This article is based on the seminar that the author presented for the Hong Kong Association for Mathematics Education at the Hong Kong Baptist University on May 22, 2008. In this article, the author differentiates (a) between institutionalized mathematics and school mathematics, (b) between ways of understanding and ways of thinking as two complementary subsets of mathematics that students should develop, (c) between sophisticated learners and passive learners, and (d) between knowledge dissemination and knowledge engagement as two modes of instructions. Harel’s (2007) pedagogical principles are discussed in relation to the design of mathematical tasks and their use in classrooms. Examples are presented to illustrate how mathematical tasks can be designed to accomplish certain learning and teaching objectives, such as provoking the need for students to learn a particular concept, promoting desirable ways of thinking, deterring undesirable ways of thinking, and assessing students’ conceptual understanding. A Chinese version of this paper (通过数学任务提高美国职前教师的数学成熟性) is published in the Journal of Mathematics Education.