THIS MANUAL WAS DESIGNED TO HELP TEACHERS TO ACQUIRE THE KNOWLEDGE AND SKILLS NECESSARY FOR THE EFFECTIVE TEACHING OF MODERN MATHEMATICS. THE SPECIFIC PURPOSE OF THIS MANUAL IS TO IDENTIFY SOME OF THE CONTENT WHICH IS FUNDAMENTAL TO THE USE OF LEARNING AIDS AND TO SUGGEST SPECIFIC EXAMPLES FOR THE USE OF VISUAL AND MANIPULATIVE DEVICES IN ELEMENTARY CLASSROOM PRESENTATIONS. MATERIALS FOR AN INSERVICE COURSE IN MATHEMATICS ARE DEVELOPED FOR (1) ARITHMETIC AS A SYMBOLIC LANGUAGE, (2) OUR OWN NUMERATION SYSTEM, (3) OTHER SYSTEMS AND BASES, (4) TEACHING THE OPERATIONS WITH LEARNING AIDS, (5) FRACTIONS, AND (6) INFORMAL GEOMETRY AND MEASUREMENT. THE PUBLICATION IS INTENDED FOR STATE SUPERVISORS OF MATHEMATICS AND CLASSROOM TEACHERS. THIS DOCUMENT IS AVAILABLE FOR $1.00 FROM THE SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE, WASHINGTON, D.C. 20402. (RP)