Mathematical Preparation of the Technical Work Force: Report of a Workshop (1995)

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KENNETH CHAPMAN

American Chemical Society

KENNETH CHAPMAN is Head of Technician Resources and Education in the Education Division of the American Chemical Society. He manages a voluntary industry standards project focused on chemical laboratory technicians and process technical operators, and is a Co-Principal Investigator for SciTeKS, a secondary-level multidisciplinary science curriculum project for school-to-work programs.

The 22 “voluntary industry standards” projects funded by the U.S. Departments of Education and Labor are developing skill standards appropriate to the workplace for a variety of occupations ranging from industrial launderers to chemical laboratory technicians. These projects are generally near their midpoint and are issuing draft reports. For example, the American Chemical Society (ACS) project titled “Voluntary Industry Standards for CPI Technical Workers” has issued a draft set of standards that addresses the two occupations of chemical laboratory technician and process technical operator.

The mission of these projects is to present an industry consensus about the knowledge and skills required for employees to perform their jobs well. The objective of the ACS project has been to describe the knowledge and skills entry-level chemical laboratory technicians and process technical operators should have when they begin work. Included are the starting-level knowledge and skills needed to provide a base for continued advancement over the first several years of work. Training for specific procedures and equipment is expected to be provided by the employer.

The voluntary industry standards projects were not provided specific templates for presenting standards nor were they restricted to specific ways of developing standards. However, most of the projects appear to have focused on identifying standards as they pertain to the workplace, with minimal attention to the impact of those standards on academe. Several projects have used surveys to collect information. Most have designed meetings to give practitioners of the occupation an opportunity to describe their work and the knowledge, skills, and attributes required to do their work well. These strategies have provided some insights about mathematics requirements; however, the technical performance skills currently presented by most of the projects, including the ACS project, may require further analysis to ascertain the full mathematics requirements.

As an example of how mathematics appears in these skills reports, we give below four lists of mathematics “standards” from the ACS project. These standards, and the related performance skills standards, will be analyzed closely over the next 18 months and presented in ways much more useful to the academic community.

Ordered from high to low by over 430 respondents to validation surveys:

• Solve simple algebraic equations.

• Calculate ratios.

• Perform chemical calculations.

• Recognize patterns from data.

• Describe accuracy and precision.

• Read and construct graphs using several scales.

• Calculate and understand the significance of means and standard deviations.

• Determine control limits.

• Solve complex algebraic equations, including quadratic and simultaneous equations.

• Statistically design experiments.

• Use differential calculus to calculate rates.

Added by respondents to the validation survey and not ordered:

• Perform unit conversions.

• Perform calculations involving exponents.

• Perform calculations involving roots.

• Perform calculations involving logarithms.

• Know standard procedures for determining and using appropriate significant figures.

• Determine detection limits.

• Evaluate propagation of error.

• Calculate gravimetric/volumetric conversions.

• Use unit analysis for checking calculations.

• Use partial differential calculus or other techniques to evaluate error.

• Be familiar with elementary statistics.

Ordered from high to low by over 430 respondents to validation surveys:

• Use decimal numbers.

• Calculate percentages.

• Recognize patterns from data.

• Read and construct graphs using several scales.

• Calculate ratios.

• Perform unit conversions.

• Solve simple algebraic equations.

• Determine control limits.

• Calculate temperature, pressure and volume relationships.

• Develop and interpret control charts.

• Perform root cause analyses.

• Conduct statistical analyses involving means and standard deviations.

Added by respondents to the validation survey:

• Perform chemical calculations

ARLA HUBER

Penn State University - York Campus

ARLA HUBER is Lecturer in Mathematics at Penn State University's York campus. She is primarily responsible for technical mathematics instruction at Penn State York and has developed postsecondary standards for the mathematics portfolio assessment of the Pennsylvania Youth Apprenticeship Program.

I believe that before we can determine the best approach to teaching mathematics that will prepare students for what comes after secondary school, we need to look at how students learn.

The learning and therefore the teaching of mathematics can be viewed as having three important components: fundamental skills; practical applications; and theory (or concepts). All of these aspects are important and interrelated. However, due to various factors, some students need more emphasis on one or two of these areas.

Students will learn these components at different paces and sometimes in different orders. Many students need to be very well grounded in fundamental skills and need to practice using those skills for practical applications before they are ready to grasp the theory. On the other hand some students are exited about theory and concepts and can master abstract ideas quickly, but may not be proficient in basic skills until forced to practice them.

By fundamental skills I mean basic arithmetic and competency in using numbers and symbols. This includes the four basic operations as well as the use of positive and negative numbers. A mastery of fundamental skills can give students the confidence and comfort with mathematics that will make learning more advanced skills and concepts easier.

Fundamental skills are first taught in elementary school. In my opinion, students should not be allowed to use calculators until they have shown proficiency with pencil and paper. Some will argue that the calculator is a tool which frees the mind to work on concepts. I agree with this, but allowing students to substitute calculators for learning the fundamentals prevents them from building a mental picture of numerical relationships and operations. If you doubt this ask any young store clerk to subtract a discount without using the cash register or a calculator. You may now think, “So what, they have tools for that.” I say it is fine to use the tools, but they should also see the numerical relationships if for no other reason than to be able to estimate as a check for errors. This is not an argument against using technology, but a warning about putting blind faith in technology.

What I see lacking in most of the technical students I teach are fundamental skills. They tend to enter my class with a fear of

mathematics and a lack of confidence in their ability to “do math.” Many times I have helped students overcome their negative feelings by stressing the practice of practical problems. I also try to give an exam every week so they can start to see success in mathematics by having “learnable chunks.”

When it comes to deciding why we teach mathematics, the answer for many students will be practical applications. Whether it is balancing a check book, calculating gas mileage, estimating the amount of materials for a building project, or determining the stress on a bridge beam, most people use mathematics for applications.

Practical applications are important because they give students a reason for putting forth effort. This is particularly true for those students who have not yet been excited or intrigued by theory and concepts—probably the vast majority of students who are headed for technical programs.

One real benefit of teaching practical applications is they can be the stepping stone between fundamentals and theory. With practical applications you can “trick” students into practicing fundamental skills and while they are not watching you can slip in some theory.

While fundamental skills and practical applications are very important, theory and concepts are what hold it all together. Each student needs to be introduced to all of these aspects of mathematics. For many, they will first need to build confidence with the things they can see before they are ready to embrace theory. Other students will soon become bored with practicing skills and applications. Students who are able to quickly grasp abstract ideas and reasoning should be given the opportunity to pursue these, but they should also be shown the importance of learning fundamental skills and applications.

It is important that all students at some point be exposed to mathematical concepts and theory because this is what empowers them to reach beyond what they have been taught and to use mathematics to solve new problems.

THERESE A. JONES

Amarillo College

THERESE A. JONES is Chair of the Division of Sciences and Engineering at Amarillo College in Amarillo, Texas. She is Principal Investigator for the Technological Sciences Academy, which presents technology-oriented classes to high school students, and Co-Principal Investigator of the Southwest Center for Advanced Technological Education, a clearinghouse for curricula and faculty enhancement.

As chairman of a division that has both transfer and technology programs, I have several concerns about technical mathematics:

1. First and foremost, exactly what is technical mathematics?

2. What is necessary to make technical mathematics fulfill general education requirements in the Associate of Applied Science degree as well as the Associate of Science degree?

3. What steps are required so that technical mathematics can be transferred directly into a baccalaureate program?

There are two parts to this question: What mathematics is needed in specific technologies, and what are the requirements for credit toward a degree? Sometimes these two concerns do not lead to consistent answers. In Texas, any

course that receives college credit must present college level material which, in the case of mathematics, is assumed to be at least at the level of College Algebra.

Yet there is a question of what is and what is not the same level as College Algebra. How can a mathematical concept be measured against something so nebulous as “at the College Algebra level”? Obviously, depth of investigation would play some role, but is “applied” less sophisticated than “theoretical”? If a history major has to abide by certain rules, does a future electronics engineering technician have to abide by the same rules? Should there be a basic technical mathematics course that all prospective technicians are required to take whether they need it or not? And remediation poses an additional consideration: are technical mathematics students to study the same remedial mathematics as other students?

In Texas, all college students seeking an Associate of Science degree are required to take a minimum of 35 general education hours, including six of mathematics. An Associate of Applied Science degree requires only 15 general education hours, at least three of which are mathematics. However, general education courses must be of general knowledge, with no discipline-specific course accepted. Therefore, technical mathematics courses are immediately in trouble. The objective of general education courses is to develop in students “those skills, understandings, and attitudes and that set of values which will equip them for effective, responsible and productive living in today's society.” It seems as if those attributes should be applied to any mathematics course, including technical mathematics. We need to persuade institutional administrators to change this policy.

The immediate goal of an Associate of Applied Science program is to provide the necessary skills and knowledge to prepare graduates for the workplace. A second objective should be to enable graduates to go on, either immediately or after some time on the job, to a baccalaureate program. To facilitate such a move, there are numerous transfer problems that need to be addressed. Since engineering and the sciences are the primary fields related to the advanced technologies, mathematics is a major concern. Technical mathematics courses need to be structured so they will fit into the requirements of the related baccalaureate majors. To meet a four-year requirement, transfer students are frequently forced to take a parallel course that contains the same topics as their technical mathematics courses, but may include more theory. Ideally, there should be an agreement by which technical mathematics would fit into the calculus sequence. Perhaps there could be a vehicle by which hours can be articulated by content, if not course-by-course. Another possibility would be a “bridge” course that would fill in any gaps between technical mathematics and the engineering calculus sequence.

CAROLE B. LACAMPAGNE

U.S. Department of Education

BARRY CIPRA

Northfield, Minnesota

CAROLE B. LACAMPAGNE is Senior Research Associate in Mathematics at the U.S. Department of Education where she co-monitors the National Center for Research in Mathematics Sciences Education. In December 1993 she convened the Algebra Initiative Colloquium which focused on four areas: algebra for all K-12 students; algebra for pre- and in-service teachers; algebra for the technical work force; and algebra for future mathematicians and scientists.

BARRY CIPRA is a freelance science writer. Before devoting full time to writing, he was a member of the Mathematics Department at St. Olaf College in Northfield, Minnesota.

Millions of jobs exist today that did not exist when the people who hold them were in school. Consequently, students' mathematics education must prepare them not just to cope with a rapidly changing world, but to participate in changing it. Because of the volatile nature of the workplace and because adolescents' interests change so dramatically throughout their education, it is a mistake to classify students according to their mathematical skills and aspirations. While it is always done in the so-called best interests of students, ostensibly

taking into account their individual abilities, tracking too often derails those in the lower tracks, shunting them aside into courses with minimal expectations.

The following recommendations are concerned with the evolving needs of the technical work force. They are drawn from The Algebra Initiative Colloquium, a soon-to-be-published report of the U.S. Department of Education.

• A CORE CURRICULUM

All students should study the same mathematics curriculum through grade 11, but not the curriculum that currently exists. Algebra should be a significant part of that curriculum, although not necessarily a discrete course.

• MATHEMATICS IN GRADE 12

All students should be encouraged to study four years of mathematics in high school. The final year should offer a selection of courses that are challenging and rewarding, as well as relevant to a high school senior's personal goals.

• MATHEMATICS FOR THE TECHNICAL WORK FORCE

All mathematics courses (K-16) should integrate preparation for the technical work force into the curriculum. Pedagogy in those courses should prepare students to become independent learners of mathematics and other technical subjects.

In order for these recommendations to be implemented, a number of issues will have to be addressed by the mathematics community.

The mathematics curriculum should be grounded in problemsolving that reflects real-world situations and offers a variety of methods of solution. Mathematics is generally seen as disconnected from the rest of the world and mathematicians, by the way they teach their subject, tend to encourage this view. However, one of the significant changes in the mathematics community in recent years has been a push toward establishing academic-industrial liaisons. More mathematicians than ever before are turning their talents toward “real world” problems—and finding interesting mathematics in return. What is called for is a close examination of the way mathematical thinking actually helps in real situations and a tailoring of the curriculum to ensure that students understand and can make use of the connections.

Students should have opportunities to participate in group work, make appropriate use of technology, and develop their communication skills, including the reading and writing of technical materials. These are the ways mathematics is used outside of school: by people working together, using whatever resources will do the job, and explaining to each other, their bosses, and their customers what they've done. The way students are currently graded in mathematics is geared toward isolated, individual study, but teachers know that students can learn far more from working together in small groups than they can by mimicking the demonstrations of an adult lecturer.

Higher education is often thought of in terms of big state universities and small liberal arts colleges, but large numbers of students attend two-year or community colleges. Many are returning students—adults who have held full-time jobs for a number of years. Much of the technical work force takes this route. Community college mathematics faculty must take the lead in meeting the real needs of these students. To do this, they must be involved in curriculum development in technical areas—health and human services, business and information management, agriculture and agribusiness, and engineering and industry. They also should take an active role in needs assessment and articulation with secondary schools and four-year colleges and universities.

There are two areas in which research is needed: assessing the value of remedial mathematics programs; and determining ways in which mathematics is used in the workplace.

Remedial instruction is big business in today's education establishment. However, it is not clear that much is gained by having a student struggle one more time with fractions. “Remedial” students may need help understanding what fractions mean, but that 's a different—and far more complicated—matter than merely “getting up to speed” on mechanical aspects of dubious value. There are many people who are proficient at the mechanics of fractions but who nonetheless are totally confused when it comes to using them. The same applies to many other parts of the traditional mathematics curriculum, but we do not yet know just what prerequisite mathematics is really needed for technical mathematics. Research is needed on whether there are alternatives to the several semesters of remedial arithmetic and algebra that many students are required to take before beginning a technical mathematics course.

Research is also needed on the kinds of mathematics students will need to be successful employees in a technical work force. Discovering how mathematics is used in the workplace is a subtle research problem that should be carried out by mathematicians who can recognize when mathematics is being used in situations that do not look like textbook examples. Mathematical thinking is not restricted to situations where numbers are involved. For example, algebra not only involves distinguishing what is known from what is not known or determining the relationships between them (generally for the purpose of figuring out the unknowns), but it also involves recognizing and representing patterns of virtually any kind, as well as comparing or manipulating such patterns. Algebraic patterns arise in making sense of the world or in making sense of other mathematics. The logical aspects of algebra carry over into areas that are not transparently quantitative.

PAMELA E. MATTHEWS

Mt. Hood Community College

PAMELA E. MATTHEWS is Associate Dean, Mathematics Division, of Mt. Hood Community College, Gresham, Oregon. She is the Principal Investigator for Application-Based, Technology-Supported, One-Track Mathematics Curriculum, a balanced, coherent entry-level mathematics program for all students (tech-prep and college-prep).

In order to function successfully in our ever-changing and highly technological society, all students must have fundamental knowledge in mathematics such as recommended by the NCTM Curriculum and Evaluation Standards for School Mathematics. Equally important is the way in which students are provided access to learning mathematics. How students are taught and what assessment tools are used play major roles in what mathematics students have opportunity to learn. In many cases, the mathematics students are exposed to is determined and limited by the amount of arithmetic skill they are able to demonstrate.

Technology is an enabling tool that provides opportunities for students to learn more mathematics earlier and apply it in a meaningful way. However, mathematical techniques must be distinguished from mathematical concepts, number facts

distinguished from number sense, and algorithmic learning distinguished from mathematical modeling. If we understand mathematics in the broader sense—as a way of communicating, thinking, reasoning, and modeling phenomena—then a strong case can be made that all students should learn and be able to apply the rudimentary concepts of algebra, geometry, probability, statistics, and number theory. These concepts are the prerequisites to a functions-based precalculus course of study as well as to college-level discrete mathematics, probability, and statistics. The demands of today's workplace require that students entering the technical work force through tech-prep programs, 2+2 programs, or youth apprenticeships learn more mathematics, and that baccalaureate-prep students learn how to apply what they know.

The purpose of high school mathematics is to provide all students with a fundamental knowledge of mathematics well beyond arithmetic skills.

By redefining the pedagogy employed in mathematics education, fully integrating the use of technology in the learning process, and varying assessment instruments, we no longer need to track students at this level. We should measure students' learning of mathematics by levels of knowledge attained rather than by grade level. This practice removes the stigma of remediation and permits natural rates of learning. Assessing a student's level of knowledge should not be compromised by constraints in time, rate, or pedagogy.

In today's world the application of knowledge is as essential as the attainment of it. “Workplace-based” mathematics can be viewed as a subset of applied knowledge. As we accept the challenge of giving definition to the mathematical preparation of the technical work force, fundamental beliefs and attitudes must change regarding what all students should know about mathematics, how it is learned, and how their attainment of mathematical knowledge and applications should be measured.

PATRICK DALE MCCRAY

SEARLE

PATRICK DALE MCCRAY is a mathematician at SEARLE, a pharmaceutical company, where he provides support for commercial systems, system software, and scientific applications. He is the Governor-At-Large of the Mathematical Association of America (MAA) representing mathematicians outside academia and is a member of the MAA Human Resources Council.

At professional conferences such as those held by NCTM, people teaching mathematics in vocational education programs and mathematics teachers should have a chance to discuss issues of common interest. They can do this, however, only if members of both communities actually do attend. From the perspective of a practitioner in industry, the fact that there are two different, non-overlapping communities is striking and indicates that communication on these issues is needed at this juncture. Ventures in business or industry that are at all non-trivial always amount to groups of people who must accurately, efficiently, and effectively communicate if the right things are to be done in the right ways at the right times.

Education benefits both the individual (local) and society (global), so it can be described from various perspectives according to who is gaining what kind of benefit. The more an

individual knows about mathematics, the more that individual can apply such knowledge in dealing with the real world. The more mathematics an individual knows, the more mathematics that individual can learn and the better that individual can appreciate the role of mathematics in his or her own culture. Quite clearly, based on evidence of people engaged in mathematical activities in business, industry, and government, mathematics does not have just one purpose.

Various specific subject areas, ranging from percents to first order differential equations, arise in the context of practical problems. The usefulness of these subjects can be seen in their direct applicability. Moreover, the mercantile-type mathematics of the shopkeeper will probably always be with us, given the human tendency to haggle and barter. It also seems clear that situations will keep arising in which specialized topics occur in a natural fashion and for which solutions will not be readily available. People in the technical work force will be confronted with them and will have to solve them or get other people to solve them.

Unfortunately for the reputation of mathematics, a lot of high school mathematics has been vilified in the past, and probably will continue to be in the future as having no practical purpose whatsoever. The same could be said of other academic subjects in high school which are not in technical education programs. Latin anyone? Taxonomy in biology class? (Why do I have to know the scientific name of a family, genus, or species?) Yet taken together, these “useless” subjects comprise the core and provide part of the benefits of a liberal education.

This is an urgent plea from an industrial mathematician to keep open as long as possible the options available to students to learn and study mathematics, both from the perspective of giving them facts to use and also from the perspective of giving them habits of mind to employ. More important than knowing facts (mathematical trivia) is the ability to reason abstractly, rigorously, and precisely. No fuzzy headed thinking wanted here! This is critical for our society and economy to survive and prosper, not to mention our planet.

Although mathematics is one subject, people are different and the ends to which they put their education are varied. Educators should therefore keep in mind that fundamental understanding of mathematical concepts derives from work with concrete examples and that the ability to work in practical contexts is much improved in individuals who possess a firm grasp of the underlying principles involved. This argues for a classroom situation in which all students are exposed to both aspects of mathematics. It also argues for an educational approach rich enough to include as outcomes pathways for people whose formal education ends at the twelfth grade (high school), fourteenth (trade school), or sixteenth (college). But certainly locking students into situations that remove their options is to be avoided at all costs. Instead:

• Students should study all kinds of mathematics. If they don't use the facts, they will use the ways of thinking about them. They should be expected to come up with their own solutions to problems.

• Students who go into the work force instead of to college don't need different mathematics; they need to really understand what has been presented to them.

• Students thrive in courses where concepts are presented in close proximity to their intended use. This is especially true in the years directly preceding the end of students' formal education. (We've seen this with our summer interns.)

The best and brightest students are only so in the context of the metrics used to identify them. Bright people should be encouraged to pursue their dreams, to do what they do best, to do what they love. With alert and active minds people will be well prepared to take on the challenges of the unknown that life and the future bring to them.

MARTIN NAHEMOW

Learning Research and Development Center, University of Pittsburgh

MARTIN NAHEMOW is a physicist who serves as a project director at the University of Pittsburgh's Learning Research and Development Center. He is responsible for curriculum and portfolio assessment system development and teacher training for the Pennsylvania Youth Apprenticeship Program.

Let's start by accepting that all mathematics is someone's workplace mathematics! I think we also can accept the axiom that part of the obligation of secondary school education is to provide all students with a set of foundation skills (as suggested by the SCANS report), one of which is mathematics. From here on it's not so easy to agree—issues of social policy, equity, tracking, etc. set in and cloud the issue.

There are those who argue that part of the purpose of learning mathematics is that it hones one's ability to reason and solve problems. This is the classical education model and certainly has validity. There are those who argue that the obligation of high school mathematics is to prepare students for college, and there are others who take the more European point of view that secondary school has an obligation to prepare all students for careers as well as for further education. Most educators will not deny the validity of any of these three positions, though they may question how much of each is good for whom.

The conundrum here is rooted in the conflict between the 65% or more of high school students who express their intention to apply to college, the 48% who actually enroll, and the 18% who eventually get a four-year degree; perhaps less than half of these (9%) take advanced mathematics courses in college. If the teaching of mathematics is co-opted by the 65% figure, the curriculum (the mathematical foundation skills) becomes a college preparatory pre-calculus skill set. Those who cannot “cut it” are tracked into “vocational math.” If on the other hand one looks at the 18% or 9% figure, those responsible for secondary mathematics are forced to address the problem very differently.

The foundation skills seem at first to need to serve two masters at the same time. For want of a better distinction let's divide secondary school mathematics into pre-calculus and non-calculus related mathematics. Set aside the possibility of an AP calculus course in high school. We can then divide each of the sets into an applied and a pure mathematics subset and ask: Do each of these four subsets contain good mathematics, and which students should be taught which subset?

At this point let's cease to be a herd of ostriches. If we take our heads out of our mathematics books and look around the world, we will see that (except for South Africa) all the industrial countries of the world have concluded that these subsets are unnatural artifacts of education and neither represent the real world nor serve the future interests of any students. To use the Japanese term, students learn “Maths” in high school—not trigonometry, algebra, business mathematics, or geometry, applied or otherwise. In Denmark, a country with among the highest educational standards in the world, the mathematics book for the last year of compulsory education for all students starts every chapter with concrete real work problems such as: What makes bridge tresses rigid? How do you evaluate the advantages and real costs of a loan? How do you set up the angles on a pool table? How do you develop DWI laws from rate equations for the time dependence of blood alcohol level? These countries realize that preengineering students need to understand the practical side of applied mathematics in high school as much as the students who will go into business, the trades, the crafts, or the service industries.

ARNOLD PACKER

Johns Hopkins University

ARNOLD PACKER is Senior Fellow at Johns Hopkins University's Institute for Policy Studies in Baltimore, Maryland. He is co-author of “Workforce 2000” and former Executive Director of the Secretary's Commission on Achieving Necessary Skills (SCANS). From 1977 to 1980 he was Assistant Secretary of Labor, and he was the First Chief Economist of the U.S. Senate's Budget Committee.

Most people who think about mathematics education propose mathematics classes that teach students to solve real-world problems. But, as yet, there is little discussion about what “reality” should be the source of those problems.

Consider today's typical high school algebra class. The problems are still likely to be the ones that were on exams a generation ago. “Find the speed of the canoe if the paddler is paddling six miles an hour upstream and the current is three miles an hour” or “Where will two planes cross if one leaves New York at 6:00 AM and the other leaves LA at 9:00 AM and both are traveling at 500 mph?”

Some might say that these problems are about real things: canoes and streams and planes and cities. By this definition, word problems are “real world” in contrast to problems that are only about disembodied x and y. But other and better definitions are possible. For example, consider the following: Secondary school mathematics should be about solving real-world problems. School problems would then resemble problems that some significant numbers of adults are paid to solve.

Using this definition, algebra students would be asked to solve problems of the following kind: “Working in teams and using computer software (database, word-processing, spreadsheet, and graphics package), write a report to evaluate buying a new piece of equipment.”

Students would establish specific work contexts for the generic problem of evaluating an equipment purchase. For example, students interested in health care might consider acquiring a new MRI machine at a local hospital. Those interested in finance might think about a new optical scanner. Students planning a career in manufacturing could learn how to justify a new numerical]y-controlled machine tool. Budding restaurateurs could ponder buying an automated chicken frying machine.

One teaching strategy would put all the data needed on an electronic database so that students, working in teams, could solve the problem in one or two class sessions. Another strategy would make this problem a six-week project, done in cooperation with English and science teachers. Students could collect data to help local administrators analyze the purchase of a piece of equipment that the hospital is really considering. They would have to determine how many of the community hospital's pediatricians, neuro-surgeons, and oncologists typically send their patients for an MRI and in what proportion. They would have to find out what Blue Cross pays for an MRI exam, how much it costs to buy an MRI machine, and how the purchase might be financed. Students would use e-mail and the Internet to obtain data, and use computer spreadsheets to simulate alternative scenarios. Finally, they would write a report containing graphs and charts that the hospital administrator would read and act on.

The goal is to have secondary school mathematics students learn three things simultaneously: the academic discipline (algebra); career awareness (what mathematics a hospital administrator uses); and “workplace know-how.” The most widely known definition of workplace know-how is the so-called SCANS competencies developed by the Secretary's Commission on Achieving Necessary Skills. The five SCANS competencies are:

1. Allocating resources (time, money, space, and staff);

2. Managing information (acquiring, evaluating, communicating);

3. Interpersonal (team work, negotiating, teaching, leading);

4. Systems (understanding, monitoring, improving, and designing);

5. Technology (choosing and applying).

An imaginative teacher might use the MRI problem to teach algebra, to make students aware of a whole class of professional jobs, and to help students acquire the workplace know-how defined by SCANS. Isn't this approach consistent with the NCTM Standards and, indeed, doesn't it show the way to making them operational?

JOSEPH G. ROSENSTEIN

Rutgers University

JOSEPH G. ROSENSTEIN is Professor of Mathematics at Rutgers University. He has been Director of the New Jersey Mathematics Coalition, the New Jersey Mathematics Curriculum Framework Project, and a number of programs for teachers.

Historically, schools in the United States were designed with a dual mission: to teach all students basic skills required for a lifetime of work in an industrial and agricultural economy and to educate thoroughly a small elite who would go to college en route to professional careers. As the needs of society have changed the balance of these two goals has shifted. Today's schools labor under the legacy of a structure designed for the industrial age misapplied to educate children for the information age. The level of literacy formerly associated with the few who entered college must now be a goal for all.

– EVERYBODY COUNTS

If we take seriously the conclusions of Everybody Counts, then we must also conclude that the mathematical preparation of the technical work force needs to be as strong as the mathematical preparation of the professional work force. Fortunately, all of the trends in mathematics education seem to be working toward narrowing the gap.

First, the idea of national standards, as expressed in the NCTM Curriculum and Evaluation Standards for School Mathematics and in standards documents currently being developed for other content areas, is that we should set high and achievable standards for all students. That means that we as a nation are thinking through the question of what we should expect every high school graduate to know, understand, and be able to do with the content encountered during her or his 13 years of K-12 schooling. As a result of this process, we should arrive at decisions about where we want to go, and then set about the task of achieving that goal. Although some have misgivings about the outcome of the process, it seems that all involved in it are committed to setting high standards, at a level far above traditional and basic skills, and to working toward ensuring that all students achieve those high standards. (“Do we really mean all students?” we are frequently asked. Our response should be that there may be exceptions, but that the exceptions should be exceptional.)

Second, the recommendations emerging from the standards development process seem to be remarkably consistent, going beyond an exclusive focus on traditional content and including significant emphasis on process. The process standards in the NCTM Standards, which involve problem-solving, reasoning, communication (of mathematical ideas), and connections (within mathematics and to other subject areas), are likely to be strongly echoed in the standards documents developed for other subject areas. (They already are echoed in documents such as AAAS' Benchmarks for Science Literacy, the National Council for Geographic Education's draft National Geography Standards, and the NRC's Draft National Science Education Standards.)

As we increase attention to these process standards, mathematics education will tend to move away from the relatively linear, abstract, formulaic, and solitary approach of traditional high school mathematics and more toward an exploratory, incontext, problem-solving, and cooperative endeavor. This is moving closer to what is normally thought of as vocational education; indeed, an often explicit assumption is that students in school should be learning and using mathematics in ways that reflect how they will be using mathematics in the workplace.

Third, within vocational education the trend seems to be away from the traditional apprenticeship perspective, where a student implements the pattern of a widget designed by someone else, to a more professional perspective, where the design of the widget is part of the student 's

task. The mathematics involved will be more substantial, including much more algebra and geometry than is currently expected from students in vocational/technical education. Thus the curricula for these students will be moving closer to that of students in non-vocational programs.

Fourth, the tendency in assessment is away from occasional examinations on isolated information and toward on-going assessment of longer-term problem situations. This also moves closer to the kinds of activities and assessments that are being developed for students in vocational areas.

Fifth, computers increasingly are used to simulate a variety of situations, so vocational students will need the conceptual understanding to interpret what they see on the screen, and non-vocational students will need the technical understanding to implement what they see on the screen.

These trends, taken together, reinforce the idea that all students should be learning more of their mathematics by looking at real-world situations, by working on large-scale projects, by constructing their own concrete solutions to problems, by focusing on conceptual understanding and problemsolving, and by working with other students and with a variety of tools. Attention to authentic problem-solving reinforces the idea that there should be a common core of mathematical experiences that all students share, which in turn would enable them to achieve the high expectations embodied in the NCTM Standards.

Having said all that, let me offer some caveats. In this short discussion, I am undoubtedly oversimplifying these trends and displaying too much optimism, but it's a pleasure to do so. It is also important to raise a question which is not really part of this discussion, but is always on everyone's mind because we are concerned about both our own children's futures and the future scientific competitiveness of our country. That question is: Will these changes undermine the achievements of the “small elite” referred to in Everybody Counts?

What this concern often boils down to is this: What will be the impact of a standards-based mathematics curriculum on the traditional precalculus sequence of courses that dominates the high school curriculum, and on the students who currently take these courses? This sequence was designed for the “small elite” that pursue scientific careers and in many ways has served them well. However, it also has ensured that all but this group consider themselves as mathematically incompetent because of their failure to overcome the hurdles that the precalculus sequence provides. As a result of the Standards, we can hope that new and rich mathematics curricula will emerge that have the opposite effect—they will enable the mathematical empowerment of all students. The result for the small elite should be that, in addition to mastering the traditional topics, they will have the common core of experiences and master the expectations (including the process expectations) that emerge from the Standards. The timing of the material in the traditional courses may be different. If the more formal approaches to algebra and geometry are reserved for those pursuing scientific careers, then the material can be covered in a much shorter time, especially if students have been using these topics in a problem-solving, applications-oriented core experience.

Thus, for example, we can envision curricula in which the four years of traditional high school mathematics are condensed into two years of coursework near the end of a student's high school career; or we can envision curricula in which more traditional approaches are provided as a parallel supplementary option. In any case, we need to first agree on the assumption that those students who can achieve more than the expectations of the Standards and the core experiences should be afforded the opportunity and encouraged to do so. We can then, in a problem-solving mode, determine how that assumption can best be implemented in a variety of school situations. There is no reason why implementing the Standards should result in a weaker preparation in traditional mathematics for those who will need it or want it, and there is no reason why it should lead to a diminution of the quantity and quality of our scientific work force.

JEAN SIMCIC

Schenley High School

JEAN SIMCIC teaches applied mathematics and physics at Schenley High School. She was part of the curriculum writing team for the Pennsylvania Youth Apprenticeship Program and has conducted workshops on integrated curriculum for Jobs for the Future.

When I first read the request for a commentary, I sat down to compose my thoughts. My first thought was, “Hey, I teach mathematics and science! Why do I have to write a 1-2 page anything? I'm not in English class!” Have I been listening to my students too long? Probably. Our students in “traditional” programs have been conditioned to sit for 42 minutes and work on one discipline and then move to the next class and the next subject. Heaven help us if a mathematics teacher asks students to write a sentence. When my students are solving physics problems, I just love the comment, “This isn't math class; why do we have to know trig? ” Or when asked to explain an answer on an essay test, they answer “yes” or “no.” Am I expecting too much? Should I “dumb it down” to meet their expectations?

I was struggling with this for several years until I became involved with the Pennsylvania Youth Apprenticeship Program (PYAP). During that time, I taught half days and spent the rest of my time writing curriculum. As part of my work with the PYAP I met with employers and teachers of vocational education. Those meetings helped me realize that I had become so “focused” on my own area that I didn't know what was going on elsewhere. It is important that I communicate with the vocational (or technology) teachers in our building so that I can show my students how the mathematics and physics concepts learned in my class become tools to be used in their technology classes.

I worry that we seem to be tracking students into two areas: college-preparatory and vocational programs. Although “college bound” students are not choosing college as their career, most academic courses do not prepare students for life after college. All students would benefit from courses teaching all aspects of industry. However, one problem facing teachers of academic classes is the pressure from parents to keep “industry” out of the curriculum. Parents, students, educators, and employers need to learn about vocational education before all students can benefit from what those programs have to offer.

Working on an integrated curriculum (academic as well as vocational) led me to find a new approach to teaching. I now am spending more time integrating work-related skills with science skills. I expect my students to apply all disciplines in my class. I am trying to reinforce their mathematics skills and provide real-life applications for the tools they are learning in other classes.

Yet I find myself frustrated by teachers who find it easier to continue doing the same old thing, believing that someone else will prepare the students for work. Employers are looking to the schools for help, but not all schools are prepared to respond to these requests. Just as we tell students that they need to learn to learn because their education is never-ending, teachers have to look at their profession as everchanging.

GARY SIMUNDZA

Wentworth Institute of Technology

GARY SIMUNDZA is Professor of Mathematics at Wentworth Institute of Technology, and a former head of the Mathematics Department. He directs an NSF-funded project involving mathematics laboratories that is an outgrowth of his interest in developing learning activities that approach mathematical concepts through technical applications.

I've always enjoyed studying and reading about history, and have probably learned as much from historical novels (not to mention movies and the occasional “docudrama”) as from texts and other nonfiction works. James Michener certainly taught me more about Hawaiian history than I' ve learned from other sources, and Herman Wouk illuminated many aspects of World War II for me. I used to think the history I learned in this way was somehow second-rate and not “honest” history, but in some ways it's actually superior, especially as regards retention and real understanding of the interrelationships among social, cultural, and political history. (Of course, the skill and research of the author matters greatly.)

Similarly, when students learn mathematics in a tech-prep rather than in a college-prep program, or study engineering technology (at the college level) rather than engineering-science mathematics, it has sometimes been argued that they are not studying “honest” mathematics. But if some students learn mathematics better (or only) when it is presented contextually, it may be because of how they learn. In many cases, an apparent distinction in ability may actually reflect the contrast between one group of students who learn things because they 're told it's good for them, and another who require a demonstration that a topic matters before being able to properly assimilate it.

Among my own engineering technology students, I see many very capable people who did learn what they were told to in high school, and who value their algebra skills highly. But I also see many others who were unable to build a strong skills base but who are capable of (sometimes startling) conceptual insights. Students in this latter group routinely make considered and sophisticated decisions in other areas (such as construction estimation), while claiming “. but I've never been good in math.” (We all have adult friends who make the same statements.)

Tech-prep and other school-to-work initiatives offer the promise of enabling students to compete on an equal footing (even in mathematics), especially with the leveling potential of new technologies that can remove some of the routine computational obstacles to conceptual understanding. I would argue that contextually embedded learning should begin in the earliest grades: primary students may be able to obtain important underpinnings for later mathematics learning from experiences in areas such as art and woodworking. I am concerned, though, that these areas are often the first to be cut in times of tight educational budgets. (Perhaps the true “core curriculum” should be broader than the one that has commonly been accepted.)

Another concern of mine is that, in an attempt to make mathematics relevant to technical students, we might narrow the focus so much that we create another set of limits. Duke University's Project CALC bases its laboratories on “problems that students can recognize as important for someone” if not for them-

selves. But technical students often are characterized by their need to see immediate utility for themselves. One drawback of many so-called technical mathematics textbooks is problems that begin, “In civil engineering, the following equation is found ” which I've never found to motivate anyone.

Encountering a true need for mathematical methods to solve meaningful technical problems in a laboratory or field setting can motivate otherwise skeptical or uninterested students both to learn mathematics well the first time and to later use it in other courses. The challenge may be to exploit the motivational advantages of contextual learning without obscuring the inherent connections mathematics makes among seemingly disparate fields. (It can be a revelation even for students at the high end of engineering technology studies to see that the same differential equation can describe the motion of the top of the Hancock building in Boston on a windy day as well as the behavior of an AC electric circuit, merely by changing the variables.)

Considering the fact that many students ultimately will work in jobs different from those for which they are trained, even jobs not conceivable at the time of their education, it is especially important that the right balance between perceived utility and breadth be found. With technological tools that help level the playing field and interesting work-based problems that make the game worth playing, mathematics for technical students can be protected from attempts to relegate it to second-rate status.

DIANE M. SPRESSER

National Science Foundation

DIANE M. SPRESSER is Program Director in Teacher Enhancement in the Division of Elementary, Secondary and Informal Education at the National Science Foundation. She is on leave at NSF from her position as Professor of Mathematics at James Madison University in Harrisburg, Virginia, where she was head of the Department of Mathematics for sixteen years.

The opinions expressed in this paper are those of the author and do not reflect those of the National Science Foundation.

Four distinct types of curricula exist in eighth-grade mathematics in the United States: remedial, standard, enriched, and algebra. These programs, which have dramatically differing mathematical content, are found in many school systems. Remedial classes are typically dominated by such elementary school arithmetic topics as fractions and percentage, with little or no exposure to algebra or geometry. Standard classes include arithmetic and some topics in algebra and geometry. Enriched or prealgebra programs give a little more attention to algebra and a little less attention to arithmetic than do standard classes. Algebra classes provide an introduction to high school algebra. According to The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective (1987), “more extensive tracking was found at this level in the United States than in any other country in the study.” Furthermore,

. not only did the distribution of opportunity-to-learn provided by the U.S. eighth grade mathematics curricula serve to limit the efficacy of individual ability, but the sorting was not efficiently done. More able students were not always placed into demanding and opportunity-laden classes. These data raise serious questions about the merits of curricular differentiation, especially in grades as early as junior high school. Under current policies, large proportions of U.S. students are being deprived of opportunities to participate in the best mathematics programs that schools have to offer (p. 111).

It is also important to note that tracking has traditionally been much more prevalent in mathematics than in the sciences.

Arguments in support of tracking seem to fall into two broad categories. First, tracking allows for curricula and pedagogical methods that are commensurate with student abilities. Second, schools can better tailor the preparation of students to their post-secondary goals and expectations (e.g., those who expect to attend college are tracked into a college-preparatory or academic program).

Opponents, however, argue that tracking curtails and perhaps eliminates for many students the full range of options for their future study of mathematics, especially when implemented as early as the eighth grade. Tracking therefore does a great disservice to many of our students: it forces decisions on students at an age when they are not developmentally ready to make choices of consequence

about their career paths; or, worse still, choices are made by school administrators or guidance counselors for the students. Frequently, choices made by schools for students are based on perceived student abilities. Research suggests that such decisions are not without bias: racial/ethnic and socioeconomic attributes seem linked to tracking patterns.

The vision espoused by the NCTM Curriculum and Evaluation Standards for School Mathematics is one of high quality mathematical experiences for all students. The Standards cite new social goals for education that include (1) mathematically literate workers, (2) life-long learning, (3) opportunity for all, and (4) an informed electorate.

These goals are not consonant with early tracking mechanisms that preclude the study of higher mathematics. For secondary school mathematics,

. the standards have been prepared in light of a core program for all students, with explicit differentiation in terms of depth and breadth of treatment and the nature of applications for college-bound students. At the same time, the mathematics of the core program is sufficiently broad and deep so that students' options for further study would not be limited. Our expectation is that all students must have an opportunity to encounter typical problem situations related to important mathematical topics. However, their experiences may differ in the vocabulary or notations used, the complexity of arguments, and so forth (p. 9).

A high quality core mathematics curriculum—such as that called for in the Standards—will help to guarantee a mathematically literate work force and an enlightened society that appreciates and values the power of mathematical thinking. For this to happen, the core curriculum needs to extend as broadly and deeply as possible into the secondary school mathematics program. While this may be a tall order, it is not impossible. Some good curricular materials, such as those of the Interactive Mathematics Project (IMP), support three years of secondary mathematics for all students, with materials to be available for a fourth year. In addition, a core curriculum must provide multiple entry points for students who make a detour to re-enter as they discover the need to study more mathematics. Schools must maximize the mathematical options open to students.

In spite of the arguments that have been made about the desirability of a core curriculum, several communities remain unpersuaded:

• Some members of minority communities have expressed skepticism about current efforts in mathematical reform. Their goals are for minority students to be prepared to jump the traditional “mathematical hurdles ” because they see such hurdles as gateways to success established by a mainstream society.

• Within the mathematics research community, there has been concern that the emphasis on “mathematics for all” compromises the mathematical environment necessary to produce the kinds of students who will eventually join the academy as active researchers in the mathematical sciences.

• Gifted students and their parents are often the most vocal critics of core curricula. They see “mathematics for all” as an equalizer that diminishes their opportunities to learn. In a sense, this argument reflects the traditional sparring between elitism and egalitarianism.

The Standards provide for student experiences of differing intensities and timelines within a core curriculum. A well-structured core curriculum, coupled with a rich assortment of enhancement activities in mathematics that include summer camps and other special programs, can provide mathematical experiences that both challenge and nurture the next generation of citizens, workers, mathematicians, engineers, and scientists.

ELIZABETH J. TELES

National Science Foundation

ELIZABETH J. TELES is Program Director for Mathematics and Lead Program Director for the Advanced Technological Education Program in the Division of Undergraduate Education at the National Science Foundation. She taught mathematics at Montgomery College, a two-year college in Takoma Park, Maryland, from 1969 until 1991.

The opinions expressed in this paper are those of the author and do not reflect those of the National Science Foundation.

Faculty and teachers in cooperation with industry are now struggling to develop a vision for the mathematics core for the technical work force that balances abstract and concrete knowledge and skills. Building this vision requires consensus for a strong and balanced core of mathematics and science skills that prepare students for employment in today's highly technical business and industry environment. At the same time, this core must assure that students have a better foundation so they can readily adapt to today's changing technological society and be prepared for lifelong learning. This is no easy challenge.

Stakeholders from many levels of the education and industrial communities are beginning to discuss this important challenge. It is vital that these communities work together to improve education rather than remain isolated from one another or, even worse, regard one another with antipathy. Most discussions today call for programs that assure that students acquire communication skills, problem-solving skills, computer and other technological skills, analytical skills, team participation skills, work ethics, and applied skills. Seldom does one hear a call for more factoring skills or more time learning how to add fractions. Yet, these skills are still being emphasized as many colleges continue adding courses to the lower end of the scale.

The following is but a partial list of the issues I see facing these communities as they work to develop the mathematics core needed by the technological work force. In addition to the curricular and technological issues that are raised below, there are additional questions about resources in terms of time, equipment, and faculty enhancement. These are not easy questions, and the list below is not meant to be complete; I offer them as illustrations of the many problems with which I see people struggling.

• Is a one-track mathematics program good for all? If not, at what point should students branch into other mathematics sequences? If students branch, how can they move into parallel mathematics sequences without starting over?

• What is the balance between the abstract and the applied? Should all mathematics problems be presented in applied terms? If so, from where do we get good applied problems? What role can industry play in helping to provide such problems? If problems represent applications from certain fields, does that disenfranchise students with other interests?

• What types of technology are appropriate, and when should they be used? How much manipulation and working with mathematical concepts without the use of technology is needed? When are graphing calculators, computers, and microcomputer or calculator laboratories most important? How much time is needed to teach students the technology itself?

• What should be the balance in mathematics programs among numerical, visual, algebraic, and verbal understandings of concepts? What is a desirable balance between the conceptual and the algorithmic? Do students need to construct their own knowledge of every concept or can they build their understanding on what others have developed?

• What are the mathematical concepts that are most important for students in technical fields? Because students spend only a limited time in formal education, which concepts best prepare them for life-long learning as well as for immediate employment?

• Should mathematics be taught as a separate course; as part of integrated science, mathematics and technology courses; or as part of fully integrated programs of study?

• What is the role of long- and short-term problems? What is the role of laboratories and experiments in mathematics? How can students acquire conceptual understandings rather than just a collection of isolated skills which they cannot use when needed?

• What is the role of mathematics in other courses that students take? Can mathematics be taught using a “just in time” concept or must students know the mathematics first?

WILLIAM THOMAS

The University of Toledo Technical and Community College

WILLIAM THOMAS is Associate Professor in the Department of Technical Science and Mathematics at the University of Toledo Community and Technical College where he also serves as the Developmental Mathematics Specialist and Workplace Numeracy Consultant. He is a member of the task force that drafted the AMATYC Standards for Introductory College Mathematics.

In the spirit of the Christmas season, I say Bah! Humbug! to teaching mathematics differently in a school-to-work context versus a school-to-college context. While the Dickensian opening may be a little strong, putting our nation's foot to such a path will lead us to a miserly future.

Why, you ask, shouldn't mathematics be taught in a school-to-work context as well as in a school-to-college context? After all, most students don't go on to graduate with a four-year degree. Many students who go to work aren't mathematically prepared to do so. Mathematics to most of these students seems stilted and not very useful.

While all of this is true, we need to consider the consequences of teaching different mathematics in different ways to different groups of students. Although the intent is to teach all students mathematics, an inevitable outcome of such an approach is tracking. One problem with tracking is that students must commit to a career decision at a point in their lives when they have trouble choosing what clothes to wear! Whatever mathematics they study ought to leave students prepared to move among a variety of career paths.

Another problem with tracking is the negative impact on equity. Disproportionately small numbers of minorities and females are in the academic track, as they are the students most likely to be “encouraged” into a school-to-work track. Once there, the options opened to them are limited. As a result, we run the risk of aggravating an already unacceptable social problem.

The effect of tracking on students in the academic track cannot be overlooked. All students will eventually work. (Parents of college students may wonder if that will ever actually happen with their daughters and sons!) They will need many of the work-specific skills that students in school-to-work mathematics learn. A broader preparation that includes workplace-based mathematics will benefit college-bound students as much as those in school-to-work programs.

What should be done? A possible answer is that mathematics should be taught as a core subject from an applied perspective for all students, both those planning to go to college prior to work and those going directly to work from school. What might such a curriculum look like?

• It would be problem-centered and problem-driven. What most businesses are looking for is not specific mathematical skills but the ability to apply

mathematics to solve problems. Unfortunately, most mathematics is not taught in that environment. It should be. Mathematical concepts should be introduced by way of real problems in which mathematics is used as a tool to solve them. The problems should vary widely and come from as many career areas as possible.

• It would allow students to explore different careers. In their exploration, students would experience the mathematics involved in various careers and be aware of the required mathematics competencies.

• It would give students a yearly “capstone” experience in which, grouped by current career choices, they would solve real problems associated with a particular career. Problem situations would be presented by a team composed of mathematics, science, applied academics, and other pertinent discipline teachers as well as people from business and industry. The adults would act as coaches for the groups.

Much is needed to bring such a vision of mathematics education to fruition. A necessary component is the dynamic and active cooperation among businesses, high schools, and two- and four-year colleges. Many changes in mathematics instruction have emphasized the system and overlooked the human resources needed. We must realize that the human resources are a significant part of the answer. Teachers and people from business and industry must become intimately acquainted with each other's areas of specialization. There must be meaningful exchange programs which allow teachers to see their subject in action and allow business professionals to experience the classroom environment. Representatives from these communities should form work teams that could bring a broadened perspective to planning and implementing a meaningful mathematics curriculum.

My view of the school-to-work issue for mathematics may not be politically correct. However, I believe it offers us a tremendous opportunity to help mathematics teachers and teachers of applied disciplines to work together with the business community to help improve mathematics education for all students.

MARGARET VICKERS

TERC

MARGARET VICKERS is Project Director of the Working to Learn project at TERC, which is developing grade 11 and 12 science curricula to complement occupational learning in the workplace, and Director of the Center for Learning, Technology, and Work at The NETWORK. She has spent the last 27years as a teacher, researcher, writer, policy analyst, and curriculum developer on school-to-work transition programs and related reform issues in secondary education.

There are many issues to talk about: mathematics for the technical work force; science for the technical work force; integrating academic and vocational education; the curriculum for workbased learning; improving the “standards” students achieve in key fields of academic and workplace competency; developing and introducing technology education; etc.

While each of these issues could be considered as a reform agenda in its own right, in practice it is difficult to keep them separate. One reason the boundaries among these agendas seem so blurred is that, even though reformers who advocate different parts of the solution march under different banners, they are mostly concerned with the same set of problems. These are:

• Too much of the mathematics and science presented in high

school is taught in an abstract, decontextualized manner;

• Too much of our vocational education is narrow and occupationally specific;

• In traditional vocational courses, students learn what to do (the operational skills) but not why the operations work (i.e., the scientific, mathematical and technological bases of standard workplace practices);

• The idea that technology education should be a key element of the general education of all young people is gaining slow acceptance, but it needs more support;

• Tracking starts too early (at grades 7-9, in contrast with the Commonwealth and the non-Germanic European countries, where pathways diverge at grade 11);

• Students in vocational tracks almost never get a solid grounding in mathematics and science (some only study earth science and consumer math);

• The aspects of science and mathematics most commonly used in the workplace rarely are used as a basis for designing key elements of K-12 science and mathematics courses.

Additional points could be added, but I think a fairly large proportion of the reformers active in this field would accede to this broad definition of “the problematique.” It would be surprising if there were equal agreement as to what the ideal solutions may be. So instead of arguing about “ideal solutions,” I am going to critique some of the reform efforts that are currently in full swing.

The School-to-Work Act and the Carl D. Perkins Act advocate that students should achieve both the academic and technical competencies demanded by the high-skill workplace. Another nation-wide reform —the Goals 2000 Act—is encouraging states to develop content and performance standards for key subject areas of K-12 education. These three reform movements —the curriculum frameworks movement, the tech-prep movement, and the workbased learning movement—should function in a complementary way. The idea of “curriculum integration” is consistent with the philosophy driving all three reforms. Unfortunately, however, in the context of the development and implementation of curriculum frameworks, curriculum integration tends to be ignored. The problem is further exacerbated by the fact that efforts to develop national occupational skills standards are occurring in yet another totally separated arena. To date, no clear discourse has been established between the people developing the national occupational skills standards and those who are developing the national K-12 academic standards.

What is missing from the current array of reform activities is a workable strategy for promoting curriculum integration. This is a term which is used so loosely that it needs definition in each particular context. Here I focus on the integration of science and mathematics with occupationally relevant learning. Curriculum integration in this context means that students will learn not only how skilled technicians solve problems, but also why those solutions work. Students will learn how knowledge from science and mathematics informs practical action. Teachers from different subject areas will explore possible complements among their different ways of framing reality. Curriculum integration provides ways for “academic” teachers to contextualize theory, and for “shop” teachers to develop a rationale for practice. Students will learn to frame issues and phenomena from different perspectives and to recognize when a concept from a particular discipline might prove most useful in solving a problem.

In what follows I describe four strategies for promoting curricular integration. I believe we need to identify sites where such strategies are being used, research the relative merits of different strategies, and commit ourselves to scaling up our support for the approaches that prove most effective.

The first approach involves developing articulated courses on related issues and teaching them

in parallel. In this approach, for example, health science courses run parallel to courses in anatomy and physiology, chemistry courses to materials science and technology, physics to engineering science, and so on. I am not suggesting that we should have special anatomy and physiology, chemistry, and physics courses for the “voc kids. ” On the other hand, I am not arguing that all the voc kids should be doing AP physics. I am suggesting that a more inclusive, more contextualized approach to high school science (and to high school mathematics) is likely to ensure that many more students will learn mathematics and science and that they will enjoy these subjects more. This transformation would benefit all students, not just those who are currently in vocational or general pathways.

A second strategy involves developing technology education programs. In the context of mathematics and science curricula, technology education changes both what is taught and how it is taught. Classroom pedagogy becomes more “hands on” as design challenges become a central learning strategy. Course content changes as students learn about the social roles of technology, and as they learn to understand, evaluate, and use technology. Applied to the vocational education curriculum, an emphasis on technology education moves the focus away from the routine practice of psycho-motor skills, such as lathing or soldering, and re-focuses the student 's attention on the more complex and responsible task of designing, building, using, and evaluating artifacts to solve both individual and societal problems.

Work-based learning links science and mathematics learning with learning at real worksites. This third strategy is one we are implementing in our project, Working to Learn. We are attempting to establish relationships between knowledge which has been constructed for and has its utility in the occupational sphere, and knowledge that has been constructed in the traditional scientific arena.

All the schools in our project have well-established work-based learning programs, but hitherto there was no specific relationship between what was learned at work and what was learned in science classes. In some cases the students were not studying any science. Working to Learn is developing articulated science courses and creating student research projects that start with a real problem that practitioners face on their jobs or a standard practice they carry out.

There are three steps to establishing a relationship between what is learned at work and what is learned in the classroom:

• HOW: First we ask how a particular practice is carried out or how a specific problem is solved.

• WHAT: Next, we ask what scientific principles lie behind this practice. We seek to understand why a solution or strategy works.

• WHERE: Finally, we create complementary learning activities for students in the classroom and at the worksite:

At the worksite

• Students gather data.

• Students learn to carry out certain procedures.

In the classroom

• Students make sense of the data they have gathered.

• Students learn why procedures work, and reflect on their work experiences.

Finally, there is the thematic approach to curriculum integration. In its most extreme form, proponents of this approach suggest that “integration” demands eliminating discrete school subjects and structuring the curriculum around themes such as an extended study of the local environment. The strength of the thematic approach is that it can help schools break through the social, intellectual, and organizational barriers that keep the different subject-matter departments so profoundly separate. We cannot, however, eliminate the academic subjects themselves, nor can we do away with the occupations and trades. Even if we do not like the structures for creating knowledge that shape both the practical and the theoretical in the adult world, we have to live with them. The best thing curriculum developers can do is to ensure that school subjects are not presented as independent and inert constellations of knowledge but rather as richly interdependent ways of making sense of the world.

SUSAN S. WOOD

J. Sargeant Reynolds

Community College

SUSAN S. WOOD is Professor of Mathematics at J. Sargeant Reynolds Community College in Richmond, Virginia. As vice-president of the Mid-Atlantic Region, she is a member of the Executive Board of the American Mathematical Association of Two-Year Colleges (AMATYC). She also is a member of the task force that wrote the AMATYC Standards for Introductory College Mathematics.

Iteach mathematics in a comprehensive multi-campus community college with many technical as well as transfer programs, continuing education, and short-term business courses. I remember poignantly a particular student some ten to twelve years ago. The student, unsure as to whether he wanted a transfer engineering or engineering technology degree, had been advised by a counselor at the college to take the mathematics placement test for the program requiring more mathematics—engineering transfer. The counselor suggested that he then make his curriculum decision based on his success on the placement test for entry into calculus.

This advice worked against everything for which I believed the community college stood. The open admissions policy, the comprehensive mission, the second-chance opportunity for students who never saw college in their future, and the equipping of students to realize their individual “American dreams” are all part of my personal philosophy of why the community college is a necessary institution in our country today. It took some doing to convince the young man that his curricular goal should be established first. The nature of the mathematics coursework would follow according to his choice of curriculum and his placement recommendation. Mathematics must open the door to options.

My college is a site for satellite courses for a university 100 miles away, offering courses leading to a bachelor's degree in technology. Students who complete a two-year engineering technology degree at J. Sargeant Reynolds Community College can complete the necessary courses leading to the bachelor's degree in a familiar environment near their jobs and families. Similarly, students in transfer programs can enhance their understanding of their chosen field by gaining work experience in a technical environment through internships or cooperative experiences. Thus the distinction between what is technical and what is transfer is becoming less clear.

The American Mathematical Association of Two-Year Colleges (AMATYC), in conjunction with other national organizations, is about to complete a standards document for lower-division collegiate mathematics before calculus. Building on a vision of reform in curriculum and pedagogy for K-12 mathematics proposed in the NCTM Curriculum and Evaluation Standards for School Mathematics, the AMATYC Standards extends this vision to the first two years of college, while particularizing its goals to meet the needs of diverse adult learners in the college setting. These Standards propose for technical programs a rich, hands-on, technology-based, applications-driven curriculum in which students will not only learn mathematics in interesting, applied contexts but also will learn how to learn mathematics to prepare for the life-long learning necessitated by the changing workplace.

As colleges work to implement curriculum and pedagogy which conform to the AMATYC Standards in technical and transfer mathematics programs, the line between these programs will become even more fuzzy. A critical issue is raised when we consider the student who, upon completion of the mathematics necessary for a technical program, decides to pursue a college transfer curriculum. It is entirely

unsatisfactory to ask that student to begin all over again in mathematics. Rather, to facilitate the student's desire to set and reach a new goal we must either strengthen the mathematics courses in technical programs or design courses that satisfy the common needs of both types of degree programs. Such courses might include individualized projects and learning contexts specific to the needs of particular groups of students.

The future workplace needs of our country require some form of postsecondary education for many students. And that postsecondary education will likely not be a one-time occurrence. Retraining within careers as well as training for new careers is becoming commonplace. (Again, note the unique role played by the community college as an institution that can respond flexibly to the everchanging needs of local business and industry!) Since the goal of our students is to become productive workers and citizens, every student is both college-intending and work-intending. Every student needs the benefits of both “tracks, ” melded smoothly into a rich, all-encompassing program: rigorous academics combined with workplace training including apprenticeship, internship, and mentoring opportunities. This broad vision needs much further study. How to make it happen effectively and how to effect the transition to such a program are but two of the obvious areas of need. Related issues of preparing teachers, developing appropriate curriculum, studying pedagogy, selecting appropriate technology, and others follow closely as priorities for further study.

DEBORAH A. ZOPF

Henry Ford Community College

DEBORAH A. ZOPF is Professor of Mathematics in the Division of Mathematics at Henry Ford Community College in Dearborn, Michigan. She has developed a mathematics course for respiratory therapy students and a shop mathematics course used at several Michigan companies.

T he NCTM Standards have fostered a link between learner-centered mathematics classrooms and school-to-work initiatives. Many of today's mathematics classes are arenas of activity where students work in teams to discover mathematical ideas, to solve problems, and to communicate their results. This type of learning environment acts as a bridge from the academic world to the work world. The work environment values teamwork, exploration, problem-solving, and effective communication skills. These same skills are now fostered in many mathematics classrooms.

The connection between learner-centered mathematics classrooms and school-to-work initiatives needs to be strengthened. Development of classroom problem-solving activities similar to those found in industry is an obvious next step, although not one that is easily attainable. Finding corporate contacts who will make time on their calendars, who care about education, and who have the communication skills necessary to discuss the technical and mathematical aspects of their work is a very difficult task.

Incorporating technology in the classroom similar to the technology used in industry is a second step. It is important for students to be able to use calculators and computers, but it is equally important for students to understand the mathematics they carry out with technology. Correct interpretation of information displayed on a calculator or computer is critical. Students must have a solid mathematics background to do this.

Creating an environment where students present reports in a business-like fashion is a third option. Requiring written reports that contain data, analysis, and interpretation in a professional form develops these skills. Formal presentations of selected projects helps to develop the ability to communicate mathematical concepts.

A strong mathematics curriculum can integrate the development of skills necessary for solid academic work and real world applications. This type of curriculum creates a strong bridge from school to work. Yet many questions remain:

1. Is this type of curriculum appropriate for all students?

2. Could a four-year program in high school mathematics with a strong focus on applications be developed that would nurture all students?

3. Might the tracking in this type of program be similar to ability grouping? Would this be desirable?