Framework for Implementing the Z. Mathematical Model to a Six Grade Class

¶ … positive outcome in the educational progress for the students resulting from applying the Z. Model framework. In Mr. Zander's classroom, the average improvement in test scores is 16.75 points. The is the rise in test scores resulting from the students taking the same standardize test, once at the beginning of the school year, and a second time after 6-7 months Z. Model application.

The baseline group data was taken form another 6th grade Mathematics' class, taught by Mr. Valree. The same test was administered at the same intervals. The difference was that Mr. Valree's class experienced standard teaching methods. The average improvement for this baseline class is only 3.25 points. The level of baseline class's improvement should be expected for no other reason that the students are completing the same test. It would be expected of the students taking the same test for these students to achieve higher scores on their second attempt. This result brings into question whether of not the students in the baseline class are experiencing any real learning during the school year.

Looking more closely at the results from Mr. Valree's class, it can be seen that his score improvement is significantly affected by the performance of 2 students. If in Mr. Valree's 5th hour, the 2 exemplary improvements are put aside as aberration in the data, the average change in the 5th hour class would be 0.6, bringing the average improvement of all students to 1.4 points. When taking into consideration that the students are taking this same test at different periods of the year, and that it can be expected that they would perform better in the second attempt, the question remains to be answered if the impact of the standard educational process during the school year was of no positive impact on the students educational progress. According to Bransford et. al. proponents of educational reform highlight the importance of creating instructional environments that encourage students' active involvement in the learning process. To be so involved, students must construct knowledge, evaluate the products of their work, and engage in the design of solutions to authentic problems. (Bransford, et.al. 1991). Their analysis of the research evidence leads to conclusions those students' thinking skills and attitudes are enhanced when they collaborate in the solution of authentic problems. The progress of the students undergoing the Z. Model clearly are being encouraged and support to create the new learning pathways and construct knowledge through the use of a project-based learning, and with the additional support of the community, particularly their parents or care givers. The students in Mr. Valree's class are not making the connection between the text materials and concrete application and understanding.

A second conclusion that can be drawn form this data, and the as an evaluation of the Z. Model is that the experimental group of students not only improved significantly, but also improved consistently. The average change improvement of Mr. Zander's classes was not due to a handful of extraordinary students who benefited from the teaching framework. Forty three out of seventy one achieved double digit improvement; 29 out of 71 demonstrated an improvement of over 20 percentage points. This is significant, if not an outstanding change in the level of academic performance. By addressing a spectrum of educational needs the students were motivated, engaged, and equipped to participate in the educational process. These students, possibly for the first time, make real time connections between their class text material and the real world. Through a project based approach, the math principles became meaningful to the group of students working on a problem, and therefore became meaningful to the individuals in the group.

Much research has been undertaken to evaluate the effectiveness of multi-media projects and equipment in the classroom. Bransford et. al. contends that educational multimedia is potentially powerful tools for constructing knowledge, especially when used in collaborative project-based instructional environments, or multimedia design projects. Their studies in the area of social studies review evidence about the efficacy of multimedia design projects in promoting students' construction of knowledge, thinking, and problem solving, and discuss some potential challenges to the efficacy of this approach. With the invasion of high tech movies, video games equipment, and personal gaming systems that have become an integrated part of the student's life, utilizing multi-media technology is a natural evolution of educational equipment.

The Z. Model, because of the interaction of the 4 domains and the constant positive pressure placed on the students toward educational improvement could be called a low tech multi-media approach to teaching. High tech multi-media equipment utilizes sound, color, the intrigue of high tech gadgets, and a fast moving environment to engage the players, or learners mind. The high tech multi-media engages the student, and with the addition of headphones, the student can become engrossed in the entire high tech multimedia world-scape. The Z. model engages the student on a multi-media approach also. The student is grabbed in the classroom with a project-based work team. He is given objective standards, solid goals he is required to meet. His class mates are engaged in the same pursuit. He is met by his parents who, on another educational 'multi media' front (the home); engage the student in the pursuit of the same goals. The tools are different between a computer with a power point lesson, but the approach, and therefore the results, are similar. In the advertising world, successful advertising campaigns are measured in viewer impressions. Again, the goal is to co-opt the consumers, attention through many mediums to deliver a consistent message. If we could combine these terms, media impressions and multi-media, the Z. Model could be similarly be called a multi-impact teaching model. It is not multi-media as the term is understood, but the goal is to develop inclusive scaffolding around the student in order to enable and encourage him to learn.

In comparison, the baseline group demonstrated a wide range of results. While many of the group improved including improving, and regressive performance. Eighteen out of fifty nine students who took both tests demonstrated regressive performance. These students scored lower on their second attempt on the test. The overall performance is "all over the map." This is a disturbing trend, and the continuing falling rates of academic achievement have been the subject of national focus. A seemingly endless stream of reports from prestigious state, regional, and national commissions (e.g., National Commission on Excellence in Education, 1983; National Commission on Social Studies in the Schools, 1989) lament the failure of educational institutions to prepare our children for the challenges of life. Most advocates of educational reform would probably agree that schools should prepare students to contribute to constructive, personally satisfying, and socially valued activities in a representative democracy. There is much less agreement, however, about the specific strategies and educational processes that are proposed to achieve this goal. For some, the focus is the acquisition of basic skills, for others, it is a national curriculum or standardized assessments, and for yet others, the student's capacity to think critically and analyze information is emphasized (Cherryholmes, 1990; Engle, 1990; Epstein & Evans, 1990; Nelson, 1990).

A national cry within the academic community for 3 decades has been for smaller classes and more money to spend on education. While the smaller classes have been achieved, it has been measured that the school districts with the highest spending per student, (Washington DC and New York) have some of the poorest performing students. In the urban setting, the ability to increase funding for schools which are already demonstrating suffering educational performance rates is very restricted. The solution is for teachers to find a method with the resources they have to engage their students, and increase their educational success rates. The Z. Model is one approach that appears to do just that.

The discipline of education is currently enjoying the unprecedented attention of parents, policymakers, and public officials. While the word 'enjoying' may be a misnomer for a process which is identifying a failing system, the positive outcome is that the educational slide is slowing, and schools are experimenting with procedures which will reverse the course of educational decay. This work is predicated upon the premise that education should encourage students' disposition toward engaged thoughtfulness and reflection (Dewey, 1916; 1933; Wiggins, 1993), and that it is possible to design learning environments that encourage students' construction of knowledge and active problem solving.

Bransford, Goldman, and Vye (1991) offered a compelling account of the issues that have shaped views about thinking and problem solving over the last decade. First, many students score poorly on tests of problem solving, essay writing, reasoning, experiment design, and the like (e.g., Gentile, 1992; National Assessment of Educational Progress, 1981; National Commission on Excellence in Education, 1983). Second, business leaders have serious concerns about workers' reading, writing, speaking, learning new skills and adapting old ones for use in the workplace, using quantitative skills to manage and produce job-related information, and assessing and generating arguments and explanations (Resnick, 1987). 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 recognition that twentieth-century skills will not be sufficient for the workforce of the next century. The Secretary's Commission on Achieving Necessary Skills (SCANS) report issued by the U.S. Department of Labor (1992) clearly identified collaborative activity, problem solving, communication, self-assessment, and competence and confidence with technology as critical to success in the twenty- first century. The emphasis on all students needing to be able to solve problems, reason, and take charge of their learning should be cause for celebration.

Unfortunately, there is considerable evidence to indicate that even the traditional standard of basic mathematical literacy is not being attained by many of our students. This phenomenon is exceedingly evident in the urban schools. Since the beginning of the National Assessment of Educational Progress (NAEP) two decades ago, media, parents, government officials, and educators have debated potential causes for the low test scores in America's schools, and have searched for ways to remedy the problem. Although American students do well on whole-number computations, they have difficulties with fractions, decimals, and percentages and with problems that involve unfamiliar, non-routine tasks. Word problems that involve two or more steps are particularly problematic for students in today's educational environment. (Kouba et al., 1988). And, although American youth possess a fairly good knowledge of the procedures associated with rational numbers, probability, measurement, and data interpretation, they lack the conceptual knowledge that enables them to apply their knowledge in problem-solving situations (C. Brown et al., 1988a, 1988b). And these trends are visible across the spectrum of economic and social groups.

The at-risk behaviors found in poor urban homes have created additional hurdles for inner city minority students to overcome for decades. However, added to the shift in educational policies and practices of the last decades, and the increased influence of electronic media, these at risk children face an even more difficult climb in order to reach a level of effective educational achievement that will enable them to change the cycle of poverty of the urban inner city. Because of the broken homes, and inconsistent support and preparation for schooling the student receives in the home these students must be taught many additional concepts, in addition to the lesson's they will find in their text books.

These students must be taught:

how to be prepared to learn when they enter the classroom

How to interact with peers in a manner that facilitates positive educational progress how to study, and learn

Then... The student arrives in a place where positive educational progress is possible.

Then... The student is ready to learn the materials they need to function in an increasingly educated society.

As stated above, because of the wide recognition of the declining educational performance, and a nation wide effort to change the trend, many new schools and educational models are coming to the forefront. For example, ten Atlanta area schools tried a new approach to school reform when school started in the fall of 2000. The schools initiated wide reaching reform initiatives that are part of the New American Schools program.

New American Schools is a proven leader in the school reform movement," said Atlanta Superintendent Beverly Hall, "and we are excited about making it a component of our whole school reform effort." In 1999, Hall announced that she would contract with the not- for-profit New American Schools and gave principals the opportunity to decide if they wanted to use the program. (Atlanta Journal, 2000)

New American Schools (NAS) is a leading provider of professional services and investments in K-12 education. As a business-led, nonprofit organization, NAS and its partners provide support to state departments of education, school districts, charter authorizers, and networks of schools. Their goal is to offer key stakeholders the tools and external assistance needed to facilitate and sustain both student and adult learning. New American Schools is committed to helping all students succeed by shaping, supporting, and sustaining system-wide innovation and improvement in learning. (NASchools.org, 2003)

New American Schools was established in 1991 by the chief executives of the country's most successful businesses. New American Schools (NAS) is a nonpartisan, nonprofit organization whose mission is to increase student achievement through comprehensive school improvement. They focus their attention with high-quality education providers to make best-in-class services available on a national scale. NAS provide a range of consulting services to state departments of education, school districts, and schools to help them design, implement, and align the essential components of a comprehensive school improvement strategy. NEW has achieved significant success in reaching their goals, and raising educational performance for the past decade, and they continue to build a financial and educational entrepreneurial foundation for future success.

The Z. Model is one such entrepreneurial educational model, one which could pursue a relationship with the NAS to further its research, and consider a wider application. According to the initial data, the Z. Model pursues educational excellence in the area of mathematical studies, and produces a higher level of practical classroom effectiveness. The z-model refuses to let the environmental and social conditioning of modern life give teacher or student an excuse not to learn. The Z-model is a proactive and an aggressive approach toward grabbing the student's attention at their existing level, and creating an environment in which they will learn.

Similarities between urban minority schools and students with learning disabilities

No one wants to connect the dots between these two areas of study, and assume that inner city children from poorer neighborhoods are learning disabled. The term disability has been greatly expanded during the past decade. While the expansion of this term may have been to the benefit of individual small groups, it has adversely affected society in general in that we are swiftly becoming a community of sub-groups which are all considered disabled to one extent or another.

However, during the research of this paper, striking similarities appeared between the needs of the learning disabled student, and what models affect positive change in their educational performance, and comparatively the approach and results of the Z. model. Recent research in mathematics instruction requires educators to rethink long-established beliefs about teaching, learning, and assessment. In particular, this research underscores the need for problem solving and higher level thinking in mathematics. Consistent with these recommendations, researchers have identified four promising themes for mathematics instruction that have emerged from research involving students with learning disabilities. These themes are providing a broad and balanced mathematics curriculum. Data driven instruction remains the anchor the successful math curricula.

The students must attain mastery of a specific and wide ranging knowledge base if they are to succeed in further mathematics' studies.

A engaging students in rich, meaningful problem tasks. As the students develop their own models, and use them to investigate math concepts, the tasks become real, and the students take ownership of the projects.

A accommodating the diverse ways in which children learn. The project base learning groups allow a measure of freedom for the students to develop their own diverse path toward a common goal.

A encouraging students to discuss and justify their problem- solving strategies and solutions. In a PBL group, one of the key factors is discussion, and a co-learner environment.

As can be seen, the format of the Z. Model facilitates a framework which addresses each of these four themes. A major goal of the Z. Model, coincides with the learning disabilities focus, are the four major themes related to higher level thinking and problem solving. These have emerged from recent mathematics studies involving students with learning disabilities. In essence, these themes embrace the philosophy that students with learning disabilities benefit from rich, challenging programs that promote mathematical thinking, as do the students studies in the Z. Model research.

As the multi-impact framework is build around the student, he increases his scholastic performance. This suggests that teaching and learning mathematics in relation to students with learning disabilities, and students of the demographics in Mr. Zander's urban school, should be evaluated for further similarities. The problems may not be a specific disability, but rather identifying a framework by which the student will learn.

The curriculum, assessment, and professional teaching Standards of the National Council of Teachers of Mathematics (NCTM; 1989, 1991, 1995) call for strategic shifts in mathematics instruction for all students. In essence, these shifts involve a movement toward higher level mathematical reasoning and problem solving, and involve rethinking long-established beliefs about teaching, learning, and curricular practices. Common practice in both general and special education classrooms still reflects a narrow emphasis on computation. This focus is also mirrored in diagnostic teaching and evaluation thrusts (Heshusius, 1991). Not only are such perspectives on instruction and assessment incompatible with the vision of the Standards, but they are also contrary to the findings of recent research about mathematics teaching and learning (e.g., Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Englert, Tarrant, & Mariage, 1992; Resnick, 1987, 1989; Thornton & Bley, 1994).

The National Council of Teachers of Mathematics (1989) proposed five goals for rethinking mathematics teaching and learning. The council held that students should learn to value mathematics become confident in their ability to do mathematics become mathematical problem solvers learn to communicate mathematically learn to reason mathematically.

To accomplish these goals, the council advocates, and teachers should decrease their emphasis on complex paper- and-pencil computation, rote memorization of rules and formulas, written practice. While these recommendations for school mathematics are grounded in constructivist theory (e.g., Cobb & Bauersfeld, 1995; Noddings, 1990) and stem from a broad research base in mathematics education (Grouws, 1992), little of this research has focused specifically on mathematics instruction. Such recommensations tend to pull the class focus away from data driven objective goals in favor of the individualized studies. For students with learning disabilities (LD) this can be a positive step. For many of the social arts classes, leaving behind a highly structured environment in favor of individualized project-based learning can facilitate student involvement. However, in the area of mathematics, there is an objective knowledge base of learned materials which must be mastered before the student is ready to progress to the next grade. It may not be of significant consequence to the student's future if a student memorizes all that state capitols and states in the union in a social studies class. It is of great importance if a student arrives in the 8th grade not knowing his times tables, and having not mastered the geometric concepts of diameter, radius, area and perimeter. The student will not be able to continue in the area pf mathematics if these basic materials are not included in their mathematics mastery. Furthermore, few studies involving students with learning disabilities have focused on higher level mathematical thinking and problem solving (Parmar & Cawley this issue; Marshall, 1988; Mastropieri, Scruggs, & Shiah, 1991).

Recent recommendations (e.g., National Council of Teachers of Mathematics, 1989, 1991; National Research Council, 1990) reinforce the need for relevant, problem-driven instruction. A central thesis of these recommendations is that all students should become confident "doers" of mathematics and, consequently, be capable and resourceful problem solvers. This requires that all students have the opportunity to explore many different types of mathematical problems and that they be both expected and encouraged to use a variety of strategies in solving them (National Council of Teachers of Mathematics, 1989).

Although problem solving traditionally has been a difficult area for many students with learning disabilities (Montague & Bos, 1986; Wansart, 1990), Bulgren and Montague (1989) reported that these students can succeed beyond current expectations if they are exposed to developmentally appropriate, meaningful problem tasks that are complemented by appropriate instructional modifications. Moreover, children experiencing difficulties with formal computation or basic fact recall should not be prohibited from engaging in more challenging problem-solving tasks (Cawley & Miller, 1989; Ginsburg, this series). In fact, a substantial body of research highlights the effectiveness of using problem solving as the vehicle for learning mathematics, including basic facts and computation (e.g., Carpenter & Moser, 1984).

This is another connection between the urban children of Mr. Zander's school, and the disabled child. As the students are exposed to the increasingly diverse spectrum of methods to reach the prescribed lessons, they assume ownership, and move into an active learning modality. The students are excited to learn as their own ideas are made part of the lessons.

Similarly when problem tasks are sufficiently complex, rich, and open-ended, they can be explored at different levels of understanding. Stenmark (1991) characterized a "rich" problem in three ways:

a) The problem leads to other problems, b) the problem raises other questions, the problem has many solution approaches.

When students are given ongoing opportunities to engage in rich problem tasks, as in this case, the results can be quite dramatic. This success is consistent with research documenting the fact that students learn what they have an opportunity to practice. Students who have had many opportunities to solve mathematical problems become better at problem solving (e.g., Carpenter et al., 1989; National Council of Teachers of Mathematics, 1989; Silver, 1985).

In another study regarding parents of children with disabilities which exposes striking similarities to the Z. Model found that when their students spent a majority of their time in general education classrooms, parental research yielded five themes:

a) Parents felt ill-prepared to help their children with homework. This may be a significant factor regarding the difficulty that urban parents feel toward their children's education. This is a facet of the Z. Model which warrants more research.

A b) parents wanted more information about the classroom teachers' expectations of their child and of their roles as parents in helping with homework. If a parent is concerned about their child's future, equipping the parent to assist the student in his homework can be a significant factor in getting the parent involved in the educational process.

A parents wanted their children to be given individualized homework assignments. This finding factors more heavily for he disabled student than the general education student. However, if a parent does become involved, and finds that their child is struggling in a subject, enlisting their involvement with personalized remedial work would be an effective means to involve the parent in the process.

A d) parents valued hands-on homework and projects in which the whole family could participate. The Z. Model students produce a wide range of individual project. These are displayed in the classroom as a reminder, and encouragement to the students of their efforts.

Be developing a new vision of parental involvement which is one of partnership, parents will not be viewed as persons themselves who are incapable of helping their children learn, and therefore requiring their own professional help. They will realize that they possess valuable knowledge and legitimate interests. (Gartner & Lipsky, 1989, p. 21)

Improving parental involvement has been a component of school reform in both general and special education over the past two decades. Definitions of involvement have ranged from the minimal legal requirements inaugurated with P.L. 94-142 (now the Individuals with Disabilities Education Act, or IDEA), through the 23 strategies used by the Follow Through program (Olmsted, 1991), to the Comer (1980) process, in which parents are seen as key policymakers. There is one traditional form of parent involvement that remains in place, however: the homework that is delivered almost daily to parents' doorsteps in schoolbags and jacket pockets.

Childre n with learning disabilities have more favorable attitudes regarding homework when assignments are made in the context of a strong support system of teachers, parents, and peers (Frith, 1991). Walberg (1984) found that involving parents in their children's homework can increase the value of homework for the children. Yet, parents are generally not asked how the partnership between home and school can be strengthened, and they often feel confused about how they can best assist their children with homework (Swap, 1993). Cooper (1989) suggested that parental support must be recognized as a mediating variable in teachers' assignment and students' completion of homework. These tendencies are especially evident in the urban school. Many parents consider their role in their children's education is to get them on the bus in the morning. The child leaves one discontinuous setting, and enters another world, the school, for 7 hours. When he leaves the temporary stetting of school, he returns to the real world. When parents are disconnected with the school, this separation is amplified. The struggle of getting students to apply lessons to real life faced in the mathematics classroom is reinforced by this perceived disconnection between home and school.

By encouraging the parents to become and remain involved in the class work, the Z. Model is building toward successful educational progress on many fronts. First, the children receive the personal support form their parent. Their self-image is built in that they are now participating in something that the parents find important enough to personally support. Secondly, the parent is moving toward a helper position. They can assist the student in the home when they are aware of the class work. They are brought across the perceived gulf that separates the school world from the home world. The parents in an urban setting may not have had successful educational careers of their own, and therefore do not know how to 'cross the gulf.' By inviting them into the classroom, the Z. Model equips parents to become involved in their child's education.

Other research which supports the construct of the Z. Model

Research conducted at more than 117 institutions of higher education with the Dunn and Dunn Learning-- Style Model documented that, like fingerprints, each child had his or her own individual learning style. Regardless of the group to which that individual belonged each student had an identifiable learning style. (Research on the Dunn and Dunn Model 2002). Like the different personality types student possess a preprogrammed means by which they are most receptive to receiving new information. Thus, it can be identified as suspicious when purported experts prescribed identical educational strategies for all students with learning deficiencies, or disabilities, regardless of achievement levels, age, gender, and cultural diversity. In such a single minded approach to meeting diverse educational needs, experts were publishing conflicting directives for teaching these children. How can teachers be advised to ensure a quiet, predictable classroom with few transitions while promoting active learning?

Proponents of the Dunn and Dunn learning style model promote different environments and educational approaches for different children based on the students' individual learning style strengths-not on the special category to which they may have been classified. Multiple studies have documented the statistically higher standardized achievement test scores earned by previously failing youngsters when their learning styles finally were addressed (Brunner and Majewski 1990; Caine and Norwood 2000; Dunn and DeBello 1999; Dunn et al. 1995; Perrin 1990).

While not specifically using the Z. Model, this approach to assisting students who are struggling in the traditional educational environment is very similar to the Z. Model. Unique approached, individualized, or adaptive teaching settings aid the struggling student to 'catch' the subjects, and begin to apply them to their own lives.

In another program which seeks to create a multi-impact environment for students, a program to enhance students' academic performance by reducing learning barriers was launched at Marchwiel School yesterday. (Timaru Herald, 2000).

The Marchwiel Project is a holistic approach to education in New Zealand which involves families and support agencies working with the school to teach the children. It aims to create an integrated culture which promotes structure and supports pupils and their families. The program also involved a community garden project so children could learn new skills and experience the physical and psychological benefits of growing plants. It would provide students with opportunities for co-operative learning with classmates and community members.

Individualism, and Accountability

Society is becoming more diverse, and with the diversity, students enter the learning environment with an ever widening range of scholastic needs, and an ever widening set of pre-existing dispositions and abilities toward learning. The highly structured classroom of the 60's and 70's is no longer suited to teach today's wired, internet enabled students. However, in the drive for individualism, and individualized learning styles, the proverbial 'baby can be thrown out with the bathwater' if the classroom becomes a place where a set of objective standards and goals are not taught and met each day, month, semester and year. When a school reinforces positive progress without correcting incorrect, or sub-standard progress, the school moved toward loosing its ability to set any standards. If a student does not have to master a specific knowledge base of information before being passed along to the next grade, then what is the purpose of grades, text books, and tests?

By eliminating a structured accountability posture in the classroom, schools have too often chosen a path which communicated to the students and parents that their child does not have to learn. If the student can have an open book test, why does he have to learn the discipline of memorization, which will be important in later education, and in the job market? If the student is encouraged to write words the way they sound, without teaching them the 34 phonetic building blocks of the English language, soon students will be writing in their own codes, and the integrity of the written communications is lost.

While this may seem like a stretch of educational wisdom, looking back over the past 30 years of declining educational progress demonstrates that as the school system departed from a more rigid accountability for learned materials, the school systems educational performance began to suffer. The process of educational decline took place over a generation and a half, so at any one time the regression was difficult to map. However, as schools struggle under the weight of declining performance, behavior problems, and funding crises, evaluating the building blocks of a successful classroom must look beyond the structure of the lesson, and whether the student is allowed a measure of individualism.

Toward this end, the Z. Model has wisely included a strong emphasis on data driven instruction. Without the measurable progress, the Z. Model would just be another theory. However, over a 16% increase in test scores over a single 6-month period is a success which should be closely examined.

Neither students nor teachers should have to choose between learning or teaching breadth and depth of content. In an electronically enhanced world, students can learn both. Students should not have to choose between memorizing a specific knowledge base, and learning how to learn. In an internet age, students need to know more specific facts and figures than ever, and they need to know how to use learning tools in order to mine information for their further educational advancement. There are specific reasons that for decades, schools taught memorization of mathematics tables in the 2nd and third grades. During this time, the student's brain development is in a specific level of advancement which make is most receptive to memorizing these facts. It is a measurable fact that student who are poor readers by the time they reach the 4th grade are most likely to remain poor readers all their lives. This is because the phonetic building blocks of the English language were bypassed at a time when the child's brain was most receptive to learning them.

In a culture which has spent the last 25 years extolling the virtues of individualism, and unleashing a slow revolution against a social structure at all levels, students have missed learning one of the most important lessons. They must also learn accountability to the teacher and authority figures in their schools in order to learn the larger scope of accountability for their actions throughout their lives. Students must learn responsibility to themselves, for themselves, and to the larger group within the social subgroup of the classroom. Without these character issues, student have no reason to grow up onto successful prosperous adults. If students are catered to, and made to feel that their progress in school is adequate regardless of the level of proficiency, they will learn that self indulgent behavior is the norm at every level.

Buy combining the 4 domains of brain development, family involvement, data-driven instruction performance, and project-based learning, the Z. Model takes all these principles into consideration. The Z. Model ultimately places the responsibility on the teacher to be the agent of change in the classroom. The Z. Model is a highly demanding framework for the teacher. He or she must develop an inner desire to see genuine change in the student's educational performance and in the school environment itself. If this model faces opposition in the classroom as it is initiated in other schools, as it should be, the opposition will likely be from teaching staff who do not want to put the additional time and energy into their classroom.

In review, the four domains, why each is important, and how the z-model affects positive change in each of them are listed below.

Brain development: Learning is a psychological, and physiological process. As a student learns growth occurs of new neural transmitter cells. This process transpires awash in a sea of chemical, and nutritional elements within the child's brain. Like any other human muscle, the brain grows as it is used, and forced to develop new thought processes. In order for the student to learn, he or she must be challenged, and mentally pushed to forge new neural pathways. This process is not automatic, and does not happen just because a student is exposed to new information. The student must be pushed, prodded, expected, and encouraged to think.

By insisting that the student create their own understanding of the mathematic principles, and create their own hands on models, the students are engaged in the process of building the new sets of connected ideas. This is the essence of a learning environment, as demonstrated in the Z. Models success, and in the research included herein. In poor urban schools, there exist significant social obstacles to creating this environment. The lack of ability of many students to generate a focused attention span, and the lack of social ability to operate in a group setting are greater than other schools. But by using the next domain, these obstacles can be effectively overcome.

Project-Based Learning programs utilize a team environment, and aid in creating the learning culture within a school. However, the student team will not lead itself. It still must be guided, and expected to learn, and produce positive results. By breaking the class into teams, and giving each of the teams some measure of autonomy in creating the hands-on tools for the lessons, the Z. Model draws each student into a positive working relationship with the teacher, his or her students, and the class lesson itself. This puts the student in a position of sensing influence over their own destiny, which is an important element of learning, and taking responsibility for moving toward the life of an educated adult.

Data driven instruction creates accountability which every school, teacher, and student needs. For a student to spend 7 hours per day, 5 days per week, and 180 days per year for 12 years in a class room, and not learn the specific knowledge base (s)he will need to succeed in the adult world is worse than a waste time. It is a criminal use of taxpayer support, and a failure to deliver the purchased product to the community. This kind of aborted education also defines a career path of frustration for teachers. The Z-Model recognizes that the students, at the end of the period, day, and year must have accomplished mastery over a set of learning materials. The students must learn how to learn. They must learn social habits within the group which will benefit them in their future years.

The single term application of the Z. Model is an extremely limited measurement of its full impact. The Z. Model could be adapted and applied throughout an entire school, or school system. By doing so, a school can have an objective means of measuring the leveraged positive affect of this learning framework. As students are exposed to this model in early elementary years, they should increase not only their academic scores but also their academic ability to learn in future years. By measuring the progress of a class that moved from 2nd grade through 6th grade under a consistent Z. Model teaching application, the full impact of the z-model will be seen.

Finally, students need to be encouraged by their teachers. They need to feel the positive peer pressure of fellow students who are learning around them in order to achieve. But most of all they need the support, encouragement, and positive expectations of their parents / custodians in order to fully accept the responsibility of learning. By involving the parents in the 4th domain of the z-model, and giving the children positive recognition for parental involvement, this framework encourages both the student, and the parent to make the connection which, in the final measure, will benefit both of them.

Additional Re-search

As discusses earlier, the research done in this educational model, and the positive results attained in the school class room are similar in approach and framework to what us used to assist mildly learning disabled students who have been mainstreamed, but are having trouble catching the educational content. Before progressing, let this researcher repeat that it is not the purpose of this research, or researcher to suggest that middle school children in urban, mostly poor school neighborhoods, should be considered learning disabled as a general category. This would be a pejorative, and baseless assumption.

However, in reviewing the literature which has been produced in this area, in particular the teaching styles and adjustments to curriculum which have been successful with learning disabled students, there exists a similarity between this literature, and the z-model framework described here-in. For example, Addison Stone proposes an enhanced framework for assisting students with mild disabilities. He suggests the metaphor of scaffolding, in which adults draw up alongside students, and build a supporting structure to assist the students with learning disabilities in comprehending the educational materials. Conclusions are drawn regarding the positive results of such instructional interventions based on research that has implemented components of the metaphor.

Stone's analysis provides an interesting description of the metaphor. The descriptive studies provided an illustrative view of the role of parental guidance during parent-child interactions. Whereas early experimental studies provided a description of what Wood, Bruner, and Ross (1976) called "degrees of calibrated assistance," Stone's works suggested the positive results of adults building a "scaffold" around the student to "support" the educational process.