Our comments on the article "A Practical Framework for Technology Integration in Mathematics Education" by Cheah Ui Hock...
Due to the progress of the society, we cannot live without technology. technology is incorporated everywhere, likle at work, home and outside for entertainment.moreover, pupils can focus on the concept and idea rather than computation. When computation is done by technology, they can know and understand the concept better by paying attention on those. in addition, technology has its advantages and disadvantages. Thus, it is up to the teacher to decide on what, how and when to integrate technology. For instance, not all the topics are suitable to be taught through technology. Therefore, the teachers need to consider the suitability of the content as well.
2) Integaration of technology (based on the article)
- AVAIBILITY
- iNTELLECTUALITY- the knowlegde
- PREPARATION
- THERE MUST BE A CONTINUOUS PROFESSIONAL DEVELOPMENT
Tuesday, October 7, 2008
Tuesday, September 16, 2008
Constructivism vs Constructionism
Constructionism evolved from constructivism. This is as Papert was a Piagetian and his theories had a foundation of Piagetian theories. Yet, what he proposed through contructionism differes slightly from constructivism propsed by Piaget.
Constructivism suggests that knowledge is not simply transmitted from teacher to student, but actively constructed in the mind of the learner.Constructionism the other hand states that constructionism suggests that new ideas are most likely to be created ideas when learners are actively engaged in building some type of external artifact that they can reflect upon and share with others.
Constructivism supports the constructivist viewpoint.However, it emphasizes the particular constructions of external artifacts shared by learners. Although learners can construct and present knowledge or meanings without producing external products, the processes of construction are more evident when learners produce through social interaction with others and share representations of their understanding and thoughts.
Contructivism is promotes science-centered and logic-oriented learning. Constructivism, too, does the same, but, to a lesser degree.
To make a long story short, both are intended for the same goal, yet through different means.
Constructivism suggests that knowledge is not simply transmitted from teacher to student, but actively constructed in the mind of the learner.Constructionism the other hand states that constructionism suggests that new ideas are most likely to be created ideas when learners are actively engaged in building some type of external artifact that they can reflect upon and share with others.
Constructivism supports the constructivist viewpoint.However, it emphasizes the particular constructions of external artifacts shared by learners. Although learners can construct and present knowledge or meanings without producing external products, the processes of construction are more evident when learners produce through social interaction with others and share representations of their understanding and thoughts.
Contructivism is promotes science-centered and logic-oriented learning. Constructivism, too, does the same, but, to a lesser degree.
To make a long story short, both are intended for the same goal, yet through different means.
Monday, September 15, 2008
Tuesday, September 9, 2008
Evaluating Applets - The Table Tree
Evaluating Applets
View SlideShare presentation or Upload your own.
Also available on:
http://www.slideshare.net/emilley/evaluating-applets-presentation
Sunday, September 7, 2008
Fun with MSW LOGO
Hey people!!! Want to learn something fun with MSW LOGO programme??? This software might strike as a complicated one for some students. Thus, it would be a good idea for you introduce this software by teaching these fun activities. They still involve geometry. Therefore, when the form is complete, you can actually discuus with your students about the geometrical shapes that are found in it. All you have to do is to just copy the following commands and paste them in your MSW LOGO and click execute.
1. repeat 100 [fd 78 lt 56 bk 67 circle 34 bk 12]
2. repeat 103 [fd 20 bk 30 rt 57 lt 50 circle 200 lt 45 fd 80]
3. repeat 103 [fd 20 bk 30 rt 57 lt 50 circle 200]
4. repeat 13 [circle 100 lt 30 fd 100]
5. repeat 50 [fd 60 rt 70 fd 23 rt 25 bk 10 lt 20 circle 20]
6. repeat 2000 [fd 90 lt 89 fd 80 rt 89]
1. repeat 100 [fd 78 lt 56 bk 67 circle 34 bk 12]
2. repeat 103 [fd 20 bk 30 rt 57 lt 50 circle 200 lt 45 fd 80]
3. repeat 103 [fd 20 bk 30 rt 57 lt 50 circle 200]
4. repeat 13 [circle 100 lt 30 fd 100]
5. repeat 50 [fd 60 rt 70 fd 23 rt 25 bk 10 lt 20 circle 20]
6. repeat 2000 [fd 90 lt 89 fd 80 rt 89]
Tuesday, September 2, 2008
Something to share with you- LOGO
What is Logo?
Logo is a computer programming language designed for use by learners, including children. It is easy for the novice programmer to get started with Logo writing programs and getting satisfaction doing so, however Logo is also powerful and can be effectively used as a mathematical problem-solving tool without limits.
Logo has been around for a long time. It was originally developed at the Massachusetts Institute of Technology in the 1960's and was intended to allow people, even small children, to use computers as a learning tool. Logo was based on LISP, a programming language used in artificial intelligence research. Powerful computer science concepts of the procedure, recursion, programs-as-data are built into Logo. It is the language for learning and is a tool to teach the process of learning and thinking.
Logo's popularity seems to have declined in the United States, however in the past few years there has been new Logo development accompanied by renewed public awareness and enthusiasm. Logo is widely used in the classrooms of Europe and Japan. In England, Logo is a mandated part of the national curriculum. In the United States, Logo's popularity peaked at the time that there was one computer in a classroom. Now that most schools have a lab such that each student can have there own computer, using Logo in the mathematics classroom can be even more powerful.
A computer scientists may not speak very highly of Logo because it is a interpreted language rather than a compiled language. This means that the computer must interpret every command separately while the program is running rather than running a program that "compiles" the entire program into machine code. To keep a long story short, a program that is "interpreted" runs more slowly, but what seems a detriment to a computer scientist is actually a plus if you want to use Logo for education. Students can try their Logo procedures immediately and make changes and additions to their work easily with immediate results.
Recently, versions of Logo have evolved that take advantage of newer hardware and the Windows operating system, such as MSW Logo. Of course the Logo philosophy and the basics of the language remain the same. You will best get a feel for the power of Logo by investigating the activities on this web site.
Why Use Logo?
Logo is not limited to any particular topic or subject, however it has a natural tendency for the exploration of mathematics concepts and for promoting mathematical thinking. Technically, Logo is a programming language, but it would be better described as a learning language that encourages students to explore and to think about the processes involved in developing mathematical ideas. Although Logo was one of the first pieces of educational software available, it is not and will not be outdated.
The very essence of Logo involves a student in thinking about process. The creation of a product becomes more important than the finished product itself. Logo is a natural for students to make the bridge from the concrete to the abstract, particularly with geometry and algebraic thinking.
The use of turtle graphics in Logo leads to natural investigations of geometry concepts with easy-to-learn commands such that the focus is on the math and not the computer. Some very sophisticated figures, concepts and ideas can be easily produced with Logo. While the visual turtle geometry is a good way to introduce logo as a mathematical problem-solving tool, it is not limited to graphics.
So often students, particularly middle school aged students, do not see a reason to use variables, algebraic expressions and equations. Logo provides an arena where students use algebra, need algebra and enjoy the productivity of algebra. After all, a numeric answer is the result of only one question, while a carefully developed algebraic expression is all the answers.
Examining the sample lessons on this web site will give better insight into the use of Logo as a unique and most effective tool for students in the mathematics classroom. There are examples of lessons in geometry, probability, data analysis and number theory. All of these lessons promote skills of number sense, estimation and algebraic thinking.
LOGO as a Programming Language for Educational Applications
by Jim Andris
The LOGO language was developed in 1967 by the Logo Group at MIT under the direction of Seymour Pappert, author of the internationally acclaimed book Mindstorms: Children, Computers, and Powerful Ideas. It was the first language specifically designed to enable children to learn by discovery. According to Billstein, Libeskind and Lott in Logo: Programming and Problem Solving, "Logo is intended to provide an environment in which learners progress through the developmental stages of learning by exploration." (p. 1)
A Language for Integrated, Experimental Learning
Originally designed to conform to the child developmental theories of Jean Piaget, it is the theories of Jerome Bruner, a great promoter of Piagetian theory, which most clearly frame the significance of Logo as a tool £or education. In Studies in Cognitive Growth Bruner and others define three modes of representing knowledge: enactive, iconic, and symbolic. In enactive representation the significance of a stimulus is in the motor reaction it produces. Iconic representation is representing external objects with imaaes, either internal or external. Finally, symbolic representation is linguistic in nature. Each of these three modes of representation has its own characteristics. Children start out with primarily the enactive mode and progress through the dominance of the iconic mode to the modal Western adult form of symbolic representation.
One of the two major capabilities which were designed into Logo is the so-called turtle graphics. A "turtle" composed of a triangle or a turtle symbol may be controlled by commands from the keyboard or from within a Logo program. For example, the command FORWARD 50 RIGHT 90 will cause the turtle to move forward 50 units and turn right 90 degrees. Complex programs for the turtle to follow are built up from a small number of primitives.
Relating turtle graphics back to Bruner's forms of representation, it can be seen why Logo is such a powerful tool for educators. The concept of a geometric figure, say a square, can be translated into enactive mode. The child may be instructed to walk forward 10 paces, turn right, walk forward 10 more paces, and so forth, until a square has been described on the floor. A connection can be made between these movements and the symbols FD 10 RT 90 FD 10 in this way, giving a concrete interpretation of a symbolic language. Finally, what the child sees as these commands are executed, is a visual image of a turtle drawing a square on the monitor--the iconic mode.
A more contemporary approach to holistic education emphasizes "whole-brain learning". Most of us are now familiar with the concepts of "right-brained" and "left-brained" thinking. The right brain tends to specialize in schematic thinking and detailed visual imagery, the left brain, in categorical and linguistic thinking. Symbols such as FD 50 RT 90 are processed by the left brain (or if they are processed by the right brain, are processed as pictures rather than symbols). Images such as a square drawn on the monitor screen are processed by the right brain. Logo turtle graphics makes it possible to have these two modes working in concert, or if a learner has a preferred modality, Logo allows that modality to dominate.
In fact, one might even think of Logo as a language which literally allows the left brain to communicate with the right brain. The learner inputs a symbolic string at the keyboard, and observes a moving turtle tracing geometric forms on a screen. This is a perfect setup for experimental learning. Add to this the fact that children love to draw, and we have a formula for learning by doing. In order to draw a certain geometric figure, the child can experiment with different strings of symbols until the desired result is achieved. With just a little prompting from the teacher, the child is on his or her own schedule of learning.
A Language for Learning Problem Solving Skills
The advantages of turtle graphics for integrated learning are almost immediately obvious to anyone observing a child use Logo. At a somewhat more subtle level, Logo is tailor-made to encourage a problem solving approach to problems. However, we need to carefully deliniate the meaning of this assertion, for there are many definitions and levels of problem solving in education.
If we look carefully at human activity, we can see that our problem solving activity grows out of our creativity. That is, we are continually conceiving of goals or broader aims that we desire, but which we have not achieved. Out of our attempts at goal-achievement grow our problem-solving endeavours. Those of us who are systematic in our goal achievement recognize the importance of a self-conscious approach to problems, i.e. the importance of being able to document the steps in our approaches to solving problems. In other words, we have a procedural outlook: we attempt to identify sets of steps which constitute solutions to our problems. On the other hand, it is important not to become so caught up in methodology that we block the generation of novel solutions to problems.
The Logo language is procedural in just this desired sense. The learner can experiment with several lists of Logo elementary commands, or primitives. When from this experimentation emerges a list of commands that solves the problem at hand (say to draw a small circle), that list of commands can be given a name. This is done in Logo by typing the word "TO" followed by the name of the procedure, followed by the list of commands, followed by the word "END". From that point on, it is no longer necessary to type in the list of commands. Simply typing the name of the procedure will execute the entire list of commands. On the other hand, used properly, Logo can illustrate that there is nothing sacred about a particular list of commands, and alternative procedures for solving the same problem can be developed and given names, e.g. CIRCLEl and CIRCLE2.
At still a deeper level of the Logo language (beyond its procedurality), we can discern still other ways in which it significantly reflects the characteristics of human problem solving. Human problem solving is recursive, general, and hierarchical. The Logo language reflects all of these characteristics, some better than others. Perhaps if we trace the solution of a problem through its phases of development we can illustrate these characteristics of human problem solving as they manifest themselves in Loao.
Recursion may be defined as repetition. For example the procedure
TO SQUARE FD 50 RT 90 FD 50 RT 90 FD 50 RT 90 FD 50 RT 90 END
will solve the problem of finding a way to draw a square in Logo. On the other hand, the procedure
TO SQUARE REPEAT 4[RT 90 FD 50] END
solves the same problem without writing out the directions for drawing each side of the square. Instead, it simply says "draw a side, four times." Incidentally, this is called finite recursion, because we are counting to a finite number. The procedure
TO SQUARE RT 90 FD 50 SQUARE END
solves the same problem with an infinite recursion. (P.S.: CTRL-S to stop!) Recursion is an extremely important characteristic of human problem solving and it is reflected in nearly all computer languages.
Another important characteristic of human problem solving is its generality. We would prefer to solve a problem for a whole set of cases rather than have to solve the problem separately for each case in the set. Translated into our square problem, we might ask, why write a procedure for drawing a square of fixed size, when one can be written for drawing a square of any size? The procedure
TO SQUARE :N REPEAT 4[FD :N RT 90] END
solves the problem of drawing a square of size N. Perhaps and even more amazing demonstration of the power of recursion and generality combined is based on the fact that the exterior angles of a regular polygon are given by the formula 360/s, where s is the number of sides of the polygon. The following procedure draws a regular polygon of side, n, and number of sides, s.
TO POLY :S :N REPEAT :S[FD :N RT 360/:S] END
Hierarchicality is perhaps the most significant characteristic of human problem solving. This is the quality of analyzing a complex task down into subtasks, and those subtasks into still other subtasks. Every complex human procedure from writing or playing a symphony to balancing a checkbook displays hierarchicality. Logo is outstanding in its ability to reflect this important quality in its procedure. The key here is the fact that if one writes the name of procedure A as a step in procedure B, procedure A is automatically executed in proper sequence. The following procedure for drawing a house demonstrates hierarchicality, in that it is broken down into a square and a triangle, which are accomplished with the procedure POLY.
TO HOUSE :N POLY 4 :N FD :N RT 30 POLY 3 :N END
http://www.donbeaty.com/ictm/default.asp
Logo is a computer programming language designed for use by learners, including children. It is easy for the novice programmer to get started with Logo writing programs and getting satisfaction doing so, however Logo is also powerful and can be effectively used as a mathematical problem-solving tool without limits.
Logo has been around for a long time. It was originally developed at the Massachusetts Institute of Technology in the 1960's and was intended to allow people, even small children, to use computers as a learning tool. Logo was based on LISP, a programming language used in artificial intelligence research. Powerful computer science concepts of the procedure, recursion, programs-as-data are built into Logo. It is the language for learning and is a tool to teach the process of learning and thinking.
Logo's popularity seems to have declined in the United States, however in the past few years there has been new Logo development accompanied by renewed public awareness and enthusiasm. Logo is widely used in the classrooms of Europe and Japan. In England, Logo is a mandated part of the national curriculum. In the United States, Logo's popularity peaked at the time that there was one computer in a classroom. Now that most schools have a lab such that each student can have there own computer, using Logo in the mathematics classroom can be even more powerful.
A computer scientists may not speak very highly of Logo because it is a interpreted language rather than a compiled language. This means that the computer must interpret every command separately while the program is running rather than running a program that "compiles" the entire program into machine code. To keep a long story short, a program that is "interpreted" runs more slowly, but what seems a detriment to a computer scientist is actually a plus if you want to use Logo for education. Students can try their Logo procedures immediately and make changes and additions to their work easily with immediate results.
Recently, versions of Logo have evolved that take advantage of newer hardware and the Windows operating system, such as MSW Logo. Of course the Logo philosophy and the basics of the language remain the same. You will best get a feel for the power of Logo by investigating the activities on this web site.
Why Use Logo?
Logo is not limited to any particular topic or subject, however it has a natural tendency for the exploration of mathematics concepts and for promoting mathematical thinking. Technically, Logo is a programming language, but it would be better described as a learning language that encourages students to explore and to think about the processes involved in developing mathematical ideas. Although Logo was one of the first pieces of educational software available, it is not and will not be outdated.
The very essence of Logo involves a student in thinking about process. The creation of a product becomes more important than the finished product itself. Logo is a natural for students to make the bridge from the concrete to the abstract, particularly with geometry and algebraic thinking.
The use of turtle graphics in Logo leads to natural investigations of geometry concepts with easy-to-learn commands such that the focus is on the math and not the computer. Some very sophisticated figures, concepts and ideas can be easily produced with Logo. While the visual turtle geometry is a good way to introduce logo as a mathematical problem-solving tool, it is not limited to graphics.
So often students, particularly middle school aged students, do not see a reason to use variables, algebraic expressions and equations. Logo provides an arena where students use algebra, need algebra and enjoy the productivity of algebra. After all, a numeric answer is the result of only one question, while a carefully developed algebraic expression is all the answers.
Examining the sample lessons on this web site will give better insight into the use of Logo as a unique and most effective tool for students in the mathematics classroom. There are examples of lessons in geometry, probability, data analysis and number theory. All of these lessons promote skills of number sense, estimation and algebraic thinking.
LOGO as a Programming Language for Educational Applications
by Jim Andris
The LOGO language was developed in 1967 by the Logo Group at MIT under the direction of Seymour Pappert, author of the internationally acclaimed book Mindstorms: Children, Computers, and Powerful Ideas. It was the first language specifically designed to enable children to learn by discovery. According to Billstein, Libeskind and Lott in Logo: Programming and Problem Solving, "Logo is intended to provide an environment in which learners progress through the developmental stages of learning by exploration." (p. 1)
A Language for Integrated, Experimental Learning
Originally designed to conform to the child developmental theories of Jean Piaget, it is the theories of Jerome Bruner, a great promoter of Piagetian theory, which most clearly frame the significance of Logo as a tool £or education. In Studies in Cognitive Growth Bruner and others define three modes of representing knowledge: enactive, iconic, and symbolic. In enactive representation the significance of a stimulus is in the motor reaction it produces. Iconic representation is representing external objects with imaaes, either internal or external. Finally, symbolic representation is linguistic in nature. Each of these three modes of representation has its own characteristics. Children start out with primarily the enactive mode and progress through the dominance of the iconic mode to the modal Western adult form of symbolic representation.
One of the two major capabilities which were designed into Logo is the so-called turtle graphics. A "turtle" composed of a triangle or a turtle symbol may be controlled by commands from the keyboard or from within a Logo program. For example, the command FORWARD 50 RIGHT 90 will cause the turtle to move forward 50 units and turn right 90 degrees. Complex programs for the turtle to follow are built up from a small number of primitives.
Relating turtle graphics back to Bruner's forms of representation, it can be seen why Logo is such a powerful tool for educators. The concept of a geometric figure, say a square, can be translated into enactive mode. The child may be instructed to walk forward 10 paces, turn right, walk forward 10 more paces, and so forth, until a square has been described on the floor. A connection can be made between these movements and the symbols FD 10 RT 90 FD 10 in this way, giving a concrete interpretation of a symbolic language. Finally, what the child sees as these commands are executed, is a visual image of a turtle drawing a square on the monitor--the iconic mode.
A more contemporary approach to holistic education emphasizes "whole-brain learning". Most of us are now familiar with the concepts of "right-brained" and "left-brained" thinking. The right brain tends to specialize in schematic thinking and detailed visual imagery, the left brain, in categorical and linguistic thinking. Symbols such as FD 50 RT 90 are processed by the left brain (or if they are processed by the right brain, are processed as pictures rather than symbols). Images such as a square drawn on the monitor screen are processed by the right brain. Logo turtle graphics makes it possible to have these two modes working in concert, or if a learner has a preferred modality, Logo allows that modality to dominate.
In fact, one might even think of Logo as a language which literally allows the left brain to communicate with the right brain. The learner inputs a symbolic string at the keyboard, and observes a moving turtle tracing geometric forms on a screen. This is a perfect setup for experimental learning. Add to this the fact that children love to draw, and we have a formula for learning by doing. In order to draw a certain geometric figure, the child can experiment with different strings of symbols until the desired result is achieved. With just a little prompting from the teacher, the child is on his or her own schedule of learning.
A Language for Learning Problem Solving Skills
The advantages of turtle graphics for integrated learning are almost immediately obvious to anyone observing a child use Logo. At a somewhat more subtle level, Logo is tailor-made to encourage a problem solving approach to problems. However, we need to carefully deliniate the meaning of this assertion, for there are many definitions and levels of problem solving in education.
If we look carefully at human activity, we can see that our problem solving activity grows out of our creativity. That is, we are continually conceiving of goals or broader aims that we desire, but which we have not achieved. Out of our attempts at goal-achievement grow our problem-solving endeavours. Those of us who are systematic in our goal achievement recognize the importance of a self-conscious approach to problems, i.e. the importance of being able to document the steps in our approaches to solving problems. In other words, we have a procedural outlook: we attempt to identify sets of steps which constitute solutions to our problems. On the other hand, it is important not to become so caught up in methodology that we block the generation of novel solutions to problems.
The Logo language is procedural in just this desired sense. The learner can experiment with several lists of Logo elementary commands, or primitives. When from this experimentation emerges a list of commands that solves the problem at hand (say to draw a small circle), that list of commands can be given a name. This is done in Logo by typing the word "TO" followed by the name of the procedure, followed by the list of commands, followed by the word "END". From that point on, it is no longer necessary to type in the list of commands. Simply typing the name of the procedure will execute the entire list of commands. On the other hand, used properly, Logo can illustrate that there is nothing sacred about a particular list of commands, and alternative procedures for solving the same problem can be developed and given names, e.g. CIRCLEl and CIRCLE2.
At still a deeper level of the Logo language (beyond its procedurality), we can discern still other ways in which it significantly reflects the characteristics of human problem solving. Human problem solving is recursive, general, and hierarchical. The Logo language reflects all of these characteristics, some better than others. Perhaps if we trace the solution of a problem through its phases of development we can illustrate these characteristics of human problem solving as they manifest themselves in Loao.
Recursion may be defined as repetition. For example the procedure
TO SQUARE FD 50 RT 90 FD 50 RT 90 FD 50 RT 90 FD 50 RT 90 END
will solve the problem of finding a way to draw a square in Logo. On the other hand, the procedure
TO SQUARE REPEAT 4[RT 90 FD 50] END
solves the same problem without writing out the directions for drawing each side of the square. Instead, it simply says "draw a side, four times." Incidentally, this is called finite recursion, because we are counting to a finite number. The procedure
TO SQUARE RT 90 FD 50 SQUARE END
solves the same problem with an infinite recursion. (P.S.: CTRL-S to stop!) Recursion is an extremely important characteristic of human problem solving and it is reflected in nearly all computer languages.
Another important characteristic of human problem solving is its generality. We would prefer to solve a problem for a whole set of cases rather than have to solve the problem separately for each case in the set. Translated into our square problem, we might ask, why write a procedure for drawing a square of fixed size, when one can be written for drawing a square of any size? The procedure
TO SQUARE :N REPEAT 4[FD :N RT 90] END
solves the problem of drawing a square of size N. Perhaps and even more amazing demonstration of the power of recursion and generality combined is based on the fact that the exterior angles of a regular polygon are given by the formula 360/s, where s is the number of sides of the polygon. The following procedure draws a regular polygon of side, n, and number of sides, s.
TO POLY :S :N REPEAT :S[FD :N RT 360/:S] END
Hierarchicality is perhaps the most significant characteristic of human problem solving. This is the quality of analyzing a complex task down into subtasks, and those subtasks into still other subtasks. Every complex human procedure from writing or playing a symphony to balancing a checkbook displays hierarchicality. Logo is outstanding in its ability to reflect this important quality in its procedure. The key here is the fact that if one writes the name of procedure A as a step in procedure B, procedure A is automatically executed in proper sequence. The following procedure for drawing a house demonstrates hierarchicality, in that it is broken down into a square and a triangle, which are accomplished with the procedure POLY.
TO HOUSE :N POLY 4 :N FD :N RT 30 POLY 3 :N END
http://www.donbeaty.com/ictm/default.asp
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