Mathematics

Mathematics

Educational Solutions Worldwide Inc.

Caleb Gattegno

Newsletter

vol. II no. 3

March 1973

First published in 1973. Reprinted in 2009.

Copyright Š 1973-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-268-8 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

Although many people are involved in teaching math, very few are engaged in the study of this activity. Among those few, we must count all of us at Educational Solutions who came together precisely because we had this common interest. The short articles which constitute the main body of this newsletter are proof of our concern and can serve as a basis for association with interested teachers.

No doubt the key to being effective in this field of research lies in shifting from interest in mathematics to interest in those activities of the mind which produce mathematics in any human being. Each of the following articles points toward a wealth of knowledge to be discovered if teachers use the opportunities in their classrooms to add to our understanding of math learning and teaching.

Table of Contents

Mathematics And Language.................................................. 1 The Set Of Our Fingers ......................................................... 5 Generalizing And Specializing .............................................. 9 Transformation .................................................................. 13 Awareness And Facility .......................................................17 Additional Information ...................................................... 19 Games For Writing Mathematics ........................................ 21

Mathematics And Language

In our work with teachers we still find many whose past experience and training have not prepared them to understand or accept on faith our assertion that algebra is properly taught before arithmetic. Yet, all of these people will readily agree that a five year old is an expert speaker of his native language. It is not as easy for them to see how he did it, or precisely what jobs he did and therefore it seems fruitful to begin our work with teachers by studying such questions as, “What, exactly, are the powers of the mind demonstrated by language learners?” When they recognize that the powers of transformation that allow someone to know that saying “I am his son” is equivalent to saying “He is my father” are the stuff of algebra itself, they will be more optimistic about engaging in the use of the four operations and fractions with kindergarten children.

“What have children done, when they are able to look at any one of a number of extraordinarily different looking shaggy or sleek, fat or thin, spotted or plain creatures and call it a dog?” Isn’t the ability that a two year old has, to understand the naming which applies to classes, a power of abstraction? If they can do this, need anyone feel diffident about offering them such an accessible abstraction as that of number?

“What allows us to call one dog Ralph or brown or cocker spaniel?” We find children know that, depending upon the criterion they use, the name for the same dog changes. When teachers see that children are in command of this, they know they will be interested in playing games where we call a number by many names.

1

Mathematics

Eight, for instance, has 128 names if we consider the criterion of its table of partitions. Of these, the partitions which are repeated additions of the same number gain another name because of this property (eight ones, one eight, four twos and two fours). The fact that there are factors other than eight and one are present in the table gives eight another name, composite, and the presence of the factor two makes it even.

Because four is two twos and eight is two fours, eight has another name, two times two times two, or two to the third power.

“What have five year olds accomplished when they know that the same person may be called a mother, wife, sister, daughter and teacher?”

Everyone has stories of how two and three year olds sort out the naming based on relationship. Since they have done it, they will be at home with the language of fractions, which is a naming game based on relationships.

“Do children learn up without down, or on without off . . . or anything without its negative?” Isn't no the most famous word in a two year old's vocabulary? And next comes not. When we see that children are comfortable with opposites, it is natural to present the operations of addition and subtraction, multiplication and division, as pairs of opposites, rather than separate as is still done in many places.

When people begin to appreciate the powers of the language learner, they are ready to distinguish mathematical thinking from other sorts.

Dr. Gattegno's textbooks I and II will at this stage be a place to find out how to teach people to be mathematicians rather than calculating machines. The language study insures that a reader can approach the books at a level of awareness where he or she will be most able to benefit from them. The questions we will ask about the lessons of Books I and II are, “Which part of what we are engaged in is actions?”

“Which part is accessible to perception?” “Exactly where does language enter in?”

The more precise one can be about what part of a lesson is language, the fewer problems one has. It is a common confusion of teachers to think that because children responded to what they heard and were 2

Mathematics And Language

able to engage in a game with Algebricks or fingers, they would be able to give the instructions to someone else verbatim. Because I could find the rod called by someone “one fourth of the brown” does not mean I have done the work of naming one rod in relationship to another. Once again, studying babies is a help. They do not say all that they understand, and talking takes practice. Having made these distinctions for themselves, teachers, when teaching language, will be aware of it and have proper respect for the price each learner has to pay Because four is two twos and eight is two fours, eight has another name, two times two times two, or two to the third power.

“What have five year olds accomplished when they know that the same person may be called a mother, wife, sister, daughter and teacher?”

Everyone has stories of how two and three year olds sort out the naming based on relationship. Since they have done it, they will be at home with the language of fractions, which is a naming game based on relationships.

“Do children learn up without down, or on with out off . . . or anything without its negative?” Isn't no the most famous word in a two year old's vocabulary? And next comes not. When we see that children are comfortable with opposites, it is natural to present the operations of addition and subtraction, multiplication and division, as pairs of opposites, rather than separate as is still done in many places.

When people begin to appreciate the powers of the language learner, they are ready to distinguish mathematical thinking from other sorts.

Dr. Gattegno's textbooks I and II will at this stage be a place to find out how to teach people to be mathematicians rather than calculating machines. The language study insures that a reader can approach the books at a level of awareness where he or she will be most able to benefit from them. The questions we will ask about the lessons of Books I and II are, “Which part of what we are engaged in is actions?”

“Which part is accessible to perception?” “Exactly where does language enter in?”

The more precise one can be about what part of a lesson is language, the fewer problems one has. It is a common confusion of teachers to 3

Mathematics

think that because children responded to what they heard and were able to engage in a game with Algebricks or fingers, they would be able to give the instructions to someone else verbatim. Because I could find the rod called by someone “one fourth of the brown” does not mean I have done the work of naming one rod in relationship to another. Once again, studying babies is a help. They do not say all that they understand, and talking takes practice. Having made these distinctions for themselves, teachers, when teaching language, will be aware of it and have proper respect for the price each learner has to pay so that he can sort out and hold a new noise (say, “plus”) and the criteria for its use.

Caroline Chinlund

4

The Set Of Our Fingers

Instead of abandoning counting on the fingers, we can recover the use of the set of fingers and see to what extent this provides a basis for doing mathematics. If we call the set of fingers the absolute, we can see that the action of putting some fingers up and the others down divides this absolute into two, easily distinguishable, complementary subsets — the set of the folded and the set of the unfolded fingers. Many complementary pairs are possible, and they are linked one to another by the action of transferring some folded fingers to the set of unfolded or vice versa.

If we start with all fingers folded and then unfold one finger for each of the noises – “one,” “two,” “three,” “four,” we will give the name “four” to this particular set of unfolded fingers. Since we could have chosen other fingers to unfold as well, we become aware that “four” is the name for a number of different but equivalent sets. Through practice in generating equivalent sets for “two,” “three,” “five,” etc., we find that it is not necessary to count when we produce with our own fingers any unfolded set. We know within ourselves which sets we have made and also recognize quickly how many fingers someone else is showing.

Since we can give a numeral adjective to the set of unfolded fingers, we can also do the same to the set of folded fingers. In this way we generate the complementary pairs for 10, one of the names for the absolute. The following questions can be asked for the pair (4, 6): a) If six (four) fingers are unfolded (folded), how many are folded (unfolded)? b) from ten fingers take away six (four). How many are

5

Mathematics

left? c) To four fingers add six more. How many are there now? By practicing all these ways we know that, for complements of ten, six always goes with four and vice versa. Naturally, this applies to all the complements in tent (0, 10), (1, 9), (2, 8), (3, 7), (4, 6), (5, 5).

Now that we have all complementary pairs in ten, we can investigate how they are linked. How do we go from the pair (6, 4) to the pair (5, 5) for instance? We can write (6, 4) (6-1 4+1) (5, 5), to indicate that we have to take one finger from the set of six and give it to the set of four since we can neither add nor subtract from the absolute. This principle of compensation applies for the transformation from any given pair to another. This means that it is necessary to know only one pair because it will generate all others.

We can now apply the dynamics of complementary pairs to numerals other than ten. First of all, we can ignore one of our fingers and consider the absolute to be nine. This will generate the pairs: (0, 9), (1, 8), (2, 7), (3, 6), (4, 5). But these are linked to the complements of ten.

For example, by removing one from (4, 6), either (4, 5) or (6, 3) is generated. We can continue and study the complements of eight, seven, etc. And always what we find is linked to what we have done so that all these pairs form a dynamic network.

To any of these we can apply the notations for addition and subtraction, and we are only adding a convention to something already known:

In addition, we can use the set of fingers to show that if we know the complements in ten, for example, we can easily know the complements in 100, 1,000, 10,000 . . . . . It is primarily a question of adding suffixes to the names from one to ten, (i.e., six, sixty, six hundred, . . .). So we can use our fingers to show complements of one hundred by calling each finger “-ty.� A set of six fingers will be called six-ty and its complement for-ty. We shall write the pairs: (0, 100), (10, 90), (20, 80), . . . (50, 50).

6

The Set Of Our Fingers

With two sets of hand we can show complements of one hundred which do not end with -ty. One person, whose fingers are called “-ty,� puts up all of his fingers to show ten-ty, or one hundred. He then puts down one thumb and pretends it is not there. He is now showing ninety and, to compensate for the ten lost, the other person puts up ten fingers.

Together they are showing ninety and ten, or one hundred. Let's say that together they put down fifty-four; then the unfolded fingers remaining will be read forty-six: 100–54=46. This principle can be ex tended to thousands and beyond by adding more people. In each case every person except the one whose fingers are named without a suffix will give up one finger. Therefore, to find the complement of 4,567 in 10,000, one has only to find the complements in nine for the first three digits and the complement in ten for the last: 10,000 - 4,567=5,433.

With this awareness, complete facility with this type of problem can be achieved after trying a few problems.

Knowing how to work with complements of powers of ten can lead to a mastery of addition and subtraction through transformations: 1

First find the complement of 4,567 in 10,000 (i.e. 5,433) then add to in the difference 2,345.

2

Children who are allowed to use their fingers in this dynamic way meet, as part of themselves, the notions of equivalent sets, complementarity, transformations, the inverse operations of addition and subtraction, and the naming of orders of numbers by adding suffixes in a way which clearly links action, perception, language and notation.

Zulette M. Catir

7

Generalizing And Specializing

When people try to characterize mathematical activity they frequently talk about the importance of generalizing. It is rare to find explicit attention given to specializing, yet it is an equally important characteristic of the way mathematicians work.

By generalizing may be meant, in geometry, finding properties that hold for all triangles — the sum of the angles, say — and not just for the particular triangles that have been drawn. But the class of triangles is a specialization of the class of polygons — the three-sided polygons — and can be specialized further, by concentrating on, for instance, right triangles or isosceles triangles or equilateral triangles. The act of specialization brings additional properties that can be studied.

Generalization is still taking place since results are obtained that hold for all right triangles, though any new theorems do not now hold for all triangles.

Generalizing and specializing are not, then, contradictory, but only different aspects of an activity which can take into account a smaller or greater number of attributes. Nevertheless since the move from one to the other requires a definite shift of the mind's attention, and since both are necessary in mathematics, it pays to give due weight to each.

In arithmetic, multiplication arises as a specialization of addition — repeated addition — and exponentiation as a specialization of multiplication — repeated multiplication.

9

Mathematics

3 + 3 + 3 + 3 + 3 ~ 5x3 3 x 3 x 3 x 3 x 3 ~ 35 The use of Algebricks dramatizes the situations so that new awarenesses are inescapable. When a train is formed of rods of the same color representing a repeated addition, it is immediately apparent that these can be stacked side-by-side to form a rectangle, and that this is a new property achieved through the specialization. The rectangle, by presenting to the perception the area of its surface rather than the length of a train draws attention to the fact that either of the numerals, which are represented by the two dimensions of the rectangle, can play the role of operator, and therefore leads to the idea of a product.

This introduces a new chapter of arithmetic in which the idea of product is generalized to the product of any number of numbers (symbolized with Algebricks by a tower of rods). A new act of specialization arises from considering the consequences of restricting towers to rods of a single color, and this decision generates a further chapter concerning powers.

We can notice in passing the part that language plays in directing our attention to the fruitful directions in which to go via specialization. We naturally speak of a train of “three light green rods” rather than of “a green and another green and another green,” or of a tower of “three yellow rods.” The introduction of numerals into the descriptions leads us away from the language of “plus” and “times” that has been associated with the reading of trains and towers — and, as a bonus, draws our attention to the similarity of the two structures.

The new operations that these acts of specialization produce exist alongside the operations already developed. Multiplication coexists with both multiplication and addition.

The equal rods forming a rectangle of rods can be replaced by trains, and if all the rods happen to be changed into trains in exactly the same

10

Generalizing And Specializing

way, the property we call the distributive law emerges to define the connection between the distinct operations of multiplication and addition: “a product of a sum is equivalent to a sum of products,” or p x (q+r)~p x q + p x r.

A tower of rods of a single color, representing a power in the mathematical language we have associated with it, can itself be symbolized or replaced by an “L” — an arrangement of two rods, the horizontal rod displacing the color of the tower and the vertical rod measuring its height. (So, for example, a tower of six yellow rods, associated with the reading “five to the sixth power” or 56, can he replaced by an “L” formed from a horizontal yellow rod and a vertical dark green.) We can bring to this situation what we know from other experience.

The vertical rod can he replaced by a train of two or more rods, introducing the association with addition. We have already associated multiplication with the action of putting rods across each other to build up a tower. Combining all these awarenesses, we arrive at statements utilizing all three operations, such as: 56 ~ 54+2 ~ 54 x 52

or, in general, a p+q+r. . . ~apxaqxar. . .

But we also know that some lengths can be replaced by trains of rods of one color — that is, that some lengths suggest a product. Combining awarenesses, and languages, in these cases will give, for instance: 56 ~52x3 ~(52)3 ~(53)2 or, in general, amn ~ (am)n ~ (an)m It is, as the reader can discover, difficult but rewarding to consider the consequences of applying the same awarenesses to the horizontal rod of an “L.” David Wheeler

11

Transformation

Transformations enable us to create and deal with complexity. In producing language we combine a small number of sounds into thousands of words which combine in many ways into all possible expressions. Without transformations only repetition of what has been said before would be possible.

Transformations are fundamental to mathematics as well. The simple transformation of adding one allows us to generate all the natural numbers of mathematics. We add 1 to 1 and get 2; 1 to 2 and get 3; and so on to produce the sequence 1, 2, 3, 4. . . . .

Reversing the operation “add one” is a transformation which in turn yields another transformation, “subtract one,” which will generate the sequence, 19, 18, 17, 16. . . . In counting aloud we learn how to make a small number of sounds and how to make transformations on these sounds to produce a name for any number we choose.

Through transformations we become able to explore many hidden possibilities within a given framework. The equivalent expressions 4+1 ~ 2x2+1 ~ 10 - 5~ 12 - 7 ~ x10 ~ x9+ x7 or (where ~ is read “is another name for”) show that seeing one thing in different forms is the essence of mathematics.

Transformations themselves can be combined to produce a host of expressions, such as:

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Mathematics

and provide numerous opportunities for mathematical fantasy within a framework of a number of mathematical operations.

We introduce addition as a transformation of a pair into a single number: 7+3~10. Then we see subtraction as another way of reading and writing the above, reversing the operation: 10-3~7 or 10-7~3. Then we see multiplication as a special case of addition where all the addends are equal, 3+3+3+3 or 4x3, the set of whose names contains 12, and from 4x3~12, by reversing, we read new statements such as: “how many three's (or four's) in twelve?”, written or : “divide 3 (or 4) into 12,” written 12 ÷ 3 (or 12 ÷4), which can also be read as, “divide 12 by 3 (or 4)” or as (or ).

All this of course independent of the magnitude of the numbers and is as true of 39x47~1833 as of 4x3~12.

Similarly, repeated multiplication yields powers and their reversal the various roots.

Clearly this makes mathematics dynamic, well connected by a web of transformations, and from this a number of interesting programs can be generated to suit the tastes and interests of student groups.

The questions to the students can be put as open questions having many answers, such as: •

Find a number of ways for going from 5 to 12 using given transformations.

•

If you know that 24x36~ 864, what else do you know?

•

Make as many equivalent expressions for 24 as you can and then ask yourself whether you made all those which are possible?

•

Given an addition, say 10+10, what transformations will generate equivalent expressions?

14

Transformation

The awareness resulting from transformations is basic for making mathematicians out of our students.

Steve Shuller

15

Awareness And Facility

In the teaching situation there are students and teachers. Do they have the same roles? Traditionally, it seems that the teacher who has some knowledge meets the students who do not have it, in order to pass it on to them. This is adequate in the areas where telling is sufficient for the transmission of knowledge — for example, that the name of 20 is twenty. But in the areas of knowhow, unless the student goes through the exercises himself, he will not acquire the knowledge.

In these areas — and mathematics is one of them — there are two phases in the process of acquiring knowledge: first, one must become aware that there is something new in the field of perception and note it: this we denote the awareness phase: and second, there is a period of dialogue between the mind of the learner and the situation, in order to yield as much as possible: this we call the period of facility.

For example, one can become aware that there are classes within classes simply by recognizing that squares are also quadrilaterals. But in order to really master the notion of inclusion of classes, one must be able to recognize, on the one hand, that an increase in the number of attributes simultaneously present in the elements of a class makes this class less extended than the first; and on the other hand, that the dynamics of inclusion are independent of the actual properties of the examples chosen. Only when this state of mind is reached can we say that facility has been reached in the study of inclusion.

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Mathematics

There are two kinds of facility: one is concerned with the content of an awareness and one with the awareness itself. A mathematician has reached both. A teacher of mathematics is only required to manage the first.

For example, when a teacher shows his students that the difference of two squares is equivalent to the product of a sum by a difference, and conversely, he has worked on their awareness. This can be tested, to see if it has become a facility when, instead of a straight multiplication by the distributive algorism of, say, 37 by 43, the student replaces the product 37x43 by (40-3) x (40+3), and this by 402 -32, which is then easily calculated to be 1591.

This restricted facility with the transformation of a product into a difference of two squares may remain an isolated ability. A new awareness of that awareness 窶田um- facility is required to move ahead and make one capable of tackling less obvious products, such as, 35x65 in the form of (50-15) x (50+15) or 502 - 152. Another awareness will be necessary to inquire which algebraic equivalences serve to transform complicated calculations into much simpler ones.

This awareness becomes second nature in the specialist mathematician. He is also the person who shifts constantly from facility to awareness and back again. Every time he enunciates a definition he describes a new awareness. But definitions are only appreciated if they lead to a new set of exercises worthy of attention.

In the way we prepare mathematics teachers for their job, the careful attention we pay in distinguishing the phases of awareness, which are the teacher's domain, from the phases of facility, which are the students' responsibility, carries with it a methodology of teaching as well as a number of alternative curricula to provide many adventures for those students who like variety and wish to go far in a short time.

Awareness and facility need to be distinguished to give (1) teachers a precise function to help react what is new, and, (2) the students the function of learning to master a skill.

Caleb Gattegno

18

Additional Information

For those who are interested in studying mathematics as a human activity, a few words are in order about what we at Educational Solutions do and have to offer in this field.

1

Many who are concerned with this study consult in public and private schools as “teachers of teachers,” a job which involves working in classrooms and holding on the spot seminars with teachers to meet the challenges as they present themselves.

2 On weekends and during school vacations intensive seminars are given where such questions as those discussed in the newsletter are investigated.

3 We have materials such as “Algebricks,” sets of colored rods, geo boards and animated geometry films as well as a new kind of textbook called “Open Books,” and other literature listed below. We are working on increasing the range of materials which make mathematics more accessible to everyone.

4 There are 16 mm. films and 8 mm. cassettes on mathematics, using the Algebricks and the fingers. In addition, video tapes have been made of actual mathematics lessons in the classroom and other teacher training tapes are planned.

5 Another aspect of humanizing mathematics is a project which will develop material to show how certain people throughout history have met particular challenges and have added to our mathematical heritage as a result.

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Mathematics

For Further Reading Gattegno, C. What We Owe Children “ For the Teaching of Mathematics “ vol. I Pedagogical Discussions “ vol.II a) Psychological Studies b) On Films vol.III Elementary Mathematics Now Johnny Can Do Arithmetic Open Books for Students “ Book I Study of numbers up to 20 “ Book II Study of numbers up to 1000 “ Book III British system of units, applied arithmetic “ Book IV Fractions, decimals, percentages “ Book V Study of numbers “ Book VI Applied Mathematics Goutard, M. Mathematics and Children

20

Games For Writing Mathematics

In contrast to the articles indicating the ways we have introduced, into schools to affect teachers by showing them that there are different and much more effective approaches to mathematics teaching, the following notes are concerned with mathematical games leading to “mathematical essays.�

I call limited domain a set of elements and operation chosen so as to determine what is to be taken as given. I give below some examples of selected choices that cover considerable ground at various grade levels and can be played in almost any classroom.

1 We put in a box a particular choice of elements and operations forming a limited domain and exemplify some of the steps, leaving to students the invention of what could follow.

2, 3, 5, x, - This means that in the equivalent expressions we make for 1, 2, 3, . . . only these signs appear (any number of times) e.g. 1~3-2, 2~5-3, 3~3x3-2x3, 4~2x2. . . . . . .

If exponents, parentheses, negative numbers are allowed, then we can 2 have expressions like 5~32-22 , 6~22 -2x2x3-(3-2), 7~2x2 - (-3).

Too many signs in the box will make the game too easy; too few may make it impossible to generate certain integers. A study of the following limited domains may teach readers how to exploit the idea:

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Mathematics

1. 2, 3, 4, -

2. 4,

3. 2, 3, 50, x, - , %

,+,á 4. 2x,

x, 3, 4, -

Of course anyone can create his own limited domain and find which are suitable for what age group and at which stage to encourage spontaneous writing in mathematics.

2 Another category of writing activities I call quicksand games.

Characterized by the repeated use of a fixed sequence of operations, the game is concluded when a particular criterion is satisfied.

Game 1: Start with any positive integer; reverse the digits and add these two numbers; repeat the procedure on the result and so on until you reach a number which is not changed when reversed (such numbers have been called “palindromic�) e.g.

(i)

67+76~l43; 143+341~ 484;

(ii)

374+473~847; 847+748 ~ 1595; 1595+5951~7546; 46+6457~14003; 14003+30041~ 44044.

Game 2: Start with a four digit number; rearrange the digits in descending order; reverse the order of digits and subtract; take the resulting difference and repeat the procedure. Stop when 6174 is reached. e.g.

Start with 3147 8640~0468~8172; 7443~3447~3996; 6642~2466~4176; 7641~1467~ 6174.

7431~1347~6084; 8721~1278~ 7443; 9963~3699~6264; 7641~1467~ 6174;

Game 3: Start with a positive integer; multiply by 3; add l; divide by 2 as often as possible with the quotient an integer; repeat the procedure; stop when 1 is reached.

22

Games For Writing Mathematics

e.g.

7; 7x3~21; 21+1~22; 22÷2~11; 11x3~33; 33+l~34; 34÷2~17; 17x3~51; 51+1~52; 52÷2~26; 26÷2~13; 13x3~39; 39+1~ 40; 40÷2~20; 20÷2 ~10; 10÷2~5: 5x3~15; 15+1 ~16; I6÷2~8: 8÷2~4; 4÷2~2; 2÷2 ~l: 1x3~ 3; 3+l~ 4; 4÷2~ 2: 2÷2 ~ 1 repeatedly.

Marty Hoffman

23

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.

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