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Mathematics: Visible & Tangible

Educational Solutions Worldwide Inc.

Caleb Gattegno


vol. IX no. 3

February 1980

First published in 1980. Reprinted in 2009. Copyright Š 1980-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-298-5 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555

This issue of our Newsletter presents some ideas which may be accessible to the public only now for the first time. Some of them occurred to people before the first World War; others forty years ago. Up to now, these ideas have not been widely understood or accepted, for the good reason that what they attempted to replace has not yet been experienced as ineffectual by those in authority. Today a change is taking place in the world at large. Twenty-five or more years of television have imposed on the public consciousness the realizations that words have limitations and that images have precious qualities. A global visual culture has been in the making, suitable for all peoples whatever their culture and history. This evolving visual culture gives people the opportunity to suggest ways of working which show, at the same time, the limitations of the verbal medium and the advantages of images. Although in this Newsletter we only use words, these words are chosen specifically as triggers for strong, organized images. Perhaps our readers will experience a new awareness about images and come to know explicitly what we mean when we speak of “tangible mathematics.� All people, except the blind, can be led to perceive a certain amount of what is present in images. This can be achieved in particular by means of animation, which can deliberately stress for the viewers what the mind of the film producer sees. Today, through television and the computer, we have entered a new phase in the evolution of the visual culture, from which much can be expected in the field of education. Auser of our films has contributed an article. News items close this issue as usual.

Table of Contents

Introduction ......................................................................... 1 1 A Point Goes Round A Circle: The Generation Of Trigonometry .................................................................... 5 2 The Star That Becomes Many Things................................ 11 3 A Lot From A Little...........................................................17 4 How Our Treatment Helps In Mathematics Studies ........ 21 5 Remarks On The Animated Geometry Films #1 And #2 ............................................................. 25 News Items ......................................................................... 31


Since for so long words have been the vehicle of school education, it is taken for granted that the way to acquire knowledge is either by word of mouth or by extracting it from books. For a long time, libraries were equipped only with shelves full of books, and tables at which one sat to rest a book and turn its pages. Recently the word “library” has started to be replaced by “Media Center,” where there are corners for viewing films or video tapes, for listening to audio tapes and for arranging multimedia displays. Behind these changes, there must be a widespread realization that these new means of communication have some functions in the gathering and dissemination of information. We have all heard that “a picture is worth a thousand words,” but much that goes on in schools, even in the early grades, is in the form of lectures, mainly relying on verbal explanations even when “aids” such as a chalkboard, overhead projectors, diagrams, chart films etc. are brought in. In Piaget’s studies of the content of children’s minds, the word is sovereign. If reliance on spoken language leads to misunderstanding on the part of the experimental subjects, the conclusions reached may not be a true description of the minds the experimenters purport to understand and describe. To reach the content of those minds, we may have to work very differently. We may have to develop approaches which by-pass the shortcomings of verbal statements.


Mathematics: Visible & Tangible

We could be helped by a well-established fact: very young children learn their native language well and for good, while the same people, ten or twenty years later, find it difficult to acquire new languages. If we could understand what babies do and older people do not do that leads babies to such an early and lasting mastery in the verbal field, we may find a way to better understand the role and place of our various learnings. In fact, when grownups do the right things, they too learn languages well. These right things are what children do with themselves, by themselves, without being taught. The innocent mind does not get distracted by preconceptions. Hence young children know that meanings come from perception and they dwell in their perceptions before they label them. They know that meanings are what lie behind words. For grownups who have lost this intuition, however, words seem to have meaning by themselves. The accepted habit of going to a dictionary to get the meaning of words further blurs the issue by implying that words are the source of meaning. In learning a new language, adults often become irritated when they attempt a verbal mental translation of what they hear or see in that language, and find it does not allow them to generate the function of some words which are in the new language but do not exist in theirs. They demand explanations and translations rather than suspending their judgment, returning to the perceptible situation and allowing the meaning to “glue” the new noises. Our work in language teaching with the Silent Way restores the direct connection to perceptible situations as a means to convey the function of words in the new language. In a similar way, the introduction of imagery at the beginning of everyone’s mathematical education restores the students’ access to meaning, which supports not only the language of mathematics but also the awareness that mathematics has its roots in the perception of relationships and their dynamics. People still speak of “teaching aids” when it is more proper to describe the effective ones as models. These models make some awarenesses possible, and open dialogues with oneself that can go on either when 2


one is alone or with others. In the following articles, examples are studied succinctly to give form to the suggestion that mathematics can become an activity which is open to all when the visual and the tangible are stressed prior to moving into the virtual, which is often called abstract.


1 A Point Goes Round A Circle: The Generation Of Trigonometry

Not all our readers have studied the high school subject called trigonometry. If they follow this narrative with pen and paper they will know something about the subject that even people who have taught it for years may not have seen. The word trigonometry was coined because at one time people wanted to study triangles (trigona in Greek). This study took a very definite form: If certain information about a triangle is given, is it possible to find all one wants to know about it? It is clear that in a triangle there are three sides and three angles, and that a triangle holds inside a portion of the plane, called its area. Thus, there are seven items that can be considered as soon as a triangle is drawn. At school, the exercises that are part of trigonometry (in its etymological sense) can be summed up as follows: 1

Do we need to be given all seven of these items to reach a definite triangle? This question becomes: If we are given only one, two, three or more of these items can we attain the same end? What is the minimum number?

2 How does one operate to get a definite triangle, or a class of triangles, from the many choices of one, two, three or any number of these seven available items? For example, what can be said about a triangle with two given sides? three given sides? a given area?


Mathematics: Visible & Tangible

3 Once these attempts to solve the above questions have led us to select some notions and ways of working, is it possible to look at these notions per se and discover properties that were not needed for the original problems? Indeed, there are a number of these properties, the consideration of which has become part of trigonometry courses. Some of these discoveries made it possible to select different introductions to this subject and, while keeping “trigonometry” as the label for the awarenesses which came from these discoveries, to propose very different ways of working. As a result, it became possible to examine which introductions are most beneficial to the learners. This we can now do, selecting what we think is the most primitive introduction, which we have found, over 30 years of testing, to be the most efficient. *** Just as we can run a finger along the edge of a coin and come back to where it started, we can conceive that a point on the circumference of a circle could move around, passing through all the points of the circumference before returning to its original position. We therefore invite students (and our readers here) to imagine this “running” point describing the circumference: 1

at various speeds;

2 alone, or with the ray which emanates from the center of the circle and passes through the point; 3 with the same ray, but not showing the point (which is now conceived as the intersection of the ray and the circumference).


1 A Point Goes Round A Circle: The Generation Of Trigonometry

Let us call each of these ways of looking a visual language. Thus, we found the language conveyed by the ray which emanates from the center of the circle as it sweeps the plane, cutting the circumference (c); the language conveyed by the couple (ray, point of intersection) in which the earlier dynamic (sweeping the plane) is transferred to the circumference, drawing attention to the motion being described by a point (b) and finally when the ray is ignored, the point’s ability to move at different speeds (a) and in both directions, making the viewers feel the independence of the point from the ray. Because we make a point of looking at the same situation in different ways, we offer, at the same time, analytically distinct descriptions and a chance to hold together some or all of them. We can specialize these very general languages by making the turning ray leave behind two tracks which we choose to contain two perpendicular diameters of the circle, one vertical and one horizontal. We call each track an axis. The cross formed by the axes gives us a frame of reference we shall consider as a source of new languages. As the point describes the circumference, the perpendicular from it to one of the axes can be easily abstracted and considered. The segment of this perpendicular between the moving point and the point on the axis on which it falls has been given a name which we shall adopt at once. The segment perpendicular to the horizontal axis is called sine (often written “sin”); that perpendicular to the vertical axis, cosine (often written “cos”). We can now look at the moving point, in terms of what happens to its sine, its cosine, or both taken together. We can also consider the moving point and the rectangle formed by the sine, the cosine, and their projections on the axes. Although this is quite simple (once drawn), it forms a richness that will make further thinking about trigonometry easier and more fruitful. Students can recapitulate the few languages already met and see how differently they affect the appearance of the situation, which itself is


Mathematics: Visible & Tangible

still integrated by the fact that the circumference is described by one of its points. We have selected only one motion of the point on the circumference, that at a constant rate, called uniform circular motion. Although this choice is not necessary, it is convenient to our purpose, and it helps our intuition and our concentration on properties belonging to a “new” foundation of trigonometry. With this choice, the motion of the foot of the perpendicular on each axis is also specified. This motion has been known historically in English as “simple harmonic motion,” and exemplifies what a pendulum does when allowed to oscillate under gravity. If we use time as a coordinate, it is possible to concentrate on the story of the sine or cosine when the point goes once around the circumference (also called one cycle). This extension over the irreversible flow of time provides three more languages: that of the socalled sine curve, that of the cosine curve, and that of their conjunction. A great deal can be said about these languages, and incorporated in detailed discussions between students at school or at home, alone or involving the teacher. Other languages can be as easily added. For example, by drawing a square using the sine as one side (called the square of the sine), we can see how it is affected by the motion of the point. While we see the square’s story molded on that of its side, we also see a future that was not apparent before. Indeed, we can draw one, then two and then three squares — the square of the sine or the cosine or the radius which joins the center to the moving point. Then we can consider together the squares of both the sine and cosine, or one of these and the square of the radius, and finally all three together. This last rich language yields the most important relationship between sine and cosine, one that is true for all positions of the point on the


1 A Point Goes Round A Circle: The Generation Of Trigonometry

circumference and for all circles in the plane. This relationship, expressed as sin2 + cos2 = 1, introduces algebra into the subject. Indeed we can see that the operations of addition and squaring have been used; therefore, we can change the appearance of that formula or relationship to read, sin2 = 1 - cos2 or cos2 = 1 - sin2. Extraction of roots becomes possible. There are other languages in store. For example, if we draw the parallels at the ends of the two diameters of the circle formed by the axes, we obtain 4 new lines which are tangent to the circle. The segments on these tangents extending from the point of tangency with the circle to the point of intersection with the rotating ray are called the tangent (on the tangent line parallel to the sine) and the cotangent (on the tangent line parallel to the cosine). Considering separately each of these segments, it is possible to generate the language of the tangent and follow the story of the tangent when the original point goes around the circumference. We now have the new descriptions called the tangent curve (often written tan) and cotangent curve. It is possible to think of some of these languages together. One such synthesis links tan to cot as reciprocal functions, written either as tan x cot = 1 or tan = cot — 1 or cot = tan — 1 or tan = 1/cot or cot = 1/tan. Another synthesis links tan with sine or cosine. The relationship cos x tan = sin, easily deduced from similar triangles, is very important, for it too reinfuses algebra into this study and makes it possible to have a large number of expressions for what cannot easily be perceived in the original simple generation of the circle by the motion of one of its points. In fact, the contents of trigonometry books can now be obtained by three mental processes: 1

algebra applied to the two fundamental formulae: sin2 + cos2 = 1 and cos x tan = sin;


Mathematics: Visible & Tangible

2 determination of the point on the circumference corresponding to a given angle formed by the ray passing through the moving point and a fixed position of that ray playing the role of a referential; 3 specialization of this angle by a) a certain measure, such as 0° or 30° or 45° or 60° or 90° (also expressed in radians, units of arc measured with the radius as the unit) or b) a relationship between angles, such as the sum of two angles or one being half another angle, so that questions like the following are possible: are the trigonometric functions (represented by the curves mentioned above) such that sin (a+b) = sin a + sin b? (answered negatively), or, are they such that sin 3 a = 3 sin a? sin a/2= ½ sin a? (also answered negatively). *** It is not our intention here to rewrite a trigonometry course which all laymen can read easily and enjoy. What we want to show is that trigonometry can become less remote and mysterious by becoming visual and tangible. * Visual, and visible too, because images are now clustered to produce such a strong impact that theorems present themselves to the looker, rather than remaining inaccessible until passed on from teacher to student. The visibility is a form of tangibility.


The films Folklore of Mathematics (1960) contain a filmic treatment of the above.)


2 The Star That Becomes Many Things

Let us consider the set of rays in one plane having one point in common and let us call it a star. Of course, there is a star associated with every point in the plane. We begin by singling out one ray R of a star S. All other rays in S can be obtained from R by a rotation around the common point O. We can sense at once that there are two choices for such a rotation: clockwise or counterclockwise. Angle can be defined intuitively as the amount of rotation imposed upon R. This definition can tell us the following: 1

Angles form an ordered set; that is, we can see whether an angle RR1 is larger or smaller than RR2 by knowing whether, in going from R to R1 and from R to R2 in the direction of rotation selected to go from R to R1, R1 is met before or after R2.

2 The largest possible angle in this rotation is the one in which R sweeps all of the plane and comes back to its starting position. Such an angle is called a complete revolution.


Mathematics: Visible & Tangible

3 By starting from a complete revolution and moving in the opposite direction, the smallest possible angle is the one in which R coincides with itself. Such an angle has been given the measure zero. Therefore any ray can be conceived as a zero angle and seen as two rays coinciding with each other. 4 It follows that we can see R in the plane as having two sides, called the “lips” of the ray. The two lips become the two sides of an angle when the angle is larger than zero and smaller than a complete revolution. 5 We can define the sum of two angles as follows. Given the angle obtained by going from R to R1 and that from R1 to R2, we can consider the angle formed by R and R2 as the sum. Let us note that we can, at the same time, define the difference of two angles as the angle to be added to one to get the other. 6 If we call positive these angles obtained by counterclockwise rotations and negative the others, addition and subtraction of angles can be merged into one operation called addition of directed angles. 7 We can therefore define multiples of angles (not the multiplication of one angle by another), first by using integers as multiples and then (reversing the operation) by using fractions of an angle of the form 1/n with n = 2, 3.… Then, by combining the two operations, we obtain multiples of an angle of the form m/n. This leads to a definition of multiples of an angle whatever the value of the coefficient. 8 Any angle can serve as a unit of measurement. If we consider a full revolution as 1, then there is meaning to the words “half a revolution” (a flat angle) and “quarter of a revolution” (a right angle). Two rays that form a right angle are said to be perpendicular to each other, or orthogonal. Two rays that form a flat angle are said to be “one in the continuation of the other,” or to form a straight line. 9 Two angles that add up to a right angle are said to be complementary to each other. Two angles that together form a flat angle are said to be supplementary. Each is called the supplement of the other.


2 The Star That Becomes Many Things

10 The addition of angles resembles the addition of numbers. Therefore, angle a + angle b = angle b + angle a (this is called the commutative property of addition). Moreover, because (angle a + angle b) + angle c = angle a + (angle b + angle c) the sum of three angles a, b, c is defined and, therefore, the sum of any number of angles. This property of extending the definition from two elements to any number is usually referred to as the associative property (of addition). Through the above work with angles, we have endowed the star with an algebraic structure, which allows us to use it as numbers are used on a straight line when segments are contemplated instead of angles. We can continue our examination of the star to generate new notions and reach new propositions the proofs of which can be intuitively ascertained. A. We can go back to what we said in number 4 above and, rather than stop our rotation of R when it comes back to its original position, proceed further. We can then define integral multiples of a full revolution, or fractional multiples with the fraction larger than one. This awareness will force us to say, every time we are given an angle, that we can only see its reality to an integral number of full revolutions. This awareness can be stated: Any visible angle O has its apparent measure plus or minus any number of revolutions. In a formula: θ ± 2kπ where 2π is the measure of a full revolution and k is any integer 0, 1, 2.… (θ is read theta, π is read “pie”). In doing this we are not making things more complicated. We are only stating the truth: we cannot say that the appearance is the reality. Indeed no one can know whether there have been a number of full revolutions of R clockwise or counter-clockwise, since they do not show. We therefore want to acknowledge our ignorance. Sometimes we hear the words: angles are defined modulo 2 π. This means exactly what we just said.


Mathematics: Visible & Tangible

B. The same awareness described above was intuited 150 years ago by a German mathematician of genius, Bernhard Riemann. He saw the rotation of the ray R as going on indefinitely in both directions, producing what is now called “the universal covering surface.� Each complete revolution produces what is called (in English) a sheet of that surface. Riemannian surfaces, composed of one or more sheets, are subjects for mathematicians working on specialized studies that have been considered demanding. With the star, however, we can bring to everybody at least the basic concepts of the sheet and of recovering. Soon we shall meet another example still more startling, though extremely common. If the readers visualize one complete revolution of R and see that the plane has been swept once, we can give to that plane the ordinal number 1. By moving R a little beyond its original location in the continuous movement of sweeping, we generate the beginning of a sheet that can be labeled number 2. When the second full revolution is completed, any pin piercing the surface will pass through one point on each sheet. These points can be joined together on the surface only by a line remaining for a while on sheet # 1 and passing on to sheet # 2 to reach the second point. The sheets have no thickness but they have two-dimensional extension. If a large number of them are considered at the same time, a pin will pass through as many distinct points as there are sheets on the surface. There will be a distance on the surface between any two of these points, but in a third dimension there will be none. By thinking of the sheets of the Riemannian surface, what coincides has been made separate, and tangibly so. Riemann gave us a process of disentangling (through a process now called uniformization) what we perceived falsely as a single entity. C. Before extending the yield of the star, let us mark a point P on R. When R sweeps the plane, P describes a circumference centered on O. If we look at what happens to P, we find what we studied in the first article in this Newsletter. If we relate what we discovered in B above to the rotation of P and of OP, we find that OP generates a Riemannian surface limited by an edge traced by the point P. Although this edge appears to be the circumference of a circle with center O and radius OP, it is really not a circumference but a universal covering line. A line


2 The Star That Becomes Many Things

that continues from sheet to sheet as a spiral or a flat circular helix. The circle is a circular universal covering surface. Although very few realize it, hundreds of millions of people carry such a surface with them all the time. Indeed, look at your (non-digital) watch and see what happens minute after minute to the trace of the seconds hand, hour after hour to the trace of the minutes hand and every twelve hours to that of the hours hand. Each of them generates a Riemannian covering surface. In time, the movement of the hands cannot go through the same spot twice, since time is irreversible. But the spatial appearance is as if they do. Time makes our non-digital watches and clocks produce Riemannian covering surfaces everywhere. Of course, this brief discussion of Riemannian surfaces does not delve deeply into the mathematical use of this notion. It does show that complex ideas can be made tangible. D. Leaving aside this discussion and readjusting our sight, let us consider what we can find by allowing the point P to appear. 1

The arbitrary choice of any point P on R tells us that, rather than one circle, we have a whole family of concentric circles. The smallest is the point O, a circle of radius zero. The largest is not conceivable.

2 Any one circle (k) of the family divides it into two subfamilies: those with radii smaller than that of (k), and those with larger radii. If we select in the first sub-family the circles whose radii are respectively r/2, r/4, r/8‌ (where r is the radius of k) and in the second those whose radii are 2r, 4r, 8r,‌ we can associate circles with radius r/n and radius nr to each other, and sense that there are as many circles inside R as there are outside it. This activity yields another experience which can refine our sense of reality beyond appearances. Consequently, on R, there will be as many points between O and P as there are beyond P. Also, every point of the plane is on one and only one ray R, and on one and only one circle k.


We stop here the listing of topics associated with a star. Many interesting ones exist which are treated somewhere else.

3 A Lot From A Little

Let us consider at the same time two stars S and S' with centers O and O' respectively. The line OO' belongs to both stars. Let R and R' be rays belonging to S and S' respectively. Let the ray Ro, with endpoint O and passing through O', be the starting position for ray R. Thus, R0 is on line OO'. There are two choices for R0', starting positions for R' on the line OO'. In the following discussion we choose the R0’ that passes through O. R can sweep S clockwise and R' can simultaneously sweep S' in the same direction. If their “rates of rotation” are the same (starting respectively from the positions R0 and R0' on the line that is defined by OO'), as soon as R and R' move away from OO' they are “parallel.” The angles θ = R0 OR and θ’ = R0' O' R' are equal. When θ goes from zero to a full revolution or 2 π radians, O' also goes from zero to 2π Here, lines covering each other are also called parallel, but not distinct. In the first revolution this happens only when θ = O, π or 2π radians. Alternatively, we can make R move clockwise around O/ and R' counterclockwise around O' with the same rate of rotation. At the beginning, R and R' overlap along OO' and form a straight line. As they move on, they cut each other as long as their angle with OO' is smaller than a right angle. All these points of intersection M are on a line called the perpendicular bisector of OO'. The segments MO and MO' are always equal.


Mathematics: Visible & Tangible

We could pair up every R with another ray of S, and every R' of S' with another ray of R' so that each pair forms a flat angle or straight line. Then we can see that as each pair of rays which forms a line sweeps S or S' there is always a point of intersection with the perpendicular bisector. This example is the simplest pairing of rays of S and S'. Any such pairing defines what we may call either a locus, a function, a graph, or by some other name, depending on the level at which we talk. Let us examine a few more of the elementary choices available to us: 1

Let us pair up R of S with a ray R' of S' which is perpendicular to it. As R rotates about O and R' about O', all the points M formed by the intersection of R and R' are on the circumference of a circle with diameter OO'. The graph of this function can be found by drawing. It can be shown to be a circle in the following way. First construct the segment Μ ω where ω is the midpoint of OO', and find that it is equal to half of OO'. Since all the points M are equidistant from ω, they must lie on the circle described above.

2 Can we replace the right angle by any constant angle θ? The answer is yes, and we find that in this case the locus of intersections of rays R and R' is made up of two equal circular arcs. If θ is larger than a right angle, these arcs lie within the circle described in#1 above. If θ is smaller than a right angle, these arcs contain this circle. As the value of θ varies from π/2 to π radians, the arcs flatten more and more, and come closer to the segment OO'. As the value of θ varies from π/2 to zero, the arcs open up. When θ is zero the “arcs” lie on the line OO', covering all but the segment OO'. 3 What happens if MO + MO' equals a given length larger than that of OO'? The graph of the function is then called an ellipse, with foci O and O'. There are, of course, as many ellipses with these foci as there are values of the given length. If the 19

3 A Lot From A Little

given length is called λ, when λ is close to the length of OO', the ellipse remains very close to the segment OO'. As λ gets bigger, the ellipse moves away form OO'. The family of ellipses generated in this manner sweeps the plane, each ellipse containing and not cutting those for which λ was smaller. 4 What happens if MO — MO' and MO' — MO remain constant? The graph of this function, called an hyperbola, has two branches: one when MO is larger than MO' and one when MO is smaller than MO'. Let the difference between MO and MO' be called λ. Then the two branches of an hyperbola H0 are defined by a value λ0 of λ. All the hyperbolae corresponding to values of λ larger than λ0 have their branches “inside” the branches of H0, which are inside the branches of any hyperbola with value of λ less than λ0. In particular, if λ = 0, the two branches become the perpendicular bisector of OO'. As K approaches infinity, they become two rays lying along line OO', directed so that neither covers segment OO. 5 What happens if the ratio MO/ MO' is constant? The graph is then a circle known since the time of the ancient Greeks by the name of Apollonius. Let the ratio of MO to MO' be called ρ. If ρ is larger than 1, the circle is on one side of the perpendicular bisector of OO'. If ρ = 1, it is the bisector. If ρ is smaller than 1, it is on the other side. As ρ becomes smaller and smaller the corresponding Apollonius circle tends towards O, reaching this point when ρ = 0. As ρ becomes larger and larger (ρ > 1), the corresponding circle again becomes smaller and smaller, coinciding with O' as ρ reaches infinity. One can easily see that no two circles cut each other. In the two sub-families separated by the perpendicular bisector B of OO', any given circle contains all the smaller ones and is contained in all the larger ones on the same


Mathematics: Visible & Tangible

side of B. But they are not concentric. Finding where the various centers are is a good challenge, requiring some study. 6 What happens if the product MO x MO' is constant? The graphs of this family of functions was studied in the 17th century by Bernoulli and Cassini. The latter gave his name to all those loci for which the constant (K) is not equal to 1. Bernoulli’s graph, also called Bernoulli’s lemniscate, looks like an 8 on its side, and corresponds to ρ = 1. The lemniscate divides the family into two sub-families: one has graphs consisting of two portions, one inside each loop of the 8; the other has graphs of a continuous closed curve containing the lemniscate, symmetrical with respect to ω, the midpoint of OO', and containing a dip which is maximum in the case of the lemniscate and becomes less and less noticeable as ρ becomes larger. This sub-family of Cassini’s ovals when the dip appears, look very much like ellipses, although none are. In this article we put together families of plane curves which challenged the best minds of the seventeenth century in Europe. The notions of the star and of pairing rays of two of them were sufficient to generate that wealth.


4 How Our Treatment Helps In Mathematics Studies

Whoever has read the preceding articles with paper and pencil at hand already knows that our approach can yield new insights, and that simple beginnings can lead to considerable harvests. Those who have a more substantial background in mathematics will be more appreciative of the proposals than those who have read the texts without comparative frames of reference. In fact, the contents of the articles show a principle for organizing a math curriculum which is totally different from those in circulation in school textbooks. Rather than constructing a stricter linear development, we suggest entering the edifice of mathematics from those entrances leading to sections of the building which are the most habitable at present, not worrying about who the neighbors are and what they do. These entrances are all on the ground floor, of easy access and well lit. The staircases for climbing up or down to other areas are soon found, and also easily accessible. Much of historical mathematics, can be called “Folkloric,” in comparison with the more naked and skeletal axiomatic approach that has been propagated during the last twenty years under the names of “the new math” or “modern math.” The Folklore of Mathematics can be saved for the generations to come, sifted and reorganized in the form given in the above three articles. In this form, it can be passed on to the next generation in a manner that is easily assimilated and enjoyed.


Mathematics: Visible & Tangible

For example, the few animated films available in the Animated Geometry and Folklore of Mathematics series provide: 1) condensed presentations of many topics in films that last only a few minutes each; 2) smooth linkages between topics, suggesting ways of interrelating them; 3) beginnings that are accessible to viewers of almost all ages; 4) aesthetic experiences that invite repeated viewings and deeper acquaintance; 5) motivation to discuss and explore the contents of the films with peers and advanced students (including teachers and mathematicians), to share experience, to give it verbal form, and to move towards proofs of the validity of what one senses to be true in the situation on film. Such an education has not been available until now because the accepted format of presentation reflected the dominance of the field by an insufficiently scrutinized tradition which did not permit the vision of alternatives. Within the framework of this tradition, intuition was tolerated only at the initial stages of the learning process and was considered not as valuable in one’s mental evolution as the adoption of the use of syllogistic reasoning. So, only the “natural” mathematicians could reach the creative stage in which intuition works all the time. The majority of the public was excluded from feeling welcome, by being led to think that mathematics is an arid area where only specially gifted people can survive. Now, however, we know how to cultivate intuition through the use of modern technological means that allow us to give students a charge. We charge them by giving them a lot in very concentrated form and by creating a desire in them to let this material settle in their minds and live its own life within them. This is possible mainly because the dynamic of the films resemble those of the mind. Intuition is defined here as the acquaintance with, and maintenance of, the whole on which one’s mind can work analytically. It is this detailed examination of the present whole that permits the re-creation of the chapters of mathematics. But the resulting mathematics is not obtained at the cost of losing one’s respect, even affection, for the mind’s 23

4 How Our Treatment Helps In Mathematics Studies

intuitive way of working. On the contrary, at the same time as one’s affectivity is mobilized by the charge, by the aesthetic feeling, all of one’s gifts can find room to work harmoniously with each other to produce the new, the unformed, that gets objectified. Computer animation makes changes in images as smooth, far-reaching and comprehensive as the transformations involved permit. As a consequence, we now have the means in every classroom to make the imagery necessary for geometrical intuition accessible to everyone. In our own clinic, we have helped numerous people, who have come to us unable to learn mathematics. Through our approach they become capable of using complex and intricate concepts in a matter of hours. Our students find in their own minds the awarenesses they need to reach the various ways of mathematizing which are involved in producing the topics they are supposed to study. Since they are working on themselves, they become independent, autonomous and responsible learners. Mastery follows, and a totally different perspective on what mathematics is and requires of them ensues. They appear to have performed a miracle, when in fact they have only done what was always passable for them but which they had never properly approached before. The confidence that normally accompanies insight (in-sight) serves as a testimony to them, their peers, and their teachers that they are at home in a field conceived until then as a foreign land. To educate intuition, using images and devices capable of promoting this in the classroom, is not only valuable, it belongs to mathematical learning and thinking. All students — required by law to be in school and to learn mathematics to “grade level” norms of national standardized tests — must be given the opportunity to discover the reality of so important a subject. This can be done most easily by presenting parts of the subject as animated diagrams that expand one’s intuition, suggest mental operations, and provide motivation to verbalize and articulate what one sees in what one is looking at, and motivation to want to convince others of the truth of one’s perceptions. Thus, mathematization is transformed into mathematical education, in the sense that one finds oneself owning dynamic mental structures 24

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which make one free to contemplate or generate ideas intimately linked to the imagery one can evoke. Making all this available to everyone surely helps them to know themselves as having at least the option to become a mathematician. In mathematics education, images have traditionally been viewed as crutches. Today, they can be seen as the very stuff upon which our intelligence reasons and works to find other realities hidden in the dynamics behind the images. In fact, films only objectify what can be reached through introspection by mathematicians wanting to objectify what they see within; clearly, before they submit a scenario to the animators, they must have found what, in their minds, can be translated into images to go on film. When examining mathematical activity, we hit upon a definition of mathematics in terms of awareness. The words that came up — “mathematics is an awareness of the dynamics of relationships per se” — seemed to encounter immediate and unanimous acceptance by all the mathematicians who heard them. It was clear that, because awareness was part of the definition, it was capable of moving educators as well. Soon it became evident that it is more proper to talk of mathematization than of mathematics, and to base teaching upon the activities that produce mathematics (mathematization) rather than upon their results (mathematics). Images and concrete manipulations have their proper place in the processes of mathematization, and they no longer come in artificially, as gimmicks appropriate only to “nonmathematical” students. Even for the working mathematician images are necessary, and in that context “concrete” simply means that which is familiar, i.e. capable of generating an inner climate in which the activities accompanying mathematical terminology and notation are grounded in fact. Our “poor” students, who are lost in this field, experience uncertainties because words and symbols remain hollow and trigger no other content besides their sounds and their written forms. As soon as images appear, meaning exists. This simple observation can become the


foundation of good teaching. Indeed, it serves as a guide in the study of mathematics learning and in the subordination of teaching to learning.


5 Remarks On The Animated Geometry Films #1 And #2

During the two weeks of November 1 - 14, 1979, we rented and showed two of the newer films of the series ANIMATED GEOMETRY, entitled A STUDY OF FAMILIES OF CIRCLES IN THE PLANE and ANGLES IN THE CIRCLE. Each film was kept for one week, the first being shown publicly five times, and the second, three times. A total of fiftytwo people saw the films and worked with them for a minimum of one and a half hours at each session. At each session, I gave a very brief introduction, describing Dr. Gattegno’s work in mathematics education and the relationship of these films to this work. I stated that our work with the film that evening might be quite different from any prior mathematical work they had done and invited those present to consider the evening an experiment, urging them to participate whole heartedly, postponing discussion of how these films might be used in their teaching until after our working session. With only one exception, everyone seemed very willing to jump right in. We then watched the film and afterwards, I asked, “What did you see?” Sometimes I took the first response and started a discussion based on that response, regardless of its content. Other times I heard everyone’s remarks without comment and then selected one to work from. Discussion usually proceeded for five to ten minutes and then a need was expressed to see the film, or a section of it, again to verify or contradict some statement. At first, I tended to refer to the film very quickly, but I learned that if I delayed re-showing the film for a few 27

Mathematics: Visible & Tangible

minutes, someone in the room would usually come up with an accurate description of the sequence in question. Only one group seemed to have difficulty noticing details or knowing what to say about them. In that case, I would ask a question, such as “What can you say about the colors used in the film?” or “How would you describe the motion in the final sequence?” As people groped with words, I sometimes encouraged them to use the blackboard or pencil and paper to illustrate or clarify their point. Usually one person in each group followed this suggestion, but most of the mathematical work was done verbally. There were none of the theorems, proofs or problems usually found in mathematics courses. I tried to constantly refer back to the contents of the film so that all our work would be based on a shared visual experience rather than prior mathematical knowledge. When the more experienced mathematicians did see a connection between the film and some mathematical result familiar to them, I urged them to explain the result in terms of the film. This turned out to be difficult for them and confusing for the less experienced participants. Such connections were usually left hanging, to be pursued by the interested individuals on their own. Reaction to the films varied. The Math Faculty at University of New Mexico immediately grasped their power and were excited by the teaching possibilities (I know nothing of any follow-up). They saw that motion could be a novel, although natural means of understanding plane geometry and that our eyes can perceive and understand a mathematical relationship in a direct way, different from the way in which a written theorem and its proof convinces us of its truth. Several faculty members thought of other specific mathematical topics they could imagine presented on film. Only one of the twelve had used mathematical films before, but many said they would like to, after seeing these films. The most enthusiastic teacher described a course he could imagine being built around such films. He also brought his Freshman Honors class to see them the following hour. The high school teachers were also very interested in this way of working with film, but were immediately concerned about how they might, or might not, fit these films into their prescribed curriculum. But at least four of these teachers said that our sessions with the films


5 Remarks On The Animated Geometry Films #1 And #2

were very stimulating and made them reflect on how unimaginative and routine-ridden their own teaching had become. Some of these teachers also vaguely “recognized� certain parts of the film, but, unlike the University of New Mexico faculty, had a tendency to fixate on those parts and could not easily be led back into the remainder of the film. The less experienced participants worked with the film willingly and with interest. They occasionally expressed frustration at not being able to verbalize what they had seen, but had no hesitation in viewing sections of the film again and again in an effort to improve their perception and description. One participant stubbornly maintained that the color was purely decorative, as if that is all color ever could be in a film. He and another teacher also felt that most of the film was intended only to be visually pleasing and that looking for mathematical content was silly. However, all the other participants, including all of the children, immediately understood that this film was deliberately constructed with specific content and ideas in mind, and that our work was somewhat like that of a detective trying to uncover these ideas. The children had no trouble working with the film and took part on an equal footing with the adults present. The two high school girls who watched the films at our house were perhaps the most responsive of all. They not only worked with the geometric content of the film, but also engaged in lengthy discussion of infinity, how the film was made and even the nature of mathematical truth. Listed below are a few particular items which attracted our interest. We probably devoted more time to their discussion than is desirable. 1 A blue circle is tangent to a fixed line at a fixed point. On the screen, the circle is to the right of the line. The radius of the circle increases without limit, finally coinciding with a straight line. The circle then reappears to the left of the line and its radius decreases in size until it coincides with the fixed point. The pro - lem here was the switch from the right side to the left side of the fixed line. Whether intended or not, this section led to intense discussion of infinity, limits, the possible curvature of space and the nature of geometric models.


Mathematics: Visible & Tangible


Points A and B are fixed, P and Q rotate along the circle. Diagonal PQ is drawn. For a while, the moving image appears flat and then it suddenly seems three-dimensional. In at least two sessions, we explored the question of why our eyes and brain make this shift and then started noticing other parts of the film that could be seen as either 2 or 3 dimensional. 3 The question about the use of color led to detailed discussion of types of motion. The difference between a circle rotating about a fixed point on its circumference (possibly also increasing or decreasing in radius) and one rolling along a line or other circle (also possibly increasing or decreasing in radius) turned out to be subtle and not immediately obvious. Discussion of this distinction led to questions like “What if the circle it rolled along were an ellipse or parabola instead?” in turn leading to mention of conic sections and hypo- and epi-cycloids. In feedback sessions, mention was usually made (with surprise) of how verbalization actually helped improve visual perception and understanding. Many said that at first they didn’t feel that they had seen or noticed anything special but that as they heard other people’s descriptions, they became more aware of their own perceptions and were better able to express them. Most of the participants were unaccustomed to such long discussions. They had thought that doing mathematics consisted of working problems, checking answers or devising proofs, silent pencil and paper activities done by oneself. It seemed that looking and talking were two operations seldom used. In addition, they spoke of never having been asked for their own personal perceptions, Mathematics, to many of them, had felt impersonal, a question of truth or falsity, right or wrong, based on criteria clear to the textbook or teacher, but not always clear to them. The idea that they could formulate their own hypotheses from the film and then test them with the film was a surprise. Furthermore, the mere act of focusing attention on circles everywhere in life. One said that a circle would 30

5 Remarks On The Animated Geometry Films #1 And #2

never again be the same. Another general awareness resulting from work with the films was an increased sensitivity to motion of all kind, not only rotation and rolling. Although I felt that all of the sessions were worthwhile and successful in terms of people’s involvement, I feel that I could have been more precise and demanding, perhaps with greater experience with the films. I still hesitate to demand the precision required by mathematics, not doubt a reflection of my own sloppiness. I was personally interested in the use of sketches as visual note-taking, but did not encourage it, since I was trying to work in Dr. Gattegno’s way, stressing verbalization. Another time, I would explore that area with more vigor. I also would have liked to work in more classroom situations. This only occurred twice, once at the University of New Mexico and the other time with a mixed class of elementary and high school students at a private school. Since this was my first time working with the films, I was not terribly confident and therefore relatively unaggressive regarding arranging more such sessions. The films also pointed up some serious shortcomings in my own background in geometry, to be remedied before or concurrently with further use of the films. The viewing and reviewing of the films had a patient, careful quality I value and often miss in regular math classes or even in private tutoring sessions. Too often, the attempt to “cover the material” means going faster and faster and never exploring the fascinating side pathways that arise when the mind is free to contemplate. I became very aware of my excessive concern with students potential boredom, actually just a manifestation of my own twitchiness. I also learned a lot about my own visual perception, of how it leaps about like a monkey, leaving great gaps unseen. When I managed to put the brakes on and work slowly and patiently, I was impressed by the wealth of discussion generated by so little actual film time. I feel as if I didn’t come close to doing the material justice, but I did what I could. Libby Palmer Santa Fe, New Mexico


News Items

1 On January 21st and 22nd, Dr. Gattegno visited the State Department Foreign Services Institute in Washington D. C. Where Dr. Earl Stevick kindly arranged for a presentation of the video tape series English the Silent Way. The purpose of the visit was to find out whether teachers who do not necessarily teach English, and come from many different countries, could find some merit in the conception and execution of this series as it relates to the teaching of language. More than this was in fact achieved in those few hours. The FSI circular announcing a lunch-hour meeting on the 21st and 22nd invited staff members to attend. Many more than expected came, and a large number stayed for two hours on the 21st and even returned on the following day. An informal discussion with 15 supervisors on the morning of the 22nd showed how much these people understood the breakthrough represented by the creation of the series. The overall impression was that, while at FSI nobody teaches English (although many have needed to take English as a foreign language to get their job), this was the right kind of testing audience for the tapes. Further, their verdict was clearly favorable. The time it took to show a scatter of samples (11 out of 140 lessons) did not permit as thorough a discussion as might have been both possible and desirable. The few words of introduction, and some brief comments by the expositor, were not sufficient to allay all the doubts existing in teachers attached to a face-to-face student/teacher relationship. Nevertheless, the evidence was there and no one could escape it. Students who had only had their


Mathematics: Visible & Tangible

own voices to listen to were producing a flow of English sounds displaying an energy distribution that to some seemed very good and to others, enviable. These results had been obtained by means the implementation of which is not generally even considered since it is commonly believed that without the teacher providing a model they are impossible. The lessons on the tapes can be conceived in any other language, and a number of people in the audience wanted to know where they might find the materials relating to their own language since tapes are at present available only for English and Hebrew. Many others saw the importance of the tapes for teacher education and for research into language learning. The session with the supervisors was more probing and, because more than an hour could be spent in examining the points raised, those involved were able to get what they came for. The most important result of the visit was not the exposure our video program received but the fact that so many language teachers are now aware that 1) the Silent Way works and, 2) there exists a further breakthrough represented by the tapes, through which learners can serve other learners and facilitate their acquisition of what was once thought to come only from knowledgeable teachers. Dr. Stevick expressed satisfaction with the preliminary effect on his colleagues of the visit he had arranged. Dr. Gattegno expressed gratitude for the opportunity to work with Dr. Stevick for several hours, the first in many years of acquaintance. 2 A weekend workshop entirely devoted to pronunciation took place at our headquarters on January 18th and 19th. In the words of the participants, the 15-hour seminar had the effect of: 1

proving that pronunciation can be taught;


News Items

2 illustrating this through a two-hour demonstration with an adult who was fluent in English but was still using Spanish sounds and melody in her speech; 3 showing why the techniques were invented, how they can be implemented, and what makes their strengths; 4 encouraging the participants to attempt to use them. 3 In December of ‘79, Educational Solutions sponsored a Conference for Bilingual Educators. Those who attended were able to learn about our concept of bilingualism, and analyze with us the state of bilingual education today. Discussion groups were formed to look at the solutions our office has offered for various problems present in bilingual education. At the same time, the Conference participants came in contact — many for the first time — with ways of working with students which take maximum advantage of the human process of learning, by making the students aware that they themselves are the producers of their own knowledge. Thus, teachers are able to utilize completely the natural capacities of the students. In sessions demonstrating our methodology, the participants took part in applying techniques conducive to the production of this awareness in their students. It was clear that, once each student attains this state of alertness, interest, the dynamics of learning, the growth of knowledge, the projection of this knowledge and its integration in wider areas all become a “natural outgrowth” of the learner, which he or she will accomplish at all times with ease, independence and responsibility. The Conference participants recognized that bilingual education in the United States is a constantly expanding field, with a growing diversity of components. They agreed that it was most urgent to meditate on the concrete problems and pedagogical needs of bilingual-bicultural teachers. A feedback questionnaire distributed among the participants indicated that there exists a concern for students who, because of their previous experience, need some sort of remedial work, especially in reading, oral expression, and ability to use a second language. At present, these 35

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students are unable to benefit the most from their own potential, and are thus placed in a disadvantaged academic position. It is obvious that there is an urgent need for remedial education, of the highest technical and professional caliber, with a precise diagnostical vision of the student and with absolute respect for her or his capacities and talents. We are now following up on the interest that a number of the participants expressed concerning our contribution to bilingual education.


About Caleb Gattegno Caleb Gattegno is the teacher every student dreams of; he doesn’t require his students to memorize anything, he doesn’t shout or at times even say a word, and his students learn at an accelerated rate because they are truly interested. In a world where memorization, recitation, and standardized tests are still the norm, Gattegno was truly ahead of his time. Born in Alexandria, Egypt in 1911, Gattegno was a scholar of many fields. He held a doctorate of mathematics, a doctorate of arts in psychology, a master of arts in education, and a bachelor of science in physics and chemistry. He held a scientific view of education, and believed illiteracy was a problem that could be solved. He questioned the role of time and algebra in the process of learning to read, and, most importantly, questioned the role of the teacher. The focus in all subjects, he insisted, should always be placed on learning, not on teaching. He called this principle the Subordination of Teaching to Learning. Gattegno travelled around the world 10 times conducting seminars on his teaching methods, and had himself learned about 40 languages. He wrote more than 120 books during his career, and from 1971 until his death in 1988 he published the Educational Solutions newsletter five times a year. He was survived by his second wife Shakti Gattegno and his four children.

Mathematics Visible & Tangible  
Mathematics Visible & Tangible  

Newsletter vol. IX no. 3 February 1980