*Vocational education is often a separate track in schools, hardly
integrated into the curriculum of college-bound students. What's your
vision of the ideal relationship between vocational and academic
education? Would you prefer a completely integrated curriculum? *

I'm certainly against the current two-track approach. My own suspicion is that it would be difficult indeed to completely eliminate units called "courses." Even in a well-integrated program there would still be courses called "mathematics," "English," etc., although the content of each of them would be more integrated both in terms of broad occupational applications and other disciplines. In contrast to what some others believe, I don't think it is possible to completely integrate courses so that disciplinary content and titles disappear. Indeed it isn't even desirable, since disciplines have been the focus of a great deal of intellectual development that should not be lost. (Howard Gardner has written eloquently about this recently.)

So a well-integrated curriculum would have two sets of pressures on every element--disciplinary content and perspectives, and interdisciplinary or applied content and perspectives--with the balance shifting according to students' and teachers' interests, students' goals, and the age and developmental stage of students.

*Mathematics teachers are often reluctant to participate in
interdisciplinary programs, especially in vocational areas where, they
believe, the mathematics is routine and unchallenging. What are the
advantages for mathematics to be taught as part of an integrated
vocational program?*

The most obvious advantage is that students might actually learn some mathematics--in contrast to the current situation where students retain very little of what they are taught. I do admit that one has to search hard for the right applications for various kinds of higher-order mathematics, and that a great deal of applied mathematics is simply arithmetic. It's also the case that many instructors may not have a good idea of what mathematics applications are appropriate.

For example, I once had an extended conversation with business teachers in
California who were aggrieved that their accounting courses did not count
as part of the University of California's mathematics requirement. This
is an absurd idea, since accounting is basically just a form of
arithmetic, and the University is looking for higher level sophistication.
But these teachers *could* have devised a higher-order business
mathematics program that introduced various kinds of optimization methods,
statistical analyses, forecasting models, optimal portfolio problems, and
the like. The upper reaches of business has many relatively complex
applications of mathematics. The search for appropriate forms of applied
mathematics is a difficult one, but I certainly think it can be done.

*For sure, there are many opportunities to use sophisticated mathematics
in business. But would a high school program that emphasized this type of
mathematics still be a vocational program? Could one justify replacing
accounting with optimal portfolio management and still achieve the goals of
K-12 vocational education?*

Sure. The point isn't to teach students every detail of accounting; they wouldn't need it for any job they could reasonably get, and much of it would be learned on the job anyway. But they do need to know how to think about business problems, and here optimization as a general approach (as well as a mathematical technique) would be much more useful than procedural detail like odd FIFO/LIFO rules (or whatever one learns in accounting).

*Has the widespread use of computers for modeling and graphical
analysis changed the kinds of mathematics that students need to
study?*

I suspect so, although as a non-mathematician I feel that I'm not especially competent to judge. There are many issues in which graphical representations are important, many of which have (I think) certain mathematical aspects in addition to spatial (visual) aspects. These problems come up in many occupational applications. For example, in computer-assisted drafting, where movement between two and three dimensions is important; in reading blueprints and diagrams of all sorts; and in what ETS calls "document literacy," which involves extracting information from various tabular and graphical presentations of information.

I'm not sure how these mathematical/spatial competencies are best taught, although nothing that I remember from school geometry is adequate to the task. I know that occupational instructors sometimes ignore these dimensions of instruction, and develop their own approaches when they find their students' backgrounds deficient in some ways. I suspect that a more sustained conversation would be valuable.

*The mathematics and science standards tend to enhance the strength of
the disciplines, whereas employers rarely stress discipline identity as a
priority for education. How would you advise schools about dealing with
these competing priorities?*

This is sometimes expressed as the effort to reconcile academic standards with skill standards (or workplace standards). The reconciliation can take place either at the level of the standards themselves--for example, by trying to make the mathematics and science standards more connected with applications--or at the level of teachers who have the facility to generate their own curricula to meet conflicting sets of standards. The latter approach is ideal, and necessary for many kinds of reforms, but also very difficult because it requires a different form of teacher preparation, and on-going support as teachers continually devise and change their own materials. If this kind of reconciliation does not take place, however, then disciplinary standards typically rule--and we will remain in our current situation, where students leave school with very little retention of the mathematics and science they have learned, and employers continue to complain bitterly about the quality of students they hire.

*Your analysis leaves few good options. Teachers rarely have the time
(or background) to develop curricula that meet conflicting standards, and
no one is seriously trying to write new standards that synthesize the
divergent academic and skill standards. Is there any hope for substantive
change on a large scale?*

Well, this is a big problem. The current way we structure high school teachers' lives--teaching 5 classes of 30-35 students--makes it impossible for them to reconstruct the teaching of anything, never mind integrating academic and vocational content in any meaningful way. That's why high schools that have been serious about reform have created other kinds of schedules for teachers (e.g., academic clusters, where they see fewer students each day, or block scheduling) and have tried to do something about professionalizing teachers. To me, this means making sure they have the knowledge and time to create curricula as necessary, rather than simply "managing" a curriculum that comes from other sources. If this makes integration hard, that's right. That's why we need a fundamental reshaping of the high school.

*W. Norton Grubb is an economist who teaches in the School of Education at
the University of California at Berkeley and is a member of the Board of
Directors of the National Center for Research in Vocational Education
(NCRVE). He can be reached by e-mail at
*`wngrubb@uclink4.berkeley.edu`.

*Last Update: * July 7, 1997