Presented are new principles and methods in the instruction of mathematics, with the primary focus at the elementary school level. The first half has the following sections: (1) The Influence of Environment on Child Behavior; (2) Pre-Mathematics; (3) The General and the Abstract; (4) The Passage from Pre-Math to Mathematics; (5) Building a Mathematics Program; (6) New Mathematics Programs; (7) The OPI Mathematics Project in Hungary; (8) Dialogue at Sherbrooke, Quebec; (9) A Visitor Looks at Sherbrooke; and (10) The New York Project. The second half is an evaluation of the three studies that are presented in the first. These investigations are referred to as the Budapest, Sherbrooke, and New York Studies. Chapter titles in this portion of the document are: (11) The Nature of Evaluation; (12) Difficulties in Educational Evaluations; (13) Theoretical Model of the Evaluation for the Dienes Programs; (14) The Structure of the Three Evaluation Studies; (15) Traditional and Modern Mathematics; (16) Fatigue (Budapest Study); (17) Attention (Budapest Study); (18) Learning Ability; (19) Creativity; (20) Perservation and Sociometric Structure; (21) Attitude; (22) Anxiety--Achievement Motivation--Effects on Personality; (23) Attitude Toward Problems; and (24) Summary. (MP)