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This booklet is a collection of the papers presented at a joint session of the Mathematical Association of America and the National Council of Teachers of Mathematics during the fiftieth annual meeting of the MAA. These papers were presented under the headings of "Geometry and School Mathematics,""High School Geometry," and "School Geometry and the Future." Eight papers are presented in their entirety. An abstract of a ninth paper, "Geometry: The Cambridge Conference View" by Edwin E. Moise is also included. Authors of the individual papers do not act as spokesmen for the MAA or NCTM but express personal viewpoints. (FL)

This document is the report of a curriculum writing session. Using the recommendations and suggestions of the planning conference (SE 012 733), the writers produced detailed outlines for the grade seven through nine curriculum. The purpose and rationale of each unit are stated, and the separate sections outlined. Sample exercises are included for most units. The aims of making mathematics more relevant to real world problems, and of avoiding excessive abstraction and rigor, are continually stressed. Also included are three general reports on geometry, real analysis, and mathematical models; several position papers on specific topics; and a comparison of the new syllabus with the "first round" curriculum. (MM)

Sixteen mathematicians from universities, schools and industries met for five days in 1966 to plan a "second round" curriculum which would place greater emphasis on the relevance of mathematics to problems of the real world. Their deliberations are reported in this document. The three main recommendations were that the curriculum should contain: (1) frequent consideration of mathematical models of significant problem situations; (2) an introduction to those mathematical concepts which are important to the general citizen; and (3) probability, logic, computing and flow charts, and the concept of function, in addition to traditional topics. A series of reports and papers on various topics contain many suggestions for putting these recommendations into effect, especially in grades seven through nine. Also included are an annotated bibliography on applications of mathematics, and a list of topics for the use of mathematical models. (MM)

This document is the report of a curriculum writing session, and covers three activities: (1) a textbook for grade seven, written using the outlines produced the previous year (SE 012 735), was reviewed and criticized; (2) the outlines for grades eight and nine, also produced the previous year, were completed in greater detail; and (3) considerable attention was given to possible sequences of courses for grades 10-12. Broad outlines for several courses at this level were written, including a deductive block; a year of vector geometry, linear algebra, and elementary functions for grade 10; a course in analytic geometry and algebra; and a course in elementary functions and calculus. Several papers treating specific topics are included in this report. (MM)

The "Television Polytechnic" program of 1966 - 1970 was a joint effort of the Polish Ministry of Higher Education and Polish Television to provide televised education for working students. The subjects taught were those normally included in the first two years of the higher technical schools--math, physics, geometry, chemistry, mechanics of materials, and electrotechnics. Each semester lectures were given by professors from the higher technical schools, and students were provided with texts and access to consultation centers. Assessment indicated that television lectures were helpful to regular students as a complement to their classroom experiences and were a useful means of providing basic instruction to those not enrolled in a higher school. Preparatory and first year television courses will remain as permanent elements in the Polish educational system and the recent implementation of a second channel will lead to an expansion of programs, including many aimed at post-graduate audiences. (PB)

A theory of psychophysics is discussed that enlarges the classical theory in three general ways: (1) the multidimensional nature of perception is made explicit; (2) the transformations of the theory are interpreted geometrically; and (3) attributes are distinguished from sensations and only partially ordered. It is shown that, with the enlarged theory and some elementary geometry, one may use a few ideas to explain many qualitative results, to give a credible and precise account of quantitative properties of data, and to design experiments which seem worth doing whether the theory is valid or not. In developing the new theory, the major criticisms of the classical theory are answered. This paper is limited to one of three aspects of the typical psychophysical measurement experiment--the relation between perceptions and attributes. Although the theory has been extended to category and rating scales, only magnitude estimation and related measurement procedures are discussed here. The scope of this paper is further limited to phenomena which have analogues in several sense modalities. (Author/KM)

This paper presents a method of encoding geometric line-drawings in a way which allows sets of such drawings to be interpreted as formal languages. A characterization of certain geometric predicates in terms of their properties as languages is obtained, and techniques usually associated with generative grammars and formal automata are then applied to the geometric framework. Section 1 of the paper specifies the geometric framework; section 2 develops the background material on formal languages, grammars, and automata; section 3 is concerned with geometric predicates; section 4 covers invariance theorems; and section 5 indicates areas for further investigations. (DT)

This NSF-sponsored conference on the teaching of mathematics at the junior high level involved mathematics teachers and teachers of the natural and social sciences. Papers written for the conference form the bulk of this report. Summaries of the papers and general discussions are organized into shorter reports to give some guidelines which could be useful to any group wishing to create curricular materials for junior high school mathematics or simply to get some ideas for direct application to the classroom. The topics of the report were: what mathematics junior high students do and do not know; new emphases in content; mathematics in geography, social science and biology; teaching strategies and styles; mathematics and language; and teaching training. (JP)

These materials were written with the aim of reflecting the thinking of Cambridge Conference on School Mathematics (CCSM) regarding the goals and objectives for school mathematics K-6. In view of the experiences of other curriculum groups and of the general discussions since 1963, the present report initiates the next step in evolving the "Goals". Three areas considered in this report are geometry, functions in preparation for calculus, and applications. Two working papers are presented on applications - probability and mechanics and slopes. One working paper on circular functions is included. Fifteen working papers are presented involving geometry and geometrical concepts. The papers on geometry include examination and description of common objectives; playing with figures, blocks, and tessellations; constructions; graphs and polygons; tessellations; dissection of figures; order; measurement; similarity and map making; symmetry, congruence, and rigid motion; transformation groups; rotations and matrices; iterated reflections in mirrors; knots; and spheres, cylinders, and torus. [Not available in hard copy due to marginal legibility of original document]. (RP)

The author describes a way of teaching a one-year geometry course which includes plane geometry, solid geometry, coordinate methods of proof, and vector methods of proof. His method is based upon encouraging students to use the properties of geometric figures that they already accept without formal proof. He employs paper folding extensively to arrive at many of the plane geometric concepts. (CT)

THE FIRST PAPER, "REPORT OF MATHEMATICS CONFERENCE," IS A SUMMARY OF DISCUSSIONS BY 29 PARTICIPANTS IN A CONFERENCE ON CURRENT PROBLEMS IN MATHEMATICS EDUCATION RESEARCH. REPORTED ARE (1) RECENT PROGRESS, PROBLEMS, AND PLANS OF CURRICULUM DEVELOPMENT GROUPS, (2) GENERAL FORMULATION OF CURRICULUM AND METHODS, (3) TEACHER TRAINING, (4) COMMUNICATION, AND (5) RESEARCH MANPOWER. THE SECOND PAPER, "ELEMENTARY MODERN MATHEMATICS FROM THE ADVANCED STANDPOINT," GIVES A RECOMMENDED SYLLABUS FOR THE FIRST FIVE SEMESTERS OF A SIX-SEMESTER COLLEGE MATHEMATICS COURSE TO BE TAKEN BY POTENTIAL HIGH SCHOOL TEACHERS. FINALLY, THE THIRD PAPER, "SCHOOL MATHEMATICS STUDY GROUP (SMSG) AND THE GIFTED CHILD," REPORTS THE REASON FOR INCLUDING EXPLICIT RECOMMENDATIONS FOR EDUCATING THE GIFTED CHILD IN ANY LONG-RANGE CURRICULUM REFORM. (RP)

This document is a compilation of abstracts of 28 research papers presented at the 52nd Annual Meeting of the National Council of Teachers of Mathematics. The major portion of the reports concerned methods of instruction; nine were related to college mathematics, ten on secondary school mathematics, and two at the elementary school mathematics level. Five reports involved research on learning theories and two dealt with teacher education. (JP)

This paper appraises recent developments in the curriculum and teaching of mathematics in the secondary schools and analyzes mathematics education in the United States with particular emphasis on Nebraska. The first section focuses upon characteristics that distinguish the modern secondary school mathematics curriculum from the traditional. The following sections emphasize the understanding and the structure of mathematics. Ability groupings and the various types of programs which have resulted from it are also discussed. Further analyses deal with a college preparatory sequence for Grades 9-12, new mathematics programs to attract and to motivate students, and new techniques in mathematics at the secondary level. Suggestions for implementing new developments are also provided. (RP)

This unit describes an experience in informal geometry that is based on work with construction paper and milk cartons. The description is mostly of work actually carried out by children in the elementary grades involving such mathematical concepts as congruence, symmetry, the idea of a geometric transformation, and some basic notions of elementary group theory. The purposes of the unit are (1) to give students experience in visualizing two and three dimensional objects, and (2) to give students opportunity to learn to raise questions, pose problems, and learn to solve them. (RP)

This booklet explains the historical background and construction techniques for various sets of uniform polyhedra. The author indicates that the practical significance of the constructions arises in illustrations for the ideas of symmetry, reflection, rotation, translation, group theory and topology. Details for constructing hollow paper models are provided for the five Platonic solids, miscellaneous irregular polyhedra and some compounds arising from the stellation process. (RS)

Assessed was the performance of subjects who had been exposed to transformational geometry treatment. The performance test consisted of geometry achievement and spacial ability measures. Second and third grade subjects were randomly assigned to experimental and control groups. The experimentals received a twelve lesson treatment and the controls a one lesson treatment concerning rigid motion and congruence concepts. Pretest and posttest results indicated that the experimentals performed significantly better than the controls on the achievement test, but not on a space test. [Not available in hardcopy due to marginal legibility of original document.] (Author/CT)

THIS PAPER IS CONCERNED WITH GRAPHIC PRESENTATION AND ANALYSIS OF GROUPED OBSERVATIONS. IT PRESENTS A METHOD AND SUPPORTING THEORY FOR THE CONSTRUCTION OF AN AREA-CONSERVING, MINIMAL LENGTH FREQUENCY POLYGON CORRESPONDING TO A GIVEN HISTOGRAM. TRADITIONALLY, THE CONCEPT OF A FREQUENCY POLYGON CORRESPONDING TO A GIVEN HISTOGRAM HAS REFERRED TO THAT POLYGON FORMED BY CONNECTING THE MIDPOINTS OF THE TOPS OF THE RECTANGLES MAKING UP THE HISTOGRAM. THE MOST IMPORTANT DEFICIENCY IN THE TRADITIONAL FREQUENCY POLYGON IS THAT THE AREA OF ANY SPECIFIC RECTANGLE IN THE UNDERLYING HISTOGRAM IS GENERALLY NOT EQUAL TO THE AREA UNDER THE FREQUENCY POLYGON OVER THE SAME INTERVAL. DUE TO THIS DEFICIENCY, DATA ARE SELDOM PRESENTED IN THE FORM OF A FREQUENCY POLYGON. (HW)

Twenty second- and third-grade children were tested on their ability to make judgements of certain area and volume equivalences. The children were then presented a conservation-of-liquid task. Of the 17 children who gave correct conservation responses, only two had solved the area and volume problems. Thus, contrary to what is suggested by learning theorist R. Gagne, knowledge of area and volume relationships is not a prerequisite for conservation of liquid. Gagne also suggested that acquisition of conservation of liquid in one type of container (e.g., rectangular) might precede conservation of liquid in another type of container (e.g., cylindrical), but no child was found who gave the correct response for rectangular containers who did not also give the correct response for cylindrical containers. The answers children give when asked to explain how they know that the quantity of liquid is the same after a transformation can be categorized as: (a) Identity ("It's the same water; nothing changed."); (b) Compensation ("It's fatter but lower."); and (c) Reversibility ("If you poured it back, it would be the same."). Identity was used by 78 percent of the seven-year-olds and 68 percent of the eight-year-olds, with 93 percent of the ten-year-olds using Compensation. These results are interpreted to mean that the initial acquisition of conservation of liquid is not dependent upon knowledge of geometry. (Author/PR)

2 STUDIES DESIGNED TO EXPLORE THE ROLE OF THE TEACHER IN THE CLASSROOM ARE DESCRIBED. THE FIRST STUDY USED ABOUT 5 HOURS OF PROGRAMED HIGH SCHOOL GEOMETRY MATERIALS. IT WAS HYPOTHESIZED THAT EFFECTIVENESS OF THESE MATERIALS WOULD BE INCREASED IF THE TEACHER ACTIVELY AUGMENTED INSTRUCTION THROUGH REPHRASING DIFFICULT ITEMS, ADDING EXAMPLES, WORKING PROBLEMS WITH THE STUDENT AND BRANCHING THE STUDENT TO DIFFERENT PARTS OF THE PROGRAM. COMPARISON OF THIS ACTIVE TEACHER ROLE, WITH A MERELY MONITORIAL ONE, REVEALED NO DIFFERENCES AMONG TREATMENT GROUPS ON A POSTTEST. RESULTS LEAD TO THE CONJECTURE THAT THE TEACHER SHOULD PROGRAM HIS OWN BEHAVIOR AS CAREFULLY AS THE INSTRUCTIONAL MATERIAL IS PROGRAMMED, IN ORDER TO BE ABLE TO AUGMENT THE MATERIAL. THE SECOND STUDY IS AN ATTEMPT TO APPLY EMPIRICAL TRIAL-AND-REVISION PROCEDURES TO IMPROVE CLASSROOM INSTRUCTION FOR FIRST-GRADE MEXICAN-AMERICAN CHILDREN. CLASSROOM OBSERVATION OF READING INSTRUCTION IN 2 ELEMENTARY SCHOOLS COUPLED WITH A TESTING PROGRAM REVEALED PARTICULAR DEFICIENCY IN USE OF SOME PARTICULARLY CRITICAL ADJECTIVES AND PREPOSITIONS. THE PLAN IS TO DEVELOP A DECISION FLOW CHART FROM USEFUL TECHNIQUES TO HELP THE TEACHER SELECT MATERIALS AND METHODS FOR DIFFERENT CHILDREN. USE OF OTHER CHILDREN AND PARENTS AS TUTORS, AND TANGIBLE REINFORCERS AS WELL AS MANY TYPES OF INSTRUCTIONAL MATERIALS ARE UNDER CONSIDERATION. PAPER PRESENTED AT THE WESTERN PSYCHOLOGICAL ASSOC. CONVENTION, SAN FRANCISCO, CALIF. MAY 6, 1967. (AF)

This is a collection of 24 papers presented or prepared by participants of the Comprehensive School Mathematics Program (CSMP) International Conference in 1970 covering a wide range of topics in mathematics and geometry education. Conference recommendations are summarized, including the use of geometry to lead into other branches of mathematics and to teach conceptual model building, abstention from using geometry primarily to teach logic, and improvement in teacher training courses. The papers are both mathematical and educational in emphasis, covering mathematical topics such as polygons, topology, geometric algebra, conics, inversive geometry, Euclidean geometry, and axiomatics. Educational topics include the relevance of geometry at the seconday level and problems in modern mathematics education, with suggestions for new approaches, particularly with respect to geometry. (JM)

In order to assess achievement in technical drawing at the secondary school level, an experimental examination was conducted using 333 boys and 2 girls from 13 schools. The objectives of the course in order of importance were: (1) Understanding and Comprehension; (2) Use; and (3) Communication. Three papers were formed: Paper I, an objective comprehension test covering a wide range of isometric and orthographic work, contained over 200 items and required right or wrong answers. Time: 1hr. 45min.; Paper II, drawing from the solid rather than drawing from the pictorial views, was "marked by impression" with four examiners sorting the entries into five grades. Time: 1hr 15min.; and Paper III tested the candidate knowledge of geometry, with seven questions and was more conventional than either Paper I or Paper II. Questions called for drawing skills or written responses. A normal marking schedule was used. Time: 2hrs. 30 min. An evaluation of the results established beyond a reasonable doubt that the two new methods employed, that is, Paper I and Paper II, are feasible and valid, and allow the examiner to change the tests over a period of years thereby avoiding the stereotyped patterns. Paper I enabled examiners to test candidates' comprehension thoroughly over a wider range than is usually possible in a single paper. Paper II was successful in that results showed overall ability was being assessed, and in Paper III more than one objective was being pursued. (Related document is TM 001 808.) (LS)

The final report of a two-year study concerned with the digital-computer processing of graphical data. Five separate investigations carried out under this study are described briefly, and a detailed bibliography, complete with abstracts, is included in which are listed the technical papers and reports published during the period of this program. The investigations consisted of (1) a study of a linguistic approach to pattern recognition, (2) the optimum two-dimensional allocation problem, (3) the hidden-line problem for quadricsurfaced solids, (4) the reconstruction problem for irregular polyhedra, and (5) the data-structure problem for computer graphics. (Author)

Reported are the results of a study which investigated the effect of temporal position of review through two experiments. Experiment I involved one review; Experiment II involved two reviews. In Experiment I, eighth grade students were pretested, presented with the learning task (four rules from algebra and geometry), and randomly assigned to one of four groups. Reviews were presented one day (Group I), one week (II), and two weeks (III) after criterion. The control group got no review. Three weeks after the original learning, a retention test was administered. Although the review groups retained more than the control group, temporal position was not significant. Seventh grade students were used in Experiment II. Reviews were made after one and two days (Group I), one and seven days (II), or six and seven days (III). The control group received no review. A three-week retention test showed that the review groups retained significantly more than the control group. Group II retained significantly more than Group I. It was concluded that one review is more effective than no review, while an early and a late review produce optimal retention over a three-week interval. (Author/RS)

This publication provides a forum for mathematics teachers who are interested in developing creative approaches for teaching mathematics. It consists mainly of articles under two heads--papers read at seminars on mathematics teaching, and original articles or reprints related to teaching and learning mathematics. There is a reference section which will be expanded in subsequent volumes--a list of symposia on mathematics teaching, a list of journals relating to teaching and learning mathematics, and a list of centers studying the problems of mathematical education. Most of the materials are printed in both French and English. (RP)

Directions are given for seven activities involving the folding of paper strips to illustrate geometric concepts. Properties of pentagons, triangles, hexagons, and Mobius bands resulting from the various foldings are discussed. (DT)

Original Russian version of this paper appeared in To Honor Roman Jakobson,'' II: The Hague, Mouton, 1967. (RS)