Term Paper: Positive Outcome in the Educational

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[. . .] Third, there is a growing awareness that rapid technological change and our nation's economic competativeness will compel workers to adapt existing skills and learn new ones to optimize performance.

As noted, interest in thinking and problem solving has been renewed, but traditional approaches to teaching these skills have met with very mixed results (Nickerson, Perkins, & Smith, 1986). It now appears that the efficacy of these efforts was determined by at least three interrelated factors: First, instructional designers often failed to appreciate the importance of domain-specific knowledge in effective thinking and problem solving (Chi, Glaser, & Farr, 1988). Programs that emphasize the use of generic reasoning and problem-solving heuristics may be theoretically general across situations, but they usually lack power in any specific situation (Anderson, 1987). Therefore, students must adapt these heuristics to the requirements of particular tasks. The research evidence shows that this is a daunting challenge that may not be met by many students with learning handicaps (Ferretti & Cavalier, 1991; Torgesen, Kistner, & Morgan, 1987).

Thinking strategies have not often been illustrated in the context of authentic problems. Authenticity involves the application of skills and knowledge to genuine problems that people must tackle in adapting to the requirements of life. According to Wiggins (1993), authentic problems are nonalgorithmic, complex, and amenable to multiple solutions; involve judgment; require the use of multiple criteria; are uncertain; involve self-regulation and the imposition of meaning; and are effortful. In contrast, many teachers rely very heavily on the use of textbook problems to illustrate thinking strategies (Brophy, 1988; Simon 1980). As a consequence, students assume that these skills are to be used for simple, school-like exercises instead of as tools for managing real-life problems. Students may be able to recall information when prompted to do so, but they are unable to spontaneously use knowledge and skills even when they are pertinent to the problem at hand. Said differently, students' knowledge and skills become inert (Cognition and Technology Group at Vanderbilt, 1990; Sherwood, Kinzer, Hasselbring, & Bransford, 1987).

There is also a growing recognition that thinking and problem solving are shaped by social context and interaction (Adams, 1990; Feuerstein, Rand, Hoffman, Hoffman, & Miller, 1987; Vygotsky, 1978). Traditional perspectives on thinking and learning emphasize the individual's performance in the absence of social supports (Resnick, 1987). In contrast, the social constructivist perspective (Carver, Lehrer, Connell, & Erickson, 1992; Harris & Graham, 1994) emphasizes the learner's active organization of meaning in a socially mediated environment. In this view, students engage in a cognitive apprenticeship (Brown, Collins, & Duguid, 1989; Collins, Brown, & Newman, 1989) with teachers who support the acquisition of strategies and skills by explicit instruction and modeling.

Through the Z. Model, many of these goals come into focus for the student, and into the teaching range of the instructor. The student, within the framework of the project-based learning environment, can focus on information of the most importance to them, set personal learning goals, and acquire information in highly individualized ways that may facilitate connections with prior knowledge (Schmalhofer & Kahn, 1988). This capability is especially important for low achievers with little background knowledge about a topic (Bransford, Kinzer, Risko, Rowe, & Vye, 1989) The previous experience of low performing students is overcome as they experience new success, and build a personal ledge on which to rest a raised level of self-esteem.

In the environment of the Z. Model, students become co-designers of their curricula, rather than mere consumers of information (Perkins, 1986). As designers, students can be involved in creating instructional environments for their peers; for younger students and gain a sense of achievement as they are able to assist other students to conquer concepts which they themselves have mastered. (Carver et al., 1992).

The potential for representing ideas to the group in the project-based learning environment increases the likelihood that students will acquire an understanding of complex information. Students can communicate the interconnectedness of concepts by creating links that connect different units of information or different and sometimes conflicting representations of the same information. Furthermore, when working collaboratively, and presenting work to their peers and other audiences, students have opportunities to discuss and defend their ideas, revise their positions, correct misconceptions, deepen their understanding, and learn to work with others in productive and mutually satisfying ways.

A third and unexpected factor that can be identified from the data is that the test score improvement is greater, and more consistent in the classes which meet in the early hours of the day than in the later periods. This may be due to a number of factors. Assuming the grade demographics are similar in each hour, the declining rates could be attributed to the framework itself. The Z. Model is a high energy teaching style. This model makes high demands of the teacher's ability to stay focused on the class. The teacher using the model must look at himself as a dog which assists the shepherd in the fields. He continually is buzzing about the room, interacting with the different groups. He must keep them on track, encourage the project teams to stay focuses, and remediate any members which are being left behind. In essence, each project-based learning group becomes a small class room. The students help keep each other focused on the basic curriculum, and the teacher is free to 'manage by exception' and give additional help to the students / groups which need it. The metaphor of a small, anxious collie barking at the heels of a flock of sheep as they travel form point A to point B. is close to an accurate picture of the time, energy, and input which the teacher will give to this class as he implements the Z. Model. This may limit the applicability of this teaching model.

The Z. Model demands that the teacher stay attentive to the needs and demeanor of the classroom all day long. The teacher must be as active, attentive, and participatory at the end of the day as (s)he is at the beginning of the day. This task is difficult for most teachers in a normal environment. Using a high energy framework makes this task even more challenging. However, choosing to accept the declining academic performance no longer means that the standards in mathematics education must also change. Quite naturally, this leads into a discussion of theories of cognition and learning that provide the psychological and educational context for thinking about basic mathematical literacy. The shift from behaviorist learning theories to constructivist and social constructivist theories (see Rivera, this series) provides an opportunity to forge a hybrid view of mathematics instruction. This hybrid embeds, or situates, important skill learning in meaningful contexts (Bransford, Goldman, & Hasselbring, 1995; Cognition and Technology Group at Vanderbilt (CTGV; 1996a). The Z. Model is an example of instructional approaches to complex mathematical problem solving that makes use of meaningful contexts, and demonstrates to the students various ways in which mathematic is important to their lives, and to their futures.

Conclusions & Application

The American educational community has suffered amidst internal controversy for many years. From the early 1980's, as educators began to shift from content and data driven instruction to a more affective socially conscious instructional framework, the test scores and academic performance of schools began a slow and steady slide. Because the performance of students and schools is measured over a period of time, and is not instantaneous, the effects of this subtle shift were not measurable until recently. During the same period, and possibly because of the shift in teaching style and priority, the declining academic performance social conditions have shifted away from the a foundation which is required to produce disciplined successful students toward an environment which places a higher priority on fun, edutainment and social engineering than climbing academic performance. This has placed American students at an incredible disadvantage in a worldwide marketplace; none more so than inner city and urban youth. It has caused educators to return to the black board, and re-identify definitions for some of the most basic questions.

Because the standards and definitions are once again up for reinterpretation as the educational community seeks to stabilize its course, one of the important issues is what it means to be mathematically literate. This debate has been stimulated by and reflected in such policy-level events as the National Council of Teachers of Mathematics' (NCTM) Standards (NCTM, 1989), as well as Goals 2000 and the New Standards (1995) project. For example, the NCTM Standards were developed in the context of glaring evidence of the need for major improvements in our nation's educational system (e.g., Resnick, 1987). Especially important has been the emphasis on the need for all students, not simply a select few, to learn to solve problems, reason, and learn on their own (e.g., Bransford, Goldman, & Vye 1991; A. Brown & Campione, 1990; Nickerson, 1988; Resnick, 1987; Resnick & Klopfer, 1989; Scardamalia & Bereiter, 1991).

Changes in conceptions of mathematical literacy are occurring in the midst of societal… [END OF PREVIEW]

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