Descriptions of mathematical thinking have an extended lineage. Sometimes accurate and sometimes not, sometimes misinterpreted and sometimes not, characterizations of mathematical thought processes have inspired and at times misled people interested in designing or framing mathematics instruction. Challenges the field faces in conceptualizing mathematics instruction include: What can be warranted as legitimate mathematical practices? Which aspects of mathematical practices are relevant and appropriate for K-16 instruction? What kinds of support are necessary? What is viable at scale? This paper provides a description of relevant history from the Western literature, bringing readers up to the present. It then addresses two key issues related to contemporary curricula: the framing of the mathematical enterprise as being fundamentally inquiry-oriented and the need for curricula and instruction to reflect such mathematical values; and the characteristics of mathematical classrooms that support students' development as powerful mathematical thinkers. An emphasis is on problem solving, a major component of "thinking mathematically." The paper concludes with a description of practices that are currently under-emphasized in instruction and that would profit from greater attention.