Mathematics permeates the high performance workplace at every step from planning and design to production processes and quality control. For example:
- A trained mechanic adjusting the tension on belts attached to the
drive shaft of a car or truck engine relies on geometric experience to
determine just how much play the belt has, first to compare this play
with specified tolerances, and then to determine how much to adjust the
drive mechanisms to bring the tension into the proper range.
- Anyone who assembles or repairs complex machinery must interpret in three dimensions information that is provided in two-dimensional illustrations such as exploded parts diagrams, blueprints, or computer assisted design (CAD) displays. "Seeing" three-dimensional arrangements in a two-dimensional illustration requires considerable geometric sophistication.
- Any retail business needs a system of inventory management to ensure that commonly requested items are readily available (either on the shelves or in nearby storage) while infrequently needed items are available by special order at a reasonable price. The balance between ordering in bulk and ordering individually involves not only careful cost accounting but also mathematical modeling to determine the optimum balance.
- Technicians in auto manufacturing facilities routinely prepare
programs for Computer Numerical Control (CNC) machine tools. A
mistake in the program can easily cause the tool to crash into and
destroy other expensive equipment. Without solid understanding of such
issues as relative coordinate systems, it is very easy for a technician
to make expensive blunders.
- Elevators used to be programmed to take each call in the order it was received. Then came zoned systems with certain elevators skipping large blocks of floors. Today's elevators use "fuzzy logic" which can balance multiple (and sometimes conflicting) objectives. The latest trend is to use neural networks that learn from past traffic what to expect and move elevators in anticipation of someone's request. Salesmen, technicians, and building managers, among others, now need to think mathematically in order to understand how elevators operate.
Workplace Conundrums
Because mathematics is so important in today's workplace, it is an important factor in hiring and promotion. In talking with employers about the quantitative and mathematical skills required of their employees, several dilemmas surface with some regularity--some of which appear to be well documented, others of which perhaps deserve more serious investigation. Examples:
- Many production workers, especially new employees, do not possess appropriate mathematical and problem solving skills. A common complaint, yet these employees somehow manage to remain employed. It is often hard to separate essential mathematical needs of a job from ideal expectations.
- Insufficient proficiency in mathematical and quantitative skills is an important factor in rejecting many job applicants. One often hears that industries have to screen dozens of candidates to find one they are willing to hire. However, the screening processes rarely employ instruments capable of assessing a candidate's ability to perform the kind of mathematical tasks that employers say they want. Often the only test of quantitative skills is a very sterile context-free test of arithmetical procedures. Such tests will rarely provide reliable indications of a candidate's ability to use mathematics appropriatley on the job.
- Industries invest extensively in upgrading workers' basic skills, especially in quantitative areas. This appears to be true of large industries that depend heavily on technical expertise, but it is not true of small businesses that employ the majority of workers. To understand the true commitment of business to worker training requires disaggregation of businesses by type and size.
- Neither employers nor employees realize the extent to which mathematical skills or quantitative reasoning are used on the job. Adults typically imagine mathematics to be only what they were taught in school, and they see little of this in their work. Mathematicians and mathematics teachers, on the other hand, have a broader view of the subject, so they can see instances of mathematical reasoning, estimation, and problem-solving where others see only common sense.
- Many jobs could be performed better or more efficiently if workers knew and used more advanced quantitative techniques. Often asserted by mathematics teachers and some problem-solving advocates in industry, this claim is especially difficult to assess because in so many areas coputers embedded in automated equipment have taken over calculation formerly required of employees. However, the success of high performance manufacturing offers compelling evidence, since improved quality has been a direct result of employees at all levels understanding and utilizing techniques of statistical process and quality control.
Educational Background
The mathematical needs of high quality ATE programs are very similar to elements featured prominently in the standards for school mathematics (NCTM, 1989) and for collegiate mathematics before calculus (AMATYC, 1995). In particular, these standards' emphasis on data, statistics, and geometry, together with their focus on active pedagogy and authentic contexts, provides a firm mathematical foundation for ATE programs.
Unfortunately, many students enter two-year colleges with significant deficiencies in mathematical preparation, not only when measured against the new standards but also in terms of traditional expectations. Students come both from a variety of educational backgrounds (e.g., college-prep, vocational, tech-prep, 2+2, and foreign) as well as from work, many having been out of school for several years. Their mathematical skills range over the entire school curriculum, from fourth grade arithmetic to twelfth grade pre-calculus.
Mathematical skills are in demand by ATE programs because they are required on many modern jobs. Indeed, most jobs in the high-paying skilled trades (electronics, manufacturing, transportation, telecommunications) require the same advanced high school preparation as do science and engineering programs in college and university. Schools, however persist in excusing "vocational" students from requirements in advanced mathematics and physics. Moreover, mathematics instruction does not typically prepare students to handle open-ended problems. Yet "the ability to manage such ambiguity in mathematics as well as in other subjects is precisely what the real world demands" [Barth].
To stress the importance of high school performance, the National Alliance of Business (NAB) has launched a campaign called "Hiring Smart" to convince employers to demand high school transcripts of job applicants: "The more your business cares about how students do in school, the more they will." Indeed, 85% of students in a recent survey said they would work harder in school if they knew that an employer was going to look at their transcripts, but only 15% of businesses actually do so.
Businesses frequently cite two problems with using transcripts. One is a fear--exaggerated according to NAB--that civil rights laws forbid use of high school grades in making hiring decisions. The other is the lack of information on the typical transcript about factors such as teamwork that are especially important to businesses. NAB points out that reasonable and relevant use of high school grades is perfectly legal, and is encouraging schools to add teachers' comments to transcripts to offer a richer picture "than just grades and numbers" of what a student can do. Such comments will also help ensure that the use of the transcript in hiring decisions is fair and legal.
Skill Analyses
In addition to the broad educational qualities that students should have
when leaving school, today's workplace expects specific skills that
match the needs of particular jobs. Accordingly, many ATE-related
curricula are rooted in goals and objectives that emerge from systematic
analysis of skills used in the workplace. The most widely used system
is DACOM, although some rely on a protocol known as KSAO (for Knowledge,
Skills, Attitude, Other), the WorkKeys system from ACT, the emerging
O*Net analysis of job skills, or the Wisconsin Instructional Design System
(WIDS).
Notwithstanding their widespread popularity, these analyses produce a pale portrait of mathematical skills that in no way reflects the discipline either as it is practiced or taught. Mathematics is an integrated subject, each piece logically connect to each other. Skill lists portray topics in isolation from each othe with no evidence of these interconnections. They shatter the intrinisic coherence of mathematics and reinforce the image of mathematics as a sterile subject isolated from the real world.
Moreover, the skills that emerge when investigators ask workers about mathematics reflects only what workers recognize as mathematics from their school experience, not the more robust mathematics of contemporary practice. Unless mathematicians do the looking, the description of required mathematical skills that emerge from these analyses is far too low to serve any legitimate educational purpose.
Finally, most mathematics-based problems can be solved in more than one way using a variety of tools from different parts of mathematics. Workplace skill lists fail to reflect this robust character of mathematics and encourage the myth that each problem should be solved by a specific, particular skill.
Specificity vs. Generality
In addition to the problems of narrowness and isolation that arise when a curriculum is built on a foundation of skill lists, there is also the problem of matching examples to the particular needs of students and their instructors. ATE programs must constantly balance the tension of specificity (which enhances relevance) with generality (which enhances versatility).
The semiconductor industry provides a good illustration of this tension. Due to competing philosophies of hiring, there have historically been no occupational skill standards in the semiconductor industry. Some companies (e.g., Intel) desire broad entry-level skills so that employees quickly become self-sustaining technicians. Other firms (e.g., Motorola) push for more specific entry-level skills suited for the particular duties required of machine operators. Education programs in the semiconductor industry, therefore, need to be individually tailored to meet local needs. However, individualized programs run the risk of non-transportability from one state (or business) to another.
Working in cooperation with the industry coalitions SEMI and Sematech, the Maricopa Advanced Technology Education Center (MATEC) provides curricular materials for current and prospective semiconductor manufacturing technicians.
The dilemma of transferability led MATEC to develop curriculum units that are both modular and micro-modular: even a single chart or diagram can be useful in a different context, and the software is written to make it easy for instructors to pull parts out of MATEC units so that they can be used in local instructional settings.
The MATEC curriculum is based on results of an analysis performed under the Wisconsin Instructional Design System (WIDS). MATEC selected WIDS rather than the more common DACUM system because WIDS was developed by educators. One outcome is a portfolio of competencies that students can use in applying for jobs.
Web-based courses developed by MATEC provide both instructional units and technical support for instructors based on a "just-in-time" philosophy. However, development of real expertise (rather than practiced procedures) remains a challenge. An example of what MATEC calls "high cognitive load" is to troubleshoot in a cleanroom. It is clearly impractical to create cleanrooms everywhere students are studying. So how can a student's performance in the demanding environment of a cleanroom be assessed? MATEC's solution: create a virtual reality clean room.
Supporting Mathematics Faculty
Many mathematics faculty have backgrounds based entirely in pure mathematics and have little acquaintance at all with mathematics outside academia. This observation led the University of San Diego to sponsor a workshop on mathematics in industry for the purpose of acquainting faculty who teach undergraduate mathematics with the world of mathematics in industry. The sponsors themselves felt a need to learn more about mathematics in industry and assumed there must be others with similar needs who would like to attend such a workshop. So many people signed up that the workshop had to be closed at about 75 participants; at least a third were from community colleges.
Six speakers from industry were scheduled in 90 minute slots to explain some mathematics they have used on their jobs, to allow participants to work on a related but simplified problem, to discuss how industry finds mathematics useful, and to identify the undergraduate preparation that would be good for students who intend to seek industry jobs. The workshop concluded with a panel discussing implications for the undergraduate curriculum. This turned out to be far too big a question to address in such a short time at the end of a strenuous workshop.
In reality, the speakers did not say much at all about specific aspects of the undergraduate mathematics curriculum. Instead, they stressed the need for communications skills, for background in computing (e.g., object-oriented programming), for a willingness to learn about other fields, and for the ability to meet people in these other fields more than half way.
References
American Mathematical Association of Two-Year Colleges. Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus. Memphis, TN: AMATYC, 1995.
Barth, Patte "Want to Keep American Jobs and Avert Class Division? Try High School Trig." Education Week, Nov. 26, 1997, pp. 30,33.
Business Coalition for Education Reform. The Formula for Success: A Business Leader's Guide to Supporting Math and Science Achievement. Washington, DC: U. S. Department of Education, 1998.
National Association of Manufacturers. The Skilled Workforce Shortage, 1998.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston VA: NCTM, 1989.