Comments on six theses about mathematics education are presented. How theories on teaching and learning mathematics are derived or developed, how philosophies are related, and their role in mathematics education are among the points discussed. (MNS)

Advances the idea that performance errors should contribute positively to the process of learning. Argues that errors do not occur randomly and that instructional theory should not condemn errors, but seek them. (PK)

Views on the need for a theory of mathematics education concern three interrelated components: meta-research and development of meta-knowledge on mathematics education as a discipline, development of a comprehensive view of mathematics education using a systems approach, and development of the dynamic regulating role of mathematics education. (MNS)

An overview to a session at the Fifth International Congress on Mathematical Education is given. Papers on research by Cooney, curriculum development by Goffree, reform in mathematics education by Stephens, and a summary by Nickson are included. (MNS)

Contributes to a reflection on the possibilities and the limits of a non-naive use of the history of mathematics for educational purposes. Discusses a problem related to the hypotheses that make it possible to confront past and modern conceptual developments. Contains 45 references. (DDR)

Discussed are the recognition and the relevance, in terms of teacher/student awareness and subsequent effective teacher treatment techniques, of cognitive disequilibrium as experienced by the student, when such disequilibrium is provoked by mathematical contradictions and/or counterexamples, within the classroom paradigm of constructivism. (JJK)

Given that mathematics is the essence of scientific and rational thinking, and that mathematics is the imprinter of both modern society and modern thought, implicit curricular proposals for both pre- and in-service mathematics teacher training are suggested for the revitalized teacher promotion of students' just and democratic behaviors. (JJK)

Discussed are categorization schema, with elaborations about attendant epistemological obstacles, concerning the twin notions of understanding and comprehension, per se, as well as the processes affecting their attainment. Included is an example of such a schema involving the concept of the limit of a numerical sequence. (JJK)

The kaleidoscope is presented as a suitable topic for a preservice mathematics teacher's first contact with a nontrivial mathematical phenomenon. Included are historical notes on the kaleidoscope, explanation of the inner mechanisms of various kaleidoscope designs, and suggestions for further student investigations. (JJK)