Five invitational addresses presented at the Fourth National Conference of the Research Council for Diagnostic and Prescriptive Mathematics are included in this document. "The Nature of the Clinical Investigation" (Romberg and Uprichard) discusses the validity of clinical investigations and suggests guidelines for conducting such studies of instructional techniques. "A Clinical Model for Diagnosing Mathematical Deficiencies, (MD) Squared, Incorporating Educational Cognitive Style" (Speer) describes a process that takes into account factors that contribute to an individual's unique learning style and content deficiencies. "Feedback in Diagnostic Testing" (Engelhardt) reviews the literature on feedback as it relates to testing. Difficulties in existing research are identified, and needed research and changes in diagnostic procedures are suggested. "Counting Performance and Achievement: Some Preliminary Observations" (Callahan) considers the rote and rational counting performance of young children, with variables and relationships discussed. Finally, "A Perspective on the Future" (Heddens) briefly poses questions concerning the direction of research efforts and provides suggestions for attaining the goals of the Research Council. Three reaction papers are also included. (MNS)