* A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community. Comments are welcome by e-mail to the project directors:
Susan L. Forman or
Lynn A. Steen. *

ATE projects enable teachers and students to look at the world differently. Most mathematics teachers have little experience either using or teaching the kind of mathematical applications that are embedded in ATE projects. So while working on ATE projects, matheamatics faculty become students again: to teach in these programs, they need to acquire new vocabulary, learn new methods, and struggle with new concepts. Through this effort they experience education once again through the eyes of their students. Likewise, ATE students become teachers: as students work in teams to solve real problems, they discover the benefits of learning from each other.

The project approach of ATE programs creates other pedagogical opportunities as well. In traditional mathematics courses dominated by homework and tests, teachers learn more about their students' weaknesses than about their strengths, more about what they can't do than about what they can do. In contrast, students working on ATE projects have much more opportunity to demonstrate their strengths--and more opportunities to work around (or remedy) their weaknesses. Notwithstanding these opportunities for creative pedagogy, many students in ATE programs find that their mathematical experiences remain as they had been in traditional courses, even as other parts of their program adopt more project-oriented, student-active, exploratory characteristics. In this section we examine some of the distinctive pedagogical challenges posed by mathematics in ATE programs.

Unfortunately, few students learn much lasting or important mathematics from worksheets. Worksheets violate most of the pedagogical principles espoused by the mathematics standards. Far too many worksheets offer little more than rote instruction for solving a particular problem or operating particular software. Students who complete worksheets may have solved the problem yet still not have learned anything about problem solving. Often, they don't even have a comprehensive understanding of the particular problem they have just solved.

The challenge of helping students move from simple, one-step problems to cognitively complex multi-step endeavors is significant, but worksheets are rarely an ideal strategy.

More generally, since ATE curricula are often unconventional when measured against traditional academic programs, students need at some point to recognize and be able to describe what they have learned using the conventional language of mathematics. Otherwise they begin to think that their friends in traditional courses are learning all the "right stuff" while they are not. ATE students need not only to solve problems but also to learn proper names for mathematical objects and procedures; to distinguish among guessing, conjecturing, solving, and proving; and to understand the the map of mathematics. Reaching this kind of mathematical closure is an important objective for any ATE program.

In response to demands for more relevance, publishers fill textbooks also with so-called *applied*problems that situate mathematical questions in some context, real or imagined:

If 8 men can do a job in 12 days, how long it will take to complete the job if two men quit?Most of these problems are contrived, and students know it. Rarely do they represent a plausible problem, and even when they do it is unlikely that a person would use school mathematics as the means to solve it. (Stephen would probably take the remaining money out of his pocket and count it.)Stephen had $24.09 in his pocket. If he spends $10.60 on a book and $3.30 on a snack, how much does he have left?

*Authentic*problems, in contrast, arise naturally in work (e.g., controlling processes on assembly lines, laying out new manufacturing facilities, preparing yield maps of a farmer's fields) and in ordinary living (e.g., understanding amounts withheld from a paycheck, planning to buy a car or redecorate a room). Because authentic problems are rooted in context, they rarely survive transplantation to generic mathematics classrooms. Since they can be properly experienced only in an environment that is hospitable to their defining context, they are rarely encountered in traditional classrooms. Even for instructors strongly committed to authenticity, finding and employing natural contexts remains a nearly insurmountable obstacle.

ATE programs can help with the problem of authenticity. Workplace settings connected with the ATE goal can be used to motivate, illustrate, and teach mathematics. For example:

- Data from the manufacuture of silicon wafers (for computer chips) can be used to introduce formal logic, sets, Venn diagrams, logic gates, and finite fields.
- Data from fast food chains on nutrition and sales (gathered from corporate reports found on the Internet) can support a project in which students analyse issues surounding the selection of a fast food chain for their campus.
- Federal guidelines from the Americans with Disabilities Act (ADA) provide an opportunity for students to experience practical trigonometry by checking campus wheelchair ramps.
- Global geography and climate (e.g., surveying, navigation, land use, heat islands, urbanization) can be used to introduce geometry in thee-dimensional contexts.
- Flow charts, decision trees, pie charts, coded maps, and business charts can be used to introduce mathematical ideas in the vocabulary of the typical workplace and with the authenticity of real data.

Educators also complain about students' lack of ability to use in new contexts skills learned in other settings. For example, science students fail to recognize in a biology class equations they learned to solve in their mathematics class. Agriculture students fail to recognize patterns in data that they learned about, albeit abstractly, in their statistics course. Students everywhere persist in believing that the mathematics they learn is of little use since they fail to recognize it when it arises outside of mathematics class.

Those who teach mathematics as part of the ATE program can aid transfer of learing by introducing students to the same mathematical concept in a different context. This approach, more self-conscious about the problem of transfer of learning, helps many students recognize that mathematical tools are meant to be used, and can be used in more than one context. But it requires an unusually high degree of coordination between the mathematics program and the ATE program--coordination that is often impeded by turf issues or articulation restrictions.

**Copyright © 1999.**
*Last Updated:* October 12, 1999.
*Comments to:*
Susan L. Forman or
Lynn A. Steen.