# JITE v32n2 - Occupationally Specific Mathematics Requirements and Application Contexts

## Occupationally Specific Mathematics Requirements and Application Contexts

## David J. Pucel

University of MinnesotaMathematics is viewed as a basic skill required by all citizens in society. Yet many employed adults and those preparing for employment do not have the minimal basic mathematics skills needed to function successfully in the workplace ( Johnston & Packer, 1987 ; Lewis & Fraser, 1984 ; U.S. Department of Education, 1983 ; U.S. Department of Labor, 1991 ). This has led to a re-examination of how mathematics is taught in elementary and secondary education programs and in programs designed to prepare people for employment ( Battista, 1994 ; Burns, 1994 ; National Research Council, 1989 ; Weber, Puleo, & Kurth, 1989 ). A central theme of the movement to revise mathematics education is that "Teaching for understanding is in; learning rote skills is out" ( Burns, 1994, p. 471 ). However, there is continuing debate about the form the new emphasis on teaching for understanding should take. There is also a debate about which mathematics skills should be taught. As knowledge of mathematics increases along with the explosion of knowledge in all fields, it is becoming clear that it is not possible to teach all people all of the mathematics skills that could be taught. Therefore, the American Association for the Advancement of Science has suggested that we "not call on the schools to cover more and more (mathematics) material, but instead recommend a set of learning goals that will allow them to concentrate on teaching less and doing it better" ( Blackwell & Henkin, 1989, p. ix ).

Both mathematics instructors and others interested in helping adults prepare for employment acknowledge that adults who have not been successful in learning mathematics in the past should not be taught again using traditional mathematics methods ( Grouws, 1992 ; Janvier, 1990 ; Pritz, 1988 ; Shelby & Johnson, 1988 ). The challenge is to adopt new approaches that have the potential for allowing adults to be more successful. Those approaches must allow people not only to learn mathematics but to be able to apply it in the workplace.

A number of approaches to teaching mathematics and other academic content to make it more relevant to work settings have evolved. They can be categorized into three major groupings: applied academics , integration of academic and vocational education , and related academics . Applied academics , in the context of preparing people for work, refers to teaching academic content around work related applications. The emphasis is on teaching the academic content. The curriculum development process starts with specifying the academic content to be taught. If the focus is on teaching mathematics, typically the content specified is traditional mathematics content. Mathematics is then taught in relation to the ways it may be used in the world of work. The belief is that if people can see how the academic content can be used, they will be more motivated to learn. The curriculum materials developed by the Center for Occupational Research and Development ( 1992 ) are good examples of such a curriculum model focusing on occupational applications. They are designed to teach academic areas such as physics, mathematics, and communications using examples from a variety of occupations.

Integrated academic and vocational education programs are designed to emphasize both academic and vocational content and to teach them together as complimentary. Both the content of the academic area and the content of the vocational area are viewed as important. The design process starts with both the academic area(s) and the vocational program area(s) listing the learning outcomes they wish to achieve. A curriculum is then developed that allows both academic and vocational education content to be taught in a manner designed to achieve the outcomes of both areas, not only the outcomes of academic education. An integrated curriculum teaches both the content of a mathematics curriculum and a vocational curriculum but does so in such a way that mathematics is taught in the context of the vocational curriculum and is reinforced in the vocational curriculum. This model is often used in magnet schools or academies ( Stern, Raby, & Dayton, 1992 ).

Related academics instruction changes the emphasis from academic education to vocational education. The driving force in selecting the academic content is the need for that content in the context of occupations. The process starts with examining the type of vocational education content to be taught and then determining the academic content that is needed to support that occupational content. This model is exemplified in the DACUM curriculum planning process ( Norton, 1985 ).

All three models are similar in that they use work contexts as a basis for teaching content. However, each is used to accomplish different goals. If the primary goal is to teach academic content, then the applied academics model is most appropriate. If the goal is to teach only that sub-set of academic content required by people in occupations, then the related instruction model is most appropriate. If the goal is to teach a more complete set of both academic and vocational education outcomes, then the integration model of academic and vocational education is most appropriate.

In preparation for conducting this study, the investigator visited a number of technical colleges to determine how they were adapting mathematics instruction to better help adults develop mathematics skills required for employment. All three of the models described above were found. In addition, considerable differences were found in perceptions of what was important in adjusting the instruction. Differences seemed to be focused on two major dimensions. First, differences existed in perceptions of whether the mathematics skills for specific occupations are substantially different or whether occupational mathematics skills are essentially the same for occupations prepared for through vocational education. Those who believed they were essentially the same argued that there is no need to design mathematics instruction more specifically for separate occupations. Those who believed they were different argued that there was a need to design mathematics instruction more specifically for separate occupations. Second, educators differed in their perceptions of whether teaching applied mathematics can be done in relation to a range of occupations or whether there are sufficient differences in the way mathematics is applied in different occupations to warrant teaching mathematics around the applications specific to a particular occupation. Those who believed that mathematics skills are applied very similarly in different occupations argued that the occupational examples used to teach mathematics did not matter. The major goal was to teach mathematics around occupational examples. Those who believed that the same mathematics skills are applied very differently in different occupations argued that it was important to teach mathematics around applications from the occupation a person is preparing to enter.

## Related Literature and Ideas

Past research and demonstration projects provided insight into new ways of developing adult mathematics instruction for employment. There is a growing understanding that mathematics on the job is used to solve workplace problems rather than doing mathematics for the sake of mathematics ( Brooks, 1991 ; LaPorte & Sanders, 1993 ). If a person is asked to take an inventory of the number of bolts throughout a store, the problem is to determine how many bolts of each type are available. The problem is not to add numbers even though addition is needed to arrive at the answer. This has led D'Ambrosio ( 1985 ) to use the term ethnomathematics to refer to forms of mathematics that vary as a consequence of being embedded in cultural activities with purposes other than just doing mathematics. He further suggests that different forms of mathematics are applied in specific contexts, such as building a house, weighing products, or calculating recipe proportions. Janvier ( 1990 ) indicates that the "crucial point is that the computation methods people use are determined by the situation or the context in which they are performed" ( p. 183 ). He indicates that when people are required to apply computations while making a living, they want to learn those that ensure success and in which they can feel confident. Crockcroft ( 1982 ) indicates that many people have a mathematics phobia and that teaching mathematics in job related contexts leads to mathematics not being thought of as mathematics, but as an integral and automatic part of the job, thereby minimizing this common phobia. This is supported by the comments of a student reported by Shelby and Johnson ( 1988 ). The student reported that mathematics no longer scared him after he realized how important it was in solving workplace problems.

These authors strongly suggest that helping adults develop mathematics skills needed on the job requires the identification of the specific mathematics skills associated with the job and the contexts within which that mathematics is applied. In other words, these authors suggest that the most effective model for teaching mathematics to adults preparing for employment is to use the related academics instruction model that focuses on mathematics as it relates to a specific occupation. They suggest that given information on specific mathematics skill requirements and their applications, curricula can be developed that have high potential for helping people develop mathematics skills needed in the workplace.

The belief that most jobs require only a subset of basic mathematical skills and that the identification of that specific subset can facilitate more efficient and effective education is supported in the literature ( Anderson & Peterson, 1983 ; Fitzgerald, 1983 ; Mark, 1984 ; Sticht & Mikulecky, 1984 ). However, they point out that methodologies for identifying specific mathematics skills are lacking and suggest that further research is needed to develop more effective techniques that practitioners can use to identify the required mathematics skills. Without such techniques they predict that most adults returning for education for employment will continue to be taught using standard mathematics courses that present the traditional range of mathematics. Such courses are often ineffective because students view many of the mathematics skills that are taught as irrelevant ( Shelby & Johnson, 1988 ; Tests of Adult Basic Skills , 1987 ). Such perceived irrelevance often causes adults to drop out of such programs and forsake their occupational preparation ( Cooney, 1981 ; Shelby & Johnson, 1988 ).

## Statement of the Problem

The primary purpose of this study was to determine if the types of mathematics skills and the applications of those skills differ substantially among occupations prepared for through vocational education. If the mathematics skills and applications are not highly different between different occupations prepared for through vocational education, then generic occupational mathematics courses may be appropriate. If there are substantial differences in the mathematics skill requirements of different occupations, and/or if the same skills are applied differently in different occupations, it may be more appropriate to tailor mathematics instruction to each occupation. More specifically the research questions were

- Do different occupations require different sub-sets of basic mathematics skills?
- Do people employed in different occupations apply the same mathematics skills differently?
A secondary purpose of this study was to develop and validate an efficient methodology that could be used to identify occupationally specific mathematics skills and the contexts in which they are used on the job as a basis for gathering data for both the study and curriculum development. One of the major barriers to modifying mathematics instruction, identified in the literature and during interviews with technical college personnel, was the inability to identify which mathematics skills to teach. Instructors indicated a familiarization with techniques used in the past, such as DACUM ( Norton, 1985 ), but indicated that they did not have the time or resources to assemble the large groups of people required and to lead them through the process. Also, they did not know how to use the relatively complex techniques. What they were searching for was a technique they could use with their limited resources.

## Instrumentation

## Assessment of Occupational Mathematics Requirements

To compare the mathematics skill requirements of different occupations, a procedure was needed to identify occupationally specific job applications that required mathematics and the mathematics skill requirements associated with those applications. A number of studies were identified that had attempted to determine the mathematics requirements of occupations. Most began by identifying the occupational applications that required mathematics and then analyzing the specific mathematics skills required.

Occupational applications requiring mathematics were most often identified by asking experts who had knowledge of the occupation to identify tasks that required mathematics and to validate those tasks through group consensus ( Anderson & Peterson, 1983 ; Baker, 1980 ; Horne, 1979 ; Norton, 1985 ). A second approach was to ask experts to envision only the job skills that required mathematics as they identified mathematics requirements ( Greenan, 1983 ; Greenan, 1984 ; Hull & Sechler, 1987 ; Moe, Rush, & Storlie, 1980 ). Both approaches to identifying the occupational applications of mathematics relied heavily on the belief that group consensus was an accurate reflection of what was needed on the job. These approaches were based on assumptions that the eventual list of mathematics skills would only be used with a particular training program, that agreement among a specific group of experts is sufficient, and there is no need for replicability ( Greenan, 1984 ). These approaches also required significant resources and knowledge of group consensus techniques.

Felton ( 1981 ) described an alternative approach for identifying job applications. It was efficient in that it did not require assembling large groups of experts, and it was more objective than the other approaches in that it did not rely solely on group opinion. Felton's approach relied on gathering actual forms and paper work used on the job. The use of real job materials provided a common basis for experts to judge mathematics requirements. A modified version of this approach was used in this study because it can be used to more clearly define the job applications requiring mathematics in a manner that can be replicated.

An investigation of the methods used to analyze the specific mathematics skill requirements of job applications also revealed differences in procedures and precision. A number of studies were identified that had developed explicit taxonomies of basic mathematics skills that could be used as a basis for analyzing mathematics skills associated with occupations ( Anderson & Peterson, 1983 ; Bernstein & Mussell, 1984 ; Cooney, 1981 ; Kawula & Smith, 1975 ; Martin, 1983 ). The mathematics skill taxonomies varied greatly in comprehensiveness and content. Some focused on mathematical operations and types of mathematics such as addition, subtraction, and fractions ( Felton, 1981 ; Holtz & Holtz, 1979 ). Some focused on mathematics processes such as calculate as well as operations ( Anderson & Peterson, 1983 ; Shelby & Johnson, 1988 ). Some focused on operations, mathematics processes, and types and units of measure, such as measure length in inches ( Adult Competency Education Kit , Part B, 1977; Adult Competency Education Kit , Part C, 1977 ; Martin, 1983 ; Smith, 1979 ; U.S. Department of Labor, 1973 ). It became clear that an internally consistent taxonomy of mathematics skills organized around categories of mathematics and increasing complexity of mathematics within categories was required. Such a taxonomy would be useful for systematically analyzing on the job applications and for organizing instruction once mathematics skill requirements were identified. An examination of how past taxonomies were used also revealed that most were written for use by mathematics experts and were not easily applied by typical instructors, trainers or curriculum developers. Therefore, in order to conduct the study, a taxonomy that could be used by mathematics experts, occupational instructors, trainers, or curriculum developers needed to be developed.

## OMRA Development

Since no existing instrument was available that met the specific needs of this study, the Performance-Based Occupational Mathematics Requirements Assessment (OMRA) was developed. OMRA was designed to a) yield an explicit list of job applications that require mathematics skills based on real job requirements identified in materials used on the job and examples of job applications supplied by industry, b) identify basic mathematics operations required to complete those applications using an explicit mathematics taxonomy, and c) be implementable by typical vocational instructors, trainers or curriculum developers ( Pucel, Feickert & Lewis, 1992 ). Completing the OMRA involved an occupational expert and a mathematics expert working together to review materials. This procedure was used because it was recognized that most occupational experts are not experts in mathematics and most mathematics experts are not experts in occupations ( Miller & Vogelzang, 1983 ; Shelby & Johnson, 1988 ; Stanier, 1983 ).

The design process started with the development of a taxonomy of mathematics operations called the OMRA Inventory . Taxonomies of mathematics skills found in the literature were reviewed in detail ( Adult Competency Education Kit , 1977 ; Martin, 1983 ; Shelby & Johnson, 1988 ; Smith, 1979 ). The OMRA Inventory was organized in the form of a matrix. The matrix presented categories of mathematics operations in order of increasing complexity across the top of the matrix from left to right (integers, fractions, decimals, percents, ratios, algebra, geometry). Mathematics operations were presented within categories in order of increasing conceptual complexity going down the column (e.g., addition before division). Increasing conceptual complexity was first judged based on whether one mathematics operation was required in order to perform another mathematics operation. If mathematics operations were not dependent upon one another they were ordered by the advisory committee, which is described later. Each mathematics operation was presented both as a sample calculation recognizable to people who work in occupations and as a written definition understandable to mathematics experts (see Figure 1). This allowed for meaningful communication among the mathematics and occupational experts and for the mathematics found in work related examples to be more easily categorized.

Figure 1 Sample cells from the OMRA Inventory

The range of mathematics operations included in the OMRA Inventory was developed for occupations requiring less than a baccalaureate degree-those typically taught through vocational and technical education.

Once the OMRA Inventory was developed a procedure for identifying the job applications that required mathematics operations was developed. The procedure was based on assumptions that the mathematics required on the job is reflected in the materials used on the job, and examples of materials that contain mathematics applications can be obtained from employers of people in the occupation. Materials were gathered from a sample who employ people in the occupation. Each employer was asked to provide copies of materials used on the job that contained mathematics calculations or verbal examples of how mathematics is applied on the job. The technique developed by Felton ( 1981 ) was modified to include verbal descriptions of mathematics applications because some mathematics is applied routinely on the job without filling out forms or writing down the calculations. Such mathematics does not show up in the materials actually used on the job.

A procedure for examining the job related materials was then developed that calls for expert teams, composed of one instructor from the occupational area under analysis and one mathematics expert, to review the materials. Occupational instructors were viewed as the occupational experts because in order to teach an occupation at a Minnesota technical college, a person must have recent journeyman level experience in the occupation. One should be cautious in using occupational instructors if they have not recently practiced in the occupation. In such cases, a third person who is currently practicing in the occupation might be added to the materials review team.

Three teams reviewed materials for each occupation being analyzed. Three teams were needed to triangulate expert judgments and to increase the validity of the results. Each expert team was asked to review each page of the materials thoroughly and first identify job applications that required mathematics. They wrote those job applications on the OMRA Applications Supplement . They then determined which mathematics operations were needed to complete that application. Finally, they estimated how many times per month a worker would actually complete that application and indicated that number after the operation in a tally box on the OMRA Inventory . For example, as shown in Figure 1, the team estimated that the subtraction of decimals would have been performed 5 times for the first job application they found that required the operation. The next time they encountered an application that required the operation they would enter a number in another box to indicate how many times that application required the operation. The technique provided a vehicle for recording the number of occupational applications that required mathematics skills and the specific mathematics operations required.

## OMRA Validation

Validation of the OMRA procedures was accomplished in three stages. First, an initial pilot test was conducted with five graduate students at the University of Minnesota who had technical backgrounds and had studied mathematics through at least algebra and geometry. The group was provided auto mechanics and aviation mechanics manuals used on the job. Each person was asked to review the same sample of materials separately and to complete the OMRA . Later, the group was assembled and suggestions were obtained for improvements. Revisions were made in the OMRA directions.

The materials were then revised and brought to an advisory committee. The committee was composed of the six experts: a community college mathematics educator, the chairperson of a mathematics department in a private college, a technical college mathematics educator, a university mathematics skills evaluator, a university mathematics teacher educator, and an occupational assessment expert from a university department of psychology. The advisory committee was selected to include expertise in assessment and familiarization with mathematics typically performed in occupations prepared for through vocational education. The goal was to develop a methodology that would be viewed as credible by mathematicians and could be interpreted by occupational experts. The advisory committee had no role in actually specifying the mathematics skills associated with a particular occupation.

The committee members were asked to review the directions for clarity and to judge whether an assessment that followed those directions would lead to valid results. The OMRA Inventory was also reviewed for inclusiveness of the mathematics operations performed in occupations trained for through vocational education, and the level of detail included in the categorizations. The OMRA Inventory and the OMRA Applications Supplement were also reviewed in terms of their usefulness for recording the mathematics operation requirements and the occupational applications requiring mathematics on the job. The committee raised questions about asking a team to estimate the number of times a mathematics operation would be performed per month for a particular application. The committee also wondered if having teams visit and observe the actual places where people were employed would not provide more valid results. After debate, it was decided that requiring teams to visit multiple job sites to observe actual employees would probably result in stronger research results but it would make the technique unusable by most schools and colleges. They also felt that the absolute accuracy of the estimates of frequency of use of mathematics operations per month was not important. The data would be used only to generate a prioritized list of mathematics operations and they felt that the average response across three teams would be accurate enough for the purpose of determining whether a mathematics skill should be taught. For curriculum development purposes, the fact that one mathematics operation is ranked 4th or 5th in priority was not considered important. The exact method of calculating this average is described later.

The materials were revised based on input from the advisory committee. Directions were rewritten, categories within the OMRA Inventory were simplified, combined, and rearranged, and the OMRA Applications Supplement was further developed to indicate which mathematics operations pertained to each application. The final OMRA Inventory contained 63 mathematics operations. The final categories of mathematics operations and the number of operations in each category can be reviewed in Table 4.

## Methods

## Sample

Two occupations were selected for study: principal secretary and electronics technician. They were selected to represent two different types of occupations that might require different types of mathematics skills. They were also selected because the Technical Colleges of Minnesota had a number of secretarial and electronics technician programs. Six review teams were assembled: three teams of secretary reviewers and three teams of electronics technician reviewers. Each team was composed of an instructor with recent occupational experience who taught in the occupational area and a mathematics instructor from a Minnesota Technical College.

## Data Gathering

A curriculum development scenario was generated that assumed that a technical college wished to prepare principal secretaries for the University and electronics technicians for electronics firms in the Minneapolis-St. Paul metropolitan area. Therefore, job related materials were gathered from four offices employing principal secretaries within the University of Minnesota and a sample of four firms employing electronics technicians within the Minneapolis-St. Paul metropolitan area. Since the goal of the study was to validate the instrumentation and to investigate potential differences in mathematics requirements between occupations, and not to develop a definitive list of principal secretary mathematics requirements or electronics technician mathematics skills that would be used for actual curriculum in the future, no attempt was made to obtain an exhaustive set of materials from a wider range of occupational settings.

Each of the three secretary review teams was presented the same set of secretarial materials. Each of the three electronics review teams was presented the same set of electronics materials. Each team was asked to read the OMRA directions carefully and to complete the OMRA Inventory and the OMRA Applications Supplement . The directions were to:

- Review the contents of OMRA and the directions.
- As a team, review each of the job related materials included in the packet for job applications requiring mathematics.
- Analyze each job application in terms of the types of mathematics operations it requires (e.g., addition of whole numbers, division of fractions, subtract decimals). Many applications require more than one mathematics operation. If an application contains two or more operations, note each separately (e.g., 40 x 40 / 680 contains that operations of "multiply whole numbers" and "divide whole numbers").
- Record each application on the OMRA Applications Supplement .
- Record the operations associated with an application and their estimated frequency of use per month on the OMRA Inventory .
Each team completed the instruments and returned them.

## Data Analysis

Data were analyzed separately for each occupation. Information from the three teams for each occupation were summarized to provide a more accurate indication of the mathematics applications and the operations required. A list of job applications requiring mathematics for each occupation was developed from the OMRA Applications Supplements . Each list included all job applications that required mathematics identified by the three teams. Redundancies among the applications identified were removed. The mathematics operations, which the teams said were associated with each application, were also recorded. Samples of the job applications listed by the secretarial and electronics technician teams that required the operation "subtract decimals" are presented in Table 1.

Table 1 Comparison of Sample Job Applications for Subtract Decimals

Principal Secretary Applications Calculate Salary Fringe Benefit Check Inventory Control Invoice Preparation Order Supplies Reconcile Budget Reconciliation Report Travel Expense Voucher Electronics Technician Applications Calculate Base Current of BJT Calculate Capacitance - Inductance Calculate Current in a Series circuit With a Diode Calculate Drain Current of a Junction Field Effect Transistor Calculate Voltage Collector to Emitter of BJT Calculate Zener Diode Power Fill Out Requisitions & Invoices Complete Work Order Calculations

Data from the OMRA inventories for each of the three teams for a given occupation were then analyzed. The OMRA inventories listed the mathematics operations required to perform all of the job applications and their average frequency of use per month. If two or more groups indicated that an operation was required in order to perform the job, it was included in the list of required mathematics operations. If an operation was listed by only one of the three teams, it was reviewed by examining the actual job materials to eliminate team bias concerning what should be taught rather than what was actually reflected in the sample materials used on the job.

Only those operations selected by more than one group, or which were selected by one group and verified against the job related materials, were included for further analysis. The operations were assembled into a prioritized list of mathematics operations for each occupation. Prioritizing was done first by how many teams selected the operation, then by the average estimated frequency of use of each operation per month across teams. If team 1 estimated an operation was used ten times per month, team 2 estimated eight times per month, and team 3 estimated six times per month, the average was eight times per month ((10+8+6)/3 = 8). Table 2 presents the prioritized list of mathematics operations contained in the sample principal secretary job materials. Table 3 presents the prioritized list of mathematics operations contained in the sample electronics technician job materials. Electronics team 2 did not follow directions when entering the number of times per month a person would perform an operation for each application. The team only indicated the number of applications that required the operation. Therefore, the results of that team were not used to calculate the average number of times per month an operation was used, but were used to judge how many teams indicated the operation was used. Operations found by electronics team 2 are indicated with an X.

Table 2 Principal Secretary Mathematics Operations

Operation Frequency

Per Month PriorityTeam 1 Team 2 Team 3 Avg f

Three Teams Selected the Operation 1-3 Add Whole Numbers 30 14 32 25.3 1 3-5 Multiply a Decimal by a Decimal 26 10 24 20 2 3-1 Add Decimals 20 20 17 19 3 4-5 Take the Percent 15 11 12 12.7 4 3-2 Subtract Decimals 15 6 11 10.7 5 1-5 Multiply Whole Numbers 19 3 5 9 6 1-4 Subtract Whole Numbers 20 3 3 8.7 7 3-6 Divide a Decimal by a Decimal 4 5 3 4 8 1-8 Add Signed Numbers 2 2 7 3.7 9 4-6 Determine the Percent 4 5 1 3.3 10 Two Teams Selected the Operation 4-2 Convert Percents to Decimals 14 0 14 9.3 11 4-4 Convert Decimals to Percents 7 7 0 4.7 12 1-7 Round Off 3 2 0 1.7 13 One Team Selected the Operation 5-3 Conversion 20 0 0 6.7 14 1-6 Divide Whole Numbers 0 2 0 0.7 15 1-1 Convert Words to Arabic Numbers 1 0 0 0.3 16 1-2 Convert Arabic Numbers to Words 0 1 0 0.3 17

## Results

## Secretary Results

The secretary teams found 34 different principal secretary applications that required mathematics (see Table 1 for a sample). Those applications used a total of 17 of the 63 mathematics operations contained on the OMRA Inventory (see Table 2). Each operation was listed by at least one of the three secretary teams. Those listed by only one team were verified as being required to perform the applications based on the job materials review. All of the operations selected by only one group were found in the job related materials. Ten operations were listed by all three groups, three by two groups, and four by only one group. Two additional types of mathematics were suggested by team members as being required: accounting concepts and determine the amount. Neither of these was a mathematics operation. They were business concepts or mathematics processes related to job applications.

## Electronics Technician Results

The electronics technician teams found 45 different applications that required the use of mathematics (see Table 1 for a sample). Those applications required 38 of the 63 mathematics operations contained on the OMRA Inventory (see Table 3). Each of the operations included in the final list was identified by at least one of the three electronics technician teams. Those listed by only one group were verified against the job materials.

A total of 55 of the 63 mathematics operations were initially listed by at least one electronics team. Twenty-eight operations were selected by only one group and were examined in relation to the sample job related materials included in the study. Seventeen operations selected by only one group were removed because they could not be found in the job related material. For example, one team included all operations pertaining to fractions. Upon review of the materials, only two fraction operations were required.

Table 3 Electronics Technician Mathematics Operations

Operation Frequency

Per Month PriorityTeam 1 Team 2 Team 3 Avg f

Three Teams Selected the Operation 1-3 Add Whole Numbers 240 X 149 194.5 1 1-5 Multiply Whole Numbers 240 X 131 185.5 2 3-1 Add Decimals 240 X 120 180.0 3 1-6 Divide Whole Numbers 230 X 112 171.0 4 1-4 Subtract Whole Numbers 240 X 40 140.0 5 3-2 Subtract Decimals 240 X 20 130.0 6 3-5 Multiply a Decimal by a Decimal 140 X 81 110.5 7 4-5 Take the Percent 150 X 63 106.5 8 6-7 Solve Equations with Fractions 160 X 3 81.5 9 6-6 Solve Linear Equations 150 X 3 76.5 10 5-2 Solve the Proportions 140 X 4 72.0 11 4-6 Determine the Percent 110 X 4 57.0 12 3-6 Divide a Decimal by a Decimal 80 X 24 52.0 13 5-3 Conversion 10 X 80 45.0 14 v5-1 Express the Ratio in Terms 60 X 4 32.0 15 Two Teams Selected the Operation 1-7 Round Off 240 X 0 120.0 16 6-5 Transpose Formulas: Solve for r 180 X 0 90.0 17 1-8 Add Signed Numbers 90 X 0 45.0 18 1-9 Subtract Signed Numbers 80 X 0 40.0 19.5 1-10 Multiply Signed Numbers 80 X 0 40.0 19.5 6-10 Find the Root 70 X 0 35.0 21 1-11 Divide Signed Numbers 40 X 0 20.0 22 3-4 Convert a Fraction to a Decimal 0 X 21 10.5 23 4-4 Convert Decimals to Percents 20 X 0 10.0 25.5 4-2 Convert Percents to Decimals 20 X 0 10.0 25.5 4-1 Convert Percents to Fractions 20 X 0 10.0 25.5 4-3 Convert Fractions to Percents 20 X 0 10.0 25.5 One Team Selected the Operation 3-3 Convert a Decimal to a Fraction 0 - 40 20.0 28 2-11 Multiply Mixed Numbers 0 - 1 0.5 29 6-8 Solve Systems of Equations: Graphically 0 X 0 X 34 6-9 Solve Systems of Equations: Algebraically 0 X 0 X 34 1-1 Convert Words to Arabic Numbers 0 X 0 X 34 1-2 Convert Arabic Numbers to Words 0 X 0 X 34 6-4 Divide Monomials 0 X 0 X 34 6-3 Multiply Monomials 0 X 0 X 34 6-1 Add Monomials 0 X 0 X 34 6-2 Subtract Monomials 0 X 0 X 34 2-1 Order Fractions 0 X 0 X 34

Of the final 38 verified mathematics operations listed by at least one team, 15 were selected by all three groups, 12 by two groups, and 11 by only one group. Six additional types of mathematics were suggested by team members as being required: Boolean algebra, finding the exponential value, base "N" operations, base conversion-binary mathematics, scientific notation, and the construction and reading of graphs. Again, upon reviewing the job-related materials, the need for these could not be verified. Follow up conversations were held with the teams that suggested the additional mathematics requirements. Team members indicated the additional mathematics should be taught because it is typically included in electronics textbooks and is always taught to people preparing for electronics technician positions. However, the team members could not substantiate the need based on examples of job applications. When asked, they could not think of real applications of the additional mathematics in any electronics technician jobs.

## Conclusions and Implications

This study provides information useful for mathematics and vocational education instructors who recognize that mathematics instruction for adults preparing for employment should not be taught again using traditional techniques used in elementary and secondary schools ( Grouws, 1992 ; Shelby & Johnson, 1988 ; Janvier, 1990 ; Pritz, 1988 ). The study was designed to investigate differences in mathematics skill requirements between occupations as a basis for developing more effective employment related mathematics programs for adults. The study had some methodological limitations, but the results clearly indicated that there are major differences in not only the mathematics skills required in different occupations but in the ways mathematics is applied in different occupations. These differences have curricular implications.

One limitation of the study was that the specific lists of mathematics skills and applications for the two occupations studied may not have been exhaustive. Limited samples of materials used by principal secretaries and electronics technicians were gathered from a limited number of sites. The number was viewed as sufficient to conduct the study, but the lists generated should be viewed as incomplete until the study is replicated on a larger range of sample materials. Second, the method of estimating the frequency of use of each mathematics operation per month required judgments that were difficult to substantiate. This was understood when the study was designed but it was not viewed as a critical limitation to the study's intent. However, some of the study participants found making such estimates distressing because they wanted to be precise, which made the overall procedure more complex than necessary. Future users of the methodology may wish to have experts rate how often an operation is used with a particular application on a more global scale (such as very often, often, and seldom) as a basis for prioritizing mathematics operations.

OMRA instrumentation was considered effective and efficient for identifying both job applications requiring mathematics and the mathematics operations required to perform those applications. A written questionnaire was sent to each of the teams asking about OMRA usability. Their responses indicated they were able to follow the directions and that they could easily provide the information required. Both the secretarial teams and the electronics teams felt they could identify job applications requiring mathematics within the job related materials as well as the mathematics operations associated with those applications.

The range of mathematics operations contained in OMRA appeared to be sufficient for occupations prepared for through vocational and technical education. All of the mathematics actually contained in the sample job related materials appeared on the OMRA Inventory . A concern was raised by some of the electronics instructors that electronics technicians would need more mathematics, but there was no evidence to support this in the literature, from the advisory committee, or from study results.

The results support the literature ( Fitzgerald, 1983 ; Anderson & Peterson, 1983 ; Mark, 1984 ) that suggests that mathematics skill requirements for different occupations and the way that mathematics is applied differ significantly between occupations. Table 4 presents a summary of these results.

Table 4 Comparison of the Number of Mathematics Operations in Each Category Required in the Two Occupations

Principal

Electronicsn Secretary

Technician

Integers 11 8 11 Fractions 13 0 2 Decimals 6 4 6 Percents 6 4 6 Ratio 3 1 3 Algebra 13 0 10 Geometry 11 0 0

Total 63 17 38

Only 17 of the 63 mathematics operations listed on the inventory were required in the sample principal secretary materials. Most dealt with operations requiring decimal numbers. Only 38 of the 63 mathematics operations were required in the sample electronics technician materials. Although all 17 of the mathematics operations required of the principal secretaries were also required of the electronics technicians, the electronics technicians required two more categories of mathematics: fractions and algebra. In addition, the electronics technicians use more mathematics in each category.

Upon comparing the job applications to which the mathematics operations were applied in the two occupations, it was obvious that they were very different (see Table 1). Therefore, even in those cases where the same mathematics operations were required between the two occupations, the applications to which they were applied were not the same. If students are motivated by seeing how mathematics is applied to the job for which they are preparing ( Brooks, 1991 ; Crockcroft, 1982 ; LaPorte & Sanders, 1993 ), this study suggests the application of mathematics in one occupation may have little relevance for people in other occupations. In addition, if mathematics is truly applied differently as a problem solving tool in different contexts ( Brooks, 1991 ; D'Ambrosio, 1985 ; Janvier, 1990 ), these results suggest that the actual application of mathematics in the solution of work related problems in different occupations is likely to be different.

This study supports the use of the related academics instruction model when teaching adults employment related mathematics. Study results indicate that only sub-sets of mathematics skills are required in different occupations and that those skills are applied differently in different occupations.

## Author

Pucel is Professor and Coordinator, Business and Industry Education, Department of Vocational and Technical Education, University of Minnesota, St. Paul, Minnesota.

## References

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Reference Citation: Pucel, D. J. (1995). Occupationally specific mathematics requirements and application contexts. Journal of Industrial Teacher Education , 32(2), 51-75.