MATHEMATICS AND THE SCHOOL CURRICULUM
Introduction
According to the Times-Picayune (Oct. 12, 1996), the 1994 Trends in Academic Progress report released by the Department of Education states that US students have continued a decade long trend of gradually improved performances in math and science. The gap between white and minority students’ performances is gradually narrowing, particularly in science, and more students are taking advanced math courses. In general, the numbers show performance in math and science returning to, or even surpassing, the levels of a quarter-century ago. The report, covering test results in science, mathematics and reading since early 1970s and writing since 1984, is considered one of the best indicators of long-term student achievement. The latest report covers tests given in 1994.
This is positive news in light of the negative statistics about mathematics education that have been the norm over the last several decades. We can only hope that the announcement by the US Department of Energy in May 1990, that it would launch America in "‘a new Renaissance period’ in science and mathematics" (Rothman, 1990), is finally showing the results expected.
HISTORICAL OVERVIEW OF MATHEMATICS
Early Beginnings
According to Aristotle, mathematics originated because the priestly class in Egypt had the leisure pleasure for its study. Over two thousand years, later exact corroboration of this remark was forthcoming through the discovery of a papyrus, now treasured in the Rhind collection at the British Museum. This ancient Egyptian papyrus from the collection of Rhind, was written by an Egyptian priest named Ahmes, considerably more than a thousand years before Christ. It contains a fairly complete applied mathematics, in which the measurements of figures and solids plays the principal role. Everything is stated in the form of problems, not in general terms, but in distinct numbers. Recent investigations of the Rhind Papyrus, the Moscow Papyrus of the Twelfth Egyptian Dynasty, and the Strassburg Cuneiform texts have greatly added to the prestige of Egyptian mathematics (Newman, 1956).
However, the history of mathematics cannot with certainty, be traced back to any one school or period, before that of the Ionian Greeks. Nevertheless, there was already a wealth of geometrical and arithmetical results treasured by the priests of Egypt before the early Greek travelers became acquainted with mathematics, through their visits to Egypt. Hence it would seem that only after the keen imaginative eyes of the Greek fell upon the ancient temples and pyramids of Egypt that they yielded up their wonderful secrets and discovered their inner nature.
Among those early Greek travelers to Egypt was Thales of Miletus, who lived from about 640 to 550 BC. He was one Greek who seized on what he had learnt in his travels, and became known to his fellow-countrymen of later generations as one of the Seven Sages of Greece. From his intercourse with the priests of Egypt he was the first to bring out something of the true significance of Egyptian scientific lore and became a popular celebrity with his successful prediction of a solar eclipse in 585 BC. It has been remarked that the geometry of Thales is the true source of algebra. Propositions ascribed to him include the following: that a circle is bisected by any diameter; that the angles at the base of an isosceles triangle are equal; and that the sides about equal angles in similar triangles are proportional. Thales never forgot the debt that he owed to the priests of Egypt, and when he was an old man, advised his pupil Pythagoras to pay them a visit. Pythagoras traveled to Egypt, gained a wide experience and eventually became even more famous than his master.
An important stage in the history of mathematics occupied the fifth and fourth centuries BC, and is associated with the Greek city, Athens. This city was the political, commercial and intellectual center of the Grecian world, and contained many great philosophers who were remarkable mathematicians and astronomers. Perhaps the greatest among them were Hippocrates, Plato, Eudoxus and Menaechmus. Each in his own sight laid some major foundation of geometry and arithmetic. Hippocrates made several notable advances. His chief result it the proof of the statement that circles are to one another in the ratio of the squares on their diameters. Today, Hippocrates' actual work survives among the theorems of Euclid, although his original book is lost. Plato was an original investigator who founded and conducted in Athens his famous Academy. Through his many books of mathematics, in which he filled with mathematical illustrations, it is said that he caused great advance in mathematics by kindling admiration for the different subjects of mathematics. It was also during this great period that the Greeks systematized their written notation by using the letters of the alphabet to denote definite numbers (a = 1, b = 2, c = 3 and so forth).
Toward the end of the fourth century BC, the scene of mathematical learning shifted from Europe to the city of Alexandria, along the Nile River, in Africa. The greatest mathematicians to emerge at this time included Euclid, Archimedes, Apollonius, Pappus and Diophantus. Euclid wrote his well-known and exhaustive account of mathematics, the Elements , a collection of thirteen books that helped to form the mathematical minds of all his successors. In particular, successors such as Archimedes, one of the greatest of all mathematicians, brought imaginative skill and insight to bear upon metrical geometry and mechanics, and even invented the integral calculus. Apollonius, another successor and one of the greatest of geometers, perfected the geometry of conic sections. Pappus wrote a great commentary called the Collection; and happily many of his books are preserved today. They form a valuable link with the lost work of Euclid and Apollonius. The work of Diophantus had a two-fold importance: he made an essential improvement in mathematical notation, while at the same time he added large installments to the scope of algebra as it then existed. The full significance of his services to mathematics only became evident with the rise of the early French school in the fifteenth and sixteenth centuries.
After the death of Pappus, Greek mathematics and indeed European mathematics lay dormant for about a thousand years. The history of the science passed almost entirely to new custodians in the Eastern lands of India and Arabia. Within this period, the decimal notation was invented by an unknown Indian genius. He was followed by the Indians ARYABHATA and BRAHMAGUPTA who made substantial progress in algebra and trigonometry. The Arabs were apt scholars who became industrious translators into Arabic of the valuable old manuscripts that their forerunners had not destroyed. The very word Algebra, is part of an Arabic phrase for ‘the science of reduction and cancellation’, and the digits we habitually use, are often called the Arabic notation (Jourdain, 1956). For long centuries they were the safe custodians of mathematical science.
Sixteen Century Mathematics and Beyond
With the sixteen century came the great intellectual and spiritual movements of the Renaissance and the Reformation in the Western world. Mathematical investigations began with renewed strength, greatly stimulated by the recent invention of the printing press. Italy led the way; France, Scotland, Germany and England were soon to follow. Scipio Ferro (1465-1526) picked up the threads where Diophantus left them and discovered a solution to the cubic equation. x3 + mx = n; and, as this solved a problem that had baffled the Greeks, it was a remarkable achievement.
Galiloe of Pisa invented dynamics and showed the importance of experimental evidence as an essential prelude to a theoretical account of moving objects. Copernicus of Germany revolutionized astronomy by postulating that the Earth and all the planets revolve around the Sun as center. Tonstall of England, published the first book on Arithmetic. Perhaps the most remarkable of all these eminent mathematicians was John Napier, of Scotland, who discovered the logarithm. This achievement broke entirely new ground, and it had great consequences, both practical and theoretical. It gave not only a wonderful labor-saving device for arithmetical computation, but it also suggested several leading principles in higher analysis.
With the beginning of the seventeenth century, the number of mathematicians increased so rapidly that it is quite impossible to do justice to all in this short survey. Three names can be singled out as representatives of their times. Descartes and Pascal of France, and Newton of England. Descartes founded analytical geometry, and changed the face of mathematics. It gave geometry a universality hitherto unattained, and it consolidated a position which made differential calculus the inevitable discovery of Newton and Leibniz. Pascal and Fermat laid the basis for probability theory, and Newton, followed a few years later by Leibniz, discovered differential and integral calculus.
The story of mathematics during the eighteenth century was centered upon the work of Leonard Euler. His delight was to speculate in the realms of pure intellect. He did great work in the problems of physics only because their mathematical pattern caught and retained his attention. There is one formula that can be quoted as an epitome of what Euler achieved:
eip + 1 = 0 .
Every symbol has its history here -- the principal whole numbers 0 and 1; the chief mathematical relations + and = ; p the discovery of Hippocrates; i the sign for the 'impossible' square root of minus one; and e the base of Napierian logarithms.
The nineteenth century was perhaps the most brilliant era in the long history of mathematics. The subject assumed a grandeur in which all that was great in Greek mathematics was fully recovered. Geometry once again came into its own, analysis further broadened its scope, and the outlets for its applications were ever enlarging. It was during this time that the pre-eminent mathematician Carl Gauss (1777 - 1855) introduced his work in the theory of numbers. Ever since the time of Gauss, mathematics has increased so extensively that no individual could hope to master the whole. Gauss was the last complete mathematician, and of him it can truly be said that he adorned every branch in the science. Gauss made an early reputation by his work in the theory of numbers, and then out of an interest in geometry grew a branch of geometry called hyperbolic geometry. Many of his later aspirations were fulfilled by his celebrated pupils including Riemann, whose thesis on geometry was the necessary prelude to that of Einstein.
The nineteenth century also brought about the discovery of the quaternions in algebra by William Hamilton of Ireland. Other prominent English mathematicians including Boole, Cayley and Sylvester also made important contributions to the new algebra inaugurated by Hamilton. As a result of Hamilton's creation of quaternions, leading mathematicians David Hilbert, Albert North Whitehead, and Bertrand Russell set out to reconstruct mathematics as a strictly formal system at the beginning of the twentieth century. But in the 1920s, mathematician Kurt Godel demonstrated that Hilbert's hope of devising a decision algorithm for all of mathematics and the Whitehead-Russell project of deducing all of mathematics from the axioms of logic were unrealizable by proving that any sufficiently complex mathematical system is necessarily incomplete (Davis, 1996).
Then in 1976, a group of mathematicians proved the century-old four-color conjecture by combining graph theory and sophisticated computing. Next, the dynamic and increasingly popular new field of nonlinear dynamics (Chaos Theory), sparked by an unexpected result from a computer-simulated weather system, opened the possibility of using computers as an investigative tool in a more "experimental" and tentative approach to mathematical research. These sorts of events helped to displace the formal proof as the primary concern of mathematical inquiry and ushered in new approaches to mathematical investigations and the ways one might go about proving particular ideas. All of this became possible with the tremendous recent advances in computer technology, coupled with the increased availability of such technology. Lynn Arthur Steen (1990) sums up recent developments in mathematics as follows:
Not since the time of Newton has mathematics changed as much as it has in recent years. Motivated in large part by the introduction of computers, the nature and practice of mathematics have been fundamentally transformed by new concepts, tools, applications, and methods. Like the telescope of Galileo's era that enabled the Newtonian revolution, today's computer challenges traditional views and forces re-examination of deeply held values. As it did three centuries ago in the transition from Euclidean proofs to Newtonian analysis, mathematics is undergoing a fundamental reorientation of procedural paradigms.
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Historical Development in Curriculum and Instruction
Drill and practice was the primary focus of mathematics education during the first 30 years of the twentieth century. Edward Thorndike (1922) was the main proponent of this approach, with his theory variously termed, "connectionism," "associationism," and "S-R bond theory." He maintained that, by means of conditioning, specific responses are linked with specific stimuli, He believed that "almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come".
But despite his strong emphasis on drill and practice, Thorndike nevertheless highlighted the importance of making arithmetic problems enjoyable and interesting for children and relevant to their everyday experiences. In this way, children would be more likely to make strong connections and be better able to make the appropriate connections at the appropriate times.
In the 1930s and 1940s, there was a shift away from meaningless drill methods to a focus on developing mathematical concepts in a meaningful way. William Brownell ( 1935, 1945) was a major proponent of the view that "meaning is to be sought in the structure, the organization, the inner relationship of the subject itself". The aim was to impart to students the structure of arithmetic, that is, the "ideas, principles, and processes" of mathematics. For example, a child who promptly gave the answer "12" to the fact "7 + 5" would not be regarded as having demonstrated a knowledge of the combination unless she understood why 7 plus 5 equals 12 and could convince others of its correctness. This would require the child to have an understanding of the mathematical principles and patterns underlying computations.
CURRENT TRENDS AND ISSUES
The Period of the New Math
The 1960s witnessed major changes to the mathematics curriculum. These changes were the result of severe criticism of the American education system in the postwar years. At a conference sponsored by the School Mathematics Study Group (SMSG) in 1959, the importance of elementary mathematics to secondary school success was recognized and an agreement was reached to revise the elementary program so that it would be supportive of the expected outcomes of the secondary program. It was considered essential that students possess a knowledge of the fundamental structures of mathematics. This knowledge would enable students to reconstruct mathematical facts should they forget them. Instead of practice in the basic skills, the New Math presented set theory including the union, complements, and intersection of sets. Different number bases and how they functioned was another important topic. The explicit teaching of the notation and algebra of sets and the general laws of arithmetic were, by their very nature, abstract.
Hence because the new curriculum was exceedingly abstract, (a major departure from the traditional one), and difficult to understand, the tension between mathematics as a commercial, practical study and mathematics as a theoretical discipline reached a fever pitch from the public and teachers alike. Students did not reach the conceptual level of understanding hoped for, standardized tests had generally not adjusted to the new curriculum, and test scores continued to decline. Criticism of the New Math was widespread. By 1974, enthusiasm for the New Math had declined to the point where schools were gradually returning to the mathematics curriculum of the first part of the century, and eventually the experimentation with the New Math system ended.
The NCTM Standards
The Curriculum and Evaluation Standards for School Mathematics. published by the National Council of Teachers of Mathematics, (1989), highlighted the importance of children being actively involved in their learning, that is, they should "construct, modify, and integrate ideas by interacting with the physical world, materials, and other children". Five general curriculum goals were established for all students at all grade levels, as follows:
The Standards called for major changes in both the content of school mathematics and in the form of mathematics instruction that, in turn, reflected changes in the underlying view of mathematics learning. For example, the Standards argued for a deemphasis on complex and tedious written calculations and on rote memorization of rules and procedures. It advocated an increased focus on conceptual development of the operations and on forming connections between ideas and procedures among different mathematical topics.
If the standards were adopted wholeheartedly, one would assume that our instructional strategies would change radically from our current practices. This would enable students to become engaged actively in a variety of problem situations, using, when necessary, both calculators and computers. Paper and pencil calculations would occur infrequently while mental arithmetic, estimation, and approximation would be frequent occurrences. Students would be transformed from passive spectators to creative problem solvers. (Longstreet & Shane, 1993).
SCENARIOS FOR THE TWENTY-FIRST CENTURY
The Technological Society
One of the main arguments for reform in our current mathematics education programs is that advances in technology and information systems have altered the needs of the future worker. The world of the twenty-first century will be a technologically advanced world, and the workplace will require more education than ever before. Modern careers will require skills that are more technologically complex, and also more interactive. Successful workers in the modern world will have to possess both an understanding of electronic technology and the ability to work cooperatively with others to solve problems of a highly intricate nature. However, school systems do not appear to be keeping up with these changes.
Leaders in business and industry claim that future employees require more than the traditional basic arithmetical skills that were once adequate for jobs requiring repetitive and routine tasks. A higher level of mathematical knowledge and skills is now needed for daily living and for effective citizenship. We need to apply these skills in analyzing and interpreting the many mathematical concepts that permeate the mass media and reside in a variety of data bases. We also need mathematical skills in making the numerous business and financial decisions that face us daily.
New goals for our society now stress the need for mathematically literate workers who are technologically competent, and can adapt to change, can formulate and solve a variety of common and complex problems, can see the applicability of mathematical ideas to these problems, and have the skills for effective life-long learning.(National Council of Teachers of Mathematics, 1989).
The caretakers of the technologically advanced world of the twenty-first century are the children of today. How well are they being prepared for the take-over of our world? The children of today are growing up with remote controls, and spending more time watching television and video tapes than reading. Toys are now filled with buttons and blinking lights. They talk and listen, and interact with children, responding to them in ways the stuffed animals and toy soldiers of the past did not. Atari and Nintendo have brought electronic entertainment right into the living room, making interactive technology as common in the home as television had been in the past. The children of today are now being raised in a world of instant access to knowledge, a world where vivid images embody and supplement information formerly presented solely through text. In today’s environment, the children are able to control information flow and access, whether through a video game controller, remote control, mouse, or touch-tone phone.
The Future of Content
In most schools, the mathematics curriculum through the third grade is relatively homogeneous. Students are not tracked for mathematics, and they receive the same overall curriculum. The quick, interested student and slower, bored student are either in the same class or are in equivalent classes. After the third grade the situation changes.
In the fourth and fifth grades both multidigit multiplication and long division have traditionally been introduced, while in fifth and sixth grades operations with fractions are introduced. These three sets of procedures take up a lot of time in the traditional curriculum. The conceptual underpinnings of multiplicative structures are often poorly taught and poorly understood by the teacher. This portion of the curriculum then becomes overwhelmingly procedurally driven. Students who do the "right" procedures quickly are considered good, while those that make careless errors or cannot follow the procedures are considered and come to believe that they are not very gifted mathematically.
In the Second International Mathematics Study (Robitaille & Garden, 1989; Travers & Westbury, 1990), 12- and 13-year-olds were assessed on the topics of arithmetic, algebra, geometry, statistics, and measurement. Students across the nations found the test items rather difficult, with the Japanese students attaining the highest scores on all five areas. Although students’ performance on items involving simple computation with whole numbers was satisfactory, their performance on items calling for higher level thinking skills was generally poor. So too, were students’ responses to the rational number examples and to the basic computational items dealing with percent, ratio, and proportion. Analyses of these assessment data have indicated that the quantity and quality of the content covered by the teacher correlates positively with achievement, as do the depth of coverage of the subject and the amount of time allocated to it. Countries that have a more rapid pace of instruction, especially in the lower grades (e.g., Japan and China), tend to have higher achievement in mathematics.
Thus it would seem appropriate to reform the curriculum content in light of these assessment data. The recommendations from the National Research Council (1990) is a prominent U.S. document presenting a philosophy for teaching mathematics and a framework for curriculum design. Its recommendations include a greater breadth of mathematical sciences, an increased use of technology, more active learning, and an increased emphasis on higher order thinking skills.
A CURRICULUM DESIGN FOR THE TWENTY-FIRST CENTURY
A Philosophy for Educational Reform
Although the schools are embedded in our culture and reflect its values, the technological changes that have swept through our society have left the educational system largely unchanged and the core of the school’s curriculum very much what it was at the end of World War II (Longstreet & Shane, 1993). In the course of 20 years, a dramatic rift has opened between the process of teaching and learning in the schools and the ways of obtaining knowledge in society at large. This rift has been made obvious by the fact that the process of teaching has not changed substantially, even in the past 100 years (David, 1990; Kolderie, 1990). The result of this rift has been an estrangement of the schools from society, and from the children who live in it.
How do we educate the "new child," raised in a world of instant information, where interactive technologies have led them to believe they can act on the world with the press of a button? What is needed is a guiding philosophy that suggests principled changes in the curriculum and effective uses of technology as part of these changes. I believe that this philosophy can be a derived under the general rubric of a child-in-society curriculum design as reflected in progressivism.
A brief overview of the progressivist views reveal its utility. Progressivism conceives of important knowledge as arising from the experiences of individuals, who are, by the very nature of humanness, members of society. The child is viewed as a source of important knowledge and the curriculum design reflecting the thoughts and decisions of each child. There is an underlying belief in each individual’s ability to deal with the great questions of the world and to be an active participant in solving the problems confronting society. A theory of cognitive growth and learning, constructivism, will underly this philosophical view of progressivism. A foundational premise of constructivism is that children actively construct their knowledge. In the process of actively constructing their knowledge, it is assumed that children will develop complexity and power in their ideas, and with the appropriate support and experience, develop critical insight into how they think and what they know about the world as their understanding increases in depth and detail. Thus the educational applications for these children will lie in creating curricula that match their understanding while fostering further growth and development of their minds.
Schools will have to start with interests and problems close to children and gradually guide them toward achieving greater "instrumental" control of abstract knowledge. Schooling will have to become experience-centered, involving "the whole child," his body and mind, his feelings and emotions. Actual experimentation that provide the children with direct, concrete feedback about the accuracy of their ideas as they work will have to be implemented. These experimentations will have to be self-structured and self-motivated processes of learning that encourage the children to reflect on their idea in ways generally not promoted by the current school curricula. Students will have to be involved in community-based activities and be allowed to reflect on their feelings, thoughts, motives, beliefs and behavior as they interact with each other in the classroom, with a values-driven direction toward the assumption an essentially unknown adult life later on in the future.
The Role of the Teacher
The educational practices that follow a child-in-society curriculum will have to be designed to facilitate the children’s learning for the future by nurturing their own, active cognitive abilities. In traditional classrooms, the teacher’s role has always been that of the sole giver of knowledge and the student, that of the passive recipient of knowledge. There will have to be a change in this relationship to provide for a resource-rich, activity-based curriculum for learning. The teacher will have to serve as a guide, rather than the source, of knowledge. The teacher will have to engage the children by helping to organize and assist them as they take the initiative in their own self-directed explorations and experiences, instead of directing their learning autocratically. Flexibility will have to be the most important feature of the new role the teacher will play in such an environment. There will be times when the teacher will have to tend toward the old model of teacher as giver of knowledge because at that particular time, students will require guidance and training in a particular task or content area. More often than not, the teacher will be obliged to move around the classroom, among groups of children, assisting individual children or the group as a whole.
Designing the New Curriculum
There are many obstacles to the design of a curriculum that tends to reconceptualize the educational practices that are deeply embedded in our cultural mindsets. Technology in and of itself will not be the focus of the future changes that are needed in American education. Simply ascribing a futures orientation to the curriculum because of a technological cultural revolution will not cause the basic disagreements about the purposes of schooling to disappear or somehow become merged in one grandiose vision of the future. As Longstreet & Shane (1993) noted "We have had a great deal of tinkering, a massive increase in testing programs, and an essentially unchanged curricular design. Curricular inertia would appear to be the dominant theme." (p. 197)
This proposed curriculum pursues the development of curricula based on enabling the young of today to deal with swiftly changing futures and the uncertainty and complexity of a society caught in an ongoing explosion of knowledge. It is a modest reform of a small segment of the school’s curriculum, leaving the overall design intact. The current curriculum design dominating our school curriculum is based on a set of core subjects, including Mathematics, Social Studies, and Science. These would remain unchanged. However, the issues and processes that make use of the knowledge from these core subjects in preparing for future challenges will be emphasized. Revisions will involve making the study of statistics a required part of the mathematics curriculum because statistics are used in many aspects of intellectual endeavor from the social sciences to quantum mechanics. Mathematical modeling will be integral to every math class as a way of allowing the students to experience the actual experimentation of their ideas as they work. Computers will be necessary in the classrooms for the statistical analysis of large quantities of data and for the visual and multiple representation of information.
Additional curriculum topics will include Communication and Information Handling where students will develop skills needed in working with and manipulating the range of modern communication devices from computers to television and radio. Skills will include reading literacy as well as computer and video literacy. This is in view of the fact that a vast amount of the information we acquire today is televised into out homes and since this is a trend likely to continue in an accelerating fashion well into the future, the development of literacy in televised communications will become extremely important.
In the Sciences, a cross-impact approach will be designed in the field of the Social Studies of Biology. The socio-ethical implications of advances in biology will be explored through scenarios of likely futures. Genetics might substitute for some of the broader studies of biology given the extraordinary developments in this field that promise to continue into the twenty-first century.
Technology will become an integral component of the curriculum by being a powerful tool for the children’s learning by doing. Traditional classroom tools – pencils, notebooks, and texts – will still be vital, but computers, video and other technologies will be used by the children to assemble and modify ideas, and to access and study information. The technology will be used to make possible the instant exchange of information between classrooms as well as individual students; to allow instant access to databases and online information services, and to provide multimedia technical resources such as interactive audio and video.
The study of Values Development and Democratic Citizenship will be added to the curriculum as a way to involve the students in the active formation of their own values as individuals and as citizens of a democracy. Students would actively engage in establishing their personal system of values and a societal based system of values that would take into account the basic precepts of democracy.
| Area of Study | Current Disciplines | Revisions |
| Mathematics | Geometry
Algebra Trigonometry Calculus, etc |
Add Statistics (possible substitute for geometry)
Mathematical Modeling |
| Science | Biology
Physics Chemistry, etc. |
Add Genetics (possible substitute for some
aspects of general biology |
| Communication
And Information Handling |
Video Literacy
Reading Literacy Computer Literacy Composition, etc. |
|
| Social Studies | History
Geography Sociology Political Science, etc |
Cross impact with biology |
| Values Development
And Democratic Citizenship |
Personal and Societal values
Governance Structures Activities of Citizens, etc. |
Evaluation
Within the new curriculum, new forms of assessment and performance measurement used to evaluate the effectiveness of the new forms of classroom learning will have to be developed. Education will become a collective effort between children, and educational methods will come to emphasize the actual process of children’s construction of new ideas. One type of assessment of the learning taking place will be the videotape recordings of student interactions as they work to reveal their grasp of the course material as well as their ability to communicate it to others. Evaluations of a portfolio that shows the evolution of a child’s work as it is created, rather than of a single completed work or a set of isolated exercises will be another form of assessment. Teams of expert teachers selected from the different schools in the district or around the state will conduct the evaluations. These alternative assessment strategies will be used to capture the qualitative different and much more detailed information about the children’s actual competence. Close-ended standardized tests will not be used for evaluation, but other assessments techniques will include the evaluation of group work, journals and student projects.
Summary
Technology has effectively revolutionized the American society. An unexpected byproduct of this revolution has been the emergence of a generation of children weaned on multidimensional, interactive media sources, a generation whose understanding and expectations of the world differ profoundly from that of the generations preceding them. However the limitations inherent in our current educational system encourage an inflexible futures-based curriculum design. This design attempts to make some changes to meet the needs of our children who will guide our futures, but the real changes will only come when we completely revise our educational practices in light of how our culture has changed.
REFERENCE