## Formative Value of Mathematics in Secondary Education

We present below a version of a lecture delivered by Pedro Puig Adam in 1951:

**The Formative Value of Mathematics in Secondary Education**

The title of this article, "The Formative Value of Mathematics in Secondary Education", could be misleading as to my opinion about the formative value of any subject. Because I understand that, more than the content of each discipline in itself and, even if you push me, more than the methods of research in each one of them, what ultimately indicates their formative value is the method followed in their teaching.

It is a proven fact that Mathematics, as well as Latin and other disciplines, can leave no formative trace or leave very different traces according to the teacher and according to the methods that have served as a guide.

A title more faithful to my thought, but also longer and pretentious, so I preferred to sacrifice it as soon as possible, would have been "Formative impression that must be demanded from the teaching of Mathematics in the Baccalaureate and methods suitable for it", and I will stay with this longer and augmented title.

**1. Narrow views of the problem and its didactic consequences**

Dismissing for the moment the utilitarian value, which we will speak about towards the end of the article, the most important mission, and for many the only specific mission, that is conferred by Mathematics from the educational point of view in second school teaching, is the cultivation and development of the logical spirit, of the art of reasoning. The didactics that consequently is advocated is the rationalism of everything concrete, with forgetfulness of the values of intuition; it does not matter much the genesis of the concepts nor of their fundamental attributes, the essential thing is to train the student to reason properly on premises made very clear and well established, without worrying about their origin.

For others, Mathematics is, above all, the science of problems. The didactics corresponding to such a conception is quantitative pragmatism at its best: solve many, many problems, that is the crux. Do not speak to such pragmatists of analysing what faculties it would require to be put into use for each of the problems proposed, nor of whether these faculties will be the same ones that the student will have to exercise in their future functions as a social and educated person. They will not understand. The automatism of the current regime of mass tests and time trials, does it not favour this conception and this didactic technique? Let us apply it, then, and we will also automatically have assured its "success" (success in inverted commas leaving each one to interpret the quotes to their liking).

More subtle people, more aware or simply more fond of contemplating things from original angles (a type frequent among essayists, some giving such a pleasant lecture) see, especially in Mathematics, either the shorthand expression of the laws of thought (dangerous evaluation of the meaning of an idea, before which any pedagogical precaution will be slight), or the cultivation of habits of "accuracy" (accuracy that I also comment on with a certain scepticism, as I will argue later), either, finally the exercise of self-criticism, of respect for the truth, from worship to disinterested knowledge.

Well, ladies and gentlemen, the role of Mathematics in the education of youth does not consist in the exclusive development of any such faculties, skills or virtues, not even in all of them added together. A good mathematical education has to demand several more values, the neglect of which has been and continues to be the frequent cause of its failure, if not before examinations, before life itself.

**2. The "Esprit Geometrique" and the "Esprit de Finesse" of Pascal**

To paraphrase Pascal, we will say that it is not enough to exercise the "esprit géomètrique" more or less integrated into the set of tendencies to which we have just alluded; it is necessary to cultivate also the "esprit de finesse", a very subtle utterance of Pascal which is very difficult to translate, but which perhaps responds more in our ears as the "fineness of spirit" version than as the "spirit of fineness". In order to realise all the roles that Mathematics must play in education, it is enough to see what it has played and plays in human progress and culture; only then will we realise the serious failure of abstract mathematical education, as practiced with disastrous exclusiveness until the beginning of this century, and is still practiced among many pedagogues of excessively rationalist formation.

**3. Opinions and criticisms about traditional mathematical teaching**

A great authority on Logistics, at the Cambridge International Congress (1912), stated that the efficacy of mathematical teaching lay simply in the development of logical meaning, and a few years earlier the instructions of the Prussian course dictated: "In all fields of this subject the object must therefore be to obtain a clear understanding of the theorems to be developed and their deductions, as well as the practice and ability to use them." Not many years ago a Spanish pedagogue wrote these words: "There are few psychological activities of the child that can be used for the rational study of Mathematics, thus justifying in a way the routine learning that has been carried out for so many centuries ... " "The ideal will be to make the whole teaching of Mathematics a continuous contest in which speed, accuracy, ease, precision and logical rigour, perfection in a word, increase successively according to the characteristics that as art and as science we have assigned."

The narrow dilemma, and at the same time the terrible leap, into which the old mathematical teaching was condensed was thus: empiricism or logicism; where the first one jumped to the second without intermediate gradations. As long as the fruits of logical reasoning could not be obtained from the child, there was no other task than to instil in it skills, arousing in the absence of any other interest, his spirit of competition and championship. But as soon as the child reached the stage of having faculties of reasoning, ah, then the time had come to overwhelm it with axioms, theorems, corollaries, and so on. All the individuals of our generation and of the previous ones have suffered the consequences of this narrow dilemma, the result of which has been the total and definitive aversion of many spirits towards Mathematics, spirits that in other fields have later proved to have great subtlety. And it is not surprising that such an aversion crystallised into notorious diatribes for its origins and for its hardness. I can not resist the temptation to name one.

Madame de Stáel said (1), for example: "The truths demonstrated with emphasis do not lead to probable truths, and these are the only truths that appear in business, in art, in society. Nothing has less application to life than a mathematical proof. A theorem about numbers is true or false; in all other matters the true and the false are mixed in such a way that only instinct can distinguish them."

Le Bon, on the other hand, thinks that Mathematics only serves to develop the taste for subtle reasoning, but it is false that it can exercise judgment, and to justify this assertion it is argued that the most eminent mathematicians often do not know how to conduct themselves in life and are disoriented in the face of minor difficulties. Huxley expresses this lamentable concept as: "Mathematics is a study that does not require observation, experience, induction, chance."

Finally, Bouasse, in one of his picturesque prologues, delivers these terrible blows: "The mathematician has a horror of the real, he abhors the particular case; abstraction and generalization are the idols to which he sacrifices good sense ... when there is nothing left of a phenomenon, it is reasoning at its widest point: the emptiness is its element, the form its god."

To soften the pathos of such an anathema, I am going to finish these examples with a characteristic trait of humour that if it is not English it seems to be so: it is a "test" (L'Allemagne, Part 1. Chapter 18), to find out if a subject has mathematical skills. They are given an explanation of how to prepare a tortilla, stating the necessary premises: among them is the precise position of the frying pan hanging from a nail on the wall, then follows details of the development of the operation, details that I omit (among other reasons not to see me in a hurry). Once the description has been completed and its perfect assimilation verified by the experienced subject, the premises are suddenly changed, and the change consists only in the fact that the pan, instead of hanging on the wall, is already on the stove. Then the question, what would you do now to make the tortilla? If the subject has a "true" mathematical spirit, they must respond quickly, according to the humorist of the joke: "Hang the pan on the nail and the problem is reduced to the previous case."

As a ridicule of the abstract deductive spirit, it must be recognized that the humorous one has a lot of ingenuity and a profound teaching. The unjust thing is that all these diatribes or mockery are directed at Mathematics or mathematicians, and not, as it should be, at the vices or clumsiness in its teaching, and I repeat that in order to realize such errors there is nothing better than to analyse the role that Mathematics has played in the history of human culture and in the very life of mankind.

**4. Logical sense and sense of application**

No one ever doubts the role that Mathematics plays in the development of the logical sense, I will say more, I will say that it is the Science most suitable for this, for the same precision and simplicity of its concepts. Nor will I attempt to lessen the importance that the developing of such meaning may have. In a world in which logical values have proper weight, such statements as picturesque as those of the United Nations which, in the name of defence of peace, denied for several years it would be possible to honourably and gallantly maintain theirs while the world was at war. It is undoubtedly in the world of politics where the failures of logic are more frequent, perhaps because the social entities it manages are so extremely complex or perhaps because in it the love of truth is so rare and boring. But leaving aside the glazed issue of government of peoples, even for self-government, there is no doubt that logical values are in many cases (not always) the rule for the best and most successful conduct. Simple generalisation to others of a demand of their own allows us to see the impossibility of a desire that our blind selfishness imposed on us with a pressing imperative. The habit of meticulous analysis of situations and the protection acquired against the fallacies of reasoning will allow us to be fairer in our judgments and more composed in our determinations.

But taking all the excellences of logic for granted, it is undoubtedly not enough for life or even sufficient for the development of science to develop the logical sense, the deductive mechanism. Without cultivating other intellectual values along side it is to condemn logic to sterility. Once again I have to repeat here a refrain that, turned into a pedagogical creed, I am emphasizing how many opportunities I am offered to touch the subject of mathematical education.

Mathematics is the filter through which man studies natural phenomena; replaces the infinite complexity of the same by the schematic simplicity of entities of reason on which one can use reasoning logic; obtains the fruits of this, then proceeds to the interpretation of the same in the field of reality. There are, then, three phases in the mathematical study of natural phenomena, a first phase of abstraction, a second phase of logical reasoning, and a third phase of translation or passage from the abstract to the concrete, an operation we shall call concretion.

Classical mathematical teaching has long been reduced to the cultivation of the second phase; mathematical concepts devoid of all real meaning, rarefied by virtue of being purified, have been transmitted from generation to generation, and hence the divorce between mathematical teaching and reality; hence the type of man of science incapable of conducting himself with good sense in life, the frequent type of engineer full of mathematical science, but unable to pose, with practical sense, the problems that the technique offers him.

If we want to achieve a complete mathematical training that enables the student to use mathematics as a living instrument in his daily life, the sense of application in its dual aspect of abstraction and concretion should not be neglected in mathematical teaching. But this is not achieved simply by putting so-called application problems after an abstract theoretical exposition (most often in these problems the application is more apparent than real). The remedy must be to attack this evil in its very origin, that is, in the stage of formation of mathematical concepts. Thus, before beginning the logical method, a rich number of observations, experiences and intuitions, made from the first years of school and accumulated in the unconscious of the child, must have accumulated in the pupil's mind to become the germ of abstract concepts.

Even though it seems paradoxical, the faculty of abstraction does not develop by reasoning in the abstract, but starting with the concrete, since if to abstract is to do without something, it is necessary that the process starts with this something that can be dispensed with. The deficiency of classical teaching on this point consists, then, in giving the complete abstractions and not teaching them to form them, which is both useful and effective.

The same can be said of the absence of the development of this faculty, which we have called concreteness, in classical teaching. Many times we have heard lamentations of university teachers about the insensitivity of the student to clearly absurd results. Is this due to lack of habit of interpretation or representation of them?

It is not to teach concrete arithmetic to add behind an abstract number the addition of a noun or an abbreviation (metres, kilos, litres, etc.). It is necessary that these concrete numbers have in the student's mind their clear representation, that the student knows how to project them at all times into the field of reality.

A few years ago I was looking through a children's Arithmetic jotter prepared in what was called a good Teaching Centre. As in all other jotters, there were in this one curious results of street lengths calculated to the millimetre, from time to work to the hundredth of a second, but what struck me the most was a number of workers given as 17.8456. I asked the child why he had calculated four decimals, and he replied that he had not been given time to compute more. Despite making him read the result followed by the word 'workers', there was no way he could understand why it was absurd. Only when I placed him imaginatively before a supposed group of workers to choose the ones he needed, the smile of the absurdity dawned on him. He ended up confessing that the teacher always gave the best score to the one who calculated the most decimals. Here is an example of the results of the competitive method.

It is not believed that the development of these faculties of application, which establish the nexus of mathematics with reality, should be relegated to the teaching of the physical sciences. I insist that mathematical concepts are rich by themselves in concrete meaning, that this meaning is precisely the one that gave them origin, and that far from denying this origin we must go to it to reproduce in the student the same evolution as these concepts have had in the human species.

**5. The Role of Intuition**

We have said that it is not enough to deduce; it is necessary to know, in addition, how to pose and to interpret. But to pose in the complex sciences of nature, let alone social sciences, is to know how to choose the variables of preponderant influence on the phenomenon, it is to guess - without effecting the experiences, often impossible to realise - that the effects of omission of certain causes will be practically imperceptible while the omission of such others would lead to a serious error. It is to predict the behaviour of a sensible reality jumping by above it, closing our eyes and seeing what happens (if that is the correct word) in the inner reality of our imagination. It is, in short, to make use of the faculty which in Mathematics we call intuition (to look inside ourselves) and that should not be confused with the faculty called intuition by some psychologists and pedagogues that hardly differs from simple perception.

It is easy to understand that it is no longer the logical values that can guide us in this previous selection of premises, since logic is only apt to act on premises previously elaborated, nor are they in many cases also logical values that ultimately determine the key ideas of the solutions of problems, but the prior internal clairvoyance of the fecundity of a certain association of ideas and the sterility of others. Even in the genesis and development of mathematical science itself it is recognised by all of us that the true beacon that illuminates and discovers new paths is intuition; the logical rigour almost always comes after, limiting itself to solidly solidifying the discoveries of that one.

The failure of many mathematicians, it would be fairer would be to say of many bad mathematicians, before the complex problems of life, failure that motivated the previous diatribes against them and against Mathematics in general, are only a consequence of a defective cultivation of this very subtle faculty in which the 'esprit de finesse' of which Pascal spoke to us mainly lies. On the other hand, this failure is not exclusive to bad mathematical specialisation, but also of any other specialisation which is defective due to incomplete education. Rey Pastor once said, with his usual humour in ridiculing defects: "The physician or the lawyer who has received only rigorously deductive and abstract instruction; without the flame of intuition capable of illuminating the dark background of the complex, can only reason when the questions are presented to them in the simplest syllogistic form; and since the patient's symptoms and witness's statements are usually disjointed and almost always contradictory, the conclusion that they will logically arrive at is the non-existence of the disease or offense."

**6. The sense of the essential**

It is now understood why the omission of the cultivation of intuition in mathematical teaching, is devitalising to the point of motivating in the learners the lack of that quality that could be called sense of the essential. Sense as indispensable in technique as in life in general. To make decisions in life it is not enough to do a thorough analysis of the circumstances that may influence the situation we want to overcome; it is necessary to have a clear intuition about which of those have greater weight and not to pretend one can achieve mathematically perfect solutions where the nature of the problem does not allow them or inhibits them by the presence of causes of opposite sign when one of them is of no importance.

Whoever in his mathematical education has cultivated the faculty of intuition, it is hoped that he has developed the sense of the essential that we are alluding to and that, knowing how to discriminate correctly the preponderant from the secondary, he does not lose himself in subtleties, useless judgment, or argue in vain, or act awkwardly in making vital decisions.

**7. The sense of approximation**

In relation to the use of the faculty of concreteness, I must emphasize the character of the alleged "exactness" that common people attribute to mathematical science, and that childishly applied to education can produce havoc that damages common sense, such as we have more clearly stated in commenting on the "exactness" of certain ten thousandths of a worker.

If the student were accustomed to constantly project the data and results of the problems into the realm of reality, absurdities of this nature would be avoided, the pupil would become accustomed to keep in mind this simple and yet so often forgotten truth that all data translating a measure of the physical world is necessarily approximate, and that, therefore, the alleged accuracy in the results is not only a pure chimera but a grotesque falsification of reality. Whoever pretending to be "exact", calculates figures and more figures without thinking if they exceed the limit of what can be achieved from the measuring devices, or the threshold of our own senses, demonstrates as much ignorance as lack of this sense of reality.

Unfortunately, this sense of approximation is so neglected in the teaching of the Baccalaureate as in the same technical education, where the offence is even more serious for doubly damaging the student: in his training and in his information. It is very frequent in the entrance examinations of our Technical Schools to see it expected that the student calculates with tables of seven decimals, formulas whose experimental data are given only with three figures. Someone assured us, on one occasion, that we had to determined the distance between two points, one of which was the spire of a bell tower, with an error less than a hundredth of a millimetre, without realizing the lack of mathematical meaning that the phrase had because of the material impossibility to specify in such an attempt being capable of precisely defining the "point" to within a hundredth of a millimetre.

**8. The method problem. The periods of its evolution**

Having concluded here the critical exposition of the educational aim to be pursued in the teaching of Mathematics in the Baccalaureate, let us add even two brief words about the didactic consequences that can be derived from such criticism in order to solve method and mode problems; that is, of the way forward and how to attain the most effective achievement of this purpose.

Without letting ourselves be carried away by simplistic and 'a priori' exclusiveness, we can say that the best methods and modes are those that, notwithstanding the stated aims, are better suited to the psychology of the scholar.

The old school of the teaching of Mathematics could be characterized by contempt, ignorance of psychological problems and consequent predominance of purely logical problems.

It forgot that the logic and interests of the child are not the same as those of the adult. And so we saw impeccable expositions of form, but not able to cultivate the analytical appetites of the child or even to develop prematurely in it habits of synthesis, since it does not develop precisely this ability giving the syntheses made. The result was only to cultivate memory once more under the false appearance of borrowed reasoning.

It forgot what some psychologists call "the intellectual realism of the child"; that is, their inability to prematurely comprehend formal or abstract logical relations. Thus we saw premature use of demonstrations by reduction to the absurd that could not be understood by students of tender age, since they are based on premises not only disconnected from tangible reality (they can only conceive that), but also contrary to reality itself.

In addition, the biogenetic law was forgotten or ignored, which in Pedagogy continues to express an interesting analogy, according to which the development of the individual reproduces in a small way the development of the species. In this way, mathematics presented itself by concealing from the child the genetic process that humanity has had in our science, dispensing with its experimental and intuitive stage and presenting it in the form of rational science par excellence, as the Greeks, who already suppose an adult civilization.

It was forgotten, finally, that the mental evolution of the child follows the same law of continuity as its physical growth; and so it jumped from the empirical procedures of the primary school to Euclidean reasoning of the venerable works of classical mathematics, without taking into account that the pupils of Euclid were definitely not children.

What was the result of all this system? Inability to adapt, incomprehension and a definitive aversion in most intelligent people, who were thus lost to Mathematics, a science for which special skills were believed necessary.

The opposite, then, of this criticism constitutes, in general lines, an indication of what must be done: (1) Compliance with the educational purpose previously studied; (2) Psychological knowledge of the student and the consequent remembering of the biogenetic and continuity laws.

These simple laws serve, for example, to justify the implantation of the cyclical methods that establish the continuity in the study of the topic without breaking them up into separate areas; justify the introduction of intuitive methods in the first years of high school to fill the gap that existed between the empiricism of primary education and the rationalism of university education, and the progressive evolution of methods that without discontinuity or sudden jumps allow one to develop the psychological activities of the child gradually from early childhood to university.

The periods of this methodological evolution and the faculties preponderantly developed in each one of them could be characterized and ordered chronologically, if not with exclusivity at least with predominant tone, as follows:

Period of observation. - Simple analysis, observation of the facts and points surrounding the child. Development of the senses.

Period of experimentation. - New facts are brought forward for analysis. Analogies are induced. Development of transduction; that is, the passage from the particular to the analogous particular.

Period of intuition. - Real and brought forward facts are replaced by imagined facts. Sensitive external reality, by the inner world of fantasy. The child begins to look within; it makes affirmations not only about what is happening, but also about what would happen if ... Consequent development of the imagination or fantasy. Development of induction.

Logical period. - It substitutes, the evidence of the senses by the logical evidence. The imagined facts by the abstract premises and their necessary consequences. Schematization of reasoning through abstract symbolism. Development of logical deduction and abstraction.

**9. The problem of the method**

In this problem I will be much more schematic, limiting myself to give some very general guidelines, and insisting that all exclusivism is disastrous, since the best way is also the one that best fits the purpose and the student.

I believe that the main guide to show the way or manner of teaching is to serve the interests of the child. I refer, of course, to interests as a willing desire for them.

In the old school the child was conceived as a reservoir to be filled with knowledge; today it is already conceived as a potential to transform through activity. Many pedagogues have not yet realized the profound transcendence that this change implies. If we could measure the amount of psychic energy that comes to us every day through the doors of our Centres of Education in the minds of our students, we would be astonished by it. To give way to these energies, to give these pupils a task that interests them and progressively educates at the same time; this is our main task and our difficult task every day.

Let us not forget that the child has a constant desire to do things, to make discoveries on its own, and the child will only listen to us insofar as we favour, stimulate and orient our explanations with his creative desires. We should not conceive of the class as a conference room, but as a workshop, and we should not feel that we are lecturers in it but teachers of that workshop.

Therefore, the mode of education more in line with these interests of the learner is the heuristic mode in which the teacher only serves as a guide for the student to discover the truth on his own or at least to make this appear to happen. As Rey Pastor keenly observes, it is not the possession of the goods, in this case cultural, but their acquisition that gives man the purest satisfactions.

The most serious drawback of this approach, in which the only text is the student's notes, is slowness. Therefore it is almost always necessary to shorten the processes and the help of a book as a vehicle of culture ends up being indispensable. But it is necessary in such a case that the technique of handling the book is adequate so that this management is not converted into simple memory exercises. The solution: analytical reading of a book and synthesis in an exercise book seems to be the formula that, without dismissing the advantages of the method of heuristic analysis, does not participate in the serious drawback of its slowness.

**10. The problem of the programme or content**

If in the problems of method and mode we have to attend to the educational purposes and to the psychology of the student, in the question of programme we must also attend to the utilitarian value of the knowledge. I do not think I have to struggle to prove the usefulness of Mathematics, which is being used more and more on a day-to-day basis with the most varied research techniques in the branches of knowledge that seemed to be furthest from it. Recall as recent examples: factorial analysis in psychology, quantitative investigations of the physiological behaviour of nerves and tissues, the mathematical study of the cyclical evolutions of certain populations of species along side each other, feeding one another, in the biological theory of the struggle for life, recent achievements in economics, sociology, statistics, etc. In recognition of the useful importance of mathematical technique, those from Tyre and Trojans agree; but if we were to specify the quantity and above all the quality of the content of the teaching programmes, the discrepancies of the different pedagogical schools would arise here.

I do not know why it was intended to present as opposites the utilitarian and formative values. On the one hand out and out utilitarian people only ask about a theory: And is this what it serves?, and of course, they do not allow more "service" than that of immediate application to daily life. On the other hand, those who are called "pure" educators neglect all usefulness. Worthy heirs of the classical Greek school, they fear that the formative value of the discipline is tainted by reality.

To the first I would say: That if people studied only that knowledge strictly indispensable to the exercise of their minuscule profession, we would all end up having nothing to say to each other because with a lack of common interests and knowledge and even common language, we could not communicate or understand each other spiritually. Hence the need for the study of the humanities, which is as much as saying the essence and the core of the culture of humanity to which we all belong.

To the latter, formalist scholastics par excellence, one should ask them if they truly believe that to reflect today's human science and art is to teach the same humanities taught in the Renaissance.

How can we combine the two utilitarian and formative tendencies without reloading the programmes with the overwhelming and unbearable weight they suffer today? I propose a very simple formula. I have already said that I do not believe much in the formative value of the knowledge itself: how formative are the methods that are followed to acquire it. Equally similar formative methods seem, therefore, appealing, whose knowledge, in addition to enriching our culture, can give us some usefulness in the demands of modern life.

Every hour has its eagerness and all programmes requires due consideration of the new and the traditional, distinguishing between tradition and routine to respect all those questions and teachings whose formative value lies in their own fecundity and discard those others whose reappearance in programs is only justified by an inherited habit. Let us proceed, on the other hand, without delay to those questions which life has imposed on us today as practically necessary, stopping us making those whimsical innovations which practice has not properly sanctioned.

**11. The problem of the testing regime**

I would gladly end the talk here if I were not afraid of its uselessness. What, then, can all the formative inspiration serve if at the end a test regime, which I deem to be inadequate, can put an end to its consequences? Let us not deceive ourselves, in the present regime of tests we have tried to resolve at the same time two problems which, even temporally linked, are in their essence different; one is the selection of abilities for higher studies, another one of the verification of the efficiency of teaching.

Leaving aside the discussion of whether the Baccalaureate has an autonomous mission or is simply training for access to the University, there is no doubt that mass tests are converted by the law of minimum effort into a technique, or worse, into an automaton that immediately requires another technique, not to say another automaton, to overcome such tests. To these tests, the attention and effort of the students and those who guide them is ultimately concentrated, and so it is that the problem of the teaching of our youth becomes a problem of preparation that has little or nothing in common with that one and which usually leads, on the contrary, to a true deformation of the student.

I pointed out the danger several years ago in a professional magazine and today I refer again to it, covered by the noble consent of José Pemartín, who knows that in making an allusion to the problem I am not moved by any criticism, but the deep and sincere concern for the formative problem, concern from which I can at no time detach myself.

It is very difficult to be a good educator and good coach at the same time. Admitted that the prestige of the Schools is involved in the success of their students in certain examinations; the teachers of the same will inevitably tend to manufacture with the raw material of their pupils an artificial product suitable to the aforementioned tests, sacrificing if necessary the authentically formative values and even the physical and mental health of the student, perhaps without realising it.

I know that this evil is very difficult to remedy. But it does not seem impossible for me to humanise the regime of evidence until the ideal of suppression of it is attainable, or what is the same, to make the entire life of the scholar a unique test.

**12. Final Summary**

And here I finish, summarising in brief review my previous conclusions:

Since the aim of the Baccalaureate is more formative than informative, the sum of knowledge acquired by the student is less important than the methods used to provide them.

The purpose of mathematical teaching in the Baccalaureate is not only the cultivation of the faculties of reasoning. To reduce mathematics exclusively to its abstract logical edifice is to forget its origin and the role it plays in the study of natural philosophy. The mathematical study of natural phenomena has three phases: The first, of formulating, schematisation, of abstraction, in a word; the second, of a logical decision mechanism; the third, of interpretation, of concretion.

To reason with already made abstractions without it being the same child who makes them is to forget the concrete origins of Mathematics, which has always progressed in trying to schematise a physical and social world of increasing complexity; is to counteract the biogenetic law, according to which the intellectual development of the individual must run in parallel with the intellectual development of the species.

If we want, therefore, that mathematical education is delivered in a fertile way for the vital future achievements of the students, let us also cultivate this double process of abstraction and concreteness, not only in the approach and circumstantial resolution of problems of a more or less practical character, but as a guideline of all mathematical teaching, beginning by accumulating in the primary groups an abundance of lived experiences (experimental processes); then let us substitute these by imagined experiences (intuitive processes) and let the abstractions settle in the subconscious of the child, to make them appear easily in the stage of rational teaching. Mathematics will thus be prevented from engendering aversion or utopia, as is the case of classical teaching on two empty spaces: empiricism and logicism.

In matters of method and mode, let us not be too carried away by petulant 'a priori' reasoning. The child is ultimately the one who points out the guidelines to follow. Their needs and their reactions will tell us when it is time to start a method or to use a special way of teaching, and such changes are always made in a gradual and continuous way since their intellectual development is gradual and continuous.

Let us also always bear in mind that the child is not an empty sack that must be filled with science, but a potential eager spirit ready for action. Let them feel the joy of discovering, of creating, of inventing; that a truth found by their own effort will have more value to their culture and to their morality than a hundred compiled truths.

In the matter of content or programme we take into account utility. If the educational efficacy of mathematical teaching lies mainly in methods, respecting them, we will have the freedom to select the knowledge that will be most useful and thus arouse greatest interest, and thus the two utilitarian and educational points of view, which have so often been presented as opposed to one another, will be joined in a simple harmonizing formula: Teaching useful knowledge with educational methods.