Our Work In

Remediation

Educational Solutions Worldwide Inc.

Caleb Gattegno

Newsletter vol. IV no. 1 September 1974

First published in 1974. Reprinted in 2009.

Copyright © 1974-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-276-3 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

Any innovation in education is tested on the most demanding situations; that is with “non-motivated” students, “culturally deprived,” or any such population in inner city schools, in poor neighborhoods. If it comes through intact it has proved itself.

In the case of our remedial work we have, by definition, a “difficult” situation. The people who come to our clinic are often rejects from other clinics after having been rejected by schools.

Our first session is called “diagnostic,” but in fact is a disguised lesson in which the client is given many chances to overcome the hurdles he has put in his own way by failing to make sense of what is considered to be his problems: reading, spelling, computation, etc. As he discovers that he can do what he was supposed not to know how to do, he rises to the occasion and lets us know that his so-called “learning disability” is something else: most often a failure to make sense of what is required of him. Our remediation is mapped out for us by the client. Often it takes a very small number of meetings to give back to our client the confidence that results from the certainty that there is nothing wrong with himself, only a dysfunction caused by having dropped out too soon, so to say.

In the articles of this Newsletter we spell out how we reach an understanding of what to do with the powers of our students to send them away sure that school tasks are rather easy compared to what they can do with themselves and have already done spontaneously in their everyday life. Even the most “deprived” child appears to us to have achieved so much that we feel confident we can put him back on this path in a few hours. This has been proved again and again.

Table of Contents

Remediation: What And How ............................................... 1

1 Reading ............................................................................. 5

What We Can Learn For Reading From Speaking ............................ 5

What We Do To Launch Our Students ..............................................8

Speaking Comprehension And Reading Comprehension ................11

How We Help Students Learn From Texts...................................... 13

2 Mathematics ................................................................... 19

A Basis For Computational Remediation ........................................ 19

Making Sense Of Written Mathematics...........................................23

How We Use Perception And Action In Remediation..................... 27

News Item .......................................................................... 31

Remediation: What And How

In school after school there are students who are making no progress at all in some fields, or making so little that they can’t get within striking distance of the work done by the majority. Their lack of success is a problem for themselves and for their teachers, and in many cases it becomes a problem for everyone who has to deal with them. It seems as if schools have had almost as little success dealing with these students as the students have had dealing with the work expected of them. Their class teachers often do not know how to find the time and opportunity to attend to them properly. Putting them with paraprofessionals in a special group, or into a special class, often gives them more individual attention but only a slower version of the teaching that failed to reach them in the first place. Making them follow an “individualized” programmed course often brings them some sense of achievement but only in a tiny closed world that is incapable of expanding into the universe of true competence. Where most of the obvious moves have failed, schools may have to consider a new approach to the problems of these students.

It is no good looking exclusively at their failings for clues to what to do since these give us too little evidence of what is wrong and too little guidance as to what we can build on to set matters right. We will get nearer to a solution by separating the task of remediation from the job of putting mistakes right.

Mistakes and uncertainties are proper to learning. Whenever anyone learns to function in a new way — for instance, to drive a car, play an

1

instrument, speak another language — clumsiness and tentativeness are to be expected and give back important information about the components of the functionings that still have to be conquered. Since much of the learning required in school, particularly at the elementary levels, concerns the acquisition of new skills, of new functionings, we can expect children learning to read, say, or learning to master some mathematics, to make mistakes and show uncertainties. We may press the point further and say that if students are not allowed to make mistakes it will make it much harder for them to achieve autonomy in their school learning since they are being denied the evidence they need in order to know their own progress from the inside.

Mistakes play a positive role in learning because they bring useful messages to the learner who knows how to read them. But when a student doesn’t understand the messages, even when his teacher interprets them for him, and when mistakes persist for a long time, there must be some dysfunction in the student that is blocking the natural course of learning. Attacking the mistakes directly in such a case will not get anywhere. What is required is the removal of the dysfunction so that the student is in a position to profit from his mistakes in the way that other learners do. We may identify this as the true task of remediation.

Expressing the problem in terms of functionings makes it clear that some remedies are not worth bothering with. It is no use trying to make someone into a cyclist by riding his bicycle for him, and it is not much more use telling him what he ought to do. The would-be cyclist and the would-be reader will only succeed if they are in touch with and in control of their own functionings. The problem of remediation cannot be solved by “show-and-tell.”

Remediation has to begin with what the student can do, to build on what is already there. If we are not distracted by the evidence of what the student cannot do in the classroom and look dispassionately at what he can do elsewhere, we find, for example, that he can

• control his voluntary muscles and direct the actions of his body,

2

Our Work In Remediation

• perceive many elements of his environment and respond to them,

• speak and make sense of others’ speech (which proves that he can deal as successfully with the abstract as with the concrete),

• imagine objects and actions and events, and make his imagination do with them what he wants.

This is enough for any remediator to start with. All he needs to know is how to recast his approach to reading, or to mathematics, so that it requires only these and similar functionings. (The other articles in this Newsletter show that this job has already been done.)

The remediator must get results fast and find a way to by-pass or inhibit the student’s habitual responses that have led to failure in the past. The key lies in presenting game-like situations to the student that are easy enough for him to get involved in straight away but complex enough to be intriguing, that only require his available functionings, and that eventually yield results that everyone can recognize, including the student, as belonging to reading or to mathematics. The astonishment felt by the student when he discovers he has done something he thought he couldn’t do, and without imitating anybody, provides a jolt which can begin to re-energize his attack on learning. The remediator repeats the experience, varying the games and the outcomes, until the course of injections has done its work — that is, until the student, although not yet knowing all that he didn’t know before, has gained a quantum jump in his awareness of what it requires to be successful in reading or in mathematics. The signs the remediator looks for are that the student now approaches his learning as if it made sense to him, and that he brings a new confidence and zest to it. With this achieved, the remediator knows that the student is ready to take over and accept responsibility for his own learning.

It seems appropriate to call the provision of this kind of remediation a “clinic:” the metaphor emphasizes that the treatment is short-term, intensive, individualized, and given by a specialist.

3

Remediation: What And How

If we find the comparison with medical care suggestive, we may be inclined to stretch it further — “preventive education” seems to be a concept with as much potential as preventive medicine. But that can be left for discussion on another occasion.

David Wheeler

4

Our Work In Remediation

1 Reading

What We Can Learn For Reading From Speaking

By the time children enter school they have begun to make sense of their world. In an attempt to test the limits of their functionings they have mastered an array of quite complicated skills. They not only demonstrate that they have developed a system of meanings but they are also capable of doing the analyses and syntheses necessary to link their system of meanings with other activities. For example, they have learned that there is a system of meanings that corresponds to a system of sounds in their environment — the by-product of which is the acquisition of speech. This relationship between the two systems we shall call “isomorphic” since in the mathematical sense this is said of two systems which have a one-to-one correspondence between the elements as well as in the product of their operations.

There is evidence then that if all children bring with them the power to break the spoken code and to make the link between two systems which are isomorphic that the teaching of reading is most easily achieved by actualizing our awareness of another relationship — namely between the system of sounds and the system of signs which are strictly “isomorphic.”

The first job in remediation then is to know which properties of speech correspond to reading and vice versa.

5

Babies who have not yet learned to speak their mother tongue but who have learned to babble already exhibit phonetic functionings which can be linked to the activity of reading. In addition to the study of which somatic adjustments yield which sounds, they also experiment with the combinations and permutations possible within the whole repertoire of sounds. We can describe the activity of combining sounds as an algebraic functioning of the mind and define it in terms of operations: four of which are addition, substitution, insertion and reversal of sound if not always of signs. Algebraic functioning occurs here at the level of phonation and later in speech, but it is also found in some other of our activities. Furthermore, these operations form the algebra which is the basis of the isomorphic relationship between speaking and writing providing we do not look at writing as a sequence of letters, in the case of English.

It would seem obvious then that once we became aware of our use of algebra in spoken speech, it would be easy to learn to read. And yet we know that many children have not seen the relationship between the two systems since in English the signs of written speech do not reflect in a uniform manner the sounds of spoken speech. For example, we know in uttering the word “eight” we make 3 sounds and yet we use five letters.

Color has been introduced as one of the solutions to this problem. The use of color makes obvious the isomorphic relation between the two systems giving to all the various spellings that correspond to one sound, one color, and giving identical spellings for different sounds, different colors. As a tool it makes easy the study of the totality of sounds and spellings in English. But more importantly it serves the student so that he can link what he says with what he sees and recognize in the activity of reading the algebraic functionings which he already owns from speech. Another means by which the activities of the mind are made accessible to the student is through the Transformation Game. The game involves the transformation of one word to another by any sequence of the four algebraic operations named above.

i.e.

6

Our Work In Remediation

We know we speak in time. We combine sounds one after the other to form words and words one after the other to form sentences. If the two systems are isomorphic then the written word which exists only in space must also possess these temporal qualities.

Alignment is the convention by which the spatial component of written speech corresponds to the temporal aspect of spoken speech. Words are placed on a line and since time is irreversible each language must have a starting point and a consistent direction on the line though the direction is arbitrary and varies sometimes with the language. Again the challenge is to make this link accessible to the student.

The pointer which has been traditionally used in the classroom to draw attention to the teacher takes on a new function here. The pointer used with colored word charts on the wall or on phonetic tables not only removes the teacher from the center of the learning, it gives learning back to the student. It is used to designate the temporal sequence necessary to form a word or a sentence. For example, how would we know which word is meant by the arrangement of these three letters in space p t a There are three possibilities in English. The game “Wait until I finish tapping” forces the student to suspend his judgment before he can come to any conclusions about the content and order of a word or a sentence. In fact is this not what we do when we hear another person speak?

Thus we have shifted from the conventional view that reading is an activity outside the individual to that of seeing it as a part of natural functioning — an extension of our intellectual and linguistic powers.

We start with where the student is and work on his awareness. Children in our clinic easily learn to read when the algebraic and temporal nature of their speech, which is already part of their

7

1 Reading

awareness, is tapped. Their reward, as it was in all stages of their learning is in the functioning now recognized by the students as being their own and that no one can take away from them.

Sally Kolker

What We Do To Launch Our Students

When we teach reading in the schools or in our clinic, we start with the learners. If they are speakers, we know among other things that they

discriminate the sounds of the language,

utter them at will,

use their ears to monitor what they say,

link their system of meanings to a system of sounds,

string words one after the other to express their thoughts and feelings,

retain what they make sense of and use it.

Because we call upon what our students already understand from learning to speak, we are certain that they will know how to use themselves for reading. Since the techniques we use and our materials are compatible with the characteristics of the language, our students easily transfer their intellectual and linguistic “know-hows” to the new universe and become readers in a matter of hours.

That we build upon functionings that students already own, rather than ask them to memorize, is illustrated in our work with beginners. The idea of introducing successive “restricted languages” makes it possible for the teacher to introduce only what the learners cannot invent. Then the teacher’s responsibility is to provide opportunities for the students to play games with what is given so that they end up knowing as much as possible about using themselves for reading.

8

Our Work In Remediation

It is possible for instance, to have a “restricted language” of only two signs (a as in at and i as in it, for instance). If the teacher puts into circulation what sound is triggered by each drawing, the learners have only two things to remember. The yield is much greater, however, if the teacher uses a pointer on these signs to generate “temporal sequences” which the student can observe in their utterances. (aa, iia, iai, iiai, are only a few of the infinite possibilities. ) The game can be expanded by putting these “words” into “sentences.” Functioning within the rules of this game, then, yields mastery of the conventions of written English as a by-product.

What remains to be done is to widen the “restricted language” so that the sounds triggered by the signs “sound strangely like English.” We know from speech that vowels and syllables are the bricks of speech. In order to keep speaking at the center of reading, then, we introduce vowels and syllables — not vowels and consonants — as the units of language. To form these syllables, we use the pointer again to join consonants with vowels — not sounding the consonants alone, but having them affect the vowels. If we add t (as in tom) and s (as in is) to a and i, for instance, the students can recognize in the possible combinations that is, as, at and it are words of their speech. By stringing them one after the other by rules already established, it is possible to produce:

- is it - at it - as it is

- it is - as is - is it as it is

Their meaning emerges when they are uttered with the melody of the language.

What takes a long time to write about may take only a few minutes in the classroom or in our clinic. The learners are functioning as readers. Within a given “restricted language” they let the signs trigger their speech; they observe the conventions for putting their utterances in print; and most important, they are independent of their teacher. Within the given set of signs (whichever it is), they read and write almost as easily, as knowingly, and as confidently as their teacher.

9

1 Reading

What remains for the learner is to meet all of the signs of English. Since in that language, the same sign can have many sounds (see, is, measure, sure) and the same sound can be written in various ways (pat, plaid, laugh). Our students are helped through color to know which sound to utter for which sign. They remain independent as they master the complex English orthography.

The materials and techniques developed by Dr. Gattegno make it possible for us to respect what each learner already knows about his speech. More important, because we begin with them, our way of working makes sense to the learners. Since they have access through perception to what we propose, our students judge very quickly that the new is compatible with and can be integrated with what already exists and is at work in them. Because we build on functionings rather than memorization, a cumulative effect of learning is possible. The learners recognize that once they have practiced one thing, they are equipped to take on bigger challenges. Since the teacher supplies only what the students cannot invent and allows them to work in ways that they already understand, our students act knowingly, from the start and all the time.

The teacher’s job is to suggest challenges and provide opportunities which are within the reach of the learners but which stretch them and which may not have spontaneously occurred to them. Reading is not seen as a separate activity of the self but as an extension of certain intellectual and linguistic awarenesses that yield skillful functioning in a new universe.

In our clinic this is done with people of different ages and therefore only the duration of the apprenticeship varies.

Katherine Mitchell

10

Our Work In Remediation

Speaking Comprehension And Reading Comprehension

In the universe of speech it is meanings, not words which are remembered. As babies, we all did a careful study of the words we found in the voices around us, but once we reached the content of speech we were no longer concerned with words. How, then does one get meaning from a silent printed page? Similarly, when one speaks there is usually no thought of composing the sentences. One somehow gives voice to a flow of words while attending to the meaning to be conveyed. How does one manage to “catch” these fleeting abstractions — words — and freeze them on paper for uses very different from oral speech?

Acquiring these two skills, composition and reading comprehension, is one of the fundamental jobs children can do in school. It is the same job whether they do it their first year in school or their last, although habits and prejudices naturally deepen year by year if students are required to function as if they had these skills before they actually do.

Each of our students speaks differently to a friend than to a teacher, or to a person of the opposite sex, or to his siblings, etc. Each understands, and when needed, acts on the different speech he hears from younger children, his mother, strangers to his neighborhood, etc., as well as the characters portrayed and people filmed on TV and in movies. Although this resourceful, subtle and imaginative language, which we find in every student we listen to, may be largely unconscious, it is far from rudimentary; and it must be taken into account if he is ever to be “at home” in the universe of print.

Our proposal is to begin with the words, phrases and sentences children already own and use in so many ways, what might be called their set of linguistic mental structures, and increase the awareness of these structures so that they serve in other dimensions than conversation. When we make a game of “catching” these bits of language, in the mind and in writing, and manipulating them so they meet certain requirements, we are stretching and refining the language

11

1 Reading

already present and transforming it into a tool fit for many tasks, including comprehending and expressing oneself.

Aural comprehension is not a smooth process but a series of small jumps. It is not as one listens that one understands, but rather, at the end of phrases and sentences one gets increments of understanding, and it is only when the utterance is complete that one can become clear about the meaning. In other words understanding is suspended and one merely holds the words in one’s mind until “the penny drops,” and, as observed earlier, as soon as the meaning is reached, the words, which are no longer needed, are forgotten.

One way this relationship between the power of the mind to synthesize meaning and the temporal nature of language can be made obvious to students is through some games played with a pointer and words printed on charts or written on a chalkboard. The words of a sentence are pointed to one by one and response is permitted only after the last word is touched. This forces students to hold the sentence in their minds and utter it all at once with the natural flow of speech. If a student gives instead a sing-song reading, the speed and rhythm of the pointer may be used to indicate the proper phrasing and melody of the sentence; or, the student may be asked to utter one phrase faster and faster until he realizes he is speaking a language he already knows.

The game can be varied to keep it interesting. Sentences can be of a particular type, true say, or sad, or commands to be carried out instead of repeated, or questions and answers, etc. Since whether or not comprehension is present is obvious from the natural way the students utter the sentences and from their non-verbal responses as well, the teacher is free to work with the specific needs of each student as they manifest themselves.

Another aspect of this game is emphasized when the number of words available for making sentences is restricted. The challenge then is to do a lot with a little. A limited field sharpens the focus and diverse and surprising links between words emerge. The fine line between sense and nonsense is explored as it is often crossed in this game. The delight felt when someone’s contribution is particularly funny, or apt, or

12

Our Work In Remediation

startling does more than spark interest, it generates an intense awareness of language as a universe in which one can grow.

In initiating students to the universe of print we give them opportunities to know themselves and each other in the act of injecting a voice “into the page,” or of “freezing” a bit of themselves in words, and we let them share their progress with one another. In offering them not dull exercises but challenging situations we find they are inspired to do the work necessary to make language their conscious tool.

Steve De Giulio

How We Help Students Learn From Texts

Once we have managed to decode all words, already known or new, to produce a flow of words dictated by those on a printed page so that the statements sound English because we know how to stress words and join them together so that an impression of spoken speech is generated, including the intonations re-required, there still remain problems for readers.

Indeed there is, besides the content of the passages we look at, the hidden involvement of writers with their own intentions and inner life. A writer may want to exalt, to edify, to castigate, to pull people’s legs or to entertain, and we have to learn to look at texts for such contributions.

In schools, including colleges, we read in order to acquire knowledge. Many students are told that they cannot read because they have not managed to discover what needs to be done to extract the knowledge deliberately deposited by authors in their texts. This kind of reading, so important for students whose knowledge is to be acquired by proxy and not through direct experience, poses different challenges to clinicians from those required for, say, enjoying reading a novel that no one has told one to read.

13

1 Reading

In our clinic we receive people who are excellent readers and who come to us because they cannot read in the way that their teacher or professor decides is required in his class. Of course the first point to make for such clients is that we are all unable to make sense of some books, written for specialists, that assume that their readers function at such a level as not to require any introduction of the concepts by the authors. The second point to make is to eliminate the stigma of illiteracy from these fluent readers who are as good as can be when reading what interests them in newspapers or the books they choose to read. After that, the remedial work is mapped out by the obvious elements that the reader has not yet mastered.

1 There is no shame in not understanding what is beyond one.

2 There is no law that says that we must understand what we read at once.

3 Reading again, stopping, reflecting, can be as much part of reading as fluency.

4 Using a pencil to illustrate what words fail to trigger in one’s imagination is also permissible.

5 Reading to retain what has been presented even when one only vaguely understands and moving ahead provisionally so as to survey a chapter in the field and recognize the recurrent themes, to return to what one has looked at and examine it more carefully, may slowly provide the climate for progress in the field.

These insights produce the affective component which permit new habits to be formed, those needed to develop a mode of thought akin to the science or the subject one is studying. Indeed the various sciences make different demands on us as readers and only if the clinician is aware of these demands can there be a way to assist the clients put in his or her care.

If we have to restore confidence in a student of mathematics taught via texts we must make sure that the reader knows that in definitions only one word can be new and its meaning must be stated in terms of other words that are supposed to be already known. If this is not the case,

14

Our Work In Remediation

earlier definitions must be examined until some are found that speak directly to the client and from them the successive ones made to make sense. It is also possible to present clusters of notions at one and the same time by linking them functionally, saving time and offering the client a view of the field much more organized and more easily retained. The notion of proof regarded as acceptable in elementary mathematics in schools has a pattern that is found again and again and can be retained as it becomes familiar. The schema is reflected in the language used and because of its recurrence the reading becomes habitual also. Weakness, if it persists no longer derives from the way the text is met but from the limitations of one’s imagination.

The remediation of fluent readers of mathematics may consist in mathematical remediation via other means than words: manipulative materials, films, drawings, etc., may be sufficient.

If we have to open books in the natural and exact sciences to learn their content we must also know that we must retain expressions, speak certain languages in certain ways (the technical jargon of those sciences), go slowly, and rather than get the meaning and forget the words, as we do reading newspapers, retain the words so that they gradually gather their technical meaning around them. Here too definitions are important but they also have perceptual support from diagrams, photographs, models, etc. Unless words trigger the proper associations in images, feelings and experiences, how could one comprehend from just reading the words concepts such as acid test, high ph, twin stars, husbandry?

Remedial reading for students of these subjects requires that the language be learned at the same time as the experience is given, i.e. in classes where instruments, materials and collections supply the perceptive content that serves as necessary support for the retained vocabulary and the usual statements in that field.

Clearly a different imagination is required for physics and for history. Although the act of reading is the same for texts in both fields the comprehension differs considerably. In the first precise images and processes must be evoked. In the second, sympathy, compassion, moral

15

1 Reading

judgments and uncertainties can creep in without conflicting with what is read. One can have partial understanding and still be considered to comprehend, for facts in history are often opinions and are in any case not totally reliable. Retention can be approximate and not lead to dangerous situations, while the wrong wiring obtained from improper reading of technical instructions can be fatal.

In our clinic we take care of the affective component and only give examples of the way technical texts are read so that the chances of comprehension be considerably increased.

C. G.

The following paragraphs are taken from a feature article, “Colour them Read” by Rosemary March, that appeared in the British daily newspaper, The Guardian, on August 15, 1974.

“Nine year-old Kevin, who has been attending an infant and junior school near Reading, Berkshire, is typical of those normal kids who slip through the net. Two months ago, his reading ability was nil. A bus ride away, Educational Explorers Ltd produces their Words in Color reading system. This was first published in 1962, and yet is still used nationally and internationally only by committed pockets of devotees. Its quiet methodology — not given to hard-sell techniques and therefore known by less than 10 per cent of our own teachers — had somehow reached the attention of Kevin’s parents. After nine one-hour lessons at the publishers’ clinic, he could master a sentence which to you and me may be laughable but to Kevin, a real breakthrough “My brother is impossible, and her mother had ten apples in a basket.” Thirteen words which meant he could never be sneered at again.

Self-confidence strengthened every week in stride with his discovery that reading is not the chaotic jumble of letters he believed. Words in Color, taking a generous view of children’s capacity for mental abstraction, is

16

Our Work In Remediation

based on the concept that whoever has learned to talk can learn to read. The color codes represent sounds, thus making the language’s spelling inconsistencies no longer the big bogy. The system makes logical the most illogical of phonetic spellings, taking away the need to remember those myriad exceptions to rules. Most important of all, it encourages a fearless, dynamic attitude to learning.

Kevin responded like a poppy to the sunshine, opening his mind at last after the darkness of four years’ confusion. ”

17

1 Reading

2 Mathematics

A Basis For Computational Remediation

Everyone who grows up speaking English knows that what he calls chair, most others will call chair. When he wants to be more precise about which chair, the language makes that possible (“that chair,” “the armchair,” “the red armchair,” “the chair next to the closet”). He knows that all chairs are not armchairs, that all armchairs are chairs, but not all armchairs are red, that not all red things are chairs, but that both chairs and armchairs may be red. The person who uses nouns and adjectives is already a master of abstraction; the algebra of classes, some structures in set theory already belong to him.

In language a general statement has a different quality from a very precise one. To say people need food to survive is quite different from saying “Henry’s little brother eats nothing but peanut butter sandwiches.”

In the first statement, each word could be replaced by many others and the sentence would still be true; in the second only a few substitutions would be possible without changing the meaning

( [Henry’s little brother] eats [nothing but] peanut butter) Peter only sandwiches

19

In both statements, the underlying relationship being commented upon is the same — that of a being to what sustains it by being incorporated.

A user of language, who understands that generalities may be hemmed in with restrictions, and conversely restricted situations may be freed up and converted into generalities is in contact with the dynamics implicit in mathematical functioning. Therefore, it is no longer necessary for a learner to be traumatized by the appearance of n or x, or a, b or c, since he can recognize that these are part of a language of generality, where relationships and their dynamics are the subject of study.

A person who uses language can come to see ordinary arithmetical computations as rather specific statements where one element has been left out. As a speaker, he is aware that in describing any situation, he has a choice about where he begins, and this choice produces sentences which may sound different but convey the same meaning, thus:

I am his sister and He is my brother

can replace each other without altering the meaning, since they are equivalent ways of describing a relationship.

In the same way 6 + 4~ 10 and 10 - 4~6 are equivalent statements describing the relationship of 6, 4 and 10. A remedial student who knows simply what he needs to add to a number in order to have 10 (its complement in 10; 2 to 8; 3 to 7; 1 to 9; 4 to 6; 5 to 5 and their reverse) can in a matter of minutes find that he can give without hesitation and without writing, the answer to 10, 000 - 6, 737, a problem which on his arrival at the session would have caused him to sigh and sharpen his pencil for a lot of painful borrowings.

The following description of how the remediator brings this about, will perhaps serve as an illustration of the psychological and mathematical strength to be gained by extending students’ powers.

20

Our Work In Remediation

1 Having made sure that the student knows the pairs of complements in 10, or can find them by inspecting the fingers of his two hands, we let him see that he also knows the complements in 9, either just because he does, or by noticing that 9 is one less than 10, so in each case the complement in 9 is one less than it was for 10.

2 We ask him to notice that if we put a zero after each numeral, we can write at once the complements in 90 and 100. Another zero yields complements in 900 and 1, 000 and we could keep adding zeros without demanding any new information.

3 We suggest that if he considers 100 equivalent to the pair of complements 90 and 10 added together, he can find the complement in 100 for any smaller number by finding complements in 90 and in 10 and adding them. Thus the complement for thirty-six is sixty-four, as thirty and sixty are complements in 90, while six and four are complements in 10. And sixty added to four is sixty-four. This makes finding any complement to 100 very easy, and by adding a zero and considering 1,000 as equivalent to 900 + 100, which is also 900 + 90 + 10, we can find quickly the complement for any number in 1, 000. For example -

four hundred seventy two requires five hundred twenty eight to yield nine hundred ninety and ten, or 1, 000

To add another zero and have four digits would be just as easy and give more practice in quickly finding the complement to a succession of 9’s and to 10. If a student reads his complement aloud, it may be necessary to make him confident about reading the larger numerals, but this game can be played silently on paper or a chalkboard, using only notation and never mentioning the word complement.

Thus far in the session, the board might contain this writing

21

2 Mathematics

10 9

1 9 8

2 8 7

3 7 6

4 6 5

5 5

which is sufficient to provide either of the numbers on the left hand side of the following equations:

34 + 66 ~ 100 seen as 90 + 10

62 + 38 ~ 100 seen as 90 + 10

536 + 464 ~ 1000 seen as 900 + 90 + 10

371 + 629 ~ 1000 and so on

4365 + 5635 ~ 9000 + 900 + 90 + 10

2114 + 7886 ~ 10, 000 seen as 9000 + 900 + 90+10

4 At this point the student can give the answer to 10, 000 - 6, 737 or any similar problem. Only one new element is needed. He simply needs to recognize that if he knows two numbers which add up to 10, 000, he can subtract either one from 10,000 and the answer is the other. (If 4

+ 6 ~ 10, 10 - 6 ~ 4) So if he sees written (the

former enemy) he can realize it is equivalent to asking what he has just done, what is the complement in 10, 000 of 6, 737, and write 3, 263.

The student who participates in this exercise has discovered that a traditionally “difficult” problem can be a very simple one. The next step the remediator may take with him will be to show him how this easy problem can be found hidden in difficult ones. For instance

22

Our Work In Remediation

contains with 12 added to the answer.

Caroline Chinlund

Making Sense Of Written Mathematics

The world of written mathematics is an unknown and unknowable language to anyone who does not have at his disposal the triggering mechanisms that associate mathematical notations with meanings. The student who comes to written mathematics for the first time can gain access to its meanings through the systems he has already developed to carry meaning — the world of his actions and perceptions, for instance, or the world of his speech — provided the links are made clear to him. And after his entry into the world of written mathematics he can give meaning to new symbols through their association with those he already knows. But where these associations have been inadequately established the student cannot “make sense” of mathematical notation, and he will either not know how to respond to it at all or will be triggered to make the wrong response.

The job of remediation, accepting the fact that the association between symbols and meanings is essentially arbitrary, is to link mathematical notation with some existing functioning in the most direct and immediately perceptible way. There is nothing in the writing:

1/9 + 2/9

that tells anyone what to do. Indeed, if a student uses his earlier experience of the writing of whole number additions and produces:

1/9 + 2/9 ~ ~ 3/18 ,

23

2 Mathematics

what he has done makes sense at the level of notation. We can help him make a different association by asking, “What have you got if you have 1 horse and 2 horses? What have you got if you have 1 hundred and 2 hundreds? What have you got if you have 1 ninth and 2 ninths?” By stressing the numerals and telling the student to listen to the words, We allow him to “fall into” an association between the new statement and other statements he already knows how to make and how to make sense of. By giving him other written examples and asking him to read them aloud he is forced to focus on the elements of the written form that correspond to the important parts of the verbal statements and that will subsequently serve to trigger a correct response. If he seems to be slipping back into his old habits we just say, “Say it and listen to it.”

The student must learn to live with the mathematical notations that everyone uses because they have been incorporated into the fabric of the language and he needs to know, for instance, that it is possible to write the instructions to carry out a particular division in several different ways:

e.g. , 30÷6, 30/6, 1/6 x 30.

The confidence to move freely between these different forms can be generated by ensuring that each one is first encountered meaningfully and unambiguously, and by introducing notational “games” that make their equivalence apparent. If the writing is associated with the spoken form, “How many 6’s are in 30?”, the usual left-to-right convention of reading is respected, and a meaning is given which is securely based in counting. The student who already knows that five 6’s together make 30 now knows that is only another way of writing 5.

The conventions of reading are also followed if we associate 30÷6 with the spoken form, “30 divided by 6,” or, “What is 30 divided into 6 equivalent parts?”

A pattern of Algebricks will show a student that if he can write 5 x 6 ~ 30 (“5 sixes are equivalent to 30”) he can also write 6 x 5~ 30 by reading in reverse. The writings ~ 5, ~6, 30÷6 ~ 5 and 30÷5

24

Our Work In Remediation

~ 6 follow at once from perception of the pattern and give meanings to the notations. The introduction at an appropriate moment of the other forms will yield additionally 1/5 x 30 ~ 6, 1/6 x 30 ~ 5, associated with the words “one fifth of 30” and “one sixth of 30.” 30/5 ~ 6 and 30/6 ~ 5, associated with the words “30 over 5” and “30 over 6.”

Awareness of this set of equivalent writings, whatever the numerals, involved, gives the student complete power over the notations.

At a more advanced stage a student may have difficulty in dealing with the notation or log3 81, for instance. These written forms cannot be understood without knowing the definitions that relate the new symbols to familiar ones. A student who “misses” the definition by being absent, or by not giving his attention at the crucial moment, or by not following it, cannot know how to handle the new symbolization. Here remediation must go back to the source of the definition to show that what is being written connects with what is known.

The student may know that a power is equivalent to a repeated multiplication and use this equivalence to calculate or transform powers into single numbers:

~ 8, ~ 25, ~ 1296, etc.

If he can do this he can find the exponent when he is given the base and the result: for example, “What power of 2 is equivalent to 64? What power of 3 is equivalent to 81?” The notation log 3 81 can now be in introduced as only another name for an exponent — the exponent obtained from the power of 3 that gives 81. So since he knows ~ 81, he knows that log 3 81 ~ 4.

Now he can similarly interpret, for example, log 2 16 or log 10 10,000 without needing the language, “the logarithm of 16 to base 2, “and when he is eventually given this new verbal formulation he will find, as we all do, that it is what we say but not what we think — since in most cases we must mentally transform the question into one about powers

25

2 Mathematics

in order to solve it, just as we must transform a power into a repeated multiplication in order to calculate it.

The problems of mathematical notation do not belong to mathematics but to language. We can effectively remediate specific notational difficulties by focusing on the right elements in the complex interplay between meanings and spoken and written language.

Remediation removes obstacles. Only the student can learn mathematics, but at least we can see to it that the job is no harder for him than it needs to be.

David Wheeler

From “For the Teaching of Mathematics” Volume One, by Caleb Gattegno

“When I was a schoolboy several of my classmates could not make any sense of the purpose of studying mathematics and many neglected it as a result. Nevertheless, they themselves considered anyone who was good at it as worthy of their respect for such intelligence and ability. I was not in the least distinguished as a schoolboy, and even though I had some ability at mental arithmetic, I showed very little sign that one day mathematics could become my full-time occupation. One day, the teacher gave us a test involving a problem of geometry, and I saw that if I dropped a perpendicular from one point to one line everything became obvious. Not one of my classmates saw this, and so the test did not count. I had, in fact, proved to all that the problem was an easy one whose simplicity resulted from a vision rather than from memory. As most people believe that learning is memorization they refuse to agree that seeing is as easy as, or even easier than, memorization.

26

Our Work In Remediation

For me (then aged 13) it was a turning point in my life. I understood that mathematics was something of the mind, resulting from the inner movement of the imagination coupled with experiences that became successively deeper and deeper.”

How We Use Perception And Action In Remediation

Looking at the figures above one might ask: what do I see in the figures? can I give them names? and so forth. But the question might also be asked: in what way do they relate to each other? The answer depends upon one’s perception of the situation, i.e., how one interprets the impacts received. One could note the common attributes of the figures —four sides, opposite sides parallel and equal, same height and length of base — without necessarily using these exact words. On the other hand one could imagine the figure on the left being stretched by having its top pushed to the right so that it is transformed into the other. This way of linking the figures is a dynamic one, and since the actions are taking place in the imagination they are called virtual actions. Through the capacity to make images, one can see in the mind’s eye a succession of transitional figures or a “film” of continuous transformation. Perhaps one may even feel within oneself sensations of stretching and displacement. In any event, whatever the experience, awareness of using oneself in a way accessible to consciousness is present although it might not easily be verbalized because it all takes place so quickly and with such complexity.

Everyone presented with situations available to perception and action has the capacity to entertain questions like those above and to

27

2 Mathematics

investigate their outcomes. However, not everyone spontaneously takes advantage of the opportunity for such an inner dialogue or knows that such personal involvements form the basis for the body of knowledge called mathematics. Indeed many students have never made the connection because they have only found themselves asked by verbal or written exhortation to manipulate symbols unrelated to realities which must first be richly experienced. Consideration of these facts prompts the conclusion that the proper role of educators concerned with teaching mathematics is to link self-education of the realization of one’s powers to the fundamental mathematical awarenesses which have generated the vast inventory of mathematical topics.

To transfigure the teaching of mathematics in this way opens the door to effective remediation of those students who have never entered this domain or just barely crossed the threshold. The task of remediation becomes simpler yet more profound; it is meeting a person as he is in order to see how mathematics must be recast especially for his reliving of it; it means acknowledging the manifestation of his uniqueness while recognizing that the layers of his experience disclose natural endowments common to all human beings; it succeeds to the extent that it not merely repairs an unhappy past but prepares for an unknown future.

This vision of remediation in mathematics recognizes the essential role of action and perception as the bases of mathematical thought and as providing “the realities which must be richly experienced.”

Today in many quarters investigators are attempting to unveil the mysteries of the action-perception-thought trinity. While ignorant of all their findings we can in living acknowledge our own participation at every moment in this mystery. Still we can say this much with absolute clarity —to see the world with the eyes of a mathematician is to become sensitive to the universe of relationships per se. Surely this suggests creating for learners appropriate situations which use action and perception to enhance the awareness of relationships.

In the light of these reflections let us return to the situation with which we began. Relationships indeed lay at the heart of our questions about

28

Our Work In Remediation

the two figures. This time however, as educators, we come with a new set of questions, not to satisfy our own curiosity but to serve others. Can the relationships implicit in this situation be made more dynamic and more accessible through the right material and actions? Can the situation be further elaborated in the realm of perception and action or significantly extended by the addition of language and notation?

In this particular case it has been done using a Geoboard, a wooden board with pins arranged on a square grid so that rubber bands can be stretched over the pins to make a great variety of shapes. The following scenario represents the result of successive actions upon the initial rubber band shape on the Geoboard:

There are many observations to make here. First the reader, so far as I know, did not use a Geoboard but accepted the drawings as a suitable substitute. The actions, here virtual for the reader but actual for the Geoboard user, can be repeated, have a definite effect and so can be designated as operations. In fact they are the inverse operations of doubling and halving, and they are employed in such a way as to transform the shape of the figure while preserving the area.

In the virtual realm the operations can be carried on indefinitely and with similar figures of sizes and proportions different from the one on the oboard. So one can glean from a dialogue with this situation, among other awarenesses, those contained in the theorems concerning the area of classes of triangles and parallelograms of equal base and altitude. The theorems may be formal statements expressed in sophisticated and specialized terms, but they can now be linked to the experience of whoever has lived this situation and, most important, can relive it as often as desired in his imagery and mental dynamics.

29

2 Mathematics

Many fertile situations have been worked out in the development of Gattegno mathematics, some using special materials, others drawn from the environment. Every learning encounter is an opportunity for new developments. Students who have been impoverished in the field of mathematics, for whatever reasons, need a rich diet of these situations to charge them with the experience and imagery which will build in them new understandings of the self and of mathematics as a human activity. For mathematical activity can be characterized by its penchant for returning again and again to the same situation to see it in a different light in order to draw out new awarenesses and extend old ones, and it leads one to the truth of William Blake’s words:

“If the doors of perception were cleansed, everything would appear to man as it is, infinite. For man has closed himself up till he sees all through narrow chinks of his cavern.”

Zulette M. Catir

30

Our Work In Remediation

News Item

1 Readers interested in our ESL mini charts can order them now at the price of $3. 50 (plus 5% postage and, in New York City, tax at 8%) per set of one mini-Fidel and three cards, each reproducing four charts, reduced to a size 8” x 5.”

Notes for users are being prepared, but teachers of English as a second language the Silent Way do not need these. The mini charts are destined for the student who wishes to be able to continue on his own the work done in the classroom. They constitute an instrument for independent learning once the color code has been mastered.

The notes will propose exercises that cover pronunciation, sentence formation for the sake of reading with the proper intonation, and a study of the structures Of English that are easily formed with the charts and some of their extensions.

This set is the first in which we have colored four of the five so-called “long vowels” with two tints indicating the two sounds that are heard when uttering a, i, o, u as they are sounded in the alphabet. As a result this material represents an advance in the use of color for English. Although such changes have not yet been carried out on the classroom materials printed years ago with the old conventions, they indicate what we shall be using in a future edition when the time comes for it.

2 The Comparative Phonetics Kit is at the printers and will be ready early in 1975. The languages in the kit are, in alphabetic order:

31

English (U. S.), English (U.K. mute R), French, German, Greek, Hebrew, Hindi, Italian, Japanese, Korean, Malay (Bahasa), Mandarin, Portuguese (Brazilian), Russian, Spanish and Thai. A second kit is being contemplated and a few of the Fidels are being prepared. Languages under consideration for inclusion are: Amarigna, Afghan, Arabic, Bengali, Burmese, Cambodian, Cantonese, Danish, Dutch, Farsi, Serbo-Croatian, Swahili, Tagalog, Turkish and Urdu.

An examination of the size of the classroom Fidels brings home at once the convenience of this new instrument that can be put into the hands of linguists and students of languages. The extensive notes being prepared for these kits will constitute a substantial addition to the arsenal of linguists.

3 The Silent Way materials for a number of languages have been completed during this last year in readiness for the printers when the number of requests from prospective users warrants it.

The Japanese, Mandarin, Hebrew, Italian, German, Portuguese, Korean, Malay, Thai, Hindi and Russian charts can be obtained as handmade copies from our original prototypes. The cost, although high, is not prohibitive and institutions may be able to afford them.

4 Dr. Cecilia Bartoli Perrault, former Assistant Dean for Faculty at Staten Island Community College, and before that Associate Professor of Italian at New York University, has joined our staff as director of our English as a second language project. She would welcome enquiries from ESL teachers interested in establishing Silent Way classes for adults or children in or out of academic institutions. Bilingual programs may find that she can be of assistance. She is assisted by Ms. Linda Warren and Mr. Steve De Giulio.

5 Concerned with the cuts in education budgets while educational problems are not diminishing, Educational Solutions is offering school systems and schools a new kind of service that could be as effective in reducing illiteracy in language and mathematics as our resident teachers-of-teachers have been in the past few years, and at a cost that can be afforded in the present difficult conditions.

32

Our Work In Remediation

For further information on our “School Reading Clinic” contact Dr. D. E. Hinman and for the mathematics clinic contact Mr. D. H. Wheeler or write for our leaflets explaining our concept.

6 During the last three months we have presented our “Absolute Visual Reading” project to three groups in Washington D. C.: in July at the Better Hearing Institute, in August at Gallaudet College and to the staff of Project Life. This project seems to have attracted the attention of a large number of teachers of the deaf all over the country. Ms. Katherine Mitchell has been given the responsibility of helping all those in the field who are interested in knowing how we have broached the problem of reading for these students. Mr. Ted Swartz will also be available for consultations.

7 The huge efforts being made in New York City to implement the agreement between ASPIRA and the Board of Education have prompted our Spanish language colleagues (Ms. Patricia Perez and Ms. Mary Seager) to investigate whether there is a place for our contribution. Bilingualism is a challenging opportunity. The readers of our Newsletter may know of the special issue on Bilingualism we put out two years ago. We believe our concepts can save a great deal of time, expense and frustration, meet the opportunity and give satisfaction. In particular our Pop-up and LeoColor programs for TV (on film), if examined seriously, will show how quickly students can put behind them the hesitations of facing two languages with the same alphabet, and use their energies to conquer their school subjects in either of these languages.

8 Our colleagues Ms. Clermonde Dominice and Ms. Shelley Kuo Burns have been working overseas. The first is in Cyprus with the International Red Cross Committee helping reunite families disrupted by the recent military actions; the second gave in June and July two intensive courses in Mandarin in Geneva (Switzerland) for a group called “Face à l’Education.”

Ms. Ann Crary is spending one year in Paris teaching English as a second language the Silent Way to classes at the Faculte de Droit of the

33

News Item

University and Words in Color to students at the American International School.

9 Educational Solutions was active during July and early August on behalf of the New York City Board of Education. The Department of Planning and Support arranged for four classes of Board of Education personnel to be taught Spanish the Silent Way for three intensive days; for one class to have an English as a second language course, and for the personnel of two new high schools to be exposed to the subordination of teaching to learning in two intensive three day seminars.

We welcomed the opportunity of showing over two hundred teachers and administrators how our research into learning and teaching can benefit all educators.

34

Our Work In Remediation

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.

www.EducationalSolutions.com