Linguistics

Mathematics education

Argument (complex analysis)

Engineering

Philosophy

Compatibility (geochemistry)

Teleology

Pedagogy

Biochemistry

Chemical engineering

Dialectic

Hegelianism

Professional development

Psychology

Mathematics

Telos

Epistemology

Sociology

Chemistry

Negation

Generalization

Mind Culture and Activity

We argue that the distinction between dialogue (after Bakhtin) and dialectics (after Hegel, Marx, Vygotsky), is of key importance to learning-teaching and to mathematics education. Some followers of Bakhtin have argued that these concepts are irreconcilable, or incompatible, since dialectics implies and dialogism implicitly denies the requirement of telos (i.e., a targeted endpoint). On the contrary, we argue for compatibility; dialogism can allow for the progress implied by dialectics, but its teleology is inherent in its efficacy in practice rather than in any pre-defined endpoint. We show how a mathematical or professional dialogue can involve dialectical negations and supersession, thus providing for progress or development, without loss of dialogism. Our case is taken from a lesson study in which progress emerging from classroom and staffroom dialogues is interpreted in dialectical terms as developmental. The connection with Vygotsky's theory of concepts in learning-teaching and the possible generalization of the argument are discussed. We conclude that the key moments on which concept development turns are: (1) the negation by multiple, lived practices, and (2) the creative, speculative, supersession of inadequate concepts, in appropriate dialogues.

On the compatibility of dialogism and dialectics: the case of mathematics education and professional development