This article presents a constructivist approach for teaching mathematics to prospective elementary school teachers in USA. This approach employs a model of a "mathematical situation," a set of physical operations and a physical language to reason about students' mathematical doings. One of the primary goals of this approach is to promote a reasoning attitude toward the learning of mathematics by prospective elementary school teachers. At the same time this approach encourages the development of their reasoning skills. This approach provides a less rigid frame for discussing mathematics "doings", [The word "doings" in this manuscript refers to the variety of mathematical activities, from thinking on a mathematical problem, using manipulatives, graphing, solving and so forth.] which is still structured enough to allow prospective teachers to appreciate the kinds of doings and argumentations found in mathematics. It provides a concrete-like basis that serves to promote the understanding of arithmetical concepts (especially fractions). In the context of teacher education, this approach adds structure and content to the usual curriculum of basic mathematics courses for future elementary and middle school teachers. It grants future teachers a higher degree of flexibility in dealing with their students' questions and learning difficulties.