Learning is much more similar to biological growth than to manufacture, where component parts are first produced, then fitted together.
W.Servais, T.Varga: Teaching School Mathematics,
A UNESCO Source Book, 1971.
During the twentieth century, mathematics experienced a real explosion of content and methods. Its modern core, however, consists of only a few principles. Acquiring these principles, understanding them, and learning the art of combining them in modelling more complex situations forms the mathematical minimum that I would like to present here.
But how to present these principles in a simple yet not simplified manner, so that the reader actually acquires them, and does not merely have the impression that she has acquired them? How to present mathematics in a way that helps the reader to develop the mathematical dimension of her humanity and not just to show her the human dimension of mathematics?
The acquisition of mathematics is not like constructing a building brick by brick, but more like the growth of a living organism in which the same principles that are present at the very beginning later develop in each phase of growth.
Instead of writing one single big book, I decided to write four shorter books. These books do not represent a big book divided into four parts, but rather something different. These little books all present the same principles of mathematics but at different levels of abstraction and complexity. They have been conceived so that you can study them one after the other until you achieve a level of understanding with which you will be satisfied, a level of understanding that will allow you to acquire the appropriate mathematical dimension for your personal activities. You do not even have to start with the first book, but rather with the book you consider to be the best starting point to fill the gaps in your mathematical knowledge. The first book already assumes that there has been a ”zeroth book” – that you have completed at least secondary school mathematics and have acquired certain calculating skills and a certain dissatisfaction with your understanding of what you had been calculating.
The first book is published and the plan is to publish a new book every six months. You can read more about the concept of the books in the preface to the entire series: Preface to the Circles Series.
You can find out more about the first book on the Math Circle 1 page.
Although the focus of these books is on understanding rather than calculating, calculating is an essential component of mathematics. Therefore, acquiring mathematical principles also requires adequate elementary calculation skills. Sooner or later, the need for more complex calculations may come along. Fortunately, this is where computers can be of significant help nowadays.
In recent decades a simple and advanced free open-source software called SageMath has been developed. SageMath is essentially a combination of all the previous big mathematical open-source software and has become a serious competitor of professional mathematical software. The SageMath tutorial and the SageMath examples for each book are available in a variety of formats on the SageMath Materials page.
We will be studying mathematics along with two different students, an Astonished guy and an Uninterested one. Each of them has one dominant psychological characteristic in the learning process, and these are present within ourselves in different ratios. My teaching will be supplemented by the Professor. He has all the characteristics of a good teacher that I lack.
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