This set of papers, selected from presentations at the Fourth and Fifth Annual Conferences of the Research Council for Diagnostic and Prescriptive Mathematics, are of primary interest to mathematics educators. In the first paper, Romberg describes a model for diagnosing mathematical learning difficulties which extends the diagnostic process beyond the mere identification of unpassed objectives. Underhill then provides a basis for distinguishing between learning hierarchies and diagnostic hierarchies, pointing out differences in methods used to validate each. Reports by Hill and Underhill, and by Underhill and Causey explain the methods used to generate diagnostic hierarchies. Novillis discusses the effect of language on learning mathematics, presenting a method for investigating children's acquisition of mathematical terms. Research studies on diagnosing mathematical learning difficulties among deaf children are reviewed by Sinatra, while Sovchik explores the effects of anxiety on mathematical learning. Crenson provides concise descriptions of several tests for assessing students' perceptual skills and critiques the role of learning style in the diagnostic process. Finally, Fennell reports on a study comparing two approaches for training preservice teachers to use diagnostic-prescriptive methods. (MNS)