ABSTRACT
This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematical-thinking). Hence, advanced mathematical-thinking can properly be viewed as potentially starting in elementary school. The definition of mathematical thinking entails considering the epistemological and didactical obstacles to a particular way of thinking. The interplay between ways of thinking and ways of understanding gives a contrast between the two, to make clearer the broader view of mathematical thinking and to suggest implications for instructional practices. The latter are summarized with a description of the DNR system (Duality, Necessity, and Repeated Reasoning). Certain common assumptions about instruction are criticized (in an effort to be provocative) by suggesting that they can interfere with growth in mathematical thinking.
