3 hypotheses were tested concerning the developmental relation between children's concepts of physical uncertainty and their comprehension of a speaker's uncertainty. 2 cognitive tasks of physical uncertainty were used to assign 56 subjects (aged 5-4 to 17-11) to 1 of 3 cognitive stages. 2 tests for comprehension of speaker uncertainty were administered to all participants. The results indicated that cognitive stage was related to (a) comprehension that a speaker could be uncertain, (b) comprehension that uncertainty could be expressed in different degrees or magnitudes, and (c) the internal consistency of judgments made about the relative degree of uncertainty conveyed by an utterance. These findings are interpreted as evidence for the position that development of cognitive stages is structurally related to comprehension of speech act uncertainty.

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Flexibility in performing mental calculations has become an important focus for mathematics educators, with research surging over the past two decades. Contemporary research results lends strong support for the development of mental flexibility in elementary classrooms. This report focuses on three interrelated themes: (1) a model of mental calculation that allows the study of flexible solution processes; (2) a review of definitions of flexibility and summary of reported research; and (3) an examination of the prerequisites for promoting mental flexibility. The approach termed Zahlenblickschulung is introduced with examples of activities that encourage students to sort numerical problems and reason about the sorting.

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