Computer Based Mathematics Education

Computer Based Mathematics Education (CBME) refers to a mathematics education method that is enriched by using computers. In order to analyze CBME in detail, the disciplines of mathematics, education, and computers including their features should be explored more and understood deeply because the domain of CBME is based on the intersection of these three disciplines and CBME rises up over this domain.
In mathematics, intersection of two different sets is defined as a set which contains elements from both sets so that intersection set demonstrates the properties of both sets. However, the situation is bit different in our case because although the features of these disciplines generate a base for the science of CBME, CBME has synthesized these disciplines and developed its own interpretation and features. This is true for all intersections as in the example of water. Water is completely a different substance than both hydrogen and oxygen.

This analysis will be performed in three stages: In the first stage, main domains and features of mathematics, education, and computers will be analyzed in order to clarify which aspects of these disciplines demonstrate what kinds of contents. In the second stage, the intersection of these main domains and their double interactions will be analyzed, and potential research areas will be described. Finally, in the third stage, a deeper analysis will be performed over the intersection of these three domains in order to explore the types of new and uncovered connections and features of this area.

First Level of Analysis
Mathematics
Mathematicians are separated into two groups when defining the features of mathematics. One group describes mathematics as a science which helps us to discover the world surrounding us whereas the other believes that mathematics is a science created by human mind. The first description suggests that mathematics spread all over the world and mathematicians are the scientists who discover these secrets (i.e Content of the book, Number Mosaics: Journeys in Search of Universals, written by Kanga is a great example for this interpretation). On the other hand, according to the second group there is no mathematics in the world instead in human mind. Human mind creates all these axioms, definitions, relations, and proofs just to advance the mathematical knowledge but not to better understand the physical world. Bereiter and Scardamalia(1996) explain this by stating that "Formal Mathematics knowledge consists of World 3 objects, constructions of human intellect."

For further reading of distinctions between the interpretations of mathematics, 11 chapter of Handbook of Academic Learning is a great source.

Education
Although there are many definitions of education in various disciplines, it is safe to classify the aims of education into three general groups: Subject matter, skills, and improving learning abilities. Subject matter refers to the subjects to be studied and therefore to be learned through a specific of education while the term of skills refers to the skills to be gained through education as problem solving skills, cognitive skills (analyzing, synthesizing, and creating), and metacognitive skills. Despite the importance of learning subject matter in all kinds of education types, educators also try to develop or advance various skills of the students in order to have them succeed in their academic life. In addition to these, improving learning abilities is another aim of education because it is developed after learning subject matter and advancing problemsolving, cognitive and metacognitive skills. The more people learn the easier and quicker they are to learn as a result of their previous experiences, i.e. people who know any second language learn the third one more quickly, fourth one even more quickly, and so on…

Computers
Generally speaking, the features of computers can be classified as storage, process, and delivery. Storage and process refer to the storing and processing information while delivery refers to the transportation of information from one place to another via diskettes, CDs and/or some cutting-edge technologies like Intranet, Internet. These three main features can not be separated with each other completely because there are some grey areas which are impossible to distinguish, and they are running together.

The data and/or the information coming from the environment to the computer will be stored in storage units either using for later purposes or processing by microprocessors. The storage and processors also work either together or one after another as well as the transportation units, because transportation and updating information processes are also in continuous progress.

However, only the related features of computers will be taken into account in this analysis because computers and computer technology is huge area of science by itself. Internet and Internet technology is the most popular feature which is commonly used in educational environment.

Second Level of Analysis
Mathematics Education
Unlike many sciences, mathematics causes permanent changes in people’s life, and affects their interpretation of the world. That is why; mathematics education has excessive responsibilities on human's life and it is not quite right to say that mathematics education consists of only subject matter. In contrast, doing and knowing mathematics improve understanding, exploring, analyzing, problem solving skills of people.
Although some of these skills help students to explore, to understand, and to discover the world around or to survive in the world they live, like problemsolving, many of these skills are gained through the process mathematics education, which is a mental process.

Mind and philosophy of mind have been great debates for centuries. Epiphenomenalism and Cartesian dualism are some theories about mind and its causes, relations with physical world. Popper defines world in a three world chema, which can be examplified in mathematics as follows:
# World 1: World of manipulatives and legos used to define mathematical objects.
# World 2: World of mental objects like numbers, symbols and events like solving equation, cancellation of polynomials.
# World 3: World of product of human mind like theorems, proofs, set theory, derivatives, integrals.

Each world has its own domain and area of interests. Bereiter and Scardamalia (1996) discuss the Popper's three-world schema, explain these differences in educational point of view in order to pose a question that which approach or approaches should be superior in school curriculum. They suggest that since each has different advantages and disadvantages school curriculum should employ a combination of all these three worlds.

Computers in Mathematics
Although use of computers in mathematics vary to some extent, and it is difficult to categorize completely, analyzing with 2x3 matrix
may provide a systematic look for us.
In case of discovery of world, database features and some office tools will be used to store the data gathered throughout discovery of the world, and data management feature of computers helps a lot to organize, reorganize and restructure of these datas. It is impossible to forget the effects of computers in exploring fractals in this stage, because the exploration journey of fractals mostly depends on the developments in features of computers.

On the other hand, whereas there are lots of examples of fractals in the real world, there some imaginary fractals improved in human's mind like 4 or n dimensions of vectors. i.e., real life examples of 4 dimension of vectors were discovered after mathematicans'. Computers are used to help mathematician in various ways:

*Mathematicians share their experiences with each others and public via internet.
*They use computers to visualize what they imagine in their mind after introducing with high level of software.
*Computers are used to generalize mathematical relations like fractals because developments in capacities of computers provide mathematicians reiterate some mathematical operations thousands time in a small amount of time.
*Mathematician use computers for calculation, evaluation of huge and complex mathematical operations, solving equations by numerical methods.

Computers in Education
Computers are used in education in a number of ways: Tutorial, Hypermedia, Simulation, Drill and Practice, Educational game, Tools and Open-Ended Learning Environment, Web-Based Learning, and Online Collaborative Environment (Alessi and Trollip, 2001).
*Tutorials: Tutorials are types of software that present information, check learning by question/answer method, judge response and provide feedback (Alessi and Trollip, 2001) and usually provide students study personally. MathMedia Educational Software (2008), Bilelim Geometry (1999), BioLab (2006) are some examples of this type.
*Hypermedia: This type of software provides students a database of information with multiple navigation methods and learning facilities, and also, a freedom of independence during the learning (Alessi and Trollip, 2001). Art and Life in Africa (1999), How your body works (1997) are some examples of this type of software.
*Simulations: This type of software provides students inteaction with simulations in order to learn as in their real life (Alessi and Trollip, 2001) and use the storage and processing features of computers. Some leading examples are ChemLab (1994), Crocodile-Clips (2003).
*Drills and Practice: Unlike Tutorials, this type of software provides only test of information and feedback but not presenting information at the beginning (Alessi and Trollip, 2001), and they look like electronic versions of drill and practice text books. Pilot Software is an example for this type.
*Educational Games: They are more like simulations and used from elementary to college students(Alessi and Trollip, 2001). The Incredible Machines (2000) is a good example of this type.

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