Henningsen and Stein (1997) begin with some background information about the (relatively) new theories about thinking and doing mathematics, which depends on educators' view of the nature of mathematics. The research shows that one theory that has been increasing in popularity is one that emphasizes exploration and has a dynamic stance of education, and holds that student learning is seen as the process in which students develop their own opinions, beliefs, and perspectives of mathematics.
Activities that promote having a mathematical disposition include the following (Henningsen & Stein, 1997, p. 525). These are the processes that students should take away from their mathematical studies, not necessarily the specific content that is covered in their classes.
- Looking for and exploring patterns
- Using available resources effectively and appropriately to solve problems
- Thinking and reasoning in flexible ways
- Conjecturing, generalizing, justifying, and communicating one's mathematical ideas
- Deciding on whether mathematical results are reasonable
In order to promote these activities, teachers must situate their classroom environments such that students engage in high-level tasks. The introduction of this article includes an in-depth discussion of the importance of high-level tasks, the difficulties that may arise from implementing such tasks, and the ways of supporting such implementations.
The authors report that tasks are essential in a mathematics lesson because they give students messages about what “doing mathematics” is. So tasks teachers pose have a major influence on how students think mathematically, and thus they have an altered perception of mathematics. This brings us to the point of developing tasks with a high level of cognitive demand. But implementing these tasks in a high-level manner is not so easy. Since students are comfortable with being told what to do, sometimes teachers lower the rating of a task by making it more simple or giving students a procedure.
To avoid implementing high-level tasks in a low-level manner, Henningsen and Stein (1997) offer some factors, as outlined in Henningsen (2000, p. 245). Teachers should focus on the meaning of the content, rather than the procedures involved in calculating it. Scaffolding is another useful tool. By using students' prior knowledge and building them up to a new concept, they will more easily connect ideas together creating a richer conceptual understanding of a topic. Other strategies mentioned are modeling and self-monitoring.
The remainder of the research in Henningsen and Stein's (1997) article is shown in their empirical study, in which they demonstrate factors that support high-level thinking, reasoning, and sense-making.
- Henningsen, M., & Stein, M. K. (). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematical Education, 28(5), 524–549.
- Henningsen, M. A. (). Triumph through adversity: Supporting high-level thinking. Mathematics Teaching in the Middle School, 6(4), 244–248.
- Smith, M. S., Bill, V., & Hughes, E. K. (). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.