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Part Five—
Epilogue

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Epilogue
The introduction to High School Mathematics at Work begins by asserting that today's world provides rich and compelling examples of mathematical ideas in everyday and workplace settings. In short, workplace-based mathematics can be good mathematics for everyone. The volume goes on to explore opportunities and challenges posed by developments in the world outside of the classroom. Several points deserve mention and special emphasis. Because this document is part of a larger reform movement, some concerns must be addressed about the reform movement in general and also about the scope of the tasks in this volume. Once again, the tasks in this volume are not prescriptions for curriculum but examples that are intended to illuminate possibilities.
Mathematics Education Reform
At the heart of some of the recent concerns about K-12 education reform efforts are issues of subject matter content: scope, depth, and levels of conceptual reasoning and technical proficiency. Concerns have been raised, for example, that some proposed revisions of curricula omit important topics and place insufficient emphasis on technical proficiency to promote understanding. High School Mathematics at Work aims neither to broaden nor restrict the scope of the high

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school mathematics curriculum. Furthermore, technical proficiency and depth of content coverage are not necessarily reduced by inclusion of workplace and everyday applications of mathematics. To the contrary, such an approach can provide meaning that increases the depth of students' understanding as well as their levels of conceptual reasoning and technical proficiency. Of course, a necessary condition for such an outcome is that students have sufficient opportunity for mathematical closure—extracting and conceptualizing the mathematics underlying the problems.
The Scope of High School Mathematics
By emphasizing connections between mathematics and workplace and everyday contexts, the mathematical content of this volume emphasizes some topics that have particularly striking, valuable, or widespread applications outside the classroom. Despite the broad range of tasks in this volume, statistics, discrete mathematics, and spatial reasoning receive little attention, and yet their relevance for today's world is without question. High School Mathematics at Work flags places in hospitals, banks, homes, and other familiar settings where important mathematical ideas are used. Many of these settings employ techniques which depend upon and lead to aspects of algebraic, geometric, and functional reasoning that have been and will always be recognized as crucial elements of a high school education.
A careful look at algebraic reasoning illustrates this point. Linear programming (a subset of algebra), for example, has many beautiful, important, and time-tested applications. That is why many textbooks already contain problems on this subject. More generally, algebraic reasoning is often addressed in High School Mathematics at Work through spreadsheets (rules for combining the entries of certain cells to produce the quantity that goes into another cell are just algebra in a new form). That some aspects of classical algebra do not appear more explicitly in High School Mathematics at Work should not be taken as a statement about their mathematical or practical value. Students heading to technical careers of any sort should understand how to use and interpret symbols. In fact, for all students, understanding of core algebraic skills and reasoning continues to be a key mathematical prerequisite.
Similar comments could be made about many other mathematical topics not explicitly mentioned in High School Mathematics at Work. Indeed, the Task Force for this volume identified quite a large number of mathematically important and delightful problems that were eventually not recommended for inclusion because their connection to workplace or everyday applications was less apparent than for others. Among the favorites not included were the following: calculating into how many regions n lines drawn at random divide the plane; and using probability calculations to see how the length of a play-off series affects the chances of the weaker team pulling off an upset by winning the series.

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A Curriculum is More Than Tasks
When one addresses concerns about reform-based materials as well as the fact that not all important high school mathematics is represented here, it is necessary once again to caution that the tasks in High School Mathematics at Work constitute neither a complete curriculum nor even student-ready curricular materials. All readers are welcome to see in these tasks potential for strengthening the mathematics education of all students, but no one should conclude that it is enough to teach these tasks or even a collection of exercises inspired by them. Any tasks need to be embedded in a coherent, well-developed mathematics curriculum that provides the mathematical understanding that a high school graduate should have. In the end, mathematics is ''more than a toolbox," as Hugo Rossi recently put it. The act of abstraction is what makes it so powerful. After completing a series of workplace or everyday problems with students, we must always remember to help them understand that what we call mathematics comes from generalizing and organizing the common features among the solutions into a coherent structure. A quality mathematics curriculum is not crafted out of tasks alone but also depends upon how these tasks are knit together and what kinds of opportunities students are afforded for abstraction and deep conceptual development.

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Appendixes

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