Contexts from within mathematics also can be powerful sites for the development of mathematical understanding, as professional and amateur mathematicians will attest.
There are many good sources of compelling problems from within mathematics, and a broad mathematics education will include experience with problems from contexts both within and outside mathematics.
There is also evidence that learning mathematics through applications can lead to exceptional achievement.
For example, with a curriculum that emphasizes modeling and applications, high school students at the North Carolina School of Science and Mathematics have repeatedly submitted winning papers in the annual college competition, Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).
His essay opens with a dialogue among employees of a company that intends to expand its business into Japan, and then goes on to point out many of the uses of mathematics, data collection, analysis, and non-mathematical judgment that are required in making such business decisions.
In his essay, Thomas Bailey suggests that vocational and academic education both might benefit from integration, and cites several trends to support this suggestion: change and uncertainty in the workplace, an increased need for workers to understand the conceptual foundations of key academic subjects, and a trend in pedagogy toward collaborative, open-ended projects.
The inclusion of tasks in this volume is intended to highlight particularly compelling problems whose context lies outside of mathematics, not to suggest a curriculum.
The operative word in the above premise is "can." The understandings that students develop from any encounter with mathematics depend not only on the context, but also on the students' prior experience and skills, their ways of thinking, their engagement with the task, the environment in which they explore the task—including the teacher, the students, and the tools—the kinds of interactions that occur in that environment, and the system of internal and external incentives that might be associated with the activity.
Research with adult learners seems to indicate that "variation of contexts (as well as the whole task approach) tends to encourage the development of general understanding in a way which concentrating on repeated routine applications of algorithms does not and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. This conclusion is consistent with the notion that using a variety of contexts can increase the chance that students can show what they know.
By increasing the number of potential links to the diverse knowledge and experience of the students, more students have opportunities to excel, which is to say that the above premise can promote equity in mathematics education.