These materials were written with the aim of reflecting the thinking of The Cambridge Conference on School Mathematics (CCSM) regarding the goals and objectives for school mathematics. Presented are plans for teaching 15 inequality lessons for above average first grade students. The discovery approach is utilized by the teacher in order to involve…

The major objective of this study was to investigate differential performances among 5-year-old children when using transitivity of matching relations. Instruments were constructed to measure the subjects' (1) knowledge of matching relations, (2) ability to conserve the relations, and (3) proficiency in making inferences using the transitive…

In this paper, a new item analysis index, e, is derived as a function of difficulty and discrimination to represent item efficiency. Item discrimination in this paper is not independent of item difficulty, and it is demonstrated algebraically that the maximum discriminating power of an item may be determined from its difficulty. Item efficiency is…

Dr. Ruth Hoffman is a mathematics professor who founded a mathematics laboratory at the University of Denver ten years ago. Here her methods of teaching the subject are demonstrated. Describes the more basic teaching aids that she promotes and why the laboratory has achieved an international reputation. (RK)

Ten recent results in pure mathematics are described, covering the continuum hypothesis, Diophantine equations, simple groups, resolution of singularities, Weil conjectures, Lie groups, Poincare conjecture, exotic spheres, differential equations, and the index theorem. Proofs are omitted, but references are provided. (DT)

The author discusses three forms of understanding statements or sets of statements in mathematics: (1) knowledge of their application; (2) knowledge of their meaning; and (3) knowledge of the relationships among theory statements. (MJB)

This study examined the ability of Israeli students to transfer mathematical concepts necessary for the tenth grade chemistry curricula. Both quantity (the percentage of students capable of successfully transferring), as well as quality (building a causal model of student and situational factors contributing to transfer of training) were…

In this paper seventh-grade pupils' ways of handling aspects of probability have been investigated. The aspects in question were embedded in a dice game, based on the total of two dice. Four different set-ups of dice were included in the situation in which they were up to explore optimal strategies for winning the game. How children understand…

In this paper, we present a didactic analysis of the mathematical concept of implication under three points of view: sets, formal logic, deductive reasoning. For this study, our hypothesis is that most of the difficulties and mistakes, as well in the use of implication as in its understanding, are due to the lack of links in education between…

In this paper, I describe how undergraduates can develop their understanding of the concept of proof by viewing the act of proving as a procedure. Such undergraduates first understand proof as an algorithm, or a step-by-step mechanical prescription for proving certain types of statements. The students can then condense this algorithm into a…

The fundamental opposition of two types of discourse, mathematical proof and argumentation to the problem of validation in mathematics is explored. The naturalistic study of interaction in classrooms suggests the possibility of a mathematical argumentation to which students have access by the practice of discussions ruled by socio-mathematical…

This textbook seeks to (1) present to the young mind, unacquainted with the methods of exact reasoning, the elementary principles of arithmetic in their simplest form and combination; and (2) explain and illustrate the various applications of arithmetic in the transactions of business, and thus make known its great practical utility.

Demonstrated are binary operations on whole numbers which are commutative but not associative, or associative but not cummutative. Also presented are sets of numbers within which the Unique Factorization Theorem does not hold true. (JP)

Mathematical Induction is used for direct proofs of many theorems; its logical equivalent, the Well-Ordering Property, replaced it to yield indirect proofs by contradiction. The author claims the later property is easier to comprehend and that it affords a non-geometrical example of proof by contradiction. (JP)