[Note: This document is the final report (excluding appendices) of a workshop organized by the Institute on Education and the Economy, Teachers College, Columbia University, on behalf of the National Center for Research in Vocational Education.]

On Nov. 7-9, 1997 the National Center for Research in Vocational Education (NCRVE) convened nearly fifty people representing three distinct communities (mathematics, industry, and skill standards policy) for a workshop on mathematics and occupational skill standards. The workshop, held at the Arden House Conference Center in Harriman, NY, was a sequel to an earlier NCRVE-sponsored conference "Integrating Academic and Industry Skill Standards."

The broad goals of the workshop were:

• To strengthen the case for both rigorous academic standards in vocational education and authentic workplace applications in traditional academic programs;
• To dispel minimalist views about the mathematics required for work by highlighting the rich mathematics embedded in everyday tasks required for tomorrow's workplace; and
• To help the mathematical and vocational education communities understand each other's standards so they can begin to work together on strategies to improve teaching and learning.

The workshop focused on one academic discipline (mathematics) and three occupational areas (retail trades, advanced high performance manufacturing, and agricultural biotechnology). Mathematics was selected because of its crucial role in school, in work, and in career training programs. The occupational areas were selected to provide a range of mathematics and career options: two (retail and manufacturing) match the first of the voluntary partnerships established by the National Skill Standards Board (NSSB), and the third (agricultural biotechnology) provides an example of a science-intensive career cluster.

Procedures

Participants in the workshop represented industry, mathematics, education, and government. They included employers, faculty at both the high school and postsecondary levels (two-year and four-year colleges); teacher educators; educational researchers; occupational skill standards developers; curriculum content developers; and state and federal policy leaders. Few people who attended the workshop knew in advance more than half the participants. Thus discussions at the workshop were designed in part to build personal connections among leaders of the different communities represented at the workshop.

In advance of the workshop, the organizers prepared a two-part, 150-page collection of resources and background reading containing:

• Information about the mathematics and occupational skill standards movements;
• Reprints of (or excerpts from) related essays;
• Examples of mathematical tasks from the three occupational skill areas (retail, manufacturing, biotechnology) and a sample of applied problems from current mathematics curricula and assessment materials; and
• Excerpts from five standards documents--the three occupational skill standards plus two from mathematics by the National Council of Teachers of Mathematics (NCTM) and the American Mathematical Association of Two-Year Colleges (AMATYC).

The workshop was organized as a series of parallel working groups followed by plenary sessions where the groups reported back to all participants. Each group in each session was given a series of questions to discuss. The first session was devoted to the three occupational skill standards areas (retail, manufacturing, biotechnology); participants were divided into six groups, two in each occupational area. The second session was divided into five groups--three focused on aspects of mathematics (everyday mathematics, essential mathematics, higher mathematics), one on pedagogy (mathematics in context), and one on building networks. The third session was divided into three constituency groups (mathematics, industry, and public policy) to focus on next steps.

In addition, NCTM President Gail Burrill reported on Standards 2000, an initiative to revise the NCTM standards for school mathematics, and invited the constituencies represented at the workshop to set up a mechanism to provide input into that process. Sally Waldron, Senior Director of Outreach at NSSB, reported on NSSB's work to establish voluntary partnerships in several different occupational clusters. Finally, participants were given time before the concluding session to write individual comments on issues of importance that may not have been adequately covered in the working group reports, or on other topics related to the theme of the workshop. (The diverse quotations that offer contrasting views throughout this report are taken from these individual commentaries.)

"We need to pay close attention to the language we use. Discussions across different groups can be improved by recognizing when we are saying similar things in different ways, or different things in similar ways."

Issues

Issues like these made the workshop sessions both vigorous and valuable. Mathematics is not a subject about which people are neutral. Everyone has studied mathematics; many have helped their own children with mathematics homework; and many have experienced difficulties with the mathematics needed at work. Workshop discussions both retraced well-trod paths (Are calculators a crutch or a tool? Should basics be mastered before undertaking more complex problems?) and explored new trails (Are standards best expressed by lists of skills or through examples of tasks? Should schools stress statistics more and calculus less?). Many issues, however, remain for future discussion.

"Colleges drive the high school curriculum by making admissions standards public and quantified. Businesses should do the same."

Industry Needs. Employees in business and industry need two kinds of mathematics not now emphasized (or taught) in schools. On the one hand, they need much stronger capabilities to recognize and use core concepts of middle school mathematics such as ratio, proportion, and percentage. Some participants argued that this is really all that most students need: they saw very little need in the workplace for much of standard high school mathematics. To be sure, many of the mathematical topics used in business are relatively elementary. However, the contexts in which they arise are often quite sophisticated.

On the other hand, employees also need to understand certain advanced topics such as statistical inference, data analysis, and process control that are hardly ever included in the required part of school mathematics. The desire for employees to have experience dealing intelligently and analytically with both observed and derived data arose in many discussions at the workshop.

Over and over, industry representatives emphasized the need for "systems thinking," for the habits of mind that recognize complexities inherent in situations subject to multiple inputs and diverse constraints. In addition, science-based fields such as agricultural biotechnology require technicians who are able to formulate a problem in terms of relevant factors and design an experiment to determine the influence of these various factors. Such systems are often so complex that the context obscures the mathematics.

"Teachers aren't prepared with experiences in business or industry, but in ivory towers that usually have little to do with the working world."

To help prospective employees know what is required for different careers, various industries developed occupational skill standards to cover broad clusters of related jobs. These standards vary greatly. At one extreme, the retail standards were specifically designed to professionalize the entry level position of Professional Sales Associate. Nonetheless, the level of mathematics specified in these standards is lower than that in many other occupational areas, and lower than what NCTM recommends for all high school graduates. Participants speculated about this anomaly, asking whether retail employment is a realistic context in which to find advanced high school mathematics. Looking to the future, they speculated about whether technology might drastically change the nature of jobs to mean that even fewer skills are needed, or whether a new kind of retail might emerge, as has happened in manufacturing, that depends on high performance mathematics skills. Those with retail experience said that concern about marginal costs would limit employers' willingness to pay for higher skilled retail workers.

These issues illustrate the special challenge of using a context such as retail in mathematics classes. Employers rarely ask retail clerks to deal with problems requiring mathematical expertise; they use consultants instead. So while retail contexts may help motivate students, and make them more alert consumers, applications of mathematics in the retail industry can not easily be justified as a career requirement. (One working group noted that the professional component of the mathematics standards may relate to employers' expectations of professional sales associates even more than the content standards, since many of the characteristics of good teaching such as listening, helping, and guiding apply also to selling.)

"The goal of schools to stock students with most of the information and skills they will need for a lifetime is unattainable. Mathematics education should instead foster a disposition to learn mathematics and the capability to learn how to learn."

The situation is very different in manufacturing, where both those who operate machines and those who manage the operators need a good understanding of one another's job. People who operate equipment need to understand what goes into decision-making, while people making decisions need to understand the reality of the factory floor. Because machine operations have become very sophisticated, most businesses form project teams with overlapping expertise. Moreover, because of weaknesses in typical vocational and general education tracks, only students who currently take the college-intending tracks graduate well prepared for high performance work. As one participant put it, "In today's factories, it is no longer OK to leave your brains outside the plant door."

In addition to setting qualifications for entry level jobs, many occupational skill standards also provide a foundation for career professionals. No one can meet all aspects of the standards right out of high school; some job experience is usually needed. Nonetheless, the occupational standards' statements about academic requirements set a visible and respected workplace expectation for school mathematics.

"We need to take responsibility for individuals who fail mathematics courses. We need to monitor their failures, motivate their efforts, and help them learn. It is not OK to fail mathematics."

School Mathematics. Everyone at the workshop agreed that too many of today's high school graduates are mathematically unprepared for the contemporary workplace. This situation leads employers to stress the basics before higher order skills, because the absence of basics is what they notice first. Most participants believed that if students mastered a curriculum that met the mathematics standards they would have the basic skills expected by employers. But of course the reality in the classroom is quite different from the vision in the standards.

Industry participants repeatedly emphasized that the way mathematics is taught is very different from the way it is used. Although students who master school mathematics learn to "do" mathematics, they rarely learn to "use" mathematics. Moreover, too many cannot even do mathematics. Participants offered a variety of conjectures to explain these failures:

• Word problems seldom resemble real-world experiences. Either they are abstract and decontextualized or artificial pseudo-applications that students easily recognize as fake. Few believed that word problems in school help prepare students for real problems at work.
• Mathematics teachers teach standard approaches--typically algebraic symbol manipulation--and resist other methods (e.g., arithmetical reasoning) that are more concrete and, for many students, easier to understand.
• Too often students' mathematics experiences are characterized by repetition rather than by reinforcement.
• Although students' intuitions differ greatly, few teachers consistently foster inventive approaches to solving mathematics problems.
• Some individuals can perform mathematics well on paper-and-pencil exercises in school but not on the job, whereas others can perform well at work (e.g., calculating mark-ups on the store floor) but not in school.

"New high school curricula that place mathematics in context are meeting public resistance because they deviate from traditional preparation for calculus. The goal of occupationally relevant curricula is orders of magnitude more difficult, and will fail unless we can convincingly connect mathematics for occupations to mathematics for college."

Several participants said that the emerging standards-inspired curricula are more closely aligned with what business appears to want than what colleges expect. "One can't help but be struck," wrote one educator, "by the strong similarities in goals (both in thinking skill objectives and in preferred teaching methods) between the mathematics reform movement and the broad policy themes of business and industry." These similarities provide an opportunity to form natural alliances in support of new models in which students gain experience tackling unfamiliar problems in more integrated, interdisciplinary settings. Perhaps industry, which thrives on selling its own products, could help educators sell a reformed context-rich mathematics curriculum to a public that is skeptical of any change in education.

Nearly everyone agreed that school mathematics should provide the mathematics that students need for careers, but that students should also be introduced to the full breadth of mathematics. There was, however, some skepticism from those outside the mathematics community about whether everyone actually can (or should) learn all the mathematics recommended by NCTM for the core curriculum. It is easy to misjudge the difficulty of solving mathematically rich workplace tasks, so perhaps fewer mathematics topics can (or need) be covered in a context-rich curriculum.

"Mathematics in school must contribute the mathematical literacy of all students. To accomplish this objective, we need to realign public policy. High school curricula that are dictated by college entrance requirements and high stakes tests do not serve us well."

What Mathematics is Essential? With few exceptions--primarily in science, engineering and finance--the mathematical requirements of work are not so different from those of daily life. The fundamental need is the ability to understand the value of quantitative information, to conceptualize problems, and to organize and interpret data in useful ways. In school mathematics this translates into data analysis, advanced arithmetic, risk analysis (probability and statistics), and financial mathematics--the latter being conspicuously underemphasized in both the NCTM and AMATYC standards. Several participants suggested that high school students should have a "capstone" course that uses newspaper clippings to help them see mathematics in everyday contexts and to lock in their problem solving skills.

How much mathematics is essential for all high school graduates? Many voices in both education and industry (e.g., NCTM, political and business leaders) argue for much more than today's youth achieve, while others both in education and industry argue that current levels (approximating eighth grade mathematics) are sufficient for most people. Workshop participants did agree that the core of high school mathematics needs to be large enough to keep students' options open and that it should include more of the essential skills needed for life and work. They recognized, however, a potential tension between a common core of mathematics for all students and the particular mathematics that may be encouraged in career academies and apprenticeship programs.

"To change high school mathematics we need to do more than align standards. We need to make contextualized workplace-relevant courses count, as college admissions officers now make AP calculus count."

Repeatedly, participants gave high priority to such topics as process control, statistical quality control, and analysis of data. The need for more statistics is real and long-standing, and is emphasized in both the NCTM and AMATYC standards. Why have the schools been so slow to respond to this need? Because political pressures and college admissions have created a "forced march" through high school mathematics to the goal of Advanced Placement (AP) calculus (which is reached by only 2% of students). Participants agreed that the premature focus on calculus in the schools is an unwise impediment to a more desired focus on statistics, probability, and discrete mathematics. "I am tired," wrote one participant, "of high school mathematics being driven by the entrance requirements of postsecondary institutions."

College is not an alternative to work, but one of several routes to work. Thus in an ideal world there would be no difference between mathematics for college and mathematics for work. The suggestion that students preparing for work need a different mathematics curriculum from those going to college is, according to one participant, "false and dangerous." Nevertheless, in today's world big differences remain. "Most academic teachers focus on goals for college-intending students," reported a workshop participant, "believing that vocational education is limited to low-level basics. The evidence from high performance workplaces shows that this is far from true. But how would colleges view such courses?"

Many participants asked how (or if) NCTM had validated the distinctions in its standards between "college intending" students and others; most found this distinction unwise. "If we continue to think of college as the only valid path for students, then all non-traditional options are cut off." However, participants did not agree on whether high school mathematics should provide more advanced topics (as it does now) or more experience in making sophisticated use of basic tools.

"Why worry about what colleges want? They will take what schools give them. Colleges need students and high schools have a monopoly on the supply."

Mathematics in Context. Several participants argued that the purpose of a career-oriented curriculum is not primarily preparation for careers, but motivation for rigorous study, both academic and vocational. Context makes problem solving and communication possible, and enjoyable. "Music students," wrote one participant, "begin by listening to whole pieces and only later learn to decompose compositions into component parts. In mathematics we learn the parts before seeing a whole into which they fit. Could mathematics be taught the other way around? Should context precede content?"

Although most workshop participants took for granted that students learn more when they learn mathematics in context, a few challenged this assumption on the grounds that there was little research to support this belief. "We need to face the fact that students who can use a mathematical concept in one context often have trouble using it in another."

How will we know if contextual learning pays off? Will it prove itself on the standardized tests that many states are now requiring? Will it prepare students well for the mathematical expectations of business and industry? Limited evidence so far has come from rather special situations--motivated teachers, experimental curricula. Some new data are becoming available from the major NSF curriculum projects. But can it work in average situations? If not, then the goal of widespread context-based mathematics instruction may be unattainable.

"In their present form the industry skill standards reveal neither the complexity of the workplace nor the expectations of workers to perform other than routine tasks. In an attempt to dissect what workers need to know and be able to do, much of the depth and complexity of their jobs has been lost."

Ironically, even as industry calls for mathematics to be taught in context, the occupational skill standards decontextualize mathematics in order to present it. As a consequence, mathematics in the occupational standards is presented as lists of topics that, on the whole, appear to be much less challenging than the vision of mathematical power conveyed by NCTM and AMATYC. However, much "hidden" mathematics is embedded throughout the occupational skill standards under other headings (e.g., problem solving, quality control, or planning). For example, to understand the role of profits in an organization requires understanding aspects of mathematics (e.g., probability and optimization) that are often not made explicit in the mathematics sections of the standards documents. Often the context of potential mathematical thinking is far less obvious. Organizing a shoe store stockroom, for example, involves subtle questions of timing, priority, and efficiency that are at their heart intrinsically mathematical. So to fully appreciate the mathematical implications of the occupational skill standards, one has to look at both explicit statements and implicit contexts.

There was widespread agreement on the need to provide teachers with effective, authentic problems. In solving such problems students would learn to ask "What mathematics do I need now?" Participants recognized that authentic problems are more difficult than standard word problems since determining what mathematics is needed is often as important (and as hard) as actually doing the mathematics. Such examples help students see that in the real world, problems are inherently complex. In fact, authentic problems often include so many other things that they don't look at all like what students expect to see in a mathematics class. It remains unclear, however, whether it is possible to package authentic problems without destroying them. Authenticity may require that teachers develop their own tasks in local contexts.

"Applied academics offer students with different ways of knowing more equitable access to information and skills on a par with their more symbolically-inclined peers."

Differing Perspectives. Two general issues about the academic and occupational standards emerged at the workshop. The first involves questions about the dangers of multiple sets of standards. Even though the academic and occupational skill standards were developed for quite different purposes, many expressed concern that two sets of standards could have the effect of reinforcing pressure for tracking in the schools.

The second issue, of primary concern to educators, is one of presentation and rhetoric. Occupational standards presented as lists of topics do not address educational issues of context, pedagogy, and motivation. Lists don't explain why things need to be learned, in what context topics may be used, how they might be taught, or how proficient students need to be. Participants agreed that scenarios and problems are more effective than lists as a means of presenting the kind of mathematics that students really need to be able to use.

"I feel very strongly that standards should be validated against applications."

Differences in perspective between industry and education were apparent in virtually every session at the workshop. Not only is the way mathematics is taught quite different from the way it is used, but industry and education perceptions of what it means to understand mathematics are totally different.

• Whereas in school students learn to domathematics, at work they are expected to useit.
• Whereas mathematicians talk about "modeling" when referring to context-intensive situations, business leaders hardly ever use that term. For them, reality isthe model.
• Whereas in mathematics class numbers have an artificial absolute reality, at work numbers are used for measurement and are always accompanied by both uncertainties and units.
• Whereas in school students rarely see mathematics used in complex ways, at work many tasks involve considerable complexity but very little advanced mathematics.
• Whereas in industry tasks involve many complex facets simultaneously, education treats concepts sequentially. There is little reason to believe that students will be able to deal with concurrent factors if they only experience concepts in a carefully prescribed sequence.
• Whereas industry envisions statistics in terms of quality control and management decisions, in school (and in the mathematics standards) the approach to statistics is more routine and academic.

"The entire school-to-work effort is pushing occupational choice much too soon. What's the rush? We don't die at 45 any more. Can't we develop a system to allow productive use of youths' energies while giving them an opportunity to explore the world of work?"

Networks and Communication. Despite the obvious need to communicate in order to build a common vision, conversations among employers and educators are rare--and when they do happen, the parties often fail to connect. Mathematics educators often avoid discussions with business because of a perception among teachers that business and industry want to dumb down the curriculum to a list of basic skills. Industry leaders seldom seek out educators, perhaps because they don't see direct benefits from time invested in this activity. Despite the extensive standards activity in both industry and education, very few people really know what is going on across the entire standards movement. Moreover, people in industry view school mathematics based on how they learned it, which is sometimes quite different from the reality of today's workplace or classroom.

This lack of coordination among groups limits their potential impact. The potential groups that need to be in communication include AMATYC, NCTM, unions, government, businesses, school administrators, parents, funders, counselors, teachers, journalists, and more. Systemic change requires involvement of all these constituencies, especially to be sure that there are shared goals and effective leadership. For example, the importance of integrated vs. separate curricula remains very confused--in legislation, in policy, and in practice. Is it important to integrate academic and vocational education, or is it better to expect that vocational programs will improve students' academic preparation and that academic programs will introduce students to the world of work?

The greatest impact may be at the grassroots level with projects that can go through a cycle of development, assessment, refinement, and adaptation (or replication). To encourage such projects, several suggestions were made: enlist the Mathematical Sciences Education Board (MSEB) as a convenor; involve business from the beginning in any education project; and write articles for journals that reach mathematics and vocational educators as well as managers in business and industry.

"Much of our discussion focused on how much mathematics was missing from the occupational skill standards. We should talk as well about how much mathematics is missing from the mathematics standards!"

Recommendations

1. Arrange a meeting of representatives of the occupational skill standards projects and the National Skills Standards Board with leaders of the NCTM Standards 2000 project to develop a process by which employer and occupational perspectives can contribute to revising the NCTM Standards. For example, NSSB could identify mathematical tools common to most industries and urge that these be situated prominently in the revised standards.
   
2. Convene a meeting of industry representatives and curriculum developers to enable business representatives to examine samples of emerging curricula that are inspired by the mathematics standards and to provide curriculum developers with access to mathematically rich workplace tasks. Such a meeting would help set new directions for emerging K-14 mathematics curricula and would help align expectations of business with those of the schools.
   
3. Encourage industry to join with educators to convince parents and universities to support curricula that stress mathematics in realistic contexts. College and university alumni employed in industry could ask their alma maters to support more flexible admissions criteria that are not so narrowly focused on calculus.
   
4. Develop a structured dialogue between mathematics teachers and those who educate them, and leaders of the occupational skill pilot projects and the new voluntary occupational partnerships. This dialogue could lead to improved industry-led programs that assist in the professional development of both teachers and teachers of teachers.
   
5. Prepare a "manifesto" on the mathematics that counts in the workplace both to support the reform movement and to encourage university admissions processes to be more receptive to high quality applied programs.
   
6. Encourage governors and state education agencies to coordinate academic and occupational skill standards in the development of state frameworks.
   
7. Validate both occupational and academic standards against external criteria. For example, academic standards need to be validated in terms of both work and postsecondary education; occupational standards need to be validated against both market circumstances and educational practicalities.
   
8. Encourage broad industry coalitions (perhaps the new NSSB voluntary partnerships) to post on the Web authentic, work-based problems and scenarios that mathematics teachers can adapt for classroom use and curriculum developers can incorporate in reform textbooks. (To ensure the usefulness of these tasks, such a project would need an effective management process to build consensus for common guidelines and to ensure both breadth and depth in mathematics content.)
   
9. Develop a mechanism to provide feedback from employers to schools on how graduates do after they enter the workforce.
   
10. Build industry experience and workplace applications into the early grades, not just in high school. This will not only help motivate students, but also provide future teachers with a vision of mathematics that is well connected to workplace applications.