The experts say in learning that the biggest effort that the human brain does in all his life is to learn to count: to be capable of relating a number (abstract concept) to a quantity of something and specially, that this quantity costs for different things, three horses or three flowers.1
All the human beings we manage to crown this port. The following important step that must do the cabecita of a lad is the division, also related to the abstraction and that returns to suppose another challenge for the mind.
“Divide and rule”
To know the concepts is simple but to know them intimately, to assume them, even to feel them, is more difficult enough.
A division is the obtaining of a relative value of the numerator with relation to a unit of denominator, in opposite case, the numerator is an absolute value, without reference; or since the dictionary of the RAE indicates:
“To verify how many times a quantity, called dividend, contains other one, called divisor”.
Saying like that, of memory, seems to be complex but, if he understands himself, the operation is very powerful, so powerful that a definition of e-DRAE of division it is:
“Way of knowing the things, which it serves to give white of egg designs of them”.
It is possible to open, with the division and the abstraction, a small window for the game of the thought. To be absent-minded, as María Moliner‘s dictionary defines it, is to separate mentally. What is to divide but to separate a thing in equal parts. What is to deal if not to do abstraction of a concept to be able to assimilate it. The animals do not deal because they are not capable of the intelligent exercise of the abstraction. To be able to divide is to have understood.
It is easy to be able to divide, but it is not easy to know what is a division. And I state that it is not easy because year after year I corroborate in the university that the pupils do not understand that the module of deformation is the division between the tension of an element and his unitary deformation, or
On the other hand yes they deal, and I it know because I ask it him as is, that the price of the oranges is the quotient between the total price that I pay for the oranges and his weight in kilos. It is enough to move this understanding to the resistance of materials to understand the module.
They understand also much better the linear behavior if, instead of explaining it of mathematical form by means of a straight line, I say to them that “the fruiterer is linear” if it applies always the same price: if they take to themselves a kilo, an Euro, if they take two, two Euros to themselves. And every kilo of more than go it is one more Euro.
On the other hand, the fruiterer would behave of not linear form when to major bought minor quantity unitary price (dependent on the drying straight line). This has several operative complications because you never know a priori all that is going to cost the following kilo (dependent on the tangent straight line) because it depends on the total bought kilos, nevertheless it has the clearest advantage for buyer (he pays less) and for seller (it sells more). In concrete this advantage is called a ductility and entity supposes other advantages that the ductile girders before breaking deform very much, they warn!
Also he would withdraw this scheme
“With oranges they understand it!”
If they one speaks about two more abstract magnitudes, as the tensions and the deformations, they do not understand the quotient, do not understand the relative measure that a division proposes. But with oranges, or with bananas, they understand it.
And the teachers do teachers understand it?
Feynman2, a famous physicist, scientist, I reward Nobel prize that dedicated an important part of his life to re-formulating the quantum theory and it departs from his scientific life to teach other one, tells that, in his famous ones (about revellers) trips to Brazil asked him in a University for the reason of which there were no Brazilian Nobel Prizes on physics and it answered that, for his knowledge of the universities, books and Brazilian teachers it could affirm that these did not know physics and that his knowledge was diminishing to the retahílas read in the books.
This image of education remembers me, between others, to the classes of religion in which it was taught by means of the litanies responsoriales of the catechisms.
I remember a class of mathematics of 2 º of career, with a few kilometric slates corresponded with an exquisite order, improper of a person who knows and explains, but own of a person who copies. The teacher, of backs, was writing and, simultaneously, reading what he was writing. In a certain moment a pupil, that still today I do not manage to understand how it could be following the class, said:
“Excuse me, I do not understand the step that has given between the end of the fifth line and the beginning of the sixth one. The teacher turned his face towards the aulario, looked at the fifth line, looked at his notes, looked at the sixth line, wrinkled the frown, arranged and, with the calm gesture, concluded: excuse I have skipped a line!”
The problem, as J.L.G. Quirós3 says, is that the science is very easily imitable. It is enough to know the involved scientific and mathematical slang to be able to fool with major or minor success. This teacher happens easily in the relation pupil, but also it happens in the high-level technology and, even in the high science. Cases of frauds have revealed themselves enough, there will be different many dancing free for the books. Surely in this text also. His readers’ labor is not to believe itself it, to criticize and to rewrite.
Is another type of education possible? I believe that yes.
An education in which the teacher understands it, transmits it emotionally and worry that the pupil receives clearly: It learns to arm a wall in less with 3 min.
Juan Carlos Arroyo (ingenio.xyz) structural engineer.
Madrid, November 2016
1 The prime numbers; Enrique Gracián, Ed. RBA. Review: Enrique Gracián is a wonderful discovery if you like the scientific spreading.
2 ¿Está Ud. de broma Sr. Feynman? Aventuras de un curioso personaje; Richard P. Feynman. Ed. Alianza editorial. 1987, 6ª reimpr. 2003 Review: Adventures of a prize nobel. The most enterteining. Do not stop reading it.
3 Several José Luis González Quirós´s articles. Review: Reading Jose Luis G. Quirós is to do a trip of pleasure for the philosophy of the science.
Ingeniero de caminos y doctor en arquitectura por la UCJC. Director de innovación y socio fundador de CALTER ingeniería, Presidente de la Asociación de consultores de estructuras de edificación. Ponente de la norma española de hormigón EHE-98.