Abstract:

What a surprising thing is progress! It seems that all the best must be preserved and transmitted to the future generations. But no: parallel to this process of accumulation values goes the process of their gradual weathering. As a result, one often see that the subsequent generations do something worse than their predecessors. Exactly this is the case with the teaching students to the problem-solving with the help of equations. When seen how this is doing in modern textbooks, you involuntarily ask yourself: Was there, Monsieur Descartes, really? Was there, Sir Newton? Maybe, they were not? In our paper, we give some examples showing that we still have much to learn from these great people.

We begin with a quotation from "Geometry" by Descartes, in which he explains how to get equations for the problem-solving, and show that in the quoted text it is said extremely clear and precisely all that teacher must teach his students on the application of equations to the problem-solving. After that, we give some examples from nowaday's educational practice to demonstrate how Descartes' words help to exclude some pedagogical mistakes in mathematics teaching.

After that, we turn to another remarkable book, "Universal Arithmetics" by Sir Isaac Newton. The ability to translate the conditions of problems expressed in words into the language of equations is considered one of the most important skills that should be mastered by mathematics students. However, it is well known that many students the skill are given with great difficulty. So useful is for each teacher of mathematics to get acquainted with the book by Newton in which he, among other things, pays much attention to the translation conditions of the problems from the "verbal" language to the language of algebra. We will give the passage from Newton's book very useful in this attitude and demonstrate the exceptional usefulness of Newton's advice on a number of examples from the modern practice of mathematics teaching.