Coming from two different perspectives, both mathematics and school-to-work reform movements shared common grounds: a belief that all students should benefit from improved and challenging curriculum and teaching practices and the need to make learning relevant to the requirements of today's world. The concept of integration bridged these two movements by linking realistic learning experiences with the power of reasoning, problem solving, and communication skills across the disciplines. The linkages between mathematics and vocational-technical education, however, have been weak. The mathematics education sector has conducted its reform efforts as an internal movement involving comprehensive reforms in curriculum, teaching, and assessment. In contrast, vocational-technical counterparts have promoted curricular changes involving integration with academic disciplines but changes in instruction and assessment appear to have been ignored.
The NCTM vision and ideas derived from school-to-work reform can provide an excellent framework to link mathematics/career curriculum, instruction, and assessment. This converging framework clarifies ideas for the development, implementation, and evaluation of authentic, integrated, standards-based learning. However, producing curriculum based on this framework is not easy and mathematics/vocational-technical instructors continue to call for specific examples. Packer (1995) offered some practical hints. In a typical high school algebra class, he indicated that students may be asked to solve the following problem: "Find the speed of the canoe if the paddler is paddling six miles an hour upstream and the current is three miles an hour" (p. 40). He argued that although this word problem reflects a realistic situation and is not about solving merely for x and y, it lacks a significant and important situational context and encourages students to solve it by using a step-by-step textbook method. To break this traditional approach, Packer suggested ways to bring authenticity to integrated mathematics/vocational-technical learning: He offered an alternative example:
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. 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 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. (p. 40)
In this scenario, students use algebra concepts in the context of decision making in hospital administration. Several SCANS competencies are addressed and instructional processes are aligned with the NCTM Standards. From High Schools that Work program experience, Bottoms and Sharpe (1996) also identified a number of ways to learn how to integrate mathematics. For instance, investigating garbage as an environmental issue, students use mathematical functions, solve problems, and study technological applications. Students are required to analyze "data on the amount of garbage generated each year and the availability of landfill space to calculate the extent of the nation's waste disposal dilemma" (p. 23). NCTM (1989) also provides examples of students' work connected to real-world situations.
An examination of these examples reveal that there is no universal model for integrating the learning of mathematics with knowledge in other fields. Integration can take many forms depending upon the level of commitment and resources available (Bottoms & Sharpe, 1996; NCTM, 1989). The NCTM vision and the framework for authentic, integrated instruction provide broad guidelines for developing problem situations that feature mathematics in realistic contexts.