Framework Of Implementing The Z. Mathematical Model To A Sixth Grade Class Thesis

Nature of the ProblemPurpose of the ProjectBackground and Significance of the ProblemBrain Development

Specific Activities to engage students

Data-Driven Instruction

Community Component of Education

Research QuestionsDefinition of TermsMethodology and Procedures

Discussion & ImplicationsConclusions & Application


The goal of present-day educational reformers is to produce students with "higher-order skills" who are able to think independently about the unfamiliar problems they will encounter in the information age, who have become "problem solvers" and have "learned how to learn, " and who are on their way to becoming "critical thinkers" and "lifelong learners." The method advocated for achieving these "higher" order skills" is "discovery learning," by which students solve problems and make decisions on their own through "inquiry" and "independent analysis" of "real-world" projects -- what Kilpatrick in the 1920s called the "project method."

The oft-repeated goal of the educational community -- to inculcate general thinking skills --is not, however, soundly based in research. The idea that school can inculcate abstract, generalized skills for thinking, "accessing," and problem solving, and that these skills can be readily applied to the real world is, bluntly, a mirage. So also is the hope that a thinking skill in one domain can be readily and reliably transferred to other domains.

Yet broad-gauged thinking abilities do exist. Most of us know well- educated people, and ones who have high general competence, can think critically about diverse subjects, can communicate well, can solve a diversity of problems, and are ready to tackle unfamiliar challenges. The belief that our schools should regularly produce such people appeals to both experience and common sense. If the goal didn' t make apparent sense, it could hardly have retained its attractiveness to the educational community and the general public. Rightly understood, then, the goal of general competence does define one important aim of modern education. The task is not to change that goal but to interpret it accurately so that it corresponds to the nature of real-world competency and can actually be achieved.

Two traditions in cognitive psychology are useful for understanding the nature of the critical-thinking, problem solving skills that we wish to develop in our students. One tradition has studied the characteristic differences between expert and novice thinking, sometimes with the practical goal of making novices think more like experts as fast as possible.( Larkin, et al.) Another tradition has investigated the differences between accurate and inaccurate thinking of the everyday newspaper-reading, bargain-hunting sort that all of us must engage in as non-experts.(Tversky and Kahneman) Both sorts of study converge on the conclusion that, once basic underlying skills have been automated, the almost universal feature of reliable higher-order thinking about any subject or problem is the possession of a broad, well-integrated base of background knowledge relevant to the subject. This sounds suspiciously like plain common sense (i.e., accurate everyday thinking), but the findings entail certain illuminating mg complexities and details that are worth contemplating. Moreover, since the findings run counter to the prevailing fact-disparaging slogans of educational reform, it will be strategically useful to sketch briefly what research has disclosed about the knowledge-based character of higher-order thinking.

The argument used by educators to disparage "merely" factual knowledge and to elevate abstract, formal principles of thought consists in the claim that knowledge is changing so rapidly that specific information is outmoded almost as soon as it has been learned. This claim goes back at least as far as Kilpatrick's Foundations of Method (1925). It gains its apparent plausibility from the observation that science and technology have advanced at a great rate in this century, making scientific and technological obsolescence a common feature of modem life. The argument assumes that there is an analogy between technological and intellectual obsolescence. Educators in this tradition shore up that analogy with the further claim that factual knowledge has become a futility because of the ever growing quantity of new facts. The great cascade of information now flowing over the information highway makes it pointless to accumulate odd bits of data. How, after all, do you know which bits are going to endure? It is much more efficient for students to spend time acquiring techniques for organizing, analyzing, and accessing this perpetual Niagara of information.

The best research on this subject shows that neither fact-filled memorization nor large conceptual generalizations are effective modes of education for higher-order thinking about...


On the other hand, it has been shown that accurate factual estimates are necessary for understanding many issues. Norman Brown and Robert Siegler summarize the underlying problem for modern education:
Faced with the issue of how to inculcate such information, educators have oscillated between two approaches. One has been to require students to memorize large numbers of quantitative facts. The other has been to de-emphasize dates, magnitudes, and other quantities, and to focus on understanding of qualitative relations. Each of these approaches has major drawbacks, however. . . . There are just too many such facts for anyone to memorize a high percentage of them. On the other hand, it is difficult if not impossible to acquire more than a superficial understanding of a domain without some degree of quantitative sophistication about it.(Brown and Siegler)

The breadth - vs. - depth issue will always be with us and will always require compromises and common sense. The particular compromise one makes will depend upon subject matter and goals. In practice, an appropriate compromise has been reached by self-taught, well-informed people and by the fortunate students of particularly able teachers. One well- tested teaching method, already followed by many good books and teachers, provides students with a carefully chosen but generous sampling of factual data that are set forth in a meaningful web of inferences and generalizations about the larger domain. Researchers have shown that such generally selective factual instruction leads to accurate inferences not directly deducible from the literal facts that were taught.

Between 1973 and 1979, Brophy and his colleagues conducted a series of studies in which they first determined that some teachers got consistently good results over the years and others consistently bad ones. They made close observations of the teacher behaviors associated, respectively, with good and bad academic outcomes. Teachers who produced the most achievement were focused on academics. They were warm but businesslike.

Teachers who produced the least achievement used a "heavily effective" approach and were more concerned with the child's self-esteem and psychic well-being than with academics. They emphasized warmth, used student ideas, employed a democratic style, and encouraged student-student interaction. The researchers further found that learning proceeded best when the material was somewhat new and challenging, but could also be assimilated relatively easily into what students already knew. The biggest contrast was not between modes of academic instruction but between all such instruction and "learner-centered" "discovery learning," which was ineffective. Paradoxically, the students were more motivated and engaged by academic- centered instruction than by student-centered instruction.

In 1982, Brophy and his colleagues summarized some of their findings on the effective teaching of beginning reading. (Brophy and Evertson) These were the most salient points:

1. Sustained focus on content.

2. All students involved (whole-class instruction dominates).

3. Brisk pace, with easy enough tasks for consistent student success.

4. Students reading aloud often and getting consistent feedback.

5. Decoding skills mastered to the point of over-learning (automaticity).

6. In the course of time, each child asked to perform and getting immediate, nonjudgmental feedback.

While the argument rages within the academic community regarding the higher priority of depth or breadth of learning required in order to attain consistently satisfactory educational outcomes, children are progressing through the educational centers throughout the country who need to have their academic needs met. Each child must be educated to a level to function proficiently in society, or in higher education pursuits by the time they leave secondary education. Each child should be educated to a level that reflects his or her intellectual, social, and psychological cognitive ability. This means the student with exceptional abilities should be educated to the extent of his or her giftedness. Students whose special needs must be educated to a level at which (s)he can reach his or her academic and vocational abilities. Anything less is an admission that the educational system is not living up to the responsibility which our society has placed in its hands.

Toward that end, and in a step away from the stalled breadth vs. depth debate, this paper proposes a new model to use as a framework for teaching middle school students. This research focuses on a particularly problematic sector of students, inner-city, low performing schools, and students. This unique framework has 4 domains, and while this study investigates the Z-model's use in the mathematics curriculum, this framework is focused on improving and enhancing students' achievement in any course of study.

The four domains involved in…

Sources Used in Documents:


Family involvement in children's eductaion. (1997). Retrieved March 23, 2003. from Federal brochure. Web Site:

Henderson, Anne T. And Nancy Berla. (1994). National Committee for Citizens in Education.

California State Board Policy. (1994). Parents infolvement in the education of their children. 89-01. Web Site: / policy89_01.htm
Institute Online. Web Site:

Cite this Document:

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