The process for implementing drastic curricular changes can be described as a restructuring act common to all four case study sites. With slight variations, what prompted institutional changes was a discouraging picture featuring high dropout rates, low academic performance, deficient employability skills, poor student attendance, and lack of parental involvement in educational matters. In some instances, high teen pregnancy rates complicated this picture. Consequently, few students were pursuing enrollment in postsecondary institutions. Students were sorted in curriculum tracks based on their academic performance with frequent over-representation of minorities in low-level and remedial classes. The teaching environment wasn't helping either. Instructors' morale was low and students had a poor attitude toward learning. Collaboration between academic and vocational instructors was practically nonexistent; vocational education was perceived as the dumping ground for low performing students; and instruction was nothing but a combination of lectures and paper-and-pencil drills emphasizing rote memorization and knowledge reproduction.
The response to this crisis was characterized by a case of leadership in action across all sites. Change was championed by individuals in different positions (e.g., superintendent, principal, and dean) who were convinced teaching and learning needed to improve. These individuals were also willing to fight key stakeholders' resistance to change because of well entrenched traditional beliefs on educational practices. First, institutional leaders set themselves to develop a "big-picture" understanding of principles underlying education reforms to provide direction for local efforts. Second, they recruited support from important stakeholders (e.g., board members and innovative teachers) by sharing with them why drastic changes were required. These actions created seed support crucial to the implementation of subsequent steps leading to restructuring curriculum. Third, an institutional assessment was conducted to determine the state of academic performance, teaching practices, administrative problems, and facilities and equipment. The purpose was to provide a frame of reference for goal setting and progress evaluation of subsequent steps.
The results of institutional assessments confirmed problems previously identified and served as the basis for plans to implement change. With the support of steering committees, working groups, or advisory boards, organizational and management strategies followed a collaborative approach. Instead of dictating change in the traditional top-down management style, site leaders involved stakeholder groups in the design and implementation of plans (e.g., faculty, administrators, parents, and business and community representatives).
In all instances, through collaborative work (e.g., participatory management, steering committees, and working groups), each site refined the nature and extent of proposed reforms, identified timelines, and built a consensus on the basic underpinnings of their efforts. This strategy created a climate favorable for implementation of change and set the stage for restructuring curriculum tracks, creating staff development programs, and reviewing instructional and assessment practices. A major task was to decide on appropriate curriculum structures upon which to ground proposed reforms. The choice was shaped by a realistic assessment of site and community resources, extent of instructional collaboration and understanding of reforms, potential impact on students' learning and career development, and public support. In all cases, sites adapted chosen models to individual circumstances and needs. CHS worked toward creating an academy program on aerospace technology to tap into the prominent presence of this industry in the community. FHS opted for a magnet-academy model with focus on public safety in response to a high concentration of area residents working in this field. SHS decided to develop career paths aligned with the Tech Prep initiative. MHCC also chose career paths to improve articulation with feeder high schools under a consortium format. The common ground was the use of occupations contexts to restructure curriculum aiming at improving students' learning and career development.
The next major task was to expand the support of instructors and other stakeholders because the academy, magnet, and career paths formats required the active participation of both vocational-technical and academic instructors. Facilitating interactions between believers and nonbelievers in curriculum changes was key to move forward with proposed plans. Since all sites were basically divided in two worlds, vocational and academic, collaboration had to be nurtured as a bottom-up approach instead of mandated moves. Plans for change involved a visionary strategy aimed at revisiting established curriculum structures, instruction and assessment practices, and instructor collaboration. Second, each site called for grounding teaching in occupational contexts to make learning activities more relevant to students. The goal was to provide students with academic and practical skills in tune with contemporary demands in the workplace. Third, local efforts showed a strong belief that all students can benefit from rigorous learning experiences linking technical and academic knowledge and skills. Finally, across all sites, there was a commitment to improve linkages between high school and postsecondary programs to ensure that all students were also well prepared for further education.
The process of promoting and managing change was not easy. Across all cases, there were accounts of intense negotiations and educational exchanges with institutional, community, and business stakeholders. Professional turf was hard to break, and getting academic instructors to collaborate with vocational-technical peers was sometimes described as "hitting a wall." Another major hurdle was having to overcome stereotyped perceptions that vocational-technical programs are of low-quality, that they only prepare students for specific jobs. Mathematics instructors were reluctant to join restructuring efforts because they feared the integrity and rigor of the discipline would be lost in integration schemes. Adding programmatic changes to accommodate restructured and integrated curriculum activities to this picture, created a juggling act that could only be managed and maintained with strong leadership, focus, and a deep sense of collaboration.
On the average, case study sites had more than 13 years of experience with integration of vocational and academic education. This experience included slightly more than seven years of formal work supported by federal or state funds and widespread instructor collaboration. The implementation of academy models and career pathways proposed by sites required restructuring coursework to reflect the new curriculum orientation. It also required changes in teaching assignments, scheduling, and instructional collaboration. Further, it involved integrated collaborative instructional work and related professional development.
Restructuring Mathematics Curriculum
Academic disciplines were well entrenched in the traditional educational system at each site and efforts to restructure academic curriculum represented an important undertaking. The following question had to be addressed: How would mathematics be aligned with a series of coherent technical courses while preserving the rigor of content? High school programs eliminated all general and remedial math courses and created a strand of applied mathematics classes (e.g., applied mathematics for the technologies and applied geometry) available to all students. The emphasis of these classes was in grounding mathematics in occupational contexts featuring hands-on, real-world problems relevant to mathematics concepts and skills used in the broadly defined technical fields. For example, at SHS, Applied Mathematics for the Technologies I, recommended for grades 9 and 10, is a "problem-solving course that makes mathematics relevant by showing how mathematics skills are used in the workplace. It teaches problem-solving by hands-on, student-centered situations. The mathematical skills include, but are not limited to, using the scientific calculator, problem-solving techniques, and measurement" (SHS, 1996, p. 25). Similarly, at CHS, the Applied Mathematics III course expected that students will "understand and apply functions, relations and graphs; understand and apply geometric properties to solve problems; and demonstrate an understanding of the structure of mathematics as it applies in the real world" (CHAAT, 1996, p. 25). Similar descriptions for restructured mathematics courses were found at FHS.
At the two-year college level (MHCC's experience), restructuring was particularly challenging because it entailed a revision of college curriculum along with the development of articulation agreements with feeder middle and high school programs. MHCC's response was the development of an applied, technology-based, one-track (ATO) curriculum linking mathematics and technical areas from middle school through the first year of college for all students (Tech Prep and baccalaureate preparation). As in the high school cases, a series of interactive mathematics courses replaced traditional courses such as remedial mathematics and intermediate algebra (see ATO chart, MHCC case study in the Appendix).[2] The interactive nature conveyed in the restructured courses was laid out by expected core outcomes for all topics included in each course. Students in these college courses were expected to actively interact with teachers and other students in learning mathematics. Further, algebra, geometry, probability, data analysis, and statistics were interconnected in each level of interactive mathematics. Also, applications grounded in technical disciplines were linked to mathematics concepts to bring relevancy to students' learning.
Curriculum Development
Changes in curriculum structures required parallel activities in curriculum development to reflect coherent sequences of courses representing broadly-defined technical fields (e.g., engineering industry, aerospace technology, and public safety) under career paths or academy formats. Course content and complementary activities were then identified along with programmatic and administrative requirements (e.g., attendance, credits, grading, and advanced coursework). This process was characterized by an interdisciplinary collaborative effort of instructors and individuals leading restructuring activities either as an internal process or guided by external consultants. Common across all sites was the gradual development of curriculum and an open attitude for trying out new things. Thus, faculty at each site were willing to experiment and make revisions as things progressed.
Available funds were used to develop applied coursework to link vocational and academic education. Technical courses in career paths and academy models were structured and sequenced and curricular activities developed in detail. In the case of mathematics, restructuring helped define applied courses, sequences, content emphasis, and teaching focus. However, the specificity of curricular activities within each course remained sketchy, and instructors still relied on commercially available resources and loosely connected student activities and projects to complement or guide instruction. To a different extent, based on time and resources available, refining curriculum is an ongoing process at each site.
Curriculum development efforts at MHCC were more successful. Led by one of their peers serving as Dean of the Mathematics Division and supported by a federal grant, mathematics faculty built a consensus on expected outcomes for all ATO core curriculum courses along with instructional strategies to facilitate the learning of mathematics. As part of this collaboration, a team of instructors completed a textbook used for Interactive Mathematics III aligned with restructured content, emphasis, expected outcomes, and instructional strategies agreed upon by all faculty. With grant support, MHCC faculty have continued to work on the development of materials and textbooks to meet the specific requirements of restructured ATO courses. The strength of MHCC curriculum development efforts lay in the detailed consensus on outcomes and instructional strategies, its collaborative approach, and commitment to produce full course materials and textbooks for restructured courses.
Integration Strategies
The career theme of each program (e.g., public safety and aerospace technology) allowed faculty to integrate concepts, ideas, and skills across various disciplines. A career theme facilitated the immersion of students in learning experiences linking academic and work skills found in a technical field of interest. For instance, at CHS, math, science, social studies, and technology instructors identified content patterns and overlaps for team teaching purposes. Mathematics content was then integrated with appropriate SCANS competencies relevant to communications, transportation, manufacturing, and other relevant topics in aerospace technology.
Integrated instructional practices grounded in career paths did not have a program-wide approach, but students were partially immersed in activities featuring broadly defined technical fields (see SHS and MHCC case studies in the Appendix). Courses were interconnected within technical fields and related to relevant topics in academic subjects. For example, mathematics topics were linked to money management emphasizing a business career path, geometry was connected to concepts in a manufacturing curriculum, and so forth. Also, across all sites, there were instances of shared integrated activities involving planning and teaching between a team of two or more cooperating instructors. Math instructors collaborated either formally or informally with technical instructors, individually or in multidisciplinary formats, to plan and teach shared concepts (e.g., data collection and interpretation and graphing). Further, all sites threaded problem solving, thinking and communication skills, and knowledge applications across vocational and academic disciplines as a prominent integration strategy. Collectively, integration strategies departed drastically from fragmented structures found in traditional settings where curriculum tracks, lack of instructional collaboration, and paper-and-pencil activities were the norm (see Fogarty & Stoehr, 1991, for a detailed description of integrated curriculum strategies).
The implementation of integration practices presents a number of practical considerations negotiated between administrators and instructors. Course scheduling is a crucial organizational factor that needs to be considered for aligning courses in the academy or career paths models. Also, a flexible teaching schedule has to be negotiated to allow teacher collaboration in team teaching, group projects, and content coordination. Therefore, shared planning time is needed during a typical week to keep mathematics/vocational-technical instructors informed of students' progress, problems, and to prepare future activities. Time management, especially in high schools, appeared to be a challenging issue for implementation of integration practices. Integrated curricula seemed to increase the demand for materials and equipment and more student-teacher contact beyond the classroom. Thus, competition for resources increases and adds another factor in management decisions as stakeholder groups lobby for their share of funds, materials, and equipment. Finally, another challenge is to maintain a climate conducive to productive collaboration aimed at designing nontrivial integrated practices involving mathematics applications in occupational contexts that also adhere to the NCTM Standards while preserving the rigor of the curriculum.
Professional Development
Individuals leading integration efforts were self-motivated and prepared themselves by reviewing relevant literature, attending conferences addressing reform topics, and seeking guidance from research and demonstration institutions. Literature on contextualized learning and documents outlining key principles for school-to-career curriculum and mathematics reforms provided initial ideas for these efforts. Site leaders promoted some strategies for professional development to early cooperative staff to expand the support base for anticipated changes.
Organizations such as the Southern Regional Education Board (SREB) and Coalition of Essential Schools (CES) appeared to be helpful in providing effective guidance and staff preparation in high school programs. In some instances, these organizations provided sites with the philosophical and management foundations of reforms along with training opportunities involving institutes and seminars on curriculum integration, authentic assessment practices, and restructuring issues. Across all sites, the strategy was to expand the support base by involving faculty reluctant to buy into restructuring efforts in these professional development activities. Early on, working in groups and engaging in reflective discussions about areas that needed improvement, faculty began to work toward future collaboration. They also learned important theoretical and practical considerations for implementing reforms since various points of view were openly and forcefully voiced by competing groups. More recently, professional development across sites are flexible programs that allow instructors to develop individual plans or decide as a group what relevant preparation is needed, instead of being required to participate in standard district programs offered to instructors at large. Further, in most cases, new hires are required to have at least one year of experience in applied instruction and be committed to continuous development of integration practices.
At a time when reforms in both vocational education and mathematics were taking shape, case study sites were already experimenting with alternative pedagogical and assessment practices. The common denominator in both secondary and postsecondary restructuring efforts was to develop program-wide emphasis on real-world applications.
In Spring 1996, case study sites reported a moderate-to-great emphasis in using the NCTM Standards to guide instructional activities. It was clear, across the board, that sites were more successful in emphasizing process rather than content-specific NCTM Standards, matching the general emphasis on skills called for by the school-to-work movement. Based on problem-solving activities, the goal was to involve students in relevant and challenging curriculum requiring teamwork, reasoning, communication skills, and concept connections. What follows is a description of how process standards were implemented at case study sites, a summary of instructional applications addressing content standards (e.g., geometry, algebra, and statistics), and assessment strategies.
Problem Solving
Authentic, integrated problem solving activities require student engagement in discovery of mathematics applications in technical contexts. The virtual learning concept typical of CHS instructional activities represented problem scenarios involving high levels of authenticity. For example, the unit "Working in Space" was based on a real-world problem situation found in aerospace technology. Students visited NASA launching facilities at Cape Canaveral and consequently considered pursuing a career in a related field upon graduation from high school. Thus, the following problem situation statement was very real to them:
NASA has asked Congress for funding to design, build, and launch a large station into the earth's orbit, and you have been selected to serve as members of the Space Station Task Force. You have committed the next ten years to the preparation of the space station, and you will be among the station's first inhabitants. Launch is tentatively scheduled for the year 2000. (CHAAT, 1996, pp. 1-2)
Several questions requiring the integration of multidisciplinary knowledge could derive from this problem scenario. Similarly, several mathematics applications could also be found. For instance, students were required to calculate the exact distance from earth the station must reach to maintain its orbit around the earth and make a scaled map of the Space Station's orbit in relation to earth. This was a nonroutine problem that caught the students' interest. The problem was nontrivial and provided opportunities for approaching solutions in various ways so that students could investigate and develop understandings of applied functions, relations, graphs, geometric properties and models, axiometric systems, relationships between plane and solid geometry, and transformational and coordinate geometry. The contextual application was linked to science and technology concepts associated with understanding orbital requirements.
The complexity of this problem was significant and produced additional questions requiring the application of more advanced mathematics concepts (e.g., trigonometric identities and the concept of limits and its applications). Mathematical models were a natural application for determining the orbit of the station.
This problem was contextualized in a genuine, significant problem scenario open to various approaches for solutions. To solve it, students had to research and understand related science and technology concepts, identify mathematical solutions at their grade level and even attempt solutions requiring more advanced mathematics. This approach represented a drastic departure from traditional mathematical problems void of realistic contexts asking merely to solve for x and y or word problems involving trivial situations.
Higher-Order Thinking
To engage students in higher-order thinking (HOT) activities, characterized by tasks fostering inductive and deductive reasoning, NCTM (1989) suggested that highly authentic instructional practices "should include numerous and varied experiences that reinforce and extend logical reasoning skills" (p. 143). An example drawn from a FHS integrated unit of study entitled "The Curious Death of Zachary Taylor" illustrates an integrated problem situation that requires the application of HOT skills.
The problem, grounded in a public safety curriculum, required the participation of students in the preparation and re-enactment of the trial concerning the death of Zachary Taylor. Students, working in teams, played different roles in the trial. To determine the possible causes of Mr. Taylor's death, data analysis and chemistry teams collected blood samples at the "scene of the crime," measured the presence and amount of toxic substances, interpreted data, drew conclusions on a toxicity report, and communicated information at the trial. In a focus group, students indicated this was a highly engaging activity. Playing various roles, students were involved in gathering evidence for different purposes. The data analysis and chemistry teams, in particular, had to identify and test hypotheses involving the presence and appropriate levels of known toxins. Students followed steps simulating real-world procedures to collect, handle, and analyze blood samples, conducting tests repeatedly to improve reliability. The results of this analysis were complemented by data from test trials on mice to determine toxicity of substances found in blood samples. Based on data presented by the chemistry team, the data analysis team summarized the information, used regression analysis to extrapolate toxicity results to humans, and produced graphs of relevant findings for reporting purposes.
In this activity, students applied inductive reasoning derived from analysis of toxicity data on mice to arrive at logical conclusions about the lethal effects of levels of toxins present in blood samples. The evaluation of available information had to be carefully conducted and conclusions well grounded since a major decision was at stake. Deep understanding of mathematics concepts and their potential applications were promoted since students had to be prepared to answer questions and justify their arguments. In comparison, in traditional classrooms mathematical problems are presented to students by using lower-order thinking instructional activities emphasizing the application of memorized formulas to solutions of problems void of relevance for students.
Communication of Ideas
Opportunities to exchange mathematical ideas and applications with peers and teachers is also central to authentic, integrated NCTM Standards-based mathematics instruction. The Working in Space curriculum developed by CHS staff, provide an illustration of instructional activities conducive to facilitating substantive communication of ideas and their relation to other concepts. Students, working in heterogeneous groups (i.e., students with a mix of academic abilities, gender, and ethnicity) were asked to make decisions to staff a space station under the following conditions:
Use as much of your two million dollar annual personnel budget as you can. You must decide which occupations are most necessary. The list of possible personnel, and the annual salary for each is [available to you]. If your station has a personnel requirement not given, you may add to the list (substantiate need and define salary). You may hire more than one person in a given occupation. Each member of the task force must prepare a different personnel plan, and then the entire team must choose, by consensus, which plan they will use. To devise the plan, each member must create a spreadsheet giving wages and calculation of social security, 8% of annual salary; unemployment compensation, 1.5% of annual salary; federal taxes, 12% of annual salary; and benefits (including health insurance, vacations, and retirement plan), 7% of annual salary. The employee's salary plus each of these additional costs represents the total cost of each employee. The final plan must include each member's spreadsheet. Remember that the entire personnel budget many not exceed 2 million. Including the families of the space stations employees, approximately 200 people will live on the station. (CHAAT, 1996, pp. 1-2)
This activity involved mathematics and business concepts requiring computer applications. First, students engaged in extended conversations discussing the need for certain services (e.g., lawyer, mayor, detective, and florist). Once this issue was settled, the conversation moved to the optimal size of staffing for each job (e.g., how many doctors?). Students began to gauge the impact of salaries on budget limits and discuss adjustments. A productive conversation was carried out about staffing, management, and organizational issues connected to budgetary decisions. As students discussed these issues, other related topics began to emerge concerning taxes and estimation of benefits and deductions. The conversation then turned to mathematics applications involving percentages, data management, tabulation systems, and use of computer spreadsheets. Collectively, these exchanges with peers and instructors appeared to help students build shared understandings of business and mathematics concepts and issues involved in payroll management and computer applications. Further, students were expected to communicate their ideas on staffing decisions orally and in writing, requiring the use of appropriate terminology and articulated presentation of information. In contrast, a low authentic instructional activity would reflect lecture formats, individual paper-and-pencil work, and little or no productive interactions in the classrooms.
Connecting Mathematics and Career Concepts
An important goal of the NCTM vision is to provide students with opportunities to develop deeper understanding of connections among mathematics topics and potential applications in other disciplines. Occupational contexts appear to be an excellent vehicle to address mathematics concepts in real-world situations and establish meaningful connections. Across all sites, this was the area where successful accounts were found more prominently, given the fact that students are basically immersed in a thematic curriculum (i.e., public safety, aerospace technology, and career paths). Even in situations where the levels of authenticity in instructional activities involving problem solving, communication, and reasoning skills were not that high, the relevance of mathematics connections was still prominent.
The academy and career paths formats seemed to weave a context beyond individual courses and lessons, carrying interrelated experiences through the curriculum. Thus, it is inevitable that students develop an appreciation for the value of mathematics applications in occupational contexts and for the ways in which mathematics topics relate to each other in certain situations. For instance, the whole program of studies in aerospace technology becomes part of the students' filter for both vocational and academic education. Under these circumstances, students soon begin to get the idea of doing instead of only knowing mathematics. Overall, there appeared to be great student predisposition for making connections between substantive mathematics knowledge and problems and personal experiences grounded in broadly-defined career fields. In contrast, low authentic instructional experiences would be characterized by both disciplines taught in separate tracks precluding meaningful connection of ideas and applied learning.
Assessing Students' Learning
NCTM (1995) defines assessment as "the process of gathering evidence about a student's knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes" (p. 3). To monitor students' learning and improve instruction, all sites showed evidence of early development of assessment plans to match changes in instruction. For example, FHS had already been using portfolios for assessment purposes by the time they were required by state law in 1990. With varying emphasis, case study sites use several alternative assessment strategies to monitor students' learning, inform instructional activities, and evaluate students' achievement. A shared belief was apparent in that assessment activities needed to be integrated with instruction since mathematics and vocational concepts had to be clearly identified within a problem situation. Further, because instructional activities were more dynamic and open to greater student exploration of potential solutions to problems, a key concern of instructors was to monitor how students progress toward set goals. The common philosophy encountered across sites was to emphasize communication with students about progress rather than whether they have correct or incorrect answers in absolute terms (learning standard focusing on multiple skills). Also, in most instances, students were presented with problem statements, expected outcomes, and criteria for evaluation (standard emphasizing use of assessment to enhance learning). See, for example, Table 1 listing broad criteria for a situation statement and problem solving activity for the problem on orbit estimation presented previously in this report.
Students may have been required to work individually or in groups, document work in portfolios, engage in classroom exchanges, research information, present reports orally and in writing, justify answers and conclusions, and take traditional quizzes and tests. Based on observations of these tasks, responses to questions, scoring rubrics, samples of work from portfolios, project products, and results from quizzes and tests, instructors informed and adjusted instructional activities to seize the learning moment (standard emphasizing multiple sources of evidence for valid inferences on learning). With variations in approaches, instructors at each site were able to determine whether the mathematics concepts students applied to problem solutions were nontrivial and whether the problem was integrated effectively in a realistic context (worthwhile mathematics standard). Finally, through class observations and exchanges, documentation of individual and group work, and presentations of findings, instructors could gauge the level of student involvement and motivation derived from instructional activities (standard focusing on setting high expectations for all students).
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General Criteria Checklist
Criteria Checklist for Applied Mathematics I, II, III, Geometry Honors, Algebra I, Algebra I Honors
Criteria Checklist for Algebra II Honors & Pre-Calculus
Criteria Checklist for Principles of Technology I & II
Criteria Checklist for Physics I Honors
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Note: Students receive full credit by completing both general and course specific criteria checklists.
The purpose and type of assessment strategies practiced at each site can give an idea of the cohesiveness of integration efforts (standard focusing on coherent process). The picture is both a source of encouragement and a reality check. It is encouraging because it is clear that these alternative strategies can influence improvements in instruction and inform curriculum revisions for integration purposes. It also appears to be an excellent framework for gauging the quality of integrated problem situations and the kind of mathematics involved. However, it is a reality check because it shows the difficulties in connecting curriculum, teaching, and assessment in a coherent fashion. Instructors struggled but continue to develop and experiment with scoring instruments and other alternative ways for assessment to account for multiple manifestations of learning elicited by problem-solving activities. Further, it appeared that comprehensive efforts were falling on the shoulders of highly motivated instructors, still a minority in most cases.
A Taste of Reality
Although case study information suggested a trend toward great emphasis in implementing the vision of the NCTM Standards, the reality of efforts may be more modest. There are areas where the NCTM Standards can be naturally emphasized as a result of program-wide integration efforts, while in more course-specific curriculum activities the shift toward the NCTM Standards remains a challenge. Promising practices were found across sites but consistency across the curricula regarding a systematic implementation of the NCTM Standards appears to be questionable. There are hints of evidence indicating that curriculum grounded in broadly-defined career fields can serve as an influential umbrella for providing a sense of purpose and meaning to mathematics learning. The career contexts seemed to facilitate the identification of worthwhile related mathematical tasks in specific integrated course activities and across the curriculum. Because students were immersed in career contexts, integrated activities reinforced the connections with mathematical applications beyond the classroom and added authenticity to learning. Hence, process NCTM Standards (i.e., problem solving, reasoning, communication, concept connections) appeared to be more easily emphasized across the board. However, addressing specific content standards (e.g., algebra, trigonometry, and discrete mathematics) in rigorous ways seemed more inconsistent.
Linking the NCTM Standards to integration efforts is not easy, and there seem to be several challenges to effective implementation. First, identifying worthwhile mathematical tasks involved in real-world problems is more difficult than it seems. Designing nonroutine, nontrivial problem situations, including significant mathematics applications, is not an easy task. Promising examples were found across sites, but they were showcases of exceptions rather than the rule. High authentic instructional practices coexisted with traditional teaching, and even those who were involved in innovative activities may have shown inconsistent pedagogical behaviors. This is perhaps more visible when it came to emphasizing HOT activities through provocative questions and challenging problem scenarios. For instance, instructors may have had trouble discriminating between guided and analytical questioning, thus limiting the quality of HOT tasks. Breaking traditional molds of instruction and faculty collaboration can be best characterized as a story of fragmented successes.
The lack of specific curricula involving mathematics integration adds to the challenges for implementation of the NCTM Standards. The majority of available curricula is restricted to lessons or units loosely structured and still in the process of development. Thus, it is not surprising to see instructors relying heavily on CORD (Center for Occupational Research and Development) and other commercially available textbook materials to guide instruction. Consequently, alternative assessment strategies are used with restricted purposes and are still evolving in the schools. Also, programmatic challenges (e.g., scheduling and resources), time demands for collaboration and planning, and other institutional issues can all contribute to the quality of emphasis on consistent and systemic NCTM Standards-based practices. It is apparent that linking the NCTM Standards to emerging vocationalism is promising but, at the same time, requires serious restructuring efforts beyond pedagogical strategies.
[3] This criteria checklist is a sample summary of various requirements listed on relevant content from different courses involved in this activity. See CHAAT (1996).