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Isometries on The Plane (Translations, Rotations and Reflections)

    This page was made to return to Internet part of all that I have learned through it for the past 15 years, at least. "Uso de Transformações Geométricas na Revigoração do Ensino de Geometria Plana" ("The use of Geometric Transformations on the Invigorating Process of Plane Geometry Teaching.") is a way of bringing the general public, studying or teaching mathematics in elementary and high school courses, techniques and philosophy involved in the use of isometries on the plane. All animations used are at the next topic. Text, in pdf (in portuguese), is at the bottom of this page. First, however, I will make some comments that I consider relevant and I did not think it would be appropriate to place on the text of the thesis, despite all the freedom atmosphere provided to me by IMPA and by my supervisor (Prof. Eduardo Wagner).

Playlist on Youtube with all 23 animations

To browse through the animations, click the  and  buttons at the bottom bar of the player below.

Yaglom Russian Textbooks

     My delight at Russian (former Soviet Union) textbooks has followed me since adolescence. In the high school, through the book "Selected Problems of Elementary Physics", I got a level of excellence on topics addressed that I certainly would not acquire with national books available.

     And what makes these books so good?    

     Russian books - in general - value the concept, much more than the calculation. Instead of training in the use of many tools, they exploit until exhaustion the using of tools that, in theory, are already known by the student. This is what I call "sustainable education": instead of learning a new tool every day, choose a good one and become great on it! It is the ecological concept of reusing incorporated into teaching. Thus, in the physics book I mentioned, many problems that were solved with elementary mathematical tools, a common course would undergo cumbersome procedures - in general - of differential and integral calculus to achieve the same results. Something similar can be found in Yaglom and the work I present here. Fagnano's problem, for example, which is practically solved without a single calculation, in a common course, would hardly be done without good knowledge of differential and integral calculus of several variables.
    In short, "cracking a nut with a sledgehammer" is avoided.

    This option on making the most of the knowledge that one has ends up being brought to day-by-day life, making student more naturally understand the need to reuse other resources that are available to him, among them, his skills, his friendships and, above all, natural resources.

WNN* Theory

     In the course of justifying "why teach it?", "why learn it?", I needed to spend some time reflecting on the raw material that needs to constitute the stuff of what a good school** should teach. Ie, what elements would I mark as the basic content taught by a good school? In my opinion, these basic elements can be divided into three major groups:

1) Pieces.

     The most ordinary and mundane concept of school necessarily involves transmission of information and training in the use of technical and scientific tools that can be used in daily life or that are prerequisite for admission of students in most popular careers. We're talking about things like percentage, interest, health programs, recent history of your country and world, political division of states, compounds, chemical or physical processes, and appliances used at home or in a common workplace. A good school, therefore, is able to select and transmit to students the most basic content from which most part of others can be obtained, and help citizens to form healthy value judgments about themes that permeate their daily lives.

2) How to connect them.

     If we stop there, actually, most of the mathematics taught is completely useless. The point is not just to give the student a lot of pieces of a huge puzzle. You need to train him in the assembly. You need to give the student the ability to connect these pieces even without knowing the whole picture, even without having in his possession all of them. And that's the best of what mathematics can do and, in my view, no other discipline is able to develop this skill as well as mathematics does.

3) Why connect them?

    This is the part of School that should exercise the use of subjectivity. Is the component of weightlessness and unpredictability that truly provides reason and motivation to our actions and makes every effort worthwhile. Not only reflecting on why to connect the pieces, we often have to drop the puzzle aside, look at the horizon and wait a bit until intuition to wake us up from the sleep of reason. It's the human side of the process; the one that we would not know how design as a computer program. This includes disciplines in the arts, such as theater, painting and music.

Everything together

     Some schools are even able to play these roles separately. But, even so, the strange feeling that everything learned is useless still remains on the student. It's more or less like giving a person wheat, eggs, milk, sugar and yeast. What are they worth for? It is necessary, at least once, to join the pieces in the correct proportions, in the proper sequence and at proper environment to see the magic happening and savor the result.

    All disciplines have each of the above three components. What differentiates disciplines each from the others is exactly the proportion that each component plays relative to the whole. Thus, for example, History would have much more of component 1) than the other components, Dance woul have more of item 3) and Mathematics would be (or should be) predominantly item 2), unless the school - or teacher - conducts in order to stress, in a specific course, the other items that are not - as a rule - prevalent. 

     This is precisely what, in general, gives the social scientist, or lawyer, high power of reasoning, despite de fact that these professions attrack professionals who generally are refractory to mathematics. The advantage of studying mathematics is that, unlike the natural training of component 2) we go through when we study in depth the other disciplines, through mathematics is possible to do this training independently of the nature of the object being modeled, discussed or analyzed. Math is actually a great laboratory of thought where the brakes imposed by the physics of real facts, in general, do not exist. There you can train without tethers or with so many strings attached as necessary to improve excellence in a particular aspect of a mental process necessary to bring us to a coherent and justified conclusion.

     Because of that, a professional, within any field os knowlege, who knows mathematics (or, more specifically, who is trained in component 2) through mathematics), will end up standing out more than the other ones that has not gone through the same training.

Immaterial Objectives

     Using the idea of groups of transformations, the game of "What is it, what is not?" and all concrete tools offered (pdf version) reduce the distance between experimentation (concreteness) and higher levels of abstraction, leading us to a confrontation with a series of discussions that will support consistent answers to questions like "Why do we study mathematics?" and "What is Geometry?". Much more than that, this line of reasoning forces the student (and teacher) to reflect on the concept of equivalence relations and classes that are what actually allows us to say that "3 = 3", or "two triangles of sides 3, 4 and 5 are congruent," or that "garden is flowering if, and only if, cat meows", or even "all are equal before the law.".

    I therefore hope this work - much more than showing "new" ways to solve old problems - helps us to stop justifying why we teach mathematics with arguments like "you study complex numbers because, one day, when studying Fourier Series in engineering course, you may need them." The main function of mathematics - even before providing technical support for professions and everyday problems - is simply giving the citizen capacities linked to argumentation, research, modeling, synthesis and analysis of complex problems. And Geometry is an excellent laboratory for all of this. Perhaps, precisely because of that, it has been taught ever since immemorial times.

     Once I'm spending this time to say things that do not fit into an academic work I'll just present another great reason to study Geometry: "studying geometry is unbelievably fun." All teachers (and many students) I know that have medium or deep knowledge in Geometry spend, with pleasure, a Sunday afternoon solving problems such as those proposed in this paper. It's a cheap, healthy and greatly optimizes your reasoning, much more than any one of the games that I know (including chess).

Text in  pdf (portuguese)