Didactics of Mathematics
In 1965 the journal Revista Española de Pedagogí had a special part dedicated to "The Spanish School System: Necessary Economic and Social Requirements". Pedro Abellanas Cebollero published the paper "Didactics of Mathematics" in that part. The reference is "Pedro Abellanas Cebollero, Didactica de la Matematica, Revista Española de Pedagogía 23 (91/92), (1965), 551-555." Below we give an English version of that paper:
- Introduction.
Teaching is subject at present to a study and analysis that is unprecedented both for its geographical extent and for its intensity. I believe that the fundamental reason for this fact lies in the need of mankind to make the most of all man's capacities as the main source of economic production. This origin of the didactic preoccupation can not in itself constitute the basis for the global approach that must be made to teaching, so it is advisable to give a word of warning, to avoid erroneous results for society. These results would be false even for the resolution of the same economic problems that humanity has raised. Let's look at a particular aspect that will clarify our ideas.
Mechanization produced unemployment of unskilled labour, giving rise to a re-adaptation to new trades that required an apprenticeship. The current prospects are of a new type, tending toward automation. This will result in the cessation of specialists of a certain type in the mechanical, electrical, chemical, etc. factories, and a new re-adaptation to other specialties will have to be carried out. But what has certainly not been noticed is the rate at which these re-adaptations will occur. This aspect is important because it seems sensible to admit that this rate of substitution of skilled labour by automatic processes is going to be such that it can cause the same individual to need re-adaptation several times throughout his life. In these circumstances, a teaching approach that only addresses the present economic problems can not achieve the intended purpose, since in the very short term a similar situation will have arisen. This shows that it can be highly detrimental to think only of current needs to adequately solve the problem of pointing to a teaching orientation that can be used in not too distant years.
We believe that the foundation upon which every system of teaching must be supported must be the harmonious development of all man's capacities. Among them I am now interested only in intellectual capacity.
For the development of the intellectual capacity serves all organized knowledge, or sciences. But mathematics has the advantage of being the most evolved of all and, consequently, the simplest. This is why it appears as a basic element in any organization of teaching.
- Analysis of the ancient teaching of mathematics.
(a) Primary education.
The fundamental subjects in this level of education are constituted by: natural numbers and their operations, decimal numbers and fractions, as far as arithmetic is concerned. Geometry consists of a set of definitions of geometric figures and calculation of perimeters of polygons, and areas of the same in simple cases. As far as arithmetic is concerned, the emphasis is on operative agility. Geometry is practically reduced to applying formulas for the areas of simple polygons, of circles and some circular figures. It should be noted that geometry has been relegated to the last term, that the definitions of geometrical figures have been abandoned and that very few pupils are able to distinguish precisely the simplest geometric figures. As for arithmetic, a statistical study of the ability of a student of nine or ten years of age to apply arithmetic operations with natural numbers to problems of ordinary life would be interesting.
(b) Middle school.
The first course includes: natural number, fractions, decimal numbers, simple rule of three, elementary geometric figures, some geometric constructions and area calculation. In the second course, numerical divisibility, revision of ordinary fractions, natural exponent powers, square root, proportionality, similarity of polygons, solid bodies. In the third course, negative numbers, algebraic expressions, first-degree equations, metric properties of figures in the plane. In the fourth year, polynomials, radicals, second degree equations, geometry of space. In the fifth course, introduction to mathematical analysis and trigonometry. In sixth year, revision of the ideas of mathematical analysis, introduction to plane analytical geometry and a slight idea of integral. The set of subjects is approximately adequate, but all courses are excessive for the number of hours available for teaching. It can be observed that in the first three courses there are lessons on fractions which result in a great number of students who do not manage to handle them correctly. Geometry is centred on the subject of numerical calculation: the Pythagorean theorem and the formulas for the areas of polygons and volumes of polyhedra constitute the centre of gravity of the whole geometric study. The orientation of teaching is directed towards the use of some mathematical techniques; this implies that those students who are thinking of orienting themselves to professions in which mathematics does not intervene feel disconnected from it and that it is believed that it is unnecessary in the upper baccalaureate of letters. On this point, it is interesting to note that at the meeting last year in Athens to study the problems of teaching mathematics in high school, it became clear that the only countries in Europe, including the United States and Canada, which did not include mathematics in the baccalaureate of letters, were Portugal and Spain, and the first of them reported that it planned to include mathematics soon.
We do not believe this is the place to be concerned with the teaching of mathematics in the system of higher education.
- Mission of the teaching of mathematics.
In order to be able to give an orientation on what the teaching of mathematics should be, it is necessary to fix the purpose that we intend to achieve through the study of mathematics. It is a mistake to assume that the problem of teaching is a curriculum problem. The previous analysis of the subjects that constitute the current curricula is not intended to discuss them but to show that they point in the wrong direction. But to achieve this it is necessary to start from a certain point and this has to be the unanimous recognition of the purpose that all studies of mathematics must have.
It has been said that numerical calculation exercises have great formative value in primary school because they suppose an excellent mental gymnastics. On the other hand, the easy handling of operations with numbers is a good starting point for the use of mathematics in its immediate application. Neither the formative aspect, which it certainly has, nor the utilitarian aspect, justify the abuse of the algorithmic part of mathematics that is done in elementary school. It is possible that, in addition to the reasons just mentioned, there is a more powerful one that is really that it has reduced the teaching of mathematics, in its elementary phase, to a simple calculator, but with the aggravating aspect of automatism. Numerical calculation is based on retention in memory of a short number of rules and a pair of tables, the student has already acquired this in a small digital calculator with four programs: add, subtract, multiply and divide. With a difference in favour of the worst calculator on the market: that the human brain has a great capacity for error, which does not happen in a calculator. There is no doubt that we need to have fixed and rigid mental schemas: the sequence of natural numbers and numbering systems are basic. A discreet skill in numerical calculus is indispensable, but from this to dedicate practically all the time devoted to mathematics to making the pupil into a bad calculating machine, there is a big difference. It is possible that fifty years ago this attitude was justified because there were many professions that needed skilful calculators. But here we have an example of a profession that was first mechanized by calculating machines, mechanical and electrical, and has now been automated. Consequently, an aptitude for calculating becomes less necessary every day. However, the management of electronic machines requires better understanding of the ideas involved in operations so that they can be programmed. We see that the reason for professional training that could be used to defend the student from automatic calculations in school is no longer valid; if anyone still doubts that, it is still being formed in centres such as banks, offices of architects and engineers, administrative centres, houses of commerce, with the number of employees whose profession is to add, subtract, multiply and divide. This fact has influenced much more than all the psychological and didactic reasons that can be used to ensure that at least in some countries with more generalized techniques the abandonment of calculative abuse in elementary school has begun (however, we know of a country that continues to think that the primary purpose of primary school is to make end-of-century accountants for colonial businesses).
We believe that calculating in primary school should be reduced considerably to give way to having time for something more interesting.
Let's go to middle school. Here, the teaching - which might be called ancient - of mathematics focuses on two things: calculation and syllogisms. The first is to attend to the aspect of practical applications of mathematics, the second for a formative aspect. Let's see what lies behind these words. To make the student see the practical utility of mathematics, it is taught how to calculate square roots, when no one needs to use the square root extraction rule: some because they will never have to calculate a square root, others because they will have to calculate so many that they will use tables or machines. Therefore, time and effort are lost. The student is also shown the usefulness of mathematics by making him calculate volumes of strange figures that will never exist or calculate with trigonometric formulas that could only be useful to a minority of the order of 1:10 000.
The practical applications of ancient mathematics are unfortunate. Let us see the syllogisms. The formative value of syllogisms has been exaggerated even more than in the case of mental calculation. Obviously a deductive capacity is important, but not the most important in the intellectual formation of the student. The logical process is totally analogous to the numerical calculation process; it is based on a set of rules and tables that can be applied as automatically as in the case of numerical calculation. For this reason, electronic calculators are able to prove theorems. I believe that an operation that a machine can carry out more correctly than a person can not be considered as formative for the human spirit.
I want to point out that the ancient teaching of mathematics is based exclusively on two pillars: calculation and syllogisms, and that neither of them consists of essential activities of the human spirit, in as much as they are performed, with less probability of error, by machines. So it is not surprising that if what is taught in mathematics is essential to the human spirit, it, in general, rejects that teaching. We sincerely believe that non-worldly view of young people to mathematics is fully justified, but in reality what I should say is that the non-worldly view of young people towards mathematics that justifies my view that the teaching that is made of it is totally false.
What remains of mathematics if you take away the calculation and the syllogisms? I believe that all of mathematics remains, which one or the other concealed. This concealment of mathematics by calculation and syllogisms is such that many scientists and even mathematicians have not arrived at discovering mathematics. So what is mathematics? If mathematics is the work of mathematicians, then we will have to observe the work of these mathematicians to know their product. The work of a mathematician always consists in the analysis of a previous situation. The result of this analysis is the observation of analogies that allow him to establish relationships, through which he can sometimes define new concepts more general than the previous ones, others can yield a law (he calls it a function) that helps him to study a problem. Possibly you will have thought of that aspect: well, but what you are defining is experimental science! I have no objection to it being given this name, but mathematics is nothing different. I have said on other occasions that there is no greater difference between mathematics and other experimental sciences than the relative position of one to the other. The experimental part of mathematics began with man and still existed between Sumerians and Babylonians, and there are other sciences that are now in the early phase of direct observation; but there is no essential difference in terms of them. Naturally, mathematics, as an older one, and having its roots in the other sciences, has better more elaborate and simpler schemes. This is why it is more useful in teaching than other sciences. Not because it is something different from them, but because its schemes, however elaborate, are simpler to introduce the novice.
An operation that a machine can never do will be to discover new relationships or to construct new concepts that demand that we give a word a meaning different from the one it had. These are the most characteristically creative operations of the human spirit. A teaching that manages this creative capacity for the spirit takes over the student irrevocably, because it focuses on a vital act of spiritual creation. Hence the happy expression that is discovered in the child when he has success from the presentation of teaching in this way. Ah! this has a drawback: this teaching is very difficult for the teacher. We embark on a problem of teacher training.
Last Updated November 2017