The International Commission on Mathematical Instruction
Bulletin No. 47 
December 1999

Reflections on the Learning and Teaching of Mathematics

Alberto P.  Calderón

The following is the translation of the "Rey Pastor" Lecture given by Dr. Alberto P. Calderón at the XXXVI Reunión Anual de la Unión Matemática and IX Reunión de Educación Matemática, held in the cities of Santa Fe and Paraná (Argentina), October 8-11, 1986.

Ladies and gentlemen,

First of all I would like to thank the Organizing Committee of this meeting for inviting me to give this "Rey Pastor" lecture.  This gives me the opportunity of giving public tribute to the teacher whose work has positioned Argentina in the mathematical world, and of whose teachings and stimuli I, like many others, have been direct beneficiary.

On occasion of this pedagogical congress, I thought it might be interesting, and perhaps useful, to share with you some of my reflections on the learning and teaching of mathematics coming from my own experiences, first as a student and afterwards as a professional of the discipline.  These reflections are aimed mainly at secondary school teachers, who have the heavy responsibility of guiding the perhaps most critical and decisive stage of the mental and intellectual shaping of youth.

Mathematics is usually conceived as a purely logical structure or building.  It is also acknowledged as having a non-negligible aesthetic value, residing in the harmony and appropriateness of parts and elements of that structure, and frequently it looks to us as an interesting intellectual game.

Even more, this discipline pursues certain goals which absolutely must be understood to attain its intelligibility.  However I believe that this is not one of its distinctive characteristics, but rather that it is shared with all the disciplines of thought.

If we look at children questioning their elders in their effort to understand the universe, we will see they ask two kinds of questions: those of the "how" or "why" type, concerning what could be called the logical structure of the universe, and those of the "what for" type, concerning its purposes or teleological aspects.  And I don't think this is a consequence of psychological traits of children, but rather a fundamental modality of our thinking.

And if this is so, it not only has a philosophical and gnoseological interest, but also a practical one because of its pedagogical consequence of fundamental importance.  If something has a purpose, the knowledge of this purpose is indispensable for its comprehension.

Let us think of a tool, for instance a screwdriver, and let us ask ourselves whether we understand what a screwdriver is.  Probably hundreds of pages could be written on screwdrivers, describing with great detail the nature and substance of their handles, the chemical, physical and crystalline structures of their metallic parts, the detailed description of their geometric forms, the processes of preparation and manufacturing of their elements, etc.  But after having informed ourselves on all these, would we know what is a screwdriver?  I would say no.  I think that to know it, it is necessary to know the answer to a fundamental question: "What is the screwdriver for?"  Until we know this answer, our knowledge of screwdrivers will be precarious.  Only this answer will make it possible to know what is important and what is secondary in their nature, whether it is important that they have a handle made out of plastic, wood, metal, etc., and that their metallic part is thin or thick, long or short, etc.  That is, understanding of screwdrivers is only possible through the understanding of their function or purpose.  The latter moreover allows us to organize hierarchically our knowledge of screwdrivers.  Without it, any hierarchy of knowledge is impossible.

An opposite example could be our understanding of the solar system.  In this case it is not necessary to ask ourselves "what for" to reach the understanding, since we know of no function or purpose.  It is possible to think of the study of the solar system without considering questions of the ?what for? type.  And there is a line of thought which to a greater or lesser extent encourages this point of view, with its pedagogical consequences.  I would like to express my conviction against such current.  I don't believe it is possible to learn mathematics properly when completely ignoring the goals it seeks.  I will come back to this point later, but let me present an image showing the reason of why this mistake is possible.  If I am traveling in an unknown country and through a long stretch of road with no crossings nor bifurcations, I may not have in mind where I am heading to.  But this knowledge will be extremely important when I get to the next crossroad.

We see why the knowledge of the purposes has tangible and immediate importance in the learning.  It is well known that one learns better and faster if the learning is done by looking for something. This particularly applies to mathematics and this fact sooner or later becomes part of the personal direct experience of those who develop an extended mathematical activity.  The presence of a more or less defined goal illuminates the way to follow, enlivens the interest and allows valuing the different aspects of the discipline, stressing what is primary and relegating what is secondary to the place it deserves.  If the goal is eliminated, progress in the discipline begins to seem to wander, its different parts start to show a monotonic and undifferentiated aspect, and this is how mathematics appears as an accumulation of difficult and horribly boring speculations.  In this regard, it is said that an important part of the solution to a problem is to state it clearly, that is, to know exactly what it is that one proposes to do.

In teaching then, it is important to make students understand, as clearly as possible, the near and far reaching goals of a theoretical development, that is, to motivate.  Of course this is sometimes difficult.  Frequently, because of lack of knowledge, the apprentice can only vaguely understand the distant goals.  But even this vague understanding is valuable, because it makes it easier to understand the selection of intermediate goals, and the distant ones delineate themselves with increasing clarity as progress is made.  The experience in this is similar to that of the explorer exploring an unknown terrain.

For the same reasons, it is clear that more important than accumulation of information is interiorizing the methods and knowing their purpose, that is, knowing where they start and where they lead. Because the methods have an orientation, a dynamics, which isolated results and theorems lack.  But it is also possible to study mathematics blindly, as far as goals are concerned, and then discover the general orientation of the theoretical development by retrospection.  This way of approaching a study is fashionable among those who deny the importance of motivation.  But I think that it is didactically incorrect, and certainly incorrect as a method to guide the research.  Finally, motivation is one of the more important sources of the student's interest, and therefore it is a powerful instrument to awake vocations.  Even though mathematics sometimes looks like a disinterested speculation, when its historical development is observed it is seen that even the more abstract chapters, apparently removed from applications, are connected to a network of interrelationships that ties them with more concrete aspects of the discipline linked to applications.  And such relationships are the ones that ultimately justify the existence of those chapters.  Mathematics is the queen of sciences, and every good queen must serve her subjects.

I said before that learning the methods is much more important than accumulating results or information, which is what the so-called encyclopedic teaching tends to.  How are methods studied?  To answer this it is necessary to keep in mind that methods are instruments for reaching a goal.  Methods are working tools, and in the same way that it is practically impossible to learn a craft or become an artisan only by studying catalogues or attending tool exhibits and it is necessary to actually use the tools, similarly it is impossible to learn mathematics well as a passive observer.  Its methods must be used and, better yet, discovered even if only partially.  In the same way tools or instruments do not of themselves make a good craftsman, but rather it is the way the craftsman uses them, as well it is true that the accumulation of knowledge doesn't make a good mathematician or scientist, but rather the way he handles or uses them.

This is why problem solving is such an important exercise.  It brings us an experience in depth, an opportunity of knowing and weighting the difficulties, of knowing the possibilities and limitations of the mathematical instruments and knowledge we have.  I am referring here of course to non-routine and non-mechanical problems whose resolution requires mental initiative and ingenuity.  It is much more valuable to be able to solve non-trivial problems than accumulating in memory various statements, theorems, proofs, etc.  For instance, if you wish to know how much plane geometry you master, take a book such as Rouché and Comberousse's and see how you perform in solving some of the many problems there, or take a theorem whose proof you have forgotten and see whether you can find one by your own means.  If you don't make it, even in relatively simple cases, I would say that the statements and theorems you have recorded in your mind are worth very little.  And this is true for all branches of mathematics, whether elementary or advanced.  Moreover, solving problems, I insist, non-trivial problems, makes us feel the power of our imagination which, controlled by rigorous reasoning, makes us discover realities whose existence we didn't suspect.  This experience impresses young minds, and it is by itself capable of awakening latent vocations.  In my personal case, it was in this way that my vivid interest in mathematics awoke.

Let me tell you this experience in some detail, because I think it might interest you from a pedagogical point of view.  Having recently started the secondary school, I was by then a little more than twelve years old, I committed a prank while my mathematics teacher was present.  The teacher approached me and told me I would be submitted to a disciplinary action because of my behavior.  He left, but very soon came back, and approaching me once more he said: "Listen, I am going to give you a geometry problem.  If you are able to solve it, your behavior will be forgiven."  The problem was the following: to construct, with ruler and compass, an isosceles triangle where the height and the sum of the base and one side are known.  After no small effort, I could reduce the problem to the following: if ABC is the sought triangle, and AB is its base, let us extend the base in both directions and take on this prolongation, on both side of AB, the points A' and B' such that A'A = B'B = 1/2 AC = 1/2 BC.

Geometric illustration

Clearly the triangle A'B'C is isosceles as well, its height is known and the length of the base is the given sum, and thus it is easy to construct.  The problem will then be solved if the point A on the segment A'B' is determined.  A property of this point is that its distance to C is the double of its distance to A'.  I then thought that the locus of points with this property could be a circle, and after experimenting graphically I reached the certitude that this was the case.  But I couldn't prove it.

I could only reach that point, and when I showed my teacher what I had done, he confirmed my certitude on this locus of points and forgave my prank.  His purpose had been to take advantage of it so as to stimulate my interest for geometry, and this he did only too well.  The problem seduced me and awakened in me an eagerness for solving more and more similar problems.  This little incident made clear what my vocation was and had a decisive influence on my life.

It also illustrates the impact that resolution of problems can have in the mental development of adolescents.  The teacher or professor certainly must be cautious when proposing problems to his/her students: these should not be too easy, since otherwise they don't fulfill their purpose, nor should they be so difficult that they prevent the student from taking a single step, thus leading to unjustified disappointment.  The teacher must try to put forward what the well-known physicist Eugene Wigner called the unreasonable and unusual effectiveness of mathematics, which is one of its greatest enchantments.

Coming back to the methods, here is another way of obtaining control over them: when studying a statement, try to find its proof by yourself.  If this is not attained after having seriously tried, look for some indications, in the books or via the teacher, of the path to follow and try again, and so on until obtaining the sought proof.  It is appropriate to do this, if not with every, at least with some of the statements.  Certainly this method of study is slower, but it is also infinitely more rewarding than the purely receptive one.  It gives us the intellectual pleasure of puzzles, and gives us, at least partially, the satisfaction of the creating act: it makes us feel that these statements, although discovered by others, are partly ours.  It also gives us a perspective of the discipline impossible to get otherwise.  All this, I think, gives importance to the method from the pedagogical point of view.  But there is more to it: in fact, more than once it has led us to find new perspectives of known results which have facilitated their understanding and that, sometimes, have been the first step in the development of new and important chapters of mathematics.  In other words, it gives an active and dynamical knowledge, which is what really matters.  It is the power of this type of knowledge that differentiates the brain from the book.  In accumulating data our brain cannot compete with paper.  But what paper can accumulate is petrified, rigid knowledge.  On the other hand, encyclopedism, the excessive accumulation of dead knowledge, goes against the good functioning of our mind.  It reduces its spontaneity, originality and creative power which are its more precious qualities, which can be cultivated and which we all possess to a greater or lesser degree.  In this regard, it is interesting to observe that frequently erudition and creative capacity are inversely related, and that it is true that the scientist is interested not so much in knowing but in discovering.  To support this I quote the biologist Szent Gyorgyi, Nobel Prize in medicine, who said: "In a library I could get in two hours a greater quantity of knowledge than in one year in my laboratory.  However I am always in my laboratory and rarely in the library."

Another aspect of mathematics, which intimidates and keeps many people away is abstraction.  I think this is so since they haven't been guided or introduced to it adequately.  To abstract is to rise up, a little like flying, and has the purpose of obtaining a panoramic vision of special facts which is impossible otherwise.  As in flight metaphor, this view may be very beautiful and useful and is in a way the reward for the effort abstraction requires.  It is important, then, to start giving this reward to the apprentice as soon as possible in order to stimulate his effort and so that he/she doesn't fall in dismay.

Let us finally consider the beauty of mathematics.  Not all its parts or chapters have the same beauty, which, as I said before, resides in the harmony and congruence of its elements and also in the effectiveness and power of its methods.  Sometimes this beauty is such that it virtually becomes an attractive seducer.  Consequently, it is not appropriate to overlook this facet of mathematics, as it also can be decisive in the awakening of vocations and in stimulating the interest of the student.

To conclude, I would like to show you an elementary geometric construction, which can be understood by the students of the final years of the secondary school and which, I believe, clearly shows the seductive power of the beauty of mathematics.

Let us consider a plane, or the plane, which we will call  E, and oriented circles in that plane.  To each of these circles let us assign a point in space obtained as follows: on the line perpendicular to E that goes through the center of a circle, let us take the point whose distance to the center equals the radius, and which is on one side or the other of the plane according to the orientation of the circle.  We will have thus assigned a point in space to each oriented circle in E and, reciprocally, an oriented circle to each point in space.  Moreover, each spatial configuration is associated to a family of oriented circles in E, and reciprocally.

Evidently, properties of spatial configurations will correspond to properties of families of oriented circles, and reciprocally.  Let us see how we can easily obtain in this way interesting statements.  To this end, let us consider first some simple spatial configurations and their corresponding families of circles.  Let us take for instance a straight line e' in space.  If e' is neither parallel nor perpendicular to E and its intersection with E is P, then the corresponding family consists of all the circles with center on the perpendicular projection e of e' on E, and at a distance from P proportional to the radius of the circle, and with one orientation or the other according to whether the center is in one or the other of the half-lines determined by P in e.  If the angle between e' and E is less than 45°, then the distances of the centers to P are greater than the corresponding radii, and the circles of the family admit two common oriented tangent lines (we will say that an oriented circle and oriented line are tangent if they are in the usual sense and the orientations coincide at the point of tangency).

Let us take now a plane E' in space, which is neither parallel nor perpendicular to E.  It is easy to see that the corresponding family of circles consists of those circles with centers at a distance to the line e, intersection of E and E', proportional to the corresponding radii and with one orientation if their centers are on one of the half-planes of E determined by e, and the opposite if they are on the other half-plane.  The families of circles corresponding to lines e' and planes E' which are parallel or perpendicular to E are described even more easily.

Let us now see a statement that in view of what I have just said is almost immediate, but may not otherwise be.

Let us consider three oriented circles in E, with different radii and mutually external one to each other.  Then the intersections of pairs of oriented lines which are common tangents to pairs of these circles are on a line.

There are other similar statements when the circles do not satisfy the restrictions we have imposed on their radii and relative position.

Let us also see how our construction leads to a transparent solution of the famous problem of Apollonius: to construct, with ruler and compass, a circle tangent to three given circles.  To do this, let us slightly modify the statement by assuming that both the given circles and the circle sought are oriented, adopting here also the convention that two oriented circles are tangent if they are so in the ordinary sense and if their orientations coincide at the point of tangency.  Thus, the classical problem of Apollonius reduces to the modified problems obtained by assigning orientations arbitrarily.

Let us first see what spatial configuration corresponds to the family of oriented circles tangent to a given one.  It is easy to see that this configuration is precisely a right cone whose vertex is the point in space corresponding to the given circle and whose generatrices are determined by the vertex and points on the given circle.

It is then clear that the family of circles simultaneously tangent to two given oriented circles corresponds to the intersection of the cones associated to each of them.  Moreover, the intersection of two right cones with axes perpendicular to E is contained in a plane.  This can be seen by reasoning with the families of corresponding circles.  To obtain this we may assume, shifting the cones vertically if necessary, that these are associated to oriented circles which are tangent in the ordinary sense but have opposite orientations at the point of tangency.  Now using the Pythagorean Theorem and the description we gave of the family of circles corresponding to a plane in space, it can be seen that the points of the intersection of the cones are in a plane that intersects E along a common tangent to the given circles.  This is an example of a statement in space geometry which can be proved with little difficulty by reasoning with families of oriented circles.  Of course, the analytic proof is almost immediate.

Thus it is now clear how to obtain the sought solution.  Let C1, C2 and C3 be the given oriented circles, and let Y1, Y2 and Y3 be the corresponding cones.  The circle or circles we are looking for correspond to points in the intersection of the three cones, which can be obtained in the following way: we construct successively the planes E1 and E2 containing respectively the intersections of the cones Y2 and Y3 and Y1 and Y3, then the line which is the intersection of the planes E1 and E2, and finally the intersection of this line with any of the cones.  All these spatial constructions can be carried on in the plane with ruler and compass, working with the corresponding families of oriented circles, so that our problem is solved.

Ladies and Gentlemen, we are all capable of inventing and discovering to a greater or lesser degree, and this active and creative aspect of our mind must be cultivated at every moment.  I would say that it even gives us the only road to achieve a deep knowledge in any discipline.  Our mind is naturally active and does not admit passiveness or inaction without running into a great danger of atrophy.