Analyzes the relationship between intuitions and structural schemata. Indicates that intuitions are generally based on structural schemata and the transition from schemata to intuitions is achieved by a particular process of compression. (Contains 38 references.) (Author/ASK)

Students react similarly to a wide variety of conceptually unrelated situations. Describes a rule that is manifested when two systems are equal with respect to certain quantity A, but differ in another quantity B. Indicates that in such situations students often argue that the same amount of A implies the same amount of B. (Contains 30…

New forms of mathematical knowledge are growing in importance for mathematics and education, including tacit knowledge, knowledge of particulars, language, and rhetoric in mathematics. Highlights the importance of attending to rhetoric and the range of rhetorical styles in school mathematics, and the need for explicit instruction in this area.…

One sentence answer to the question in the title is that the ability to prove depends on forms of knowledge to which most student are rarely, if ever, exposed. Presents more detailed analysis, drawing on research in mathematics education and classroom experiences. (Contains 44 references.) (Author/ASK)

Considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum, such as children's arithmetic shows divergence in performance. Explains how students cope with the transition to advanced mathematical thinking in different ways, leading once more to a diverging…

Illustrates distinctions among kinds of knowing. Distinguishes knowing-to from other forms of knowing, and explores implications of that distinction for teaching and learning mathematics. Proposes that knowing-to act in the moment depends on the structure of attention at the moment, and on what one is aware of. (Contains 75 references.)…

Addresses issues related to the ways teachers learn mathematics and the teaching of mathematics, and the relevance of those ways to their professional development. Mathematical experiences need to be congruous with the kind of teaching expected of the reflexive, adaptive teacher. Presents both practical and theoretical considerations of how these…

Discusses what important ideas about forms of knowing mathematics should be included in mathematics methods courses for preservice teachers. Proposes ideas related to Shulman's framework of teacher knowledge. Provides a brief discussion of the implications each idea holds for teaching mathematics, and makes some suggestions about experiences that…

Describes a course in Cantorian Set Theory relating to prospective secondary mathematics teachers' tendencies to overgeneralize from finite to infinite sets. Indicates that when comparing the number of elements in infinite sets, teachers who took the course were more successful and more consistent in their use of single method than those who…

Examines an attempt to encourage integration of knowledge learned in the academy with knowledge learned in practice as a means to challenge educational practitioners' (i.e., teacher leaders and inservice teacher educators) existing conceptions and beliefs and promote intellectual restructuring. (Contains 16 references.) (Author/ASK)

Offers an outline and a characterization of the didactics of mathematics--alias, science of mathematics education--as a scientific and scholarly discipline. Discusses why its endeavors should be of interest to research mathematicians. Presents a number of findings in the discipline, including the astonishing complexity of mathematical learning and…

Presents a study on the meanings of average and variation displayed by pediatric nurses. Traces how these meanings shape and are shaped by nurses' interpretations of trends in patient and population data. Suggests a theoretical framework for making sense of the data that compares and contrasts nurses' epistemology with that of official…

Examines a theoretical perspective on the ways in which children progress in learning mathematics. Suggests that there is difficulty in associating teaching discourses with the mathematics they locate that can result in an incommensurability between alternative perspectives being offered. Resists attempts to privilege any particular account but…

Presents a theoretical framework based on interviews with undergraduate students and written questionnaires to study how undergraduate students cope with abstract algebra concepts. Indicates that students' responses can be interpreted as a result of reducing the level of abstraction. Examines the theme of reducing abstraction based on three…

Develops a model to examine teachers' instructional practice in secondary school mathematics in relationship to their underlying cognitions. Indicates that the value of the model lies in its usefulness as a guide for enabling teachers to reflect on their practice and underlying cognitions in a structured, comprehensive manner. (Contains 54…