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There are several ways in which mathematics in school classrooms misses elements that are vital to mathematicians' practice. Here, we wish to emphasize processes such as selecting between or devising new representations, looking for invariances, observing extreme cases and typical ones to come up with conjectures, looking actively for counterexamples, estimating quantities, approximating terms, simplifying or generalizing problems to make them easier to address, building on answers to generate new questions for exploration, and so on. In terms of content area and the methodology of content creation, it may be hard to mirror the discipline of mathematics in the school classroom, but we suggest that bringing these processes into school classrooms is both feasible and desirable. This not only enriches school mathematics but can also help solve problems that are currently endemic to mathematics education: perceptions of fear and failure, and low participation. By way of illustration, we offer examples of classroom interactions that show such processes making for lively mathematical exploration.

Mathematical knowledge as object of teaching-learning processes undergoes changes in its epistemological status. In primary and secondary schools:

• mathematics teaching does not aim at training mathematical experts but contributes to the students' general education to become politically mature citizens (expert knowledge vs. knowledge in everyday settings)

• mathematical knowledge cannot be conveyed as a ready made product but it develops in a genetic manner by students' own activities (Mathematics as Product vs. Process, Hans Freudenthal)

• mathematical concepts (i.e. number, probability) cannot be introduced by formal definitions, consistent axioms or defining equations, but receive their meaning by referring to (different embodiments of) structures, patterns and relationships The epistemological particularities of mathematical knowledge in teaching-learning processes will be elaborated by using elementary examples of basic mathematical concepts.