
I hear and I forget; Mathematics in the schools serves multiple purposes, including preparation for work, for citizenship, for general education, and for higher education. Of these, preparation for productive work takes priority in the minds of parents and the public. Indeed, in this era of advanced technology, global competitiveness, and electronic telecommunications, more students than ever before need to learn more mathematics than anyone has every tried to teach on so massive a scale. Despite differences in educational traditions, all nations now face a similar challengeof preparing students for a rapidly changing world economy. In the United States, this conviction is creating strong pressure for more pragmatic workoriented programs in the schools. Indeed, all students will eventually be employed. This pressure offers mathematics education a rich opportunity to rethink the traditional school curriculum in order to better integrate academic and workrelated mathematics. But it also runs a serious risk of deleterious side effects, notably the rebirth of secondclass tracking under the guise of preparing students for twentyfirst century jobs. To begin to address these issues, the Mathematical Sciences Education Board recently convened at the U.S. National Academy of Sciences a meeting of vocational and mathematical educators with mathematicians, industrial leaders, and policy makers to discuss key issues such as articulation, tracking, student needs, and core curriculum. The issues that are unfolding through these discussions in the United States provide a useful case study that contains lessons from which others can learnand which might serve as a prelude to some future ICMI Study.
Preparing for an International EconomyIn a recent study entitled 'Preparing for the Workplace,' the National Research Council reports that only one in five U.S. workers holds a (fouryear) baccalaureate degree. The other 80% enter the world of work either directly out of high school, after beginning but not completing a fouryear college program, or after completing some other form of postsecondary education in institutions ranging from community colleges to employersponsored institutes. Ironically, increasing numbers of university degree holders cannot find employment based on their degree, even as industry claims to be unable to hire workers with the technical skills needed for many jobs.The jobs available to this "other 80%" depend increasingly on mathematical thinking. Many corporations worldwide are moving to a flat management structure in which workers are responsible for their own quality control and for their own management through small employeeled teams. Workers no longer have the luxury of just doing what they are told. They are now expected to develop their own problemsolving strategies, often using statisticalor mathematical tools suitable to a datarich high technology work environment. What used to be called 'blue collar' jobs are quietly but inexorably being transformed by technology into 'white smock' jobs. In response to this challenge, U.S. officials now strongly advocate greater emphasis on educational programs that prepare students for technical careers in fields ranging from telecommunications to manufacturing, from nursing to agriculture. These programs fall under the general titles of "techprep," "schooltowork," "education for employment," or "advanced technological education." Many combine the last two years of high school with the first two years of postsecondary education into "2+2" programs; some expand to form "2+2+2" programs. In addressing a similar set of issues, the Topic Group on 'Mathematics for Work' at ICME7 distinguished two levels of education for work: that for "qualified workers" and that for professionals. The current U.S. discussion about preparing an internationally competitive technical workforce is more about the former than the latter, but the major educational challenges lie in building effective transitions from one to the other.
Ensuring OptionsEducational traditions in the United States differ in many respects from those in other nations, not least in how we handle transitions from school to work. In nations with wellestablished apprenticeship systems, the route from study to employment is transparent to students and their parents: everyone can foresee the consequences of choices made at various stages in the educational system. In the United States, however, linkages between school and work are virtually absent, and the various options are opaque not only to students and parents, but also to educators and employers.In addition, the multicultural character of the United States imposes perhaps more heavily than in other nations a demand that education be structured so as to keep student options open for as long as possible. It is an axiom of U.S. educational policy, repeatedly asserted by U.S. Labor Secretary Robert Reich, that all educational programs should prepare students for postsecondary study. Yet by increasing the emphasis on preparation for work, schools may create a new and apparently more 'respectable' form of trackinga deeply entrenched tradition of U.S. schools that relegated large numbers of students, disproportionately minorities, to deadend courses in which they received little challenge and even less education. New careerbased tracking will be more seductivewho can resist the allure of hightech occupational preparation?but equally capable of perpetuating socioeconomic class distinctions. Decisions made in early high school yearsespecially decisions about mathematics coursescan program students into a techprep or collegeprep curriculum with little opportunity for switching. Yet students' disposition for change between ages 14 and 20 virtually guarantees that no inflexible system of early tracking can be personally or educationally sound.
Setting StandardsAnother way in which the United States differs from many other nations is in how it goes about setting educational standards. In most nations, a central governmental authority sets broad standards for education in mathematics and other subjects. In the U.S., however, it was not the government but mathematics educators acting through the National Council of Teachers of Mathematics (NCTM) who developed national Standards for school mathematics. These Standards (a) call for a broader curriculum that encompasses the full reach of mathematics in practice; (b) stress active, engaged learning of mathematics situated in authentic contexts; (c) argue against deadend tracking that disenfranchises some children by denying them access to important mathematics; and (d) urge widespread use of technology.These NCTM priorities are remarkably similar to those identified in the ICME7 Topic Group on "Mathematics for Work": the need to adapt mathematics teaching to a technological environment; the need to focus instruction on 'situated knowledge'on mathematics embedded in the context of work; and the need to avoid abuses such as deadend tracking and inappropriate use of mathematics as a filter for jobs. At the level of rhetoric, these international expectations appear entirely consistent with those articulated for the United States by the NCTM. However, educators are not the only (and perhaps not even the primary) developers of standards. Industrial associations in the U.S. are now creating occupational standards in 'skills clusters' to provide portable national credentials that will fit the workplace better than the traditional high school or college diploma. These industrydriven expectations have a different character than those developed by educators: they stress general workplace expectations such as punctuality, reliability and teamwork, and technical expectations such as precision and zeroerror performance. In many ways, true rigor emerges when students seriously prepare themselves to meet industry standards.
Preparing for WorkComparison of mathematics in the workplace with mathematics in the classroom reveals a disjuncture that is disconcerting to anyone who believes that a primary purpose of school is to prepare students for work. School mathematics lives in a decontextualized ether, employing data that are without blemish and language that is devoid of ambiguity. In contrast, real problems are embedded in concrete tasks, use data that are often illdefined or inaccurate, and rely on language that is often imprecise and misleading. In the world of work, mathematics is collaborative rather than individualistic; accuracy is defined by the situation rather than given by the textbook; and mathematical processes are used rather than studied. The new challenge is to seek common ground among these very different traditionsof mathematics for and from the workplace and of mathematics as preparation for further study.One resolution of the dilemma of tracking would be a common mathematics program that could serve equally well as preparation both for college and for skilled work. All students could benefit from the broadening effects of such a high school preparation, yet there are currently few good models of curricula that serve both agendas. Another approach would be to develop a new form of vocational and technical education, with status equal to the academic track, that would simultaneously prepare students for the world of work and for further study in postsecondary institutions. U.S. educators who are concerned about vocational education debate both the desirability and feasibility of such a "separate but equal" track.
Achieving Equal StatusEfforts to bridge the vocationalacademic gap often founder on matters of status. Because of their low status in the U.S., vocational programs often attract the least able students and teachers. Public support for vocational programs is rooted in the conv iction that they are always for someone else's children. Neither teachers nor parents nor students believe vocational programs are or can be of comparable value to the regular "academic" curriculum.Indeed, as currently constituted, vocational courses frequently provide students with only a thin veneer of mathematical understanding. All too often, such programs operate within a narrow band of educational goals, providing little of value that transfer s to a variety of jobs. The working knowledge that students need involves much more than technical skills; a welldesigned educational program requires a learning trajectory in which students move from simple to more complex tasks, and from activities tha t are specialized to those that are central to the work enterprise. One important trend that addresses these needs is the shift from preparation for particular trades (e.g., carpentry) to education about all aspects of an industry (e.g., construction), including economics, history, science, and, of course, mathematics. In order to avoid the stigma imposed by the academic caste system, many in the U.S. now argue that vocational education should be seen not as a path to a different goal, but as a different path to the same (academic) goal. According to this view, the cent ral purpose of vocational education is not to train students for a particular job, but to offer students an alternative path to prepare for advanced academic training. This interpretation, which emphasizes vocational education for all students, not just a trisk students, might avoid many of the weaknesses often associated with occupational programs. These changes are important since, if it is to be accepted as central to schooling, vocational education needs to remain close to school reform,
Cognitive ApprenticeshipsAn interesting contrast emerges in comparing the NCTM Standards with educational practice in strong vocational programs. The Standards call for both broad mathematics and active pedagogy. Among mathematics teachers one can more readily find examples of the former than of the latter, whereas among vocational teachers the reverse is true: vocational pedagogy almost always exemplifies the ideals of the Standards, while content too often lags behind.Historically, apprenticeships such as carpentry and tailoring involved physical activities that could be easily observed and learned by imitation and practice. Today, many trades (e.g., medical technicians) operate at symbolic and cognitive levels that depend on reasoning strategies ordinarily hidden from view. The teacher's job in this new environment is to develop classroom practices that put students into "cognitive apprenticeships" in which they come to understand rather than merely imitate the work of experts. To be effective as preparation for lifelong learning, vocational programs must include abstract principles, varied situations, and general procedures that allow learners to adapt to new situations.
Responding to the ChallengeEducational institutions, being inherently conservative, have great difficulty responding to the multiple challenges posed by a rapidly changing workplace. Fortunately, in the United States, this challenge has been taken up with imagination and flexibility by twoyear or community colleges.These colleges operate within commuting distance of 95% of the population of the United States and often serve a population of older, minority, and low socioeconomic students. Typically, they offer two types of programs: "associate" degrees that provide either a midpoint milepost for students who intend to continue on for a four year degree or an applied credential for students preparing for jobs in technical areas; and "certificates" that respond to employers' needs for specialized programs. Twoyear colleges have grown in enrollment by over 250% during the past two decades, and offer some of the most exciting new routes into the emerging economy. These schools are the prominent shapers of tomorrow's frontline workers in occupations such as dental hygiene, marketing, engineering technology, and agribusiness. Twoyear colleges currently serve as the hub of a growing system of vocationoriented education that links high schools, industry, and universities. Indeed, the National Science Foundation has targeted these colleges for major new funding since they offer an effective crucible for the necessary curricular development. Mathematics is a central yet underdeveloped component of these rapidly growing programs.
Next StepsMathematicians who think about curricular issues typically focus on the "academic track" that leads to scientific, engineering, and mathematics courses at the university level. Failure by university mathematicians and mathematics educators to recognize and employ the significant mathematics embedded in workplace applications has led to divergence between the kind of mathematics taught in schools and that which is useful in the workplace.Similarly, those who develop curricula for technical and vocational programs rarely work with mathematicians or mathematics educators to ensure consistency with the expectations and standards of mathematics education. This has created an urgent need to im prove communication among the leaders of the different communities, since most of the schooltowork curriculum leaders have little or no background in mathematics education and few mathematics educators have any experience in work outside of education. The increasing political pressure for more utilitarian alternatives to the traditional preuniversity curriculum provide mathematicians with a marvelous opportunity to demonstrate the universal applicability of their disciplineto show that it is good for something other than just preparing more university professors. An examination of these issues may be useful for ICMIperhaps as a prelude to some future ICMI Study on Technical Education. The issues are vast, encompassing aspects of mathematical liter acy, adult education, and the very nature of school mathematics for all students.

Copyright © 1994.  Contact: Lynn A. Steen  URL: http://www.stolaf.edu/people/steen/Papers/work.html 