DNR perspective on mathematics curriculum and instruction, Part I: focus on proving

This is the first in a series of two papers whose goal is to contribute to the debate on a pair of questions: (1) What is the mathematics that we should teach in school? (2) How should we teach it? This paper addresses the first question, and the second paper, to appear in the next issue of ZDM, addresses the second question. The two questions are addressed from a particular theoretical framework, called DNR-based instruction in mathematics. The discussions in the current paper are instantiated mainly in proof-related contexts. The paper offers a definition of mathematics as a union of two categories of knowledge: ways of understanding and ways of thinking. The latter are generalizations of the notions, proof and proof scheme, respectively. The paper also discusses cognitive-epistemological and curricular implications of this definition, focusing mainly on the inevitable production of narrow or faulty mathematical knowledge and the asymmetry in educators’ attention to ways of understanding and ways of thinking.