Teaching Mathematics in ATE Programs

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.

Worksheets

Since all students need help learning how to approach complex problems, ATE curriculum developers and instructors generally provide stepping stones that bridge the gap between students' prior experience with narrow exercises and the ATE requirement of working through broad projects. The most common means of providing this help is a worksheet that lays out a precise plan in a succession of small steps with clear instructions for completing each step. Worksheets can be especially helpful as an aid for enabling students with spotty mathematical backgrounds to successfully complete large-scale projects. With worksheets, virtually any conscientious student can complete the required work, and everyone involved--teacher and student--feels greater confidence that challenge of problem-solving has been met.

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.

Highlighting Mathematics

Although mathematicians often focus on formal proof as an essential aspect of mathematics, in real life careful reasoning is more important than formal proof. Effective ATE programs help students learn how to make a persuasive case for the analysis they have performed. Learning how to make decisions based on evidence and how to convince others of the soundness of one's own work is not only essential training for successful employment, but is also very good practice for understanding formal proofs which a student may encounter in subsequent mathematics courses. ATE students need not only to learn to reason, but they also to recognize how reasoning in applied contexts compares with reasoning in mathematical contexts.

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.

Authentic Problems

Mathematics problems that students encounter tend to come in three flavors: pure, applied, and authentic. Pureproblems--the majority found in school classrooms--present mathematics naked, without clothing or context. What is 25% of 73? Solve 3x + 7 = 15. What is the area of a triangle with sides 3, 5, and 7? Even though problems such as these represent the core of mathematics and are essential to any uses of mathematics, they fail to pique the interest of many students.

In response to demands for more relevance, publishers fill textbooks also with so-called appliedproblems 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?
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?

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.)

Authenticproblems, 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.

Transfer of Learning

An important practical and political impediment to ATE programs is the well known difficulty of transferring mathematical skills and knowledge from one context to another. Many parents and policy leaders oppose applied and vocationally oriented programs such as ATE because of a belief that skills learned in such programs will be applicable to only one type of job. They are well aware that today's students must be prepared to change both jobs and even careers several times throughout their working lifetimes. Moreover, mathematics is seen as a subject that is supposed to be useful in many fields, so the idea of teaching it in the specific context of a single ATE applicaiton strikes many as problematic.

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.

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Supported by the Advanced Technological Education (ATE) program at the National Science Foundation. Opinions and information on this site are those of the authors and do not represent the views of either the ATE program or the National Science Foundation.