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Ordering theory was used ro empirically determine hierarchies of developmental precedence for Piager's 16 logical combinations. In two studies carried out in the Netherlands, subjects were tested in individual interviews. In Study 1, 94 subjects responded to statements about odd and even numbers; in Study 2, 30 subjects responded to statements about line segments. For each study, a hierarchy of operations was produced from the responses. The two hierarchies were similar to each other and to a hierarchy from a previous study in which a nonmathematical task was used. Different linguistic forms for the same logical operation appeared to yield greater differences in performance than differences in tasks.

Two recent trends in the teaching of mathematics can often go hand in hand. These two are the spiral approach to content development and the discovery-teaching/learning method. This paper will present some topics that illustrate how these approaches to content and method may complement each other. Space will not permit the full explication as it might occur in a classroom, but it is hoped that both the discovery approach to mathematical conjecture as well as the spiral nature of the topics will be evident. The specific topics discussed here range from those for the upper elementary school pupil to those for the high school senior—the exact placement depends not only on the topic itself, but on the language and notation as well.