Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →
60
HOW TO SOLVE A PROBLEM!
What is better - to solve the problem yourself or read its solution? For a math lover, there can be no two answers to this question; the most interesting and useful thing is to find the solution yourself. Well, what if you spend a long time working on a problem, but you stubbornly fail to achieve a solution? It seems that both this and that have been tried, but still it is not possible to find the path leading to the goal.
However, advice on what to do if a problem “cannot be solved” is usually not given to mathematical problems. You have to open the ready-made solution placed at the end of the book and read it. At the same time, I involuntarily felt a feeling of dissatisfaction, annoyance with myself - after all, everything was clear, even simple, some small sense was missing. Now, if only someone could tell me where to start, what additional construction to make, then I could complete the decision myself! It’s even worse if, along with the solution, only the answer is included in the book: then you remain stuck with a treacherous task without any help…
Numerous observations (and, in particular, the experience of applied university exams) show that many students do not imagine how to attend to the solution of a problem if it is not an exercise on a “known type”, but is presented in a rather unusual way, if its formulation differs from the canons accepted in textbooks. Great difficulties are presented by tasks in which the center of gravity lies not in formal calculations, but in consistent logical reasoning, requiring the complex use of various sections of the school course.
Yungeyel will find such problems in sufficient quantities in the new problem book by E. B. Vakhovsky and A. A. Ryvkin on elementary mathematics*). There are 477 problems in total. Almost each of them requires an individual approach and a fresh approach, i.e., it provides an opportunity to test a small independent study, and not just apply some standard solution algorithm.
But what’s interesting about the book by E. B. Vakhovsky and A. A. Ryvkin is that the author did not just select various problems and provide them with solutions. To the traditional formula “problem condition - solution” an intermediate link is added - “instruction”, which has become its goal to give the necessary “additional impetus” to those readers who experience difficulties in solving a particular problem.
The authors, as it were, put themselves in the place of the person solving the problem, try to see and understand the source of its occurrence, direct his efforts in the most natural direction, dividing the solution into several stages (many tasks are given two instructions, and for some even three!), each of which the student can carry out himself. By skillfully and naturally helping the reader, leaving him a reasonable share of work and therefore maximally stimulating his independence, Youngheim allows amateur mathematicians to develop their “mathematical understanding”, to accumulate that experience that will later help to find approaches to new problems.
There is no doubt that the new problem book, both in its content and thanks to the “semi-programmed” system for presenting materials, will be very useful to students, especially those who live in rural areas, towns and small towns where there are no physics and mathematics schools.
The book by E. B. Vakhovsky and A. A. Ryvkin is no less