## Rafael Laguardia - A Space for Basic Sciences

On 18 November 1934 Rafael Laguardia wrote to the Dean of Secondary Education, Eugenio P Baroffio, giving his views on teaching mathematics at secondary level. We give a version of an English translation of this letter below:

**A Space for Basic Sciences**

With my greatest respect,

I was recently appointed to the commission responsible for preparing a programme for the fourth year of Mathematics, a show of confidence which I appreciate, understanding that it requires the desire to know through me the opinion of a core of respectable teachers. As I have not been able to agree with my colleagues, I believe it is my duty to explain to you Dean and to other Councillors what my points of view are:

First of all I must refer to the plausible tendency that animates the presidency of the Council in the sense of removing from Mathematics its traditional and aristocratic pretension of isolation.

Science, historically rooted in the concrete, born to satisfy material needs, only by a long process of abstraction and generalization has been able to constitute an imposing and harmonious conceptual monument that usually both attracts and frightens at the same time.

I can only congratulate myself when I see the Council's concern to channel the elementary education of this subject towards its applications, trying to link it with the other subjects that make up the secondary curriculum. But I am firmly convinced that this orientation will only be fruitful if the connection between theory and practice is made step by step, at each stage of teaching. The teaching of every subject that contains a novelty must be preceded by a series of reasons and living examples that show the student the convenience of mastering the new method. And the acquisition of knowledge must be immediately followed by frequent examples and application problems. I disagree with my colleagues because I believe that, when lessons are taught in this way, a fourth year of applied mathematics is unnecessary and will be reduced to a monotonous parade of unconnected problems, completely devoid of the cultural value that, I believe, middle school should have. It is not for the professor of mathematics to artificially bring up the laws of Physics or Chemistry to transform them into a motive for an algebraic exercise that masks their content, but that the teachers of these subjects must show the student, when in the development of the course the opportune moment arrives, the advantages of a mathematical formulation.

Does this mean that I am against the implementation of the fourth course with mandatory character? Not at all. I have always believed, and the few years that I have been teaching affirm me in that belief, that mathematical truths must be the object of a slow and mature reflection. It is counterproductive to pretend to cram knowledge into the brains of adolescents in cases like this one in which, more than memory, one must address the intellect. That is why I have been a consistent defender of short programmes, which leave enough time for gradual, slow and mature assimilation. Many times Physics or Chemistry teachers complain about the inability of their students to apply mathematical notions to these subjects, attributing it to a too theoretical and formal teaching. Without ignoring the truth that this judgment contains, I believe nevertheless that the essential cause lies in the intensive mathematical culture. For these reasons, I advocate teaching which is limited to the essentials taught over a sufficiently long period of time.

Consistent with the ideas I have just explained, I can not support in any way the 4th year program proposed by my fellow commissioners, because it just suffers from those vices which I try to combat. Instead of alleviating the current programmes, in which the corresponding content of the third course (as had been foreseen by Professors Beltramo and Sarachaga) has not been fully developed by almost any teacher, its implementation would mean a new aggravation of the current circumstances. I refer in particular, and by way of example, to the excessive importance granted by trigonometry, a branch of very little cultural value whose knowledge is primarily of interest to the specialist and his learning should not go beyond the most elementary notions.

Accompanying this report is a programme to be developed in four years, prepared in collaboration with Professor Sarachaga, and which I submit for the consideration of the Councillors, without hiding that it contains, as it is easy to see, numerous concessions to professors who do not share our ideas. Its main novelty, which is not in other places, is to treat the geometry of space in the second year, understanding that it has to be done intuitively, as Rey Pastor and Puig Adam do in their well-known text.

We have been moved by the conviction that a large part of the spatial properties that can be taught in Secondary School are more elementary and understandable than some of plane geometry. For this reason we have postponed until the third year the systematic study of proportional lines, the notion of similarity and the theorem of Pythagoras, which is in accordance with the intellectual development of the student.

Being loaded with tasks, I could not write, as I wanted, the instructions that would complete that programme.

I greet Mr Dean with my most pointed consideration.

Rafael Laguardia