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 PROGRAMMING
Seymore 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 EDUCATION
Plato: 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 EDUCATION
Three 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.
- Study of psychology shows that
some events are more memorable when unique surroundings
and experience occurs.[Return]
- BIBLIOGRAPHY
- Abbot, Chris. “ICT: Changing Education.”
London: RoutledgeFalmer, 2001.
- Bornas, X. & Llabres, J. “Helping Students
Build Knowledge.” Information Technology in Childhood
Education Annual, Association for the Advancement of Computing
in Education (AACE), 2001.
- Bozeman, C.W. & Baumbach, J. D. “Educational
Technology: Best Practices from America’s Schools.”
Florida: Eye On Education, Inc., 1995.
- Clements, H.F. & Nastasi, K.B. “Metacognition,
Learning, and Educational Computer Environments.”
Information Technology in Childhood Education Annual, AACE,
1999.
- Davis, B.R., Maher, A.C. & Noddings, N.”Constructivist
Views on the Teaching and Learning of Mathematics.”
Journal for Research in Mathematics Education, Virginia:
National Council of Teachers of Mathematics, Inc., 1990.
- Druin, A. & Solomon, C. “Designing Multimedia
Environments for Children.” Canada: John Wiley &
Sons, Inc., 1996.
- Forcier, C.R. & Descy, E.D. “The Computer
as an Educational Tool.” 3rd edition, New Jersey:
Pearson Education, Inc., 2002.
- Geisert, G.P. & Futrell, K.M. “Teachers,
Computers, and Curriculum: Microcomputers in the Classroom.”
3rd edition, Needham Heights, Allyn & Bacon, 2000.
- Jonassen, H.D., Peck, L.K. & Wilson, G.B.
“Learning With Technology: A Constructivist Perspective”
New Jersey: Prentice Hall, Inc., 1999.
- Maddux, D.C., Johnson, L.D. & Willis, W.J.
“Educational Computing: Learning with tomorrows technologies.”
3rd edition, Needham Heights: Allyn & Bacon, 2001.
- Papert, S. “Mindstorms: Children, Computers,
and Powerful Ideas.” New York: Basic Books, 1980.
- Yelland, N. “Reconceptualising Schooling
With Technologies for the 21st Century: Images and Reflections.”
Information Technology in Childhood Annual, AACE, 1999.
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 PROGRAMMING
Seymore 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 EDUCATION
Plato: 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 EDUCATION
Three 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.
- Study of psychology shows that
some events are more memorable when unique surroundings
and experience occurs.[Return]
- BIBLIOGRAPHY
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