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The computers we know today are electronic devices that can receive a set of instructions, or program, and then carry out this program by performing calculations on numerical data or by manipulating other forms of information (Dell, 1992).

The modern world of high technology could not have come about except for the development of the computer. Different types and sizes of computers find uses throughout society in the storage and handling of data, from secret governmental files to banking transactions to private household accounts. Computers have opened up a new era in manufacturing through the techniques of automation, and they have enhanced modern communication systems. They are essential tools in almost every field of research and applied technology, from constructing models of the universe to producing tomorrow’s weather reports, and their use has in itself opened up new areas of conjecture. Database services and computer networks make available a great variety of information sources. The same advanced techniques also make possible invasions of personal and business privacy. Computer crime has become one of the many risks that are part of the price of modern technology (Kohn, 1999).

            Seemingly the most important contribution to the humanity of the advance that is the computer revolves around its role in education.

            In education, an assumption of how children learn called social constructivism has gained recognition around the globe. 

            This paper deals with the subject of using computers as tools in teaching mathematics in a social constructivist way of learning. The discussion of the topic starts with a succinct description of the nature of social constructivism and the role computers play in its preponderance. A discussion of the theories of learning will then proceed. Social constructivism in mathematics will then be characterized, so do the relationship of computers with mathematics. The paper ends with integration on the relation between the theories of learning and social constructivism, computers and mathematics.

Social Constructivism: An Overview

            A cognitive psychologist by the name of Lev Vygotsky shared many of Piaget’s assumptions about how children learn, but he placed more emphasis on the social context of learning. Piaget's cognitive theories have been used as the foundation for discovery learning models in which the teacher plays a limited role. In Vygotsky's theories both teachers and older or more experienced children play very important roles in learning.

In contrast to the individual-cognitive constructivist, the socio-cultural constructivist locates the mind in the individual-in-social action. Learning, then, is primarily a process of enculturation into a community of practice.

Vygotsky's constructivist theory, which is often called social constructivism, has much more room for an active, involved teacher than cognitive constructivism, although the two have a great deal of overlap. For Vygotsky the culture gives the child the cognitive tools needed for development. The type and quality of those tools determines, to a much greater extent than they do in Piaget's theory, the pattern and rate of development. Adults such as parents and teachers are conduits for the tools of the culture, including language. The tools the culture provides a child include cultural history, social context, and language. Today they also include electronic forms of information access.

Although Vygotsky died at the age of 38 in 1934, most of his publications did not appear in English until after 1960. There are, however, a growing number of applications of social constructivism in the area of educational technology.

We call Vygotsky's brand of constructivism social constructivism because he emphasized the critical importance of culture and the importance of the social context for cognitive development. Vygotsky's “the zone of proximal development” is probably his best-known concept. It argues that students can, with help from adults or children who are more advanced, master concepts and ideas that they cannot understand on their own.

General Implications of Social Constructivism

If Vygotsky is correct and children develop in social or group settings, the use of technology to connect rather than separate students from one another would be very appropriate use.

A constructivist teacher creates a context for learning in which students can become engaged in interesting activities that encourages and facilitates learning. The teacher does not simply stand by, however, and watch children explore and discover. Instead, the teacher may often guide students as they approach problems, may encourage them to work in groups to think about issues and questions, and support them with encouragement and advice as they tackle problems, adventures, and challenges that are rooted in real life situations that are both interesting to the students and satisfying in terms of the result of their work. Teachers thus facilitate cognitive growth and learning as do peers and other members of the child's community.

All classrooms in which instructional strategies compatible with Vygotsky's social constructivist approach are used don't necessarily look alike. The activities and the format can vary considerably. However, four principles are applied in any Vygotskian classroom:

1. Learning and development is a social, collaborative activity.
2. The Zone of Proximal Development can serve as a guide for curricular

and lesson planning.
3. School learning should occur in a meaningful context and not be separated from learning and knowledge children develop in the "real world.".
4. Out-of-school experiences should be related to the child's school experience.

Computers as Teaching Tools

Teaching Machines or tools are mechanical devices employed, especially in the United States, to present systematically a programmed sequence of instruction to a student.

The first teaching machines were designed by the American

psychologist Sidney Leavitt Pressey in the 1920s to provide instantaneous feedback for multiple-choice tests. Immediate correction of errors served a teaching function, enabling students to practice test items until their answers were correct.

Computers as teaching machines offer much greater potential than early

linear teaching machines that could not judge the student's response nor, indeed, even determine that the student had responded; they simply presented the correct answer on demand, providing a chance for the student to inspect that feedback before proceeding or Branching teaching machines which uses multiple-choice questions, sent students to different next frames, providing either remedial information and a chance to try again, or confirmation of success and the next step in the sequence.

Computers can be programmed to judge student input and to tailor lessons to each individual's level of mastery. In a tutorial mode, computers can present instructional input and require mastery of each step in ways that were not possible in the early machines. The sensitivity of the instructional designer to alternative patterns of student learning is the necessary key to full use of this advance in machine capacity. Simulation—using the machine to model a real situation—enables even greater sophistication, allowing realistic reactions to student input. Well-designed intellectual games can provide patient environments in which to practice important problem-solving skills.

Many teachers, however, regard this type of “programmed” education as extremely limited in scope, although perhaps valuable for specific skills such as basic numeracy, where some drill and practice have been found to improve skills and the confidence of the individual. Integrated Learning Systems (ILSs) are the modern equivalents of the earlier teaching machines, but some systems now provide considerable flexibility and allow teachers to produce tailor-made programmes of study for the individual student. An advantage of ILSs is seen to be the fact that they enable each student to work at his or her own pace.

Social Constructivism focused on Computers

Technology such as computers provides essential tools with which to accomplish the goals of a social constructivist classroom. Below are a few examples of the way information technology can support social constructivist teaching and learning:

  • Telecommunications tools such as e-mail and the Internet provide a means for dialogue, discussion, and debate -- interactivity that leads to the social construction of meaning. Students can talk with other students, teachers, and professionals in communities far from their classroom. Telecommunications tools can also provide students access to many different types of information resources that help them understand both their culture and the culture of others.
  • Networked writing programs provide a unique platform for collaborative writing. Students can write for real audiences who respond instantly and who participate in a collective writing activity.
  • Simulations can make learning meaningful by situating something to be learned in the context of a "real world" activity such as running a nuclear power plant, writing up "breaking" stories for a newspaper, or dealing with the pollution problems of local waterways. 

The development of educational pedagogy has interesting parallels with the development of personal computer technology. Centralized and autocratic, mainframe technology (and, in the public schools, similarly designed Integrated Learning Systems) distributed a CAI (computer-assisted instruction) approach to education which was strictly content-based and driven by behavioral objectives. With the onslaught of personal computers came the popularity of constructivist approaches to educational technology, where open-ended environments provided individual students with tools to experiment and build their own learning constructs. In the last few years, as the internet and World Wide Web have matured; the social aspects of learning as described by Vygotsky have become useful for those looking to design educational projects involving a distributed but intercommunicating audience.

Theories of Learning

            Learning is the change or modification of the knowledge or skill as a result of experience. The changed behavior may be overt, or observable, as when a boy swims for the first time after a number of training sessions in a pool. On the other hand, the changed behavior may be convert, or hidden, as when a person changes his mind after seeing a baseball game and decides that he prefers one team from another ( Ellis, 1995). 

            Human beings are heavily dependent on learning. Such simple animals as the earthworms behave reflexively or instinctively in any given situation. They have probably behaved in this way for millions of years. Man, however, is able to modify his behavior through learning. From his learning he is often able to change his environment, and this in turn causes further learning to occur. For instance, the modern businessman who flies in  a jet plane starting one city to another or who speaks long-distance via satellite to a colleague on another continent is behaving quite from the business man 500 years ago, or even from the businessman 25 years ago. The differences are the result of accumulated learning (Hilgard, 1994).

            A person is not always aware that he is learning. A little girt putting her doll to bed is unaware that se learned this behavior, probably by seeing her mother act as this way toward a little sister or brother.

            Among the most important learning are attitudes and beliefs. Very often an individual learns these without being aware of it.  A person who dislikes mathematics, for example, may not be aware that his attitudes was caused by the faculty way he was taught in arithmetic school. This unconsciously learned attitude may, in the long run, seriously affect the individual’s progress.

            Learning is a process that occurs over a period of time. It is complex and continuous, and any analysis of it is likely to give an oversimplified picture of the large number of factors involved. If for some reason some part of the process is missing or faulty, then efficient learning is unlikely to take place. There are a number of ways  that learning can be analyzed or divided so that it can be studied more closely (Travers, 1993).

            One method breaks the process down to four elements: drive, cue, response and reinforcement. Drive is the initial motivation; that is, the person must want something or want to do something. The cue is his awareness that there are elements and situations in his surroundings that make the task possible to achieve. The response is the action he takes to accomplish the task. Whether overt or covert, it is the actual attacking of the task. The reinforcement is the satisfaction that the person gains from his success.

            Another way of analyzing learning, from the point of view of learning himself, divides the process into seven segments: (1) the situation; (2) personal characteristics; (3) the goal; (4) interpretation; (5) action; (6) the result and (7) the reaction (Hill, 1993). 

            One model that  has been used intensively to study learning is the famous experiments described by the Russian physiologist Ivan Pavlov. In the experiments that have been conducted , dogs were conditioned, to salivate to a neural stimulus,  such a the sound of a bell. This was done by having the bell sound just before food was given to the dog. Of course, food naturally caused the dog to salivate, and within  a few trials the sound of the bell was also having this effect. Generalizations occurred when a somewhat similar bell was rung and the dog salivated with this new sound. Discrimination occurred when a quite different sound was than the bell was heard and the dog did not salivate. Extinction occurred when the bell was sounded on a number of trials without food being presented to the dog. Parlov discovered that, in such experiments, the dog would cease to salivate when the bell was sounded,

            The American psychologist John  Watson showed that, in the same manner, a child will learn to fear objects that initially did not cause the fear reaction to occur. He had a white mouse introduced to an infant boy, and as the boy reached for the mouse, a very loud, frightening voice was sounded. Very soon the boy showed great alarm at the sight of the mouse (conditioned fear) and in fact, was also frightened by any small, white fluffy object (generalization).

            Not all learnings occur this way, although some psychologists thinks that conditioning describes how attitudes are learned. Certainly, most psychologists would agree that a large amount of animal learning can be described in terms of conditioning (Hilgard, 1996).

            Learning is a complex process to study because there are so many factors or variables that affect the course of learning. Each of these variables can cause learning to be rapid or slow, efficient or inefficient, and remembered or forgotten. These variables include task variables, learner variables and situation and method variables.

            Animals, including humans either learn everything they do (from “nurture”), or they know what to do instinctively (from “nature”). Neither extreme has proved to be correct (Microsoft Encarta Encyclopedia, 1999).

            Until recently the dominant school in behavioral theory has been behaviorism, whose best-known figures are J. B. Watson and B. F. Skinner. Strict behaviorists hold that all behavior, even breathing and the circulation of blood, according to Watson, is learned; they believe that animals are, in effect, born as blank slates upon which chance and experience are to write their messages. Through conditioning, they believe, an animal’s behavior is formed. Behaviorists recognize two sorts of conditioning: classical and operant.

In the late 19th century the Russian physiologist Ivan Pavlov discovered classical conditioning while studying digestion. He found that dogs automatically salivate at the sight of food—an unconditioned response to an unconditioned stimulus, to use his terminology. If Pavlov always rang a bell when he offered food, the dogs began slowly to associate this irrelevant (conditioned) stimulus with the food. Eventually the sound of the bell alone could elicit salivation. Hence, the dogs had learned to associate a certain cue with food. Behaviorists see salivation as a simple reflex behavior—something like the knee-jerk reflex doctors trigger when they tap a patient’s knee with a hammer.

The other category, operant conditioning, works on the principle of punishment or reward. In operant conditioning a rat, for example, is taught to press a bar for food by first being rewarded for facing the correct end of the cage, next being rewarded only when it stands next to the bar, then only when it touches the bar with its body, and so on, until the behavior is shaped to suit the task. Behaviorists believe that this sort of trial-and-error learning, combined with the associative learning of Pavlov, can serve to link any number of reflexes and simple responses into complex chains that depend on whatever cues nature provides. To an extreme behaviorist, then, animals must learn all the behavioral patterns that they need to know.

In contrast, ethnology—a discipline that developed in Europe—holds that much of what animals know is innate (instinctive).

Social Constructivism in Mathematics

            Mathematics is the study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or the generalization of these two fields, as in algebra. Towards the middle of the 19th century mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic— the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for transforming primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself: evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.

Many authors offer some perspectives on mathematics education from an instructional design viewpoint. The authors of  “Instructional Design Prospective on Mathematics Education with Reference to Vygotsky’s Theory on Social Cognition” do this in a somewhat eclectic fashion, beginning with an overview of the ideological "paradigm wars" within the instructional design community. Alternative philosophies of mind, including Vygotsky's emphasis on the social origins of cognition, have implications for the teaching of mathematics, as well as for instructional design generally. The authors conclude with some recommendations for the instructional design of mathematics education curricula that are consistent with a Vygotskian framework.

Computers and Mathematics

            At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at Göttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert's address at Göttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the “Hilbert problems” has been solved, mathematicians all over the world await the details of the story with impatience (Desmond, 1994).

Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics—namely, the invention of the programmable digital computer. Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbage's imagination outran the technology of his day, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois (Microsoft Encarta Encyclopedia, 1999).

            Calculators and computers are examples of excellent mathematical compensation tools that allow less gifted students to deal with advanced topics. Algebraic Calculators, for instance,  are graphic calculators which offer features that so far were available only on computers. These calculators can simplify expressions, differentiate, integrate, and plot functions, solve equations, manipulate matrices, etc. In short: They can do most of what we teach in mathematics at schools and colleges (Berkeley, 1959).

 It goes without saying, that the ultimate goal in mathematics teaching is to weed out all weaknesses in skills that are regarded essential. A physically challenged person need not be tied to a wheel chair for the rest of his life. A physician will endeavor to repair a patient’s physical disability as much as possible, using an individual therapy. Similarly, a teacher should endeavor to repair a student’s intellectual/mathematical disability with a proper, individual therapy. In both cases we will facilitate the patient’s „daily life with the disability“ – i.e. the time outside the therapy – by providing an appropriate compensation tool (wheel chair, calculator).

Connection Between Theories, Computers and Mathematics

            The central element in the social constructivist theory Vygotsky is the role played by the society in a child’s learning process. In this research, the writer was able to discuss the different areas of Vygotsky’s theory.

            Computers, as a communication and learning device, are of much use in the social constructivist arena. Computers facilitate the “social” learning process of a child. A child, through interaction with classmates, teachers and mentors by means of computers, may be able to learn to the fullest, as the theory of social constructivism suggest.

            Mathematics teaching through computers is therefore a very plausible and promising area in education. Many computer-based mathematical teaching packages are currently available in the market and seemingly, this industry will go a long way in that it supports child learning through social constructivism to the fullest.  




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