A two-part communication model for teaching elementary mathematics is proposed. Part I delineates symbols commonly used in teaching arithmetic. Parts of mathematical language (mathematical objects, relations, operations, expressions, and sentences) are compared to analogous parts of the English language. Part 2 is a conceptualization of strategies that teachers can use in the communication process. The four distinct components of the model (manipulatory verbal stage involving the concrete, manipulatory verbal stage involving the concrete followed by symbolization, symbolization as an instruction to manipulate the concrete, and symbolization without reference to the concrete), each of which constitutes a particular sequential communication strategy, are described, and particular features of the model (individualization of instruction, identification of prerequisites, remediation, individual differences, computation, open sentences, and problem solving) are considered. Also, a problem solving model (derived from the communication model and incorporating nine heuristics of inquiry--enumeration of specific cases, deduction, inverse deduction, analogy, preservation of enabling principles, variation, continuity of form, existential counter example, and determination of limits) is presented. (This document and SP 002 155-SP 002 180 comprise the appendixes for the ComField Model Teacher Education Program Specifications in SP 002 154.) (Author/SG)