Eleven research reports in the area of models of learning mathematics are presented in this publication of the Mathematics Education Reports series. The papers represent a mixture of theories, viewpoints, and references to other areas. Content areas addressed range from preschool to college levels. All the papers are concerned with the learning of mathematics and all are attempts to clarify and apply theory through the mechanism of models. The papers rely on observations of students and have possible implications for classroom instruction. Topics include: (1) modeling and theorizing in mathematics education; (2) experimental models for the child's acquisition of counting and of addition and subtraction; (3) a logical-cognitive view of learning and teaching mathematical concepts; (4) instructional and theoretical implications of a mathematical model of cognitive development; (5) a learning model for some algebraic concepts; (6) abstraction, generalization, and the design of mathematical experiences for children; (7) towards a model for the teaching of mathematics as goal-directed activity; (8) preliminary steps toward a cognitive model of learning primary mathematics; (9) intraconcept analysis of rational number addition and subtraction, indirect validation studies; (10) an organizational model of abilities related to inference in mathematical contexts; and (11) logical reasoning learning hierarchies. (MP)