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In this article considered are the following questions:
- Why we must learn math theory?
- How can training programs help in studying theory?
The difference between mathematics and all other sciences is that math is based on axioms. What is an axiom and what is an axiomatic method of constructing a theory? The essence of axiomatic method consists in the following:
- The basic concepts are defined
- A set of facts, connecting the defined concepts, is postulated without proof and is called a set of axioms
- On basis of axioms, all other relations of a given theory are proved
Algebra, for instance, is based on axioms of real numbers. Also, what is learnt at schools under the name of the distributive law of multiplication is one of axioms. The same theory can be based on different systems of axioms. For example, geometry can be constructed on basis of axioms of Euclid, but also on basis of axioms of vector space. However, we shall leave this for the experts. It is enough to understand that in mathematics, after axioms are postulated, everything should be proved. To study mathematics means to study various proofs and substantiate these proofs.
Now, let us make sense of a theorem of the theory. Essentially, there is no difference between the theorem and any solved task. A task gets the status of theorem when it is frequently used. The use of the theorem structure considerably facilitates various calculations. Imagine that the formula for solution of square equation is not singled out as a theorem. In this case, every time that you solve the square equation you have to carry out the proof of this formula.
Let us go back to the question: why does one need to learn theory in mathematics? Firstly, if you are not capable to prove a theorem, you will never learn how to use it; at best, you can be brought to apply it to a small number of tasks. Secondly, theorems are typical tasks of the studied theory. By studying theorems, you simultaneously learn methods and techniques basic for a given theory. This is the necessary basis for solving problems on your own. And thirdly, by applying a theorem without being capable to prove it, you are engaged in anything but studying mathematics.
I assume that the necessity of studying math theory is now realized. Let us consider how the computer technologies can help in this uneasy business. "Theoretical material" consists of:
Definitions
- Axioms
- Theorems and formulas
- In relation with these categories, the following educational tasks can be considered:
To learn
- To study
- To apply
- To learn
How do we learn? We write, we repeat again and again, until we know the subject "by heart". How can the computer help in this process? The help of computer can be very substantial:
It can check your knowledge so many times as necessary and it does not get tired at all
- It can prompt you the correct answer if you doubt
- It takes a part of your work: you do not need to write formulas, the computer writes them for you
- It selects the material for study
- If you forget the proof of the formula you are learning, it provides you with the complete proof and with all definitions used in this proof
- To study
This task refers to theorems and formulas. How can the computer help in studying? This is the list of computer's options:
It provides the detailed proof, step by step
It provides all definitions, formulas and theorems used in the proof. Every step is explained at large. Your questions are answered
- It provides proofs of all auxiliary formulas and theorems appearing in the course of the parent proof
- It provides definitions, formulas and theorems used in the proof of auxiliary theorems
- In pedagogics, the entire process above, provided by a computer, is called "revealing of internal cause-and-effect relationships". These relationships you need to study and learn.
- And finally, well, estimate how much time and patience you would need to do the same job using paper-books.
To apply
This is extremely easy. Just choose a formula, press the button and you get a set of problems in which the studied formula is used. With regard to each problem, the program will write down the detailed course of solution and will give the complete substantiation of this solution.
Do you want to try? Load the free-of-charge program "EMFormulaLight".
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