**HYMAN BASS**

*Professor of Mathematics, Columbia University*

*Chairman, Mathematical Sciences Education Board*

**W**elcome to this workshop on Mathematics for the Technical Work Force, organized by the Mathematical Sciences Education Board here at the National Research Council with the generous support of the Alfred P. Sloan Foundation, through the good offices of Hirsh Cohen.

The National Research Council (NRC) is playing an increasing role in the national effort to improve mathematics and science education. In mathematics these efforts are spearheaded by the Mathematical Sciences Education Board (MSEB) in partnership with the National Council of Teachers of Mathematics (NCTM) and the Mathematical Association of America (MAA). The early focus was on K-12 education, and the policy frameworks have been the NCTM *Standards* for curriculum, teaching, and assessment. The concept of national education standards has now been firmly embraced in professional and government circles, though perhaps not yet in the scientific community and the broad public. In particular, the *National Science Education* *Standards* are near completion at the NRC, and are soon to be released.

Apart from some scholarly interest to the social sciences, K-12 education has not historically been a prominent concern of the NRC. One might well ask what justifies its present involvement. Its interest in the issue stems from the fact that improvement of mathematics and science education is a national problem of great urgency, one whose solution has a great bearing on the well-being of our scientific enterprise and technologically based economy and to which the national community of scientific professionals potentially can contribute significantly. The NRC's special qualifications for playing a leading role in these efforts derive from several characteristics: it is a politically non-partisan organization of high public trust; it has access to people of the greatest expertise, experience, and balanced judgment; and it can bring diverse interests and points of view to the table in the debate and resolution of issues.

In fact, it is in this latter regard that the NRC has been particularly valuable to education reform. While there are notable cultural divisions in the scientific communities, such divisions are much deeper and more pronounced in the world of education. Since advancement of the nation's education agenda must be based on a broad consensus, it is vital to establish and nurture robust dialogue among all of the important communities involved. The NRC is especially well suited to do this. Indeed, today's workshop is an excellent example of such fostering of needed connections.

**HYMAN BASS** *is Professor of* *Mathematics at Columbia University.* *He is Chair of the Mathematical Sciences* *Education Board, has served on the* *Executive Committee for the American* *Mathematical Society, and was* *Chair of the Board of Trustees for the* *Mathematical Sciences Research* *Institute, Berkeley. He is a member of* *the National Academy of Sciences.*

In its *Curriculum and Evaluation Standards for School Mathematics,* the NCTM emphasizes a strong core curriculum in mathematics for all students. They are vigilant in guarding against the past insidious effects of tracking, conceding only some special curricular materials for

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 9

MATHEMATICAL PREPARATION OF THE TECHNICAL WORK FORCE: REPORT OF A WORKSHOP
WELCOMING REMARKS
HYMAN BASS
Professor of Mathematics, Columbia University
Chairman, Mathematical Sciences Education Board
Welcome to this workshop on Mathematics for the Technical Work Force, organized by the Mathematical Sciences Education Board here at the National Research Council with the generous support of the Alfred P. Sloan Foundation, through the good offices of Hirsh Cohen.
The National Research Council (NRC) is playing an increasing role in the national effort to improve mathematics and science education. In mathematics these efforts are spearheaded by the Mathematical Sciences Education Board (MSEB) in partnership with the National Council of Teachers of Mathematics (NCTM) and the Mathematical Association of America (MAA). The early focus was on K-12 education, and the policy frameworks have been the NCTM Standards for curriculum, teaching, and assessment. The concept of national education standards has now been firmly embraced in professional and government circles, though perhaps not yet in the scientific community and the broad public. In particular, the National Science Education Standards are near completion at the NRC, and are soon to be released.
Apart from some scholarly interest to the social sciences, K-12 education has not historically been a prominent concern of the NRC. One might well ask what justifies its present involvement. Its interest in the issue stems from the fact that improvement of mathematics and science education is a national problem of great urgency, one whose solution has a great bearing on the well-being of our scientific enterprise and technologically based economy and to which the national community of scientific professionals potentially can contribute significantly. The NRC's special qualifications for playing a leading role in these efforts derive from several characteristics: it is a politically non-partisan organization of high public trust; it has access to people of the greatest expertise, experience, and balanced judgment; and it can bring diverse interests and points of view to the table in the debate and resolution of issues.
In fact, it is in this latter regard that the NRC has been particularly valuable to education reform. While there are notable cultural divisions in the scientific communities, such divisions are much deeper and more pronounced in the world of education. Since advancement of the nation's education agenda must be based on a broad consensus, it is vital to establish and nurture robust dialogue among all of the important communities involved. The NRC is especially well suited to do this. Indeed, today's workshop is an excellent example of such fostering of needed connections.
HYMAN BASS is Professor of Mathematics at Columbia University. He is Chair of the Mathematical Sciences Education Board, has served on the Executive Committee for the American Mathematical Society, and was Chair of the Board of Trustees for the Mathematical Sciences Research Institute, Berkeley. He is a member of the National Academy of Sciences.
In its Curriculum and Evaluation Standards for School Mathematics, the NCTM emphasizes a strong core curriculum in mathematics for all students. They are vigilant in guarding against the past insidious effects of tracking, conceding only some special curricular materials for

OCR for page 9

MATHEMATICAL PREPARATION OF THE TECHNICAL WORK FORCE: REPORT OF A WORKSHOP
“college-bound” students. They urge that mathematics learning be contextualized, embedded in convincing problem-solving settings that relate to real world experience. Mathematical motivation, it is argued, derives from its concrete and particular uses, and these should always precede general principles.
WORKPLACE SETTINGS FURNISH COMPELLING YET ACCESSIBLE MATHEMATICS PROBLEMS OF A KIND THAT MOST CONVINCINGLY RESPONDS TO THE RHETORIC OF THE NCTM STANDARDS.
While I agree in general terms with this philosophy, I think it is too often interpreted in a simplistic manner. Indeed, some of its early implementations have been over-zealous and naive in ways that have some bearing on the theme of this workshop. Many of the examples of allegedly contextualized and externally motivated mathematics problems I have seen lately are no less contrived than the precursors that are so maligned by reformers.
Meanwhile, programs to educate students who will directly enter the technical work force are being developed by communities and in environments that are virtually disjoint from those of the standards-based reform movement. While mathematical skills must be an essential underpinning of the needed work force skills, there has been no serious effort to compare and reconcile the mathematics furnished in these “tech-prep ” programs with that proposed under the NCTM Standards. One of the prominent questions to be considered here is whether, and to what extent, a common core curriculum can serve all students, including those planning to directly enter the work force, perhaps after two years of college. And if these mathematics programs must diverge, how can we ensure that such divergence does not limit students ' future options?
As a newcomer to this arena, I have been struck by some of the novel thinking and experimental practice in the work force preparation community. While past attitudes held such programs in low esteem, I find some of the new programs intellectually exciting, and with potentially much to offer to our core mathematics curriculum. Some of them are based on holistic study of a large but well defined technological sector, thus offering a focused context in which to learn a great deal of synthesized mathematics and science, put to direct use. Other programs are based in part on direct immersion in an authentic work environment, in which students are required to do quantitative multi-step problemsolving and bring it to closure by demonstrating successful performance. These settings furnish some of the most highly compelling and complex, yet accessible, mathematical problems I have seen, the kind that most convincingly respond to the rhetoric of the NCTM Standards. So it is quite possible that we may see well-conceived work force preparation programs driving up the quality of standards-based curricula, as much as the converse.
Mathematics is taught to everyone because, at a rudimentary level, it permeates our lives—and increasingly so with the growth of technology. It is a tool for conceptually organizing and interpreting the world we inhabit, and for precise quantitative communication. But why does it exist as a self-standing discipline, rather than as one that is taught and invoked in context each time the need arises, and only to the extent appropriate to serve that need, at that moment? It should be clear to any educated person that something essential is lost when mathematics is taught on an “as needed” basis, and that indeed it is impractical to do so. This approach to mathematics loses one of its

OCR for page 9

MATHEMATICAL PREPARATION OF THE TECHNICAL WORK FORCE: REPORT OF A WORKSHOP
most distinctive features—its generality. Each mathematical method and formalism can serve a variety of contextual models. A significant amount of time must be accorded to focus, intermittently, on the purely mathematical issues, so as to see general principles in their disengaged simplicity. These include the basic properties of numbers and their representation; the geometry of space; the mathematical equivalence between much of algebra and geometry; methods of solving commonly encountered types of equations; and mathematical reasoning.
IN MATHEMATICS THE ROLE OF PROOF IS TO PRODUCE CONVICTION OR VALIDATION. IN THE WORKPLACE, CONVICTION IS ESTABLISHED BY SATISFACTORY PERFORMANCE OF A COMPLETED PRODUCT.
The last item is now often disputed, with the claim that general students have no need to learn mathematical proofs. Given the way that proofs are often taught in the schools, as strictly formal two-column lists following pedantic rules that even the teachers rarely understand properly, I can sympathize with such sentiments. However, I see this claim as missing the essential nature of mathematical reasoning. The role of proof is to produce conviction, validation of a claim or a product. This is in fact not so far from the spirit of the workplace, where conviction is established by satisfactory performance of a completed product. The laws of nature and the properties of the materials used define the rules of the game, and they must be respected.
In science, proof is consistency with reproducible experiments. Similarly, a computer programmer must strictly adhere to the grammar and syntax of the programming language and know the scientific principles to be embodied in the algorithms designed, if the program is to perform as desired. Such programming has many of the features of executing a mathematical proof. Students should learn mathematical reasoning more as practiced by working mathematicians, where heuristics, estimation, computational experiment, analogy, and other devices can be used to establish various levels of plausibility, often sufficient for a use at hand. But they also need to learn, at least in principle, how such reasoning can be brought to rigorous closure. These thinking skills and strategies are themselves a powerful form of problem-solving that can benefit more than mathematicians.
With that let me conclude these welcoming remarks. I look forward to a very stimulating and informative workshop.

OCR for page 9

MATHEMATICAL PREPARATION OF THE TECHNICAL WORK FORCE: REPORT OF A WORKSHOP
This page in the original is blank.