How we wrote the planimetry course

school dream

When I was in high school, the most difficult subject for me was geometry. No, other subjects were also difficult, like English or Russian, but I understood that the difficulty in mastering them could be solved by mass reading of fiction. I didn’t particularly like fiction (the exception was the works of Jules Verne), and there was simply no other literature. In the mid-90s, only a select few had the Internet, and it was possible to find and freely download a book only after 2005. In general, the humanities did not particularly excite me, and only geometry was the subject, the lessons of which made me sad and despondent. At the lessons of geometry, I had a dream: to find and get even with the author of the geometry textbook, and then write my own “understandable” textbook. As you can guess, I could not fulfill the first part of the dream for the reason that the author of the textbook had not been alive for a long time, but I simply forgot about the second part.

From dream to action

I again remembered my dream when my wife and I had a child a few years ago. Then we began to look for and select literature on pedagogy, as well as to study modern school textbooks in all school subjects. Since we already had an engineering and mathematical education behind us, we began our study with textbooks in the natural sciences.

First, physics textbooks were analyzed, then they moved on to algebra textbooks, and then to geometry. We started our work by opening our old school textbooks. When you are still a schoolboy, you have no time to talk about the merits or demerits of a textbook. The student simply takes what is given to him and “crams” what he does not understand. He doesn’t really have a choice. The sword in the form of an “exam” no longer hung over our necks, so we could calmly and freely analyze.

Analysis

Before establishing the criteria for analysis, we tried to answer a simple question: “why study geometry?”. In voluminous studies, starting from the end of the 18th century, this question was raised more than once, and in most of them the answer sounded something like this: “a schoolboy needs geometry to teach him to reason.” A wonderful answer, after which a new question is immediately posed: how and to what extent do you need to learn geometry in order to learn how to reason? My wife and I could not answer this question on our own, so we turned to the study of pedagogical dissertations and reviews of published geometry textbooks. Further, we singled out the most significant, as it seems to us, criteria for the analysis of the textbook:

1. The timeliness and expediency of introducing new divisions.

2. The absence of conceptual “omissions”.

3. logical sequence.

4. Detailed and easy-to-understand solutions to problems and explanations of theorems.

5. Availability of definitions.

6. Usefulness and feasibility of tasks and exercises for independent solution.

Let’s explore each criterion with examples.

Timeliness and expediency of introducing new definitions

If on the first pages of the textbook we are talking about the arc of a circle and the sector of a circle, and interesting tasks for these figures appear only on the penultimate ones, then there is no particular expediency in introducing these concepts at the first stages of studying geometry. Overloading with unworkable definitions only tires the child’s brain and does not solve any educational goal. This is especially evident in the classification of angles. Often, after defining a right angle, definitions of acute and obtuse angles are given in the very first sentences. For what? How can a student distinguish an acute angle from an obtuse one without a protractor? It is clear that there can be no answer to this question on the first pages of the textbook, then why introduce this definition? We believe that the definition of acute and obtuse angles must be introduced when we are able to construct an angle equal to the given one, and also know how to construct a right angle. Only in this case can we distinguish an acute angle from an obtuse one. Also, when studying angles, we usually immediately get acquainted with the angle bisector. At this stage, the student is not yet able to bisect the angle with a compass and ruler. It is much more reasonable to define the angle bisector when he can already divide the angle in half without a protractor.

Regarding the protractor, a special conversation. We noticed that the older the textbook, the longer the author delays acquaintance with this device. It is true, because the theorem that the magnitudes of central angles are proportional to the magnitudes of their corresponding arcs appears somewhere in the middle of the textbook. It is logical to introduce the protractor only after the proof of this theorem, then the child will not have a feeling of “magic” of this device.

Lack of conceptual “innuendo”

It is interesting that in the textbook it can be shown how to measure a segment or measure an angle, but few of the authors define what “measure a value” is. This definition is given in old textbooks.

The concept of “inner area of ​​the corner” is introduced, but few people say how to determine it if this area is not explicitly set. This leads to difficulties in comparing angles. If we can’t compare angles, how can we tell an acute angle from an obtuse one?

logical sequence

The lack of logical sequence is seen in the example of the interior angles of triangles. All textbooks give a theorem that the sum of the interior angles of a triangle is 180 degrees. That’s good, but a triangle has three corners. It is logical to first find out the value of one angle of the triangle (greater than, less than or equal to), then determine the value of the two angles of the triangle, and only then proceed to the sum of the three internal angles.

There was also a consequence of the Pythagorean theorem that the hypotenuse of a right triangle is greater than any of the legs. This is true, but this is a consequence of the fact that the larger side of the triangle lies opposite the larger angle of the triangle, and not from the Pythagorean theorem.

Detailed and easy-to-understand solutions to problems and explanations of theorems

All tasks solved on the pages of the textbook should be analyzed in as much detail as possible. There should be no omissions. In most textbooks, the tasks are analyzed in sufficient detail. Reference in this regard are the pre-revolutionary textbooks of geometry.

Availability of definitions

Definitions should be submitted in the language that is most accessible to the student. At the same time, scientific character naturally suffers, but a school textbook, in our opinion, is not a scientific work. The most “accurate and scientific” geometry textbooks have already been written, and they are beyond the power of the “average” student.

Usefulness and feasibility of tasks and exercises for independent solution

Tasks should be useful and feasible. Their number should not be large, since the textbook is not a problem book. Too easy tasks do not teach anything, and after solving them, the student has a feeling of wasted time.

Analysis result

After analyzing about five Soviet and modern geometry textbooks, as well as about five pre-revolutionary textbooks, we came to the conclusion that A.Yu. Davidov and “Manual of Geometry” by A. Malinin and F. Egorov. However, these tutorials are a little outdated and require significant editorial revision.

Action

Then we decided to sit down at the keyboard on our own and write our own “ideal textbook” of geometry. It sounds too loud, so in the end we got a manual for the school planimetry course, rather than a textbook. We started typing work at the end of 2022. The general plan of the manual and the structure of some chapters were formed in the course of a two-year analysis of other textbooks and manuals. Naturally, we did not write the course from scratch. Some of the evidence we came up with on our own, some we borrowed from other guides. The structure of the material was thought out independently, based on Davidov’s textbook. Almost all tasks are taken from the manual of Malinin and Egorov. We did not focus on assignments from the USE, because we believe that the goal of learning to solve geometric problems and the goal of passing the USE are two different goals.

Tools

Typing was done in the TeX Studio environment. The compiler is PdfLaTeX. Lvovsky’s manual on PdfLaTeX was used as reference literature. The drawings were made in Ipe. This program develops the skill of computer geometric constructions.

Result

As a result, we published a 176-page manual. Here is an incomplete list of differences between our manual and currently accepted textbooks:

• We refuse lengthy introductions and historical introductions. There is a reference literature for this, which can now be easily accessed.

• As early as possible, we introduce the circle.

• We give the definition of a geometric construction.

• The right angle is set as a measure of other angles. We introduce the degree measure only on the last pages, and in the same place we talk about the protractor.

• We actively use construction tasks with the help of a compass and ruler. We believe that geometric constructions best develop the skill of analyzing and synthesizing a problem.

• As early as possible, we introduce the perpendicular bisector.

• The first criterion to be proved for the equality of a triangle is the “third” criterion, as the most understandable and obvious.

• Before establishing that the sum of interior angles is equal to two right angles, we thoroughly work on the topic of relationships between the angles and sides of a triangle.

• Set the value of the sum of two angles of a triangle adjacent to one of its sides. This fact allows us to prove the criterion for two lines to be parallel.

• We give the fifth postulate of Euclid in the following form: the perpendicular and the oblique of one straight line will intersect with sufficient continuation.

• We try not to introduce new terms unless absolutely necessary.

• We are familiar with the parallelogram, square, rhombus, rectangle and trapezoid after Ptolemy’s theorem.

• If we are talking about scale, then we are talking about how to build it and how to use it. The described level can be made at home from improvised means.

• Points must be marked on the drawings.

• We believe that there should be a separate textbook on trigonometry, as it was before the middle of the 20th century, and it is better to get acquainted with vectors in the lessons of physics and algebra.

hidden text

The result of our work can be downloaded here: http://www.stepanov.top/planimetry.pdf

What’s next?

After the transfer of the layout to the printing house, work began on the second edition. We decided that it is necessary to add as many parsed tasks to the build as possible. Solutions to these problems must be explicitly divided into analysis, synthesis, proof, and research. It is worth explicitly mentioning the main methods for solving construction problems. Small drawings will be enlarged.

PS One copy was given to a school teacher. She said that she would use the book in her work. We hope that none of the students will dream of getting even with us for the fact that the book is written incomprehensibly.