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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)

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)

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 consists of test questions used in three state high schools teaching the new Matriculation pure mathematics course (approximately grade 12). This material was circulated to all schools teaching this course as a teacher resource. The questions are arranged in 14 papers of varying structure and length. Most questions are of the essay type. The topics covered include integral and differential calculus, algebraic and trigonometrical functions, the binomial expansion, linear algebra, fields and complex numbers. (MM)

An easier way for students to draw families of conics.

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

THERE are many well-known phys ical representations of convergent se quences. A standard example is to traverse the length of a room by walking half the remaining distance each time a "step" is taken. One sequence corresponding to this model (0, y2, %, %, i^e, . . .) has 1 as its obvious limit. There is a sequence (see table 1) that also has a simple and interesting physical model but whose limit is not so obvious. And precisely because of these facts, the sequence and its physical representation can be used as the basis and motivation for several interesting lessons. The physical model requires the student (or the teacher) to fold a strip of paper according to the directions given in the next section. One lesson that can be based on this exercise involves discovery of a mathematical description of the resulting sequence and finding the limit of this sequence. This is done in the section called "Mathematical Description of the Se quence." In the last section, "Other Re lated Lessons," some other lessons that can be built around the physical model are suggested.

The proof of Theorem 1 in the paper “On Krull’s conjecture concerning valuation rings” (vol. 4 (1952) of this journal) is not correct. We want to give here a corrected proof of the theorem: From p. 30, l. 14 to p. 31, l. 7 should be changed as follows.

The paper is entitled Solutio facilis problematum quorumdam geometricorum difficillimorum, and is printed in Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom. xi., pp. 103–123. The volume is for the year 1765; the title-page is dated 1767.

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)

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Results of A. Feldzamen on semi-similarity of operators are proved here using matrix methods. The use of these methods yields simpler proofs, the formulations of the theorems assume a more transparent form.