# Math Theory in Training Software

Let us answer two questions: Why we must learn math theory? How training software helps in studying math theory?

The difference between mathematics and all other sciences is that math is based on axioms. The essence of an axiomatic method of constructing a theory can be described as follows:

• The basic concepts are defined
• A set of facts, connecting the defined concepts, is postulated without proof. This set is called a set of axioms
• All other relations of a given theory are proved based on these axioms

Algebra, for instance, is based on axioms of real numbers. Also, what is learnt in schools under the name of “distributive law of multiplication” is one of the axioms. The same theory can be based on different systems of axioms. Geometry, for instance, can be constructed on the basis of axioms of Euclid and on axioms of vector space. However, we shall leave this subject for the experts. For an ordinary user is enough to understand that, after axioms are postulated, every single mathematics statement should be proved. Proved!

## Studying mathematics means studying various proofs.

What is a theorem? Essentially, there is no difference between a theorem and any solved task. A task gets the status of a theorem when it is frequently used. The theorem structure considerably facilitates various calculations. Imagine that the formula of square equation solution is not singled out as a theorem. In this case, every time that you solve a 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 math theory? 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 theorems without being capable to prove them, you are engaged in anything but studying mathematics.

I assume that the necessity of studying math theory is now accepted. Let us consider how computer technologies can help in this uneasy business. “Theoretical material” consists of:

• Definitions
• Axioms
• Theorems and formulas

In relation with these categories, the following assignments can be considered:

• To learn
• To study
• To apply

### To learn

How do we learn? Basicly, we learn best through repetition. We write, we repeat again and again, until we know the subject “by heart”. The help of software in the learning process can be very substantial. The software:

• Can check your knowledge as many times as necessary and does not get tired at all
• Can prompt you the correct answer if you doubt
• Takes a part of your work: you do not need to write formulas, the computer writes them for you
• Selects material for study

### To study

How can software help in studying theorems and formulas? Here is the list of 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. All your eventual questions are answered
• It provides proofs of all auxiliary formulas and theorems occurring in the course of the parent proof
• It provides definitions, formulas and theorems used in the proof of all auxiliary theorems
• In pedagogics, the learning process above is called “the disclosure of the internal cause-and-effect relationships”. And finally, please estimate how much time and patience you would need to do same job without the computer.

### To apply

With the help of software, this job is extremely easy. Just select 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 provide you with complete theoretical substantiation of this solution.