: None.
849
P. LANGUAGE
IN WHICH MATERIALS: 1. Were originally written: English. 2. Have been or will
be translated: Undetermined.
Q. COUNTRIES
IN WHICH MATERIALS ARE USED: United States.
R. PROJECT
IMPLEMENTATION: 1. Total number of teachers using any of the materials: Over
Over 2500 teachers. 2. Total number of students using any of the materials: Not
known. 3. Total number of schools using any of the materials: Approximately
75. 4. Number of teachers who have adopted the entire program: Not known. 5.
Number of students involved: Not known. 6. Number of schools involved: Not
known. 7. The totals stated in 1 and 3 are estimated. 8. Name and location of
selected schools where the program is in use: Seameo Regional Centre for
Education in Science and Mathematics, Penang, Malaysia; Perkins School for the
Blind, Watertown, Massachusetts; City College New York, New York; Fairfax
County Public Schools, Bailey's Crossroads, Virginia; Model Cities In‑Service,
New Bedford, Massachusetts; Shrewsbury Public Schools, Shrewsbury, Massachusetts;
Principia College, Elsah, Illinois; Watertown Public Schools, Watertown,
Massachusetts; University of North Carolina, Chapel Hill, North Carolina;
University of Virginia, Falls Church, Virginia; Burlington Public Schools,
Burlington, Massachusetts; Marywood College, Scranton, Pennsylvania; Lesley
College, Cambridge, Massachusetts.
S. TEACHER
PREPARATION: 1. Consultant services available for teachers using the materials:
Upon request the project can arrange for former project staff members and
others familiar with the course to serve as consultants. 2. Activities
conducted for pre‑service and in‑service teacher training: The
project no longer conducts institutes; the course materials are designed to be
fully effective without specially trained instructors. (See No. 3 below.) 3.
Available pre‑service and/or in‑service teaching materials for
science educators to use in preparing teachers: The project's principal
activity has been to produce pre‑service and in‑service materials
which are contained in the course described here. Costs will be determined by
the publisher of the materials when one is selected. For costs during the interim
distribution period write to: University of Illinois Arithmetic Project,
Education Development Center, 55 Chapel Street, Newton, Massachusetts 02160.
T. PROJECT
EVALUATION: Effectiveness of the materials has been evaluated by the project
staff.
850
U. PROJECT
PUBLICITY:
1. Science
editorial, 18 May 1962.
2. NCTM
Bulletin for Leaders, September, 1968.
3. Arithmetic
Teacher December, 1967.
4. Arithmetic
Teacher November, 1968.
5. American
Mathematical Monthly March, 1970.
6. Grade
Teacher February, 1971.
7. Local
newspaper items.
V. BRIEF
SUMMARY OF PROJECT ACTIVITIES SINCE 1970 REPORT: Some
editing
of the course and simplification of format have been
carried out
in response to experiences in ongoing institutes.
W. PLANS
FOR THE FUTURE: The project has concluded its major
course‑production
activity at EDC.