# Level of Substantiation in Math Training

What is a substantiation of solution? This is a complete theoretical explanation of each solution step. An ability to substantiate solutions is an indispensable condition for independent problem solving. Let us consider a practical question: how to use computer technologies to learn substantiation skills?

To clarify the matter, let us perform a small mental experiment. Let us write down the complete substantiation of square equation solution. This means that you have to provide objectives for all solution steps, write down all rules and definitions related to solution and prove all formulas and theorems used in the course of solution. I can ensure you that this job will result in a number of written pages. Is it really necessary to substantiate each solution completely? Certainly not, but under one condition – that you know how to carry out this substantiation. If you do not know how, then the learning process reduces to “one-type-task-training”, when you remember not the cause-and-effect mathematical relations, but a concrete technique applied to one particular structure. As soon as this structure slightly changes, you cannot solve anymore. To build up the robust problem-solving skills one needs to develop an ability to substantiate solutions – that is to be able to explain which formulas, rules, definitions and theorems were used in solution. Don’t look for such level of substantiation of each example in paper books! Books cannot provide this information simply due to limited power of printing industry.

What this has to do with computer technology? Everything! The point is that computers enable solving this problem. Technical implementation is hardly interesting – who is interested in the cycle of Karno? We care for horsepowers, petrol consumption, driving system and – let’s go for a spin!

## About “horsepower” or what you can squeeze out of the EMTeachline mathematics software:

• Step by step solution of each example
• Objective for each transformation step
• Definition and rule used at each transformation step
• Formulas and theorems used in the solution
• Complete proof of used formulas and theorems
• Analysis and evaluation of your performance
• Methodical recommendations for error correction

## “Petrol consumption” or overflow of information

It seems that the EMTeachline software offers too much. Don’t worry. Be happy! If you do not need some details, just do not call them. However, should you require complete information – it is always under hand, in a button click.

## About “driving system” or how difficult is the EMTeachline software control?

The program control is reasonably straightforward. An extensive help is provided.

Do you want to try? Download free trial versions of the EMTeachline mathematics software. Or, alternatively, check our online demos.