Designing questions that develop understanding

Although the development of an understanding of mathematical concepts is usually one of the stated goals of instruction, the types of exercises most frequently given to students do not appear to contribute to this goal. Standard exercises ask students to carry out mathematical procedures, not to think about them. A model of what it means to understand mathematics is presented, followed by examples of exercises which are designed to develop understanding in terms of the model in specific ways. It is assumed that understanding is more likely to develop, if it is the direct focus of exercises rather than if the development is left to chance.