|
| Stasz, C., & Brewer, D. J. (1999). Academic Skills at Work: Two Perspectives (MDS-1193). Berkeley: National Center for Research in Vocational Education, University of California. |
In this section, we consider how communities of practice regard mathematical knowledge and how that knowledge relates to practice in a broader sense--for example, how math requirements have changed within a profession in which changes in technology or other factors have taken place. We consider the level of mathematics required in work tasks, from simple arithmetic to the use of high-level mathematics such as trigonometry. We also consider the centrality of mathematics to work: Is math essential for doing one's job or is it required only for certain tasks or technology applications? Using specific examples to illustrate these aspects of math at work, we show that mathematics in practice differs considerably across these jobs.[13]
Overall, although all the frontline workers in our study used math on the job, mathematics is clearly more central to some jobs than others (see Table 3.1). In the traffic management business, electronics is the core discipline, and mathematics is a basic tool of electronics. As one manager told us, "traffic signal management is all about electronics, so techs need math to understand the electronics." When traffic signal technicians discuss math, it is in reference to electronic examples or problems.
| Skills | Task Examples | |
| Construction Inspector |
|
|
| Survey Inspector |
|
|
| Traffic Signal Technician |
|
|
| Test-Cell Technician |
|
|
| Equipment Technician |
|
|
| Home Health Aide |
|
|
| Licensed Vocational Nurse |
|
|
| Note: Skills in bold are used more often; that is, they are more central to the work. |
The test cell and equipment technicians at the semiconductor plant also require basic electronics knowledge for their jobs. This knowledge is important for operating sophisticated equipment, but it is not viewed as a defining characteristic of their work. For these technicians, although they may use mathematics in their job, the essential aspect of their work is keeping up with technological equipment which may have a floor life as short as six months.
Mathematics is also essential to survey inspection but in an entirely different context. Survey inspectors, particularly the party chief who heads a survey crew, must make complicated calculations and measurements. A large part of the chief's job is calculating the exact horizontal and vertical placements from objects in the field from two-dimensional plan specifications. Depending on the situation, these calculations can be straightforward or very complex, as illustrated in some examples that follow. Survey inspectors' mathematical expertise is also recognized by others; the party chief reports that contractors or other craftsmen at work sites depend on surveyors to check or make mathematical calculations. The following excerpt from the fieldnotes illustrates this interdependency:
A civil engineer called on the radio to ask a question about a staircase at a particular station. The top of a staircase was going to fall one inch short of the platform, and the engineer needed help in understanding why, so he could craft a solution. Over the radio, without performing calculations on paper, the chief was able to understand the problem being described. He formulated a hypothesis, and made some simple calculations to confirm it. He told the observer that he recognized the problem from the description, constructed a picture of it "in his head," and used calculations to confirm his solution. [TA fieldnotes]
While mathematics is also important in the other jobs we studied, it is not a defining factor of those occupations; rather, mathematics is necessary because of its connection to certain tasks or technology.
The level of mathematics needed in the jobs we cite in this chapter varies considerably. In this section, we provide several examples, beginning with jobs and tasks requiring fairly basic to more advanced mathematics skills (see Table 3.1). We define "basic" math as calculation using addition, multiplication, subtraction, or division. In these examples, it is also apparent that mathematics is firmly linked to task and technology.
The home health aides and LVNs need only basic mathematics skills. Each is licensed to carry out particular tasks, with LVN work requiring more mathematics knowledge. Home health aides and their supervisors report little need for math on the job; at most, they use a calculator to figure out their reimbursable mileage or do simple calculations for recordkeeping. LVNs, however, have responsibilities for patients' medications, take their vital signs, and work with various kinds of equipment. When asked about the technical skills required in her job, an LVN gave the following example:
You have some [patients] that are being monitored through a pump. You have to know how to calculate your rate per minute, the volume that the machine is going to deliver, and how many cc's per minute. So you have to know how to set that up. If the doctor says, well I want the patient to have 24, well say 2,000 cc's of fluid over a period of six hours, you would have to know how to set that up. And most patients, if that patient is having that kind of a drip, you would have to know how to set up your equipment.
This example shows how mathematics is embedded in setting up particular equipment, in this case a pump apparatus that dispenses fluids intravenously. The LVN considers this to be a technical task and, although it requires mathematical calculation on her part, she does not describe the mathematics concepts or operations involved. Rather, the explanation is tied to the delivery of fluids through the pump, as prescribed by the doctor.
The mathematics "embedded in the task" is also tied to the social context of the LVN's work. She participates as a member of a managed care team, in which authority and tasks are specified for each job. She communicates with doctors or other health professionals and carries out their prescriptions for patient care when she visits a patient in the home. In the following examples, she discusses mathematics again, also in the context of describing technical skills:
Technical is doing your diabetic teaching. You have to know as far as your units, if the doctor says to give the patients 20 units of NPH, you have to know what--how is he using, what are his units? You have to know units in comparison to cc's or milligrams or what have you.
This example shows that doctors may define a "unit" in different ways. Part of her task, then, is first understanding how a particular doctor defines a unit, as this will affect the calculation she needs to make.
When asked whether math skills were required for their jobs, most respondents in the HA discussed math in the context of specific tasks:
It's helpful. Matter-of-fact, you really should have a lot because you know, you're constantly dealing with your drugs, your patient's medications. And you have to be able to recognize the milligrams, what have you. Or if you're doing your diabetic teaching, you have to know as far as your units, as far as your insulin, so you need quite a bit of math skills. [LVN]
These examples illustrate that within this community of practice, mathematics is embedded in technical aspects of the job. Furthermore, the meaning of a term directly related to doing the mathematics can be highly specific. For example, "unit" may be defined differently by different doctors. In this case, understanding the meaning is essential to properly carrying out the task.
Several jobs involve electronics and require enough mathematics to deal with particular tasks or technology. In the traffic management world, technicians, supervisors, and managers universally agree that technicians need basic mathematics skills and algebra. In their view, any mathematics associated with electronics is considered "basic." Trigonometry, geometry, and calculus were less frequently mentioned but can also be required from time-to-time.
Traffic signal technicians use mathematics to repair and maintain equipment both in the shop and out in the field:
It's not just in the shop that you're doing this [math] either. If you were working in an intersection to where you have a service point, then you have your cabinet, and it needs to get the power. You know you have a certain type of wire that has a certain amount of ohmage or amount of resistance per foot. Does it make sense that you have 120 volts here and 100 volts here? What's the problem? Why the loss? Well, I've got 400 feet times how many ohms per foot. That's within reason. OK, so I've got to pull a different type of wire there. You can't just sit there and say it won't work. You have to do some figuring. [TM manager]
Algebra comes in handy for technicians working with test equipment:
You need algebra because you have to do a lot of plotting, a lot of percentages and stuff like that, [and] a lot of decimal numbers. All the pressures are negative pressures, so you've got to know a little about a timeline and stuff. I'd say, you don't need calculus, but you need to know how to plot, how to read a graph. [TS technician]
Use of different types of technologies also figures heavily into mathematics requirements on the job. Traffic signal technicians, for example, work with oscilloscopes, which requires some understanding of trigonometry:
Trigonometry is really tied with electrical theory and how power is calculated. That building block is essential . . . because you need to understand vectors. Our specifications call for a certain part of the sign wave--which is the electrical symbol for opening a current--where the different components will flow, or will turn on and off. They have to be able to read what a sign wave looks like and what the degree marks are. Where do you get that exactly in a mathematical background? It's not just basic math. [TM manager]
Depending on the technology available in a particular traffic management department, traffic signal technicians may need to set controller timing at traffic intersections. One technician described the task as follows:
We've each been given a calculator watch because the way signal systems were done previously was by percents. If you had a 90 second cycle and that equals 100%, you need to know if you want to make what they call a 45-55% split--you want to split the 90 seconds. You want to give 45% of it to one direction and 55% to the other. Then you need general math skills and a calculator.
In jurisdictions with newer technologies, however, signal control may occur from a centralized control room, which sends signals to microprocessor-based controllers located at street intersections (see description of this system in Appendix I). In this case, technicians won't use math to set timing, but need to "know how the monitors really work--how the programming actually relates to what actually happens in the field." As discussed further below, the shift from electromechanical to digital technology in traffic management has had a widespread impact on skill needs.
In comparison to other jobs, survey inspectors' work requires the most advanced mathematics, including algebra, geometry, and trigonometry. As the "chain man" on the survey crew described math skills, "Trigonometry is what we do." To become qualified even as an apprentice, surveyors must take an examination that tests the applicant's ability in both trigonometry and algebra. The crew chief in our study also occasionally uses least-square regression and other statistical techniques that go beyond the normal mathematics required for surveying. He does this both to keep his skills up and as a means of independently testing some of his earlier calculations.
The following excerpt from TA fieldnotes provides a typical example of the mathematics involved in survey work:
There's a lot of calculating to do in this job, most of which seems to be the responsibility of the chief. He compares copies of design drawings with his charts of "control points," which he has previously checked for height, azimuth (the horizontal distance from a fixed control point), and elevation, checking and rechecking correct survey points to ensure that the object being placed will be correct. The surveyor must calculate "northings" and "eastings" (placement in the vertical and horizontal dimensions of a plane) and the elevation (height) of the object. This is not just rote calculation.
The transportation work observed in this study was somewhat unique as the geometry of the rails and tracks introduces specific construction and inspection challenges such as spiral curves. A spiral curve is a parabola: one track is higher than the other to accommodate faster train speeds at a curve. While calculating spirals is a challenge to even a skilled surveyor, it is also a fairly rare occurrence; the supervisor estimates that a surveyor will only run into such problems on one percent of jobs. Spirals also introduce new terminology in the surveyors' lexicon that is only used in rail work.
The chief described the procedure for calculating a "very substantial" spiral. The crew first establishes the horizontal, then measures the spiral off the horizontal to ensure that the relative heights of the rail are correct at each point. Spirals are similar to curves, but whereas curves have a constant radius and a fixed center point, spirals do not. Thus, the crew is unable to use in their calculations the relatively simple formulas for circles. More complex calculation is required.
Surveyors also feel that future technologies such as the global positioning system (GPS) may significantly impact their work, although the underlying mathematics knowledge remains fairly constant. In this field, as in others, the worker needs to understand the underlying concepts behind the technology application and how that relates to the work setting.
[13]For confidentiality reasons, we use pseudonyms for proper names appearing in examples. We identify sources of information with titles or site locations.
|
| Stasz, C., & Brewer, D. J. (1999). Academic Skills at Work: Two Perspectives (MDS-1193). Berkeley: National Center for Research in Vocational Education, University of California. |