High development in computer technology brings with it a wide range of applications used as support in education. Constructivists believe that the knowledge is actively constructed by the learner in their process of interpreting the new aspect based on their current knowledge and system of reasoning.
The learners make meaning of the situation by actively engaging in their own thought process and thereby producing their idea of the new knowledge. Process of learners constructing their model that represents their new knowledge can be made more efficient with the use of computers.
Some applications have shown to be very useful to learn and teach mathematics in a constructivist way. Incorporating these new tools in schools would make the process of education more enjoyable and efficient.
HISTORY OF COMPUTER APPLICATIONS FOR EDUCATION
Most applications have been separated according to functions that they provide.
Drill and practice software provides students with repetitive exercises that allow them to practice what they learned previously. Students are provided with immediate feedback of the results of their practice. There are no new concepts presented to the student.
The research has shown that students are more excited when using a computer over a workbook, and so it is motivating. By offering corrections and hints for improvement, it disallows students in making same mistakes.
Tutorial software expands on drill and practice by attempting to introduce new concepts, ideas and skills. This software presents new information that is practiced and tested for understanding. It requires student’s input to proceed to a higher level of instruction thereby allowing students to progress at their own pace.
These two approaches are classified as Direct-Instruction approaches, which are based on behaviorist theory of learning. They can be further divide in Skills-Based Direct Instruction, which is a strategy to help students master basic or foundation skills, and Just-in-Time Direct Instruction delivered at the point of need (Maddux, Johnson and Willis: 2001).
Another category of computer applications used in education fall under the term CMI, which stands for Computer-Managed Instruction. Here the computer plays an important management role as it provides tests for students to engage in and provides feedback.
This application is rather used as a tool to help the instructor manage the clerical and assessment work. CMI applications correspond to applications being management in nature. The largest category of CMI applications is called Integrated Learning Systems (ILS).
This is a complex system providing hardware, software, training and technical support for delivering of a school curriculum through technology. These systems are very successful and are continuing to grow.
However they are mostly accepted by those holding the behaviorist view of learning and are inadequate for the constructivist approach (Maddux, Johnson and Willis: 2001).
CONSTRUCTIVISM AND MATHEMATICS
The principles of constructivism are based on the assumption that we learn through experiences we engage in and that the knowledge we form is based on our so far developed knowledge or beliefs, and our way of reasoning. Instead of just absorbing what is being presented learners actually form their own model of the situation from which they construct their own idea of the phenomenon presented. Three main principles behind constructivism are that: “Knowledge is shaped as part of social interaction and negotiation; what is learned cannot be separated from how it is learned; the learner learns meaningful material only when s/he has an intrinsic desire to create meaning.” (Geisert & Futrell: 2000). According to constructivists there exist cognitive structures that are activated during the construction. They are under continual development, and they account for the construction as they explain the result of a cognitive activity. Purposive activity induces their change as the environment presses the organism to adapt (Davis, Maher & Noddings: 1990). Even though children form ideas through reasoning on their knowledge, it is still socially constructed. It is compared to other views and beliefs in order to determine its validity. Learning is formed through interaction with the environment and with other people, which form a part of that environment. Mathematics nearly always builds on top of the current knowledge. Previous knowledge is necessary for the progress in some field of mathematics. If some concept is poorly understood difficulties are encountered with new topics constructed on top of this concept. This is why constructivist theory of learning goes hand in hand with learning of mathematics. In order to accommodate to this way of meaningful learning we have to consider what constitutes effective mathematics teaching. In the research in mathematics education, constructivists have adopted the following views: • Mathematics is invented or constructed by human beings; • An interpretation of mathematical meaning as constructed by the learner rather than imparted by the teacher; • Mathematical learning occurs most effectively through guided discovery, meaningful application, and problem solving; • The study and assessment of learning through individual interviews and small-group case studies; • Effective teaching through creation of appropriate learning environments, thereby encouraging the development of diverse and creative problem-solving skills (Davis, Maher & Noddings: 1990).Learning mathematics is viewed to be more of an internal process where the learner tries to construct the meaning of the topic, rather than just absorbing what is being presented without forming some model of the representation, wherefrom the learner abstracts the meaning. According to Robert Davis a constructivist, in order to think about a mathematical situation one must: • Build a representation for the input data; • From this data representation, retrieve or construct a representation of relevant knowledge from memory to be used in solving the problem; • Construct a mapping between the data representation and the knowledge representation; • Check these mappings and constructions for their correctness; • If they appear satisfactory, use technical devices (or other information) associated with the knowledge representation in order to solve the problem (Davis, Maher & Noddings: 1990). This cycle makes one represent a mathematical situation in a certain way and this process itself builds some new knowledge. Mathematical learning involves active manipulation of meanings in order to be convinced to form or adjust a certain belief or knowledge about the mathematical phenomena in question. Learning environments should be as such as to promote student’s creativity, motivation and own way of dealing with mathematical problems. The traditional view on learning mathematics assumed that learning facts and algorithms would eventually lead to their application in appropriate situations. Constructivism is opposed to this and suggests that students must be helped in achieving more powerful ways of reasoning. Suggested way of doing this is by supplying students with some tools to aid in their process of understanding. With current development in computer technology many of these tools are implemented as application software. Students and teachers interact with these tools and many have shown to be useful in process of learning mathematics.TECHNOLOGY SUPPORT FOR CONSTRUCTIVISM Technology use in education should be used to promote meaningful learning and support constructivists theory of learning. As learners assign meaning through experience the computer tools should work towards improving problem-solving skills. Sensory experiences come through perceptions in order to be further organized to form understanding. Computer tools can be very effective in presenting data in such way that many of our sensory experiences are enhanced through the use of quality graphics, sounds and different environments. This increases motivation and makes experience of learning more memorable.[1] Some tools that support mathematical education allow learners to proceed at their own pace, thereby having the opportunity to grasp onto some concepts better than when working at the instructor’s pace. Different ways of reasoning occur amongst learners and certain individuals find some topics harder and more time consuming than others. They will spend more time on harder topics and less on the topics they easily understand. This makes the process of learning more efficient to the individual as they will not waste time on easy topics, and also not fall behind with new concepts until the basis is properly understood. Thanks to our understanding of learning theories we can evaluate effective software according to following guidelines: • Software must simulate a high degree of interest in the learner; • Software must contribute to developmental learning; • Software must be based in concrete experience to enhance understanding; • Software must make optimum use of the visual and, where appropriate, the aural sensory channels to strengthen the reality of the experience (Forcier & Descy: 2002). Computer tools can improve learner’s reasoning and forming of logical connections from their so far constructed knowledge. Learning in constructivist means consists of 5 types of learning. Through computer tools learners engage in: • Active learning, where they explore the technology-based environment and get familiar with the outcomes of their actions upon it; • Constructive learning, where they articulate their knowledge and construct its meaning in a larger social and intellectual contexts; • Intentional learning, where they perform their activities according to the goals they have set; • Authentic learning, where they examine and attempt to solve complex, ill-structured, and real-world problems; • Cooperative learning, where they interact with others and socially negotiate their constructed meanings (Jonassen, Peck & Wilson: 1999). Internet is another way computers can prove to be useful in promoting meaningful learning. Knowledge gained on Internet will be formed by social negotiation, and meaningful learning will result from the desire to interact with the environment provided by the Internet. Meaningful learning will be promoted by the curiosity, puzzlement, and desire to gain knowledge and understanding of various aspects of the social and intellectual world encountered on the World Wide Web (Geisert & Futrell: 2000). Even though computer tools seem to be useful in many ways, there are people who believe that the traditional way of teaching mathematics is still the best way of learning mathematics. This caused lots of research and comparison of the two techniques. One study by Elliot and Hall has grouped children into different groups and teaching was performed in different ways amongst the groups. Group A carried out math activities on the computer and received strategic, meta-cognitive advice by teachers. Group B carried out the same activities on the computer but without advice by teachers, and group C carried out the activities in a workbook while engaging in some other non-math related activity on the computer. The results, measured with standardised test of math aptitude showed that computer aided teaching was more effective than the traditional teaching, and strategic support by teachers leaded to better results in the test. In another study a similar procedure was done to teach multiplication tables, one group receiving instruction from a teacher and other from the computer. The percentage of higher scores in tests was significantly higher for the group instructed by the teacher. Teacher worked at a higher speed and had more time left for exercises to strengthen new knowledge. Computer could not substitute the role of being a guide as good as the teacher (Bornas & Llabres: 2001). Computer can be used as an efficient tool to learn mathematics especially when advised by an instructor on the progress of the activity. Teachers are still better instructors, but they could improve their way of teaching with these computer tools encouraging construction of meaningful knowledge.CONSTRUCTIVISM THROUGH COMPUTER PROGRAMMINGSeymore Papert was the initiator of the idea of computer programming being used as an aid in learning mathematics in constructivist way. He developed a unique programming language called “LOGO” which he presented in his book entitled “Mindstorms: Children, Computers and Powerful Ideas” in 1980. Paperts aim was to use computers in such a way so that difficult and abstract ideas can be made more concrete in order to be easier understood by children. For Papert, any learning is easy if you can assimilate it to your collection of models, and individual learning will depend on the models the individual has available. Therefore Logo was designed as a thinking tool providing children with powerful models allowing for better understanding of new concepts. As opposed to other views holding that during computer-aided instruction computer is being used to program the child, Papert writes in his book about this process as: “…the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building” (Papert: 1980). Papert believed that learning to communicate with computers should be a natural process. When we make a computer to be mathematics-speaking entity, we should attempt to make communication between children and computers easy and enjoyable for children to do. This way, children can learn mathematics as a living language occurring to them naturally. According to Papert, mathematics of space and movement and repetitive patterns of actions is what most comes to children naturally. He introduces the term “Turtle Geometry”, which he defines as computational style of geometry, and by which he believes aspects of geometry will naturally occur to children. Papert uses model of children as builders of their own intellectual structures introduced by Jean Piaget whose theories have greatly contributed to the development of educational psychology in general. Computer programming could be an aid in the development of these intellectual structures. Papert emphasises this point by stating what occurs in the LOGO environment is: “The child programs the computer. And in teaching the computer how to think, children embark on an exploration about how they themselves think. The experience can be heady: Thinking about thinking turns the child into an epistemologist, an experience not even shared by most adults” (Papert: 1980). Programming of computers would provide more effective learning. Children would adapt to certain new ways of thinking and reasoning through the programming of computer to perform some tasks. They would obtain more information on applicable aspects of programming, thereby adapting to a new way of thinking or reasoning when put in those situations. Through teaching the computer how to think in order to achieve the desired outcome, children themselves become aware of their own way of thinking. This is a great achievement for the development in learning at such an early stage. Being aware of their own way of thinking and reasoning, children would also become aware of the certain faults that occur when an undesired outcome is achieved from their process of thought. Children would learn quicker from their mistakes by being aware of what has brought the mistakes into existence. Early work with LOGO showed beneficial effects on the measurement of ‘metacognition’. Metacognition has two meanings assigned to it: “The first being conscious and purposeful reflection on various aspects of knowing and learning, and second being the unconscious regulation of knowledge structures and learning that some information processing theorists posit to be under the control of “executive” processes” (Clements & Nastasi: 1999). During this study it is shown that educational environment facilitate the process of metacognition and its growth. LOGO being an educational environment has shown to facilitate the role of the teacher as a mediator of metacognitive experiences. Furthermore LOGO encouraged collaborative problem-solving occurring in the social context, where different ideas are met and negotiated amongst learners.OTHER COMPUTER APPLICATIIONS IN EDUCATIONPlato: The System The Plato system was designed in 1972 by professor Robert Davis. Its aim was to be a large time-share or multiple-user environment for teaching any courses. It had to accommodate to the needs of all educational institutions and elementary school teachers, and handling different teaching styles. It therefore had flexibility of expression. Plato could be used as an interactive textbook concentrating on the creative and exploratory aspects of mathematics and forms of social cooperation and interaction. Many people thought that a child using a computer would fail to develop a ‘meta’ language for mathematics, but Plato could not be accused of producing that behavior. Children talked to the computer during their interaction with its environment, as they could talk to other three classmates at terminals and later share their experiences with others. Courseware encouraged sharing ideas and celebrating diverse solutions with classmates. All these things contribute to better memorising and constructing of the new knowledge. The courseware was divided in three parts: whole number arithmetic; fractions, mixed numbers, and decimals; and graphs, variables, functions, and equations. Most impressive part of this curriculum was its overall adherence to the integrity of design. Plato was very popular and was used and talked about by children throughout day, always rising new ideas and further structuring their knowledge. Despite all these positives the commercial costs of Plato were still too high to support its use in schools (Druin & Solomon: 1996). Inner London Education Authority’s (ILEA) schools used a range of software, which defines the state of computer use in the UK during the 1980s. ILEA Primary Discpack 1 was issued in 1986 and contained eight pieces of software. Two pieces of software intended as support to learning mathematics were Maths with a Story (MWAS) 1 and a set of programs called Smile 2 – The Next 17. MWAS was a set of numerical games or puzzles. Smile 2 was a part of the SMILE (Secondary Maths Individualised Learning Experiment) project in London. The Smile programs were intended for use in secondary schools. These programs consisted of mathematical adventures, which took participants on a journey while solving the puzzle (Abbot: 2001). Computer Supported Intentional Learning Environments Project (CSILE) was developed at the Ontario Institute for Studies in Education. Schools participated in collaborative problem-solving ranging from making animals to solving complex mathematical problems. Children are able to access each other’s notes that are contained in a networked hypermedia system. Furthermore children have on-line discussions of the progression of ideas. Many features of this environment promote development of a community of knowledge builders who use higher order thinking skills during their interactions and problem solving. Some of these features are: projects are initiated by children; communication initiated and maintained by the children; children have high level of control over their learning; shared ideas and processes of solving problems are kept in a database; minimised competition by encouraging collaboration (Yelland: 1999). These are some of the best practices used in America’s schools. Project IMPACT (Increasing Mathematics Potential Through Access To a Curriculum Enriched by Technology): Goal of this application was to increase the potential of high-school students by affording teachers and students, access to a state-of-the-art technology to facilitate their learning. It is an innovative approach where improvement is expected by having computer labs equipped with mathematics software and multimedia equipment. Teachers themselves are equipped with the latest technology in order for them to be able to present the traditional way of teaching in new and exciting ways. Students go to labs for individualised instruction as well as cooperative learning activities. However this attempt did not experience great success. Housing of the lab was the main problem, and many schools did not have the finances or the structure to allow for this approach.Project Reach (Research, Equate, Accelerate, Cooperate, Help) Overall goal was to improve academic achievement and social skills of at-risk and gifted students by addressing the special areas of mathematics, science, social studies, language arts, computer science, higher order thinking skills, and career awareness. It is an interactive, integrated program in which different type of students cooperatively work to produce a deeper insight on some topic. This approach resulted in improved grades, higher self-esteem, developing of higher order thinking skills and other advantages compatible with the topic.Project TOPS (Technology Optimises Performance in Science) Some of the goals were to improve student achievement and knowledge in science, develop critical thinking and problem-solving for the student, stimulate interest and motivation in science and demonstrate to teachers the benefits of using technology to support science instruction. TOPS has been used to integrate technology in different ways and is used by other educational agencies to conduct research. Surveys have shown increased understanding and appreciation of science and technology by the students (Bozeman & Baumbach: 1995).IMPACT OF COMPUTER USE IN EDUCATIONThree multinational studies whose research influenced the view of psychological impact of school computer use are: Association for the Evaluation of Educational Achievement Computers in Education Project (IEA CompEd), the Information Technology in Education and Children Project (ITEC), and the Young Children’s Computer Inventory Project (YCCI). Three studies taken together, account for the impact of computer use in: elementary, middle school and secondary education. The IEA is a non-governmental international organisation of professional educational research centers, from more than fifty national educational systems. Surveys from this organisation are towards overall computer use in countries and the differences amongst countries in computer-related knowledge among students. ITEC research was based on classrooms as research unit ecosystems using Vygotskian theoretical foundations and mixed research methods. The researchers have categorised behaviors seen during computer use as an evidence of metacognitive development: • Relating a problem to previous problems; • Formulating appropriate questions; • Trying alternative approaches; • Evaluating one’s actions; • Analysing problems; • Recognising relationships; • Generating new ideas; • Observing central issues and problems; • Comparing similarities and differences. YCCI project began in 1990 as a longitudinal study of childhood computing in school. The project started as a collaborative effort to search for all positive effects of computer use in education without any negative effects. Major conclusions drawn from the research are that computer use in primary school has a strong positive impact on attitudes toward computers, as well as positive impact on motivation and study habits (Collis, Knezek, Lai, Miyashita, Pelgrum, Plomp & Sakomoto: 1996). When children are using computers they may feel that the computer is their new playmate or toy. A research which observed interactions of six kindergarten children using the program LOGO for three months, remarked that children often spoke to computers considering them as they playmates. Children have much desire and curiosity to explore new things, and they may simply enjoy using them without any fear that might be shown in adult use. Computers technology has been incorporated into daily practice of education in virtually every corner of the world, and major findings from the research of the three organisations mentioned all relate to the following observation: “While style of interaction and teaching and classroom organisation varied widely, the image of happy and engaged children, working with and around computers in a productive and confident way, talking to each other, working not independent of the teacher, but on the path that the teacher set for them, comes out again and again” (Collis, Knezek, Lai, Miyashita, Pelgrum, Plomp and Sakamoto: 1996). The only possible negative effect encountered through the research is the actual computer overuse to the point of infringing traditional explorations. They would experience the lack of engaging in certain events such as cooperative play and normal physical action. In the extreme case this would prove to be unhealthy and unwise. However this negative effect only applies to the case not expected of occurring when computer tools are used in an appropriate way.Using computers as a tool of teaching mathematics in a social constructivist way has shown to be a very powerful tool. With the continual development in computer technology there are always new ways offered to present knowledge in an exciting and useful way. It keeps the students highly motivated in performing mathematical activities when they are presented in an interesting way. Furthermore there are many different types of learning with computers, and some promote self awareness by observing the process of thoughts. This occurs in computer programming where the programmer (in our case child) is required to make the computer perform some operation and making the computer think, which allows them to get familiar with their own way of constructing knowledge. This does not suggest that computers should replace teachers, but rather that teachers can use computer as a powerful tool in promoting high structured meaningful learning. Teachers as well as students benefit greatly from these applications, and there would be no need of letting education halt when everything else has been changed and improved in some way by the use of computers.
High development in computer technology brings with it a wide range of applications used as support in education. Constructivists believe that the knowledge is actively constructed by the learner in their process of interpreting the new aspect based on their current knowledge and system of reasoning. The learners make meaning of the situation by actively engaging in their own thought process and thereby producing their idea of the new knowledge. Process of learners constructing their model that represents their new knowledge can be made more efficient with the use of computers. Some applications have shown to be very useful to learn and teach mathematics in a constructivist way. Incorporating these new tools in schools would make the process of education more enjoyable and efficient.
HISTORY OF COMPUTER APPLICATIONS FOR EDUCATIONMost applications have been separated according to functions that they provide. Drill and practice software provides students with repetitive exercises that allow them to practice what they learned previously. Students are provided with immediate feedback of the results of their practice. There are no new concepts presented to the student. The research has shown that students are more excited when using a computer over a workbook, and so it is motivating. By offering corrections and hints for improvement, it disallows students in making same mistakes. Tutorial software expands on drill and practice by attempting to introduce new concepts, ideas and skills. This software presents new information that is practiced and tested for understanding. It requires student’s input to proceed to a higher level of instruction thereby allowing students to progress at their own pace. These two approaches are classified as Direct-Instruction approaches, which are based on behaviorist theory of learning. They can be further divide in Skills-Based Direct Instruction, which is a strategy to help students master basic or foundation skills, and Just-in-Time Direct Instruction delivered at the point of need (Maddux, Johnson and Willis: 2001). Another category of computer applications used in education fall under the term CMI, which stands for Computer-Managed Instruction. Here the computer plays an important management role as it provides tests for students to engage in and provides feedback. This application is rather used as a tool to help the instructor manage the clerical and assessment work. CMI applications correspond to applications being management in nature. The largest category of CMI applications is called Integrated Learning Systems (ILS). This is a complex system providing hardware, software, training and technical support for delivering of a school curriculum through technology. These systems are very successful and are continuing to grow. However they are mostly accepted by those holding the behaviorist view of learning and are inadequate for the constructivist approach (Maddux, Johnson and Willis: 2001).
CONSTRUCTIVISM AND MATHEMATICSThe principles of constructivism are based on the assumption that we learn through experiences we engage in and that the knowledge we form is based on our so far developed knowledge or beliefs, and our way of reasoning. Instead of just absorbing what is being presented learners actually form their own model of the situation from which they construct their own idea of the phenomenon presented. Three main principles behind constructivism are that: “Knowledge is shaped as part of social interaction and negotiation; what is learned cannot be separated from how it is learned; the learner learns meaningful material only when s/he has an intrinsic desire to create meaning.” (Geisert & Futrell: 2000). According to constructivists there exist cognitive structures that are activated during the construction. They are under continual development, and they account for the construction as they explain the result of a cognitive activity. Purposive activity induces their change as the environment presses the organism to adapt (Davis, Maher & Noddings: 1990). Even though children form ideas through reasoning on their knowledge, it is still socially constructed. It is compared to other views and beliefs in order to determine its validity. Learning is formed through interaction with the environment and with other people, which form a part of that environment. Mathematics nearly always builds on top of the current knowledge. Previous knowledge is necessary for the progress in some field of mathematics. If some concept is poorly understood difficulties are encountered with new topics constructed on top of this concept. This is why constructivist theory of learning goes hand in hand with learning of mathematics. In order to accommodate to this way of meaningful learning we have to consider what constitutes effective mathematics teaching. In the research in mathematics education, constructivists have adopted the following views: • Mathematics is invented or constructed by human beings; • An interpretation of mathematical meaning as constructed by the learner rather than imparted by the teacher; • Mathematical learning occurs most effectively through guided discovery, meaningful application, and problem solving; • The study and assessment of learning through individual interviews and small-group case studies; • Effective teaching through creation of appropriate learning environments, thereby encouraging the development of diverse and creative problem-solving skills (Davis, Maher & Noddings: 1990).Learning mathematics is viewed to be more of an internal process where the learner tries to construct the meaning of the topic, rather than just absorbing what is being presented without forming some model of the representation, wherefrom the learner abstracts the meaning. According to Robert Davis a constructivist, in order to think about a mathematical situation one must: • Build a representation for the input data; • From this data representation, retrieve or construct a representation of relevant knowledge from memory to be used in solving the problem; • Construct a mapping between the data representation and the knowledge representation; • Check these mappings and constructions for their correctness; • If they appear satisfactory, use technical devices (or other information) associated with the knowledge representation in order to solve the problem (Davis, Maher & Noddings: 1990). This cycle makes one represent a mathematical situation in a certain way and this process itself builds some new knowledge. Mathematical learning involves active manipulation of meanings in order to be convinced to form or adjust a certain belief or knowledge about the mathematical phenomena in question. Learning environments should be as such as to promote student’s creativity, motivation and own way of dealing with mathematical problems. The traditional view on learning mathematics assumed that learning facts and algorithms would eventually lead to their application in appropriate situations. Constructivism is opposed to this and suggests that students must be helped in achieving more powerful ways of reasoning. Suggested way of doing this is by supplying students with some tools to aid in their process of understanding. With current development in computer technology many of these tools are implemented as application software. Students and teachers interact with these tools and many have shown to be useful in process of learning mathematics.TECHNOLOGY SUPPORT FOR CONSTRUCTIVISM Technology use in education should be used to promote meaningful learning and support constructivists theory of learning. As learners assign meaning through experience the computer tools should work towards improving problem-solving skills. Sensory experiences come through perceptions in order to be further organized to form understanding. Computer tools can be very effective in presenting data in such way that many of our sensory experiences are enhanced through the use of quality graphics, sounds and different environments. This increases motivation and makes experience of learning more memorable.[1] Some tools that support mathematical education allow learners to proceed at their own pace, thereby having the opportunity to grasp onto some concepts better than when working at the instructor’s pace. Different ways of reasoning occur amongst learners and certain individuals find some topics harder and more time consuming than others. They will spend more time on harder topics and less on the topics they easily understand. This makes the process of learning more efficient to the individual as they will not waste time on easy topics, and also not fall behind with new concepts until the basis is properly understood. Thanks to our understanding of learning theories we can evaluate effective software according to following guidelines: • Software must simulate a high degree of interest in the learner; • Software must contribute to developmental learning; • Software must be based in concrete experience to enhance understanding; • Software must make optimum use of the visual and, where appropriate, the aural sensory channels to strengthen the reality of the experience (Forcier & Descy: 2002). Computer tools can improve learner’s reasoning and forming of logical connections from their so far constructed knowledge. Learning in constructivist means consists of 5 types of learning. Through computer tools learners engage in: • Active learning, where they explore the technology-based environment and get familiar with the outcomes of their actions upon it; • Constructive learning, where they articulate their knowledge and construct its meaning in a larger social and intellectual contexts; • Intentional learning, where they perform their activities according to the goals they have set; • Authentic learning, where they examine and attempt to solve complex, ill-structured, and real-world problems; • Cooperative learning, where they interact with others and socially negotiate their constructed meanings (Jonassen, Peck & Wilson: 1999). Internet is another way computers can prove to be useful in promoting meaningful learning. Knowledge gained on Internet will be formed by social negotiation, and meaningful learning will result from the desire to interact with the environment provided by the Internet. Meaningful learning will be promoted by the curiosity, puzzlement, and desire to gain knowledge and understanding of various aspects of the social and intellectual world encountered on the World Wide Web (Geisert & Futrell: 2000). Even though computer tools seem to be useful in many ways, there are people who believe that the traditional way of teaching mathematics is still the best way of learning mathematics. This caused lots of research and comparison of the two techniques. One study by Elliot and Hall has grouped children into different groups and teaching was performed in different ways amongst the groups. Group A carried out math activities on the computer and received strategic, meta-cognitive advice by teachers. Group B carried out the same activities on the computer but without advice by teachers, and group C carried out the activities in a workbook while engaging in some other non-math related activity on the computer. The results, measured with standardised test of math aptitude showed that computer aided teaching was more effective than the traditional teaching, and strategic support by teachers leaded to better results in the test. In another study a similar procedure was done to teach multiplication tables, one group receiving instruction from a teacher and other from the computer. The percentage of higher scores in tests was significantly higher for the group instructed by the teacher. Teacher worked at a higher speed and had more time left for exercises to strengthen new knowledge. Computer could not substitute the role of being a guide as good as the teacher (Bornas & Llabres: 2001). Computer can be used as an efficient tool to learn mathematics especially when advised by an instructor on the progress of the activity. Teachers are still better instructors, but they could improve their way of teaching with these computer tools encouraging construction of meaningful knowledge.CONSTRUCTIVISM THROUGH COMPUTER PROGRAMMINGSeymore Papert was the initiator of the idea of computer programming being used as an aid in learning mathematics in constructivist way. He developed a unique programming language called “LOGO” which he presented in his book entitled “Mindstorms: Children, Computers and Powerful Ideas” in 1980. Paperts aim was to use computers in such a way so that difficult and abstract ideas can be made more concrete in order to be easier understood by children. For Papert, any learning is easy if you can assimilate it to your collection of models, and individual learning will depend on the models the individual has available. Therefore Logo was designed as a thinking tool providing children with powerful models allowing for better understanding of new concepts. As opposed to other views holding that during computer-aided instruction computer is being used to program the child, Papert writes in his book about this process as: “…the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building” (Papert: 1980). Papert believed that learning to communicate with computers should be a natural process. When we make a computer to be mathematics-speaking entity, we should attempt to make communication between children and computers easy and enjoyable for children to do. This way, children can learn mathematics as a living language occurring to them naturally. According to Papert, mathematics of space and movement and repetitive patterns of actions is what most comes to children naturally. He introduces the term “Turtle Geometry”, which he defines as computational style of geometry, and by which he believes aspects of geometry will naturally occur to children. Papert uses model of children as builders of their own intellectual structures introduced by Jean Piaget whose theories have greatly contributed to the development of educational psychology in general. Computer programming could be an aid in the development of these intellectual structures. Papert emphasises this point by stating what occurs in the LOGO environment is: “The child programs the computer. And in teaching the computer how to think, children embark on an exploration about how they themselves think. The experience can be heady: Thinking about thinking turns the child into an epistemologist, an experience not even shared by most adults” (Papert: 1980). Programming of computers would provide more effective learning. Children would adapt to certain new ways of thinking and reasoning through the programming of computer to perform some tasks. They would obtain more information on applicable aspects of programming, thereby adapting to a new way of thinking or reasoning when put in those situations. Through teaching the computer how to think in order to achieve the desired outcome, children themselves become aware of their own way of thinking. This is a great achievement for the development in learning at such an early stage. Being aware of their own way of thinking and reasoning, children would also become aware of the certain faults that occur when an undesired outcome is achieved from their process of thought. Children would learn quicker from their mistakes by being aware of what has brought the mistakes into existence. Early work with LOGO showed beneficial effects on the measurement of ‘metacognition’. Metacognition has two meanings assigned to it: “The first being conscious and purposeful reflection on various aspects of knowing and learning, and second being the unconscious regulation of knowledge structures and learning that some information processing theorists posit to be under the control of “executive” processes” (Clements & Nastasi: 1999). During this study it is shown that educational environment facilitate the process of metacognition and its growth. LOGO being an educational environment has shown to facilitate the role of the teacher as a mediator of metacognitive experiences. Furthermore LOGO encouraged collaborative problem-solving occurring in the social context, where different ideas are met and negotiated amongst learners.OTHER COMPUTER APPLICATIIONS IN EDUCATIONPlato: The System The Plato system was designed in 1972 by professor Robert Davis. Its aim was to be a large time-share or multiple-user environment for teaching any courses. It had to accommodate to the needs of all educational institutions and elementary school teachers, and handling different teaching styles. It therefore had flexibility of expression. Plato could be used as an interactive textbook concentrating on the creative and exploratory aspects of mathematics and forms of social cooperation and interaction. Many people thought that a child using a computer would fail to develop a ‘meta’ language for mathematics, but Plato could not be accused of producing that behavior. Children talked to the computer during their interaction with its environment, as they could talk to other three classmates at terminals and later share their experiences with others. Courseware encouraged sharing ideas and celebrating diverse solutions with classmates. All these things contribute to better memorising and constructing of the new knowledge. The courseware was divided in three parts: whole number arithmetic; fractions, mixed numbers, and decimals; and graphs, variables, functions, and equations. Most impressive part of this curriculum was its overall adherence to the integrity of design. Plato was very popular and was used and talked about by children throughout day, always rising new ideas and further structuring their knowledge. Despite all these positives the commercial costs of Plato were still too high to support its use in schools (Druin & Solomon: 1996). Inner London Education Authority’s (ILEA) schools used a range of software, which defines the state of computer use in the UK during the 1980s. ILEA Primary Discpack 1 was issued in 1986 and contained eight pieces of software. Two pieces of software intended as support to learning mathematics were Maths with a Story (MWAS) 1 and a set of programs called Smile 2 – The Next 17. MWAS was a set of numerical games or puzzles. Smile 2 was a part of the SMILE (Secondary Maths Individualised Learning Experiment) project in London. The Smile programs were intended for use in secondary schools. These programs consisted of mathematical adventures, which took participants on a journey while solving the puzzle (Abbot: 2001). Computer Supported Intentional Learning Environments Project (CSILE) was developed at the Ontario Institute for Studies in Education. Schools participated in collaborative problem-solving ranging from making animals to solving complex mathematical problems. Children are able to access each other’s notes that are contained in a networked hypermedia system. Furthermore children have on-line discussions of the progression of ideas. Many features of this environment promote development of a community of knowledge builders who use higher order thinking skills during their interactions and problem solving. Some of these features are: projects are initiated by children; communication initiated and maintained by the children; children have high level of control over their learning; shared ideas and processes of solving problems are kept in a database; minimised competition by encouraging collaboration (Yelland: 1999). These are some of the best practices used in America’s schools. Project IMPACT (Increasing Mathematics Potential Through Access To a Curriculum Enriched by Technology): Goal of this application was to increase the potential of high-school students by affording teachers and students, access to a state-of-the-art technology to facilitate their learning. It is an innovative approach where improvement is expected by having computer labs equipped with mathematics software and multimedia equipment. Teachers themselves are equipped with the latest technology in order for them to be able to present the traditional way of teaching in new and exciting ways. Students go to labs for individualised instruction as well as cooperative learning activities. However this attempt did not experience great success. Housing of the lab was the main problem, and many schools did not have the finances or the structure to allow for this approach.Project Reach (Research, Equate, Accelerate, Cooperate, Help) Overall goal was to improve academic achievement and social skills of at-risk and gifted students by addressing the special areas of mathematics, science, social studies, language arts, computer science, higher order thinking skills, and career awareness. It is an interactive, integrated program in which different type of students cooperatively work to produce a deeper insight on some topic. This approach resulted in improved grades, higher self-esteem, developing of higher order thinking skills and other advantages compatible with the topic.Project TOPS (Technology Optimises Performance in Science) Some of the goals were to improve student achievement and knowledge in science, develop critical thinking and problem-solving for the student, stimulate interest and motivation in science and demonstrate to teachers the benefits of using technology to support science instruction. TOPS has been used to integrate technology in different ways and is used by other educational agencies to conduct research. Surveys have shown increased understanding and appreciation of science and technology by the students (Bozeman & Baumbach: 1995).IMPACT OF COMPUTER USE IN EDUCATIONThree multinational studies whose research influenced the view of psychological impact of school computer use are: Association for the Evaluation of Educational Achievement Computers in Education Project (IEA CompEd), the Information Technology in Education and Children Project (ITEC), and the Young Children’s Computer Inventory Project (YCCI). Three studies taken together, account for the impact of computer use in: elementary, middle school and secondary education. The IEA is a non-governmental international organisation of professional educational research centers, from more than fifty national educational systems. Surveys from this organisation are towards overall computer use in countries and the differences amongst countries in computer-related knowledge among students. ITEC research was based on classrooms as research unit ecosystems using Vygotskian theoretical foundations and mixed research methods. The researchers have categorised behaviors seen during computer use as an evidence of metacognitive development: • Relating a problem to previous problems; • Formulating appropriate questions; • Trying alternative approaches; • Evaluating one’s actions; • Analysing problems; • Recognising relationships; • Generating new ideas; • Observing central issues and problems; • Comparing similarities and differences. YCCI project began in 1990 as a longitudinal study of childhood computing in school. The project started as a collaborative effort to search for all positive effects of computer use in education without any negative effects. Major conclusions drawn from the research are that computer use in primary school has a strong positive impact on attitudes toward computers, as well as positive impact on motivation and study habits (Collis, Knezek, Lai, Miyashita, Pelgrum, Plomp & Sakomoto: 1996). When children are using computers they may feel that the computer is their new playmate or toy. A research which observed interactions of six kindergarten children using the program LOGO for three months, remarked that children often spoke to computers considering them as they playmates. Children have much desire and curiosity to explore new things, and they may simply enjoy using them without any fear that might be shown in adult use. Computers technology has been incorporated into daily practice of education in virtually every corner of the world, and major findings from the research of the three organisations mentioned all relate to the following observation: “While style of interaction and teaching and classroom organisation varied widely, the image of happy and engaged children, working with and around computers in a productive and confident way, talking to each other, working not independent of the teacher, but on the path that the teacher set for them, comes out again and again” (Collis, Knezek, Lai, Miyashita, Pelgrum, Plomp and Sakamoto: 1996). The only possible negative effect encountered through the research is the actual computer overuse to the point of infringing traditional explorations. They would experience the lack of engaging in certain events such as cooperative play and normal physical action. In the extreme case this would prove to be unhealthy and unwise. However this negative effect only applies to the case not expected of occurring when computer tools are used in an appropriate way.Using computers as a tool of teaching mathematics in a social constructivist way has shown to be a very powerful tool. With the continual development in computer technology there are always new ways offered to present knowledge in an exciting and useful way. It keeps the students highly motivated in performing mathematical activities when they are presented in an interesting way. Furthermore there are many different types of learning with computers, and some promote self awareness by observing the process of thoughts. This occurs in computer programming where the programmer (in our case child) is required to make the computer perform some operation and making the computer think, which allows them to get familiar with their own way of constructing knowledge. This does not suggest that computers should replace teachers, but rather that teachers can use computer as a powerful tool in promoting high structured meaningful learning. Teachers as well as students benefit greatly from these applications, and there would be no need of letting education halt when everything else has been changed and improved in some way by the use of computers.
