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Two Of Our Breakthroughs

Educational Solutions Worldwide Inc.

Caleb Gattegno


vol. XI no. 3-4

February/April 1980

First published in 1982. Reprinted in 2009. Copyright Š 1982-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-308-1 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555

This Newsletter over the years has taken to its readers studies of topics which could be conceived as attractive for different reasons, possibly mainly because awareness was at the center of our concerns. Occasionally, we gave the space to a less broad issue, but only very rarely did it seem proper to invite our readers to take a look at a product of ours, as was the case five years ago when The Silent Way Video program was completed. On that occasion a corporate effort was written about by various people who had occupied different positions in the project. It happens that these last weeks we reached the final stages in two projects that have occupied us for the last two years almost exclusively. To our eyes they represent major achievements for any corporation, and for us — who could not count for support on size or capital — even more. For that reason alone two news items have become main articles. In fact, the two products we consider here represent educational breakthroughs with consequences that we believe go far beyond the mere fact of their availability. It is not necessary to be a mathematics specialist to see why we call them breakthroughs. In fact, their users are young children in the case of one, and elementary school teachers and their students in the other case, and none of these are at present math specialists. (This may change as a result of the existence of these products.) Although they were conceived separately, they have merged in our mind as it became clear that the computer programs could provide more easily and lastingly the awarenesses we wished to develop through the printed word in the Mini-Tests and make them more profound and, that the sheets of the Mini-Tests could provide the practice and generate the extensions, that could not easily and at a reasonable cost, be developed for the computer. They are still offered to the public separately as their presentations stress different functions and aim at different people. Today it is still uncommon that the administrators — who make the decisions of equipping their schools with educational aids — think both in terms of electronic and of printed materials. Perhaps this issue will narrow the gap.

Table of Contents

Visible And Tangible Math.................................................... 1 A Few Notes On Trials By Students ..................................... 25 The Mathematics Mini-Tests............................................... 27 Addition And Subtraction ................................................... 31 Multiplication And Division................................................ 35 Fractions, Decimals And Percentages ................................. 39 Directed Numbers .............................................................. 43 Notes On The Math Mini-Tests............................................ 45 News Items ......................................................................... 49

Visible And Tangible Math

A microcomputer courseware designed and directed by Caleb Gattegno, Dr. Phil. We have just completed a ten diskette courseware which has been tested and is still being tested, with children of age 5 and above and with adults. In our proposal, accepted by the National Science Foundation granting committee two years ago, we outlined the objectives which were: to develop a computer program for our well-tested subordination of teaching to learning in mathematics for beginner students and for remedial cases, limiting ourselves to introducing children to mathematics and finishing with the algorithms of addition and subtraction. The narrative in the proposal outlined the main chapters of the course contemplated. The first would be about numeration from one to billions (in the common base and in others). The second would be dedicated to complementarity in numbers which are powers of 10. The third concerned the concepts of equivalence, and the fourth to transformations as needed to work out a new algorithm for addition and subtraction (in any base). This curriculum — which makes as much sense to mathematicians and epistemologists — differs radically from the current basic curricula


Two Of Our Breakthroughs

used in elementary schools anywhere in the world, including the United States. *** For the readers it may be more important to select those points in our treatment of the subject-matter that can serve as criteria for a computer courseware in mathematics than to tell in detail from the point of view of contents what we did to make our programs effective. With respect to the appearance of words on the screen, from the start we restrained ourselves i.e. we used devices other than words to let young students know what they had to do in order to obtain from the computer that it performed some of the required changes, like erasing figures or asking for a particular transformation. We instructed the computer to do only what students could not invent and never to take their place. We did not instruct the computer to congratulate students that answer correctly any challenge presented to them. Instead, either another problem is offered them or they are moved to the next level of work established in the courseware. When students make mistakes the computer does not refer them to those sections of the course they seem not to have properly assimilated. Instead, the courseware itself has been so structured that each student will reach mastery of the necessary steps, leading to a broadening mastery integrating all that which had been done to that point and subordinating it for use in the next leap ahead. Another feature of the structure is that students decide when they have had sufficient practice and when to use the “next� command to move to the next challenge. In this way students will always have as much time as they need to feel on top of what they are invited to do when interacting with the computer. The people with whom we tested the courseware included a few five and six year olds who were not yet literate as would be the case of many children of that age if the program were to be introduced in


Visible And Tangible Math

kindergartens or in first grades. We found that the manner we adopted to make them use our proposal was right for them. The others — who could read — did not object to that approach. At the very beginning of the program we introduce on the right-hand side of the TV screen four rectangles 1 cm x 2½ cm; on each of these is written one word: “next,” on the top rectangle, “talk,” on the second, “print,” on the third, and “erase,” on the last. The area of the rectangles is lit and therefore capable of activating a lightpen (a joystick cursor can also be used) whose coordinates are interpreted by the computer in this program to move to the next module or to sound (talk), or to take out of the scratchpad what is on it and print it out somewhere else on the screen, or to erase part or the whole of what has been put on it. Calling attention is done by either flashing alone, or by flashing and with a voice (produced by a super talker out of sound files stored on the disks) or by a voice alone, saying the command-words “touch” or “print.” In our experience, these commands are immediately learned by anyone, whether literate or not, speaking English or not. The numerous games that are part of this courseware are essentially played with the student touching (with a lightpen or a joystick) items on the screen which are made to move to or disappear from, some area to go to another; or to serve to compose sequences of digits with or without other arithmetical signs (commas, addition and subtraction signs; equal sign; a line in the vertical notation of an operation; arrows; stars to count; brackets); or on graphics created to convey definite ideas. There are 122 modules in this courseware organized in 9 chapters: 1

Numeration; diskettes # 1 and #2. # 1, up to 1000, #2, thousands, millions and billions (in different bases, but mainly in the common system where 10-1=9).


Ordering numerals, cardinals, and counting; diskette #3.

3 Complementarity; diskette #4. 4 Introducing complementary number pairs; diskette #5.


Two Of Our Breakthroughs

5 From number-pairs to addition and subtraction; diskette #6. 6 Transformations of operations; diskette #7. 7 Equivalent additions and subtractions; diskette #8. 8 The algorithms of addition and subtraction; on numbers of up to three digits; diskette #9. 9 Addition and subtraction of numbers of any length and in any base; diskette #10. Essentially the curriculum contained in this courseware aims at giving every beginner student of mathematics the most solid basis for future work in the field. This we shall examine now. *** Since students entering their first grade generally bring with them a good command of their spoken language and since numeral adjectives — both cardinal (i.e. used for counting) and ordinal (i.e. used to convey an order, generally in time) — are current occurrences in the vocabulary of most environments, in this courseware, we treat numeration as language. This means that we present the study of numerals both as a notation and as a set of sounds and their corresponding English written forms. This allows us to work in a manner that leads to mastery of recognition, implying knowledge of the sounds as triggers of signs (digits or words) and of signs as triggers of sounds and images. Modules I-1, I-2, I-3, I-4 concern themselves with one digit numerals (or 1, 2, .… 9) and their written forms (one, two, .… nine). We then can immediately add the numerals (100, 200, .… 900) — only in figures — in which 00 will trigger the word “hundred” and conversely. But this permits us to introduce the eighty-one composite numerals in sounds, from one hundred-one through nine hundred-nine and in figures from 101, 102, …. to 109, then to 201, .… to 909 presenting them dogmatically as they are conventionally written — but clearly. Two programs and one test are given to that task of mastering 99 numerals in notation and words.


Visible And Tangible Math

Because numeration is a challenge, within the language the next modules of diskette #1 are structured by the English language. That is to say that the regularly formed names are treated first, and each of the irregular ones is treated separately. Hence there is one module dedicated to 40, 60 - 90 (which are regular: their name being that of a digit followed by “ty”) and the integration of the previous modules provide five hundred new numerals. The following module concerns the three irregular ‘ty’s” (20, 30, 50) producing three hundred additional numerals. The last module treats: ten, the teens (in two stages: regulars first, irregular second) and the remainder of the 999 numerals from 1 to 999. This program #1 ends with a test that will show whether the students know the language of this field, without any doubt in their mind about what to say and how to write these numerals. The pace for the above study is in the students’ hands since the tools for showing their knowledge is a lightpen that has to be put exactly in front of the proper item on the screen in order to be accepted. (A joystick can also be used.) This allows students to move as slowly or as fast as they are able to on the set of problems offered them. The second diskette is also concerned with numeration but now extended to thousands, millions and billions. The numerical language and the recognition of the names of numerals of up to twelve figures in a row, is acquired through a number of interactive games. We need to introduce zero as a digit even though it has appeared as the notation for “ty” (and in ten) and, when doubled, as the notation for “hundred.” We then have to introduce the comma whose name is “thousand” when there is only one; “million” when a second one is used; and “billion” if a third is used. Practice with the language will ensure retention and recognition. The meaning of meaning at this level is precisely that. Hence when one hears a string of sounds (say three thousand two hundred seventy five) and writes 3, 275 and reads this in that way, we must conclude that the spoken and written languages are available to that person. We do not aim at another meaning at this stage. Because we are also producing a math courseware, we endow numeration with an “order structure” which will provide the ordinal


Two Of Our Breakthroughs

adjectives. We do it for the first 1000 numerals and in other bases as well. From our experience we believe that young children take to bases easily, they will know from the start the important fact that the appearance of a two-digit numeral called 10 does not necessarily follow the numeral 9 except in one system of numeration: the common one. Hence, this diskette is usable by beginners and advanced students alike. Its games require concentration, fortify retention and because they are fun, they carry students ahead swiftly and safely. It is in this diskette (#3) that the notion of cardinal numerals appears, and as the invariant associated with a set, and in reply to the question: “how many?” Answering that question is a practical activity, called counting. But the mathematical awareness behind this activity is that there is no element that must be considered as the first element in a set of objects; there are as many beginnings as there are elements in the set. But if we observe the following rules and •

no item is pointed at twice,

no numeral in the sequential order defined earlier, is said twice, and

neither an item of the set, nor a numeral in the sequence, are omitted

we obtain the definition of counting. Hence, counting generates a sound or a sign (or both) associated with every set which must be “countable” and exhausted by the process of counting. This sound or sign is called the cardinal of the set. It is an invariant with respect to the choice of how the counting is done. The sound of the last numeral of the sequence of numerals used in counting a set starting from 1 is “unique” and is the name of the cardinal of that set. Clearly because this sound is unique (in each particular base of numeration used in the counting) and comes from sounding (or writing) that numeral and from nothing else, the “nature of the items” of the set does not come into consideration.


Visible And Tangible Math

Therefore all sets are mathematically equivalent if they have the same cardinal. Since counting — which generates the cardinals — implies the one-one correspondence above, between the items of a set and the numerals in the ordered set of numerals, one-one correspondences become activities we can consider separately and see them as links between sets. It has become customary to begin with one-one correspondences when introducing students to set theory. *** At the level of diskette #3, the merging of numeration and counting has taken place. We are therefore allowed to say that all the actual mental structures which give meaning to cardinals of any magnitude (up to say one trillion minus one, in the common system) are owned by those students who went through the first three diskettes systematically. We can now enumerate the mathematical awarenesses which have been singled out and brought to the students in the programs of these three diskettes: 1

the language of everyday life serves as a universe of discourse for the formation of the universe of numerals;

2 the nomenclature of numerals is organized in a manner which can be systematic and accommodate the few irregularities thus reducing considerably the burden on memory; 3 the set of one digit numerals does not have to stop at 9 one can go beyond or stop earlier; the choice of that station determines the base of numeration of which there are as many as we wish; 4 the set of numerals can be ordered; 5 this order determines in the set of numerals any number of ordered subsets in which the first and the last and therefore the jump in between, are arbitrary. 6 there are two orders on the ordered set of numerals: one can be called increasing and the other decreasing


Two Of Our Breakthroughs

according to whether (i) the order coincides with the one of the whole set, or (ii) goes counter to it; 7 the everyday life question “how many?” (objects) are in a given set, can be answered by an activity called counting which presupposes: •

a. the set of numerals as defined above;

that this set is ordered — and is used in its increasing order;

that one starts with 1 and moves on on the ordered set, uttering each successive numeral as one shifts from one item of the set to another, seeing to it that none is counted twice, nor that any numeral is uttered unless simultaneously an item is pointed at.

8 through counting, each set gains an attribute called its cardinal which is independent of the order in which its elements are counted. This means that cardinals are invariant with respect to this transformation (i.e. which element is selected to be the first, and that going from one element to another does not imply an order); 9 the one-one correspondence of counting — defined in #7 above — tells that the “nature” of the element of a set has no effect upon the cardinal of the set and therefore that any number of sets can have the same cardinal. They are then said to be equivalent. This is a direct connection between sets, and it can be seen as by-passing counting, giving an independent status to one-one correspondence (also called bijection). 10 since there are 9 one-digit numerals (in the common system of numeration) and a 0 (zero) that has been introduced by the name of “ty” when it was alone on the right of any one of these 9 digits (except 1), by the name of “hundred” when two contiguous ones follow the digits on the right, and — since there are so many more numerals than these ten digits — some of them are part of each of the sequences of 2 or 3 digits met. The notion of place value has been introduced and the same digits can be called “ones,” “tens” or “hundreds;” 11 by the artifact of separating digits in groups not exceeding three and starting the grouping in a string of 8

Visible And Tangible Math

numerals from the right, it becomes necessary to develop a nomenclature for strings of digits beyond three. The last three-digit numeral in the sequence is written 999, the next numeral is called one thousand and written 1,000. This means that “commas” come into the notation and can be used to separate groups of three digits, starting the count with the first one on the right. Their names are “thousand,” “million,” “billion,” etc., allowing the use of the mastery of the subject matter in diskette #1, to name any string of digits and to know exactly the place value of each of the digits in a string. By asking to recognize long numerals when their names are given, or to write a string with one, two or three commas, one can provide the opportunities needed to master at the same time place value and the reading and writing of long strings of digits; 12 thus far it has not been necessary to assume the existence of a universal and unique unit. Numerals are locators and counting assimilates elements even if they are unequal, for, to ask the question, “how many?” does not require the identity of items, only their separability so that they can be recognized as distinct or/and distinguishable and so that a numeral can be associated to each of them; 13 odd and even numerals can be defined by the act of •

starting with one, and only uttering the alternate ones,

starting with two, and using the same procedure.

14 naming evens in succession is also called counting by twos; odds are the set of numerals starting with one and counting by twos; 15 the array formed on top by the 9 digits studied at the beginning and then of groups of ten numerals in rows each starting with one of the successive digits followed on the right by 0, 1 or by 2 .… or by 9, and each successive row then by being numbered from 0 on, allows us to count by tens. 16 this can be extended as far as one wants to counting by hundreds, thousands, millions, billions, and so on. ***


Two Of Our Breakthroughs

To be able to shift from numerals and cardinals to numbers, we need to involve operations. From diskette #4 to diskette #10 we develop chapters of the algebra of integers. An algebra is a dynamic structure on a set which allows two items of the set to be replaced by another item. A feel of an algebra is a feel of the presence on a set of operations which transform a given pair of elements of the set into another of its elements. This with the notation * as an unspecified operation, and a, b, c as specific elements of a set looks like this: a * b → c (or in words a star b yields c). The star is represented in diskettes 6 through 10 either by (+) addition or (-) subtraction. These two operations are one the opposite of the other hence there is need for simultaneous definition. In this courseware we arrive at this definition through a more general operation called complementarity which we introduce in diskette #4. We use a pair of hands and the physical operation (or action) of folding or unfolding some of the fingers. We therefore begin by partitioning a set into two subsets which can be examined for what they can offer. And they can offer a lot. The main ideas behind the development of this study are: •

the whole set is visible and is made of the (10) fingers of two hands;

the partition is visible since it is easy to perceive which fingers are folded or unfolded, and

whether we start with the unfolded set and fold some fingers or start with all fingers folded and unfold some of them, we give ourselves a dynamic which changes the appearances.

If we have at our disposal a voice we can add to the visual perceptions above some sounds which indicate more precisely what students must concentrate on. The voice can direct the mind to reach some specific awareness on which the program can elaborate later. To develop interaction between the student and the program, the TV screen shows two stylized hands with 8 rectangular fingers and two thumbs and the computer will ask the students to do on their own hands what it makes the fingers do: fold or unfold. The random function of the computer


Visible And Tangible Math

can be used to randomize the choices of the fingers affected so that the likelihood of a repetition of a situation is reduced. The voice sounds: “do what I do,” on some modules, and “do what I don’t,” on some others and both, in still others. Once the students understand that the action of folding (or unfolding) fingers is the source of two subsets of fingers upon the totality of fingers of two hands, it is possible to suggest: 1

that there are different choices of fingers available to fold (or unfold) and which still respond to the sounds of 1, or 2, or 3, .… or 9. This is a new form of awareness of cardinality: instead of counting a set by using a random starting item and a random order, the fingers selected to be folded (or unfolded) are also random. Since there are 10 choices for 1 (and for 9); 45 choices for 2 (and for 8); 120 choices for 3 (and for 7); 210 choices for 4 (and for 6); and 252 choices for 5 — although we do not include this particular study in our questionings of students — the mathematical awareness resulting from the use of the random function is that there are stores of choices of subsets in the set of fingers.

2 that the two subsets, generated by folding, are in a special relationship which is experienced as a partition if the mind stresses the existence of so many folded and so many unfolded, and as complements if the mind stresses that together the two subsets constitute the permanent set of (ten) fingers. This shifting of awareness by stressing this (partition) or that (complements) is the greatest gain of the interactions of students with these modules. The techniques used to force these awarenesses can be put in the following words: •

once we know that there is a way of partitioning the original set into two complementary subsets, by folding (or unfolding) a few fingers, we can shift from telling, “do what I do,” or “do what I don’t,” to asking, “how many are up,” and “how many are down,” The first was learned on one’s own hands when what the computer did on its own (schematic) hands was also managed by the viewer; the second is learned on two pairs of hands side by side on the screen, one pair belonging to the computer — and only affected by it and not reachable to inputs from the student — and


Two Of Our Breakthroughs

one pair being the student’s and responsive to what he does with a lightpen or a joystick. •

the actions of the students and the images resulting from such active involvement with the program are being translated into writing two numerals in parenthesis, the first indicates how many fingers are up and the second how many are down. In the beginning the hands are seen but soon a screen covers them and the entry must be done entirely from what one knows. It is then that we know that each numeral of the pair triggers the other one or — which is equivalent — that the eleven pairs of complements (0&10, 1&9, 2&8, 3&7, 4&6, 5&5 and their reverse) are thoroughly known by the students.

all through the courseware the computer has been programmed, (i) to receive correct answers and then move to a new challenge, or (ii) to let students have a chance to try again and again until they get the required entry, only repeating the question in between trials or keeping question marks flashing on the screen. A short delay has been placed after each exercise to allow students to examine the acceptable answer and to get some feedback from their success. ***

Diskette #5 extends the knowledge of the complements in 10 to those in 100 and then to those in 1000, using simultaneously two pairs of hands for 100 and three pairs for 1000. The mathematical awarenesses dominating these sections are: 1

that if we rename the fingers in certain ways all that was learned when working on diskette #4, remains valid while yielding new facts: tenty or one hundred can for example generate (forty, sixty) or (40, 60) where (four, six) or (4, 6) were met before adding the sound “ty” after those of the digits and yielding immediately eleven complementary pairs (100, 0), (90, 10), (80, 20), (70, 30), (60, 40), (50, 50) and their reverses. Likewise, replacing “ty” by “hundred” we generate immediately the eleven complementary pairs (1000, 0), (900, 100), (800, 200), (700, 300), (600, 400), (500, 500)


Visible And Tangible Math

and their reverse ones. 2 that if we begin with “tenty” and substitute for one thumb (equivalent to “onety”) a pair of hands producing ten fingers, we have “ninety ten” to work on to produce the sequence from 1 to 100. Hence folding fingers on those four hands — which to start with had all fingers extended — we generate the pairs of complementary numerals (0, 100), (01, 99), (02, 98). . . . (49, 51), (50, 50), (51, 49). . . . (100, 0). 3 with three pairs of hands (the “ty” pair and the “hundred” pair having lost one thumb each as above) we find the complementary pairs in “nine hundred ninety ten” or one thousand and learn to determine the numerals to enter in parentheses to represent 1000. (0, 1000), (001, 999), (002, 998). . . . (499, 501) (500, 500), (501, 499). . . . (1000, 0) 4 it becomes clear from the absence of one thumb on one hand of a pair or two thumbs when three pairs of hands are on the screen that if “a-hundred b-ty c” are folded the name of the complement in ten hundred will be “(9-a)hundred (9-b)-ty (10-c)” This much has been learned from complementarity; i.e. from the model of pairs of hands and from the existence of two subsets obtained by folding some fingers in the set of named fingers (ty and hundred). *** Diskette #6 has been dedicated to making the previous awareness take the form of the two operations of addition and of subtraction. Because of the use of the hands — as was done in the previous two disks — we can only concern ourselves with complements in the numbers 10, 100 and 1000. But because we use complementarity we can see that our way of looking at the pair of numerals may lead us to distinctive notations that distinguish them. In one of them, we link the two numerals in the brackets — a folded and b unfolded — as forming


Two Of Our Breakthroughs

together the totality of the set of fingers (and their names). This we shall translate into a notation holding the pair together and related by the sign + (read, plus) like a + b. In the other way of looking, we link to the initial whole (10 or 100, or 1000) one of the subsets, seeing, for example, that folding takes away from the whole the folded fingers a and leaves visible the unfolded ones b. So, we can translate this awareness as 10-a (or 100-a or 1000-a ) using the sign — (read, minus). What we see can be mentally acted upon and treated at the same level as what we can evoke. “The whole” can be evoked and used as if it were present in the mind, photographed from one displayed on the screen before any fingers are folded, whether we want to see the situation as a “take away” one, or as: the two present subsets “adding up” to the whole. On the screen, the same finger configuration is then translated into either a + b = 10 10-a = b

(or 100, or 1000) or into (or 100 -a = b or 1000 -a = b)

Hence, complementarity is truly the source of addition and subtraction. An awareness that we can force independently is that the pairs of subsets of fingers — which we formed until now as an “ordered pair” when we agreed earlier to place on the left of the parenthesis the subset of “up” fingers and on the right of the subset of “down” fingers — can be considered both ways and no longer as an ordered one. This is done on the screen by the computer and is known as the commutativity property of the pair and hence of addition. From now on, every addition a+b can also be read b+a. Subtraction which requires the evocation of the whole as well, and not only of the pair cannot be commutative; each pair generates two different subtractions.


Visible And Tangible Math

In this diskette (#6) the horizontal and vertical notations are introduced, practiced and recognized as translatable one into the other. Once additions to, and subtractions from 10, 100, 1000 have been mastered, extension of the operations to 20, 30. . . . 90 and to 200, 300. . . .900 follow easily. This is done in three phases having examples in which — •

the first terms are ordered but the second terms are constant;

the first terms are random but the second terms are constant;

both terms are random.

The next diskette (#7) is devoted to the generation of classes of equivalence and aims at leading to the definition of numbers as they live in the minds of mathematicians. The device used is the formation of classes of pairs of complementary numerals by partitioning a small given set of stars. Distinguishing stars by either printing them or poking them on the screen, and limiting ourselves to the partitions that are pairs, each numeral a can now be associated to a class of pairs of the numerals preceding a in the sequence of numerals. Thus 4 is equivalent to (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), these numerals being the cardinals of the set of stars and of some of its subsets. When all the dual partitions of sets of stars of 9, 8, 7. . . .2 stars are formed and commutativity is used, a table can be formed which contains all the pairs needed to extend the work of diskette #6 to numerals without zeros. This is done in disk #7. With this diskette (#7) we learn to generate eight equivalent relations for each pair, 4 additions and 4 subtractions. There are other ways of extracting more mathematics from the properties of partition, commutativity and complementarity, but they are not attempted in this courseware.


Two Of Our Breakthroughs

What we concentrate on is the recognition that what we look at contains more than meets the eye: here, that a pair of complementary numbers contains hidden in it at least eight relationships. To see them as implied in any one of them, is equivalent to acquiring a mathematician’s eye even when one is a very young beginner. To make them explicit will increase the computational powers of those same children. In particular, the notion of inference which we can also meet in the situations of the modules of the third program of this diskette, shows that if one knows one thing one knows many others, or if one knows a little one knows a lot. For example if x+y is known, so is xo+yo and xoo+yoo etc; if x-y is known, so is xo-yo, xoo-yoo and so on. *** We reserve to the eighth diskette a thorough study of transformations as these appear when considering addition and subtraction of integers. The main lesson is that a given problem can be transformed into an easier one. The easier one is normally one which yields its answer by sight or through a process suggested by the new form. We then see additions as reading, if one digit in every column is a 0. Hence the transformations suggested are the shifting of some units from one of the addends to the other to produce a 10, a fact well practiced that far. Shifting is an operation which is a subtraction on one digit and an addition of the number taken away “to the corresponding digit in the other number.” In the vertical notation these words become: “to the one above” (or below). So shifting is studied thoroughly in some modules of this diskette leading to additions answerable by sight. In the case of subtraction, what produces equivalent ones is not shifting units but adding or subtracting the same number from both terms. This is called equi-addition or equi-subtraction. Their practice


Visible And Tangible Math

will provide the flair for how to take the proper initiatives so as to obtain, at first, simpler problems — and later, the simplest possible problem — equivalent to the given one. In particular it is in this diskette that 1

the unit is introduced, by showing: •

that shifting requires that what is taken away from one term is equal to what is added to the other and conversely, and

equi-addition or equi-subtraction just means that, say, 9-8, 8-7, 7-6, 6-5, 5-4, 4-3, 3-2, 2-1 are all equivalent to 1 or that 1=2-1 = 3-2 = 4-3. . .

2 we see from this last form that also there are an indefinite number of subtractions equivalent to 1 or, for that matter to any number; 3 when in the mind the end of the process is left unbound the indefiniteness just mentioned becomes known as infinity. This notion is what makes mental work mathematical and from now on numbers are seen as infinite classes of equivalence by subtraction and finite classes of equivalence by addition; definite classes corresponding to definite numbers. A whole number is the class of equivalence of all the partitions which add up to that number. Thus as we already mentioned 5 = 4+1 = 3+2 = 2+3 = 4+1, and substituting for 2, 1+1, for 3, 2+1, and 1+2, for 4, 3+1, 2+2, and 1+3, we get the whole number 5 equivalent to 1+1+1+1+1 = 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 1+1+1+2 = 2+2+1 = 1+2+2 = 2+1+2 = 3+1+1 = 1+3+1 = 1+1+3 = 3+2 = 2+3 = 4+1 = 5. The larger the number, the larger its class of equivalence by addition — but it will always be a finite class. But in terms of subtraction, infinity is the property of all the classes whatever the magnitude of the number.


Two Of Our Breakthroughs

In this courseware these matters are barely hinted at, certainly not developed. But the germ being there, it is possible to expand the study with older students or adults. The students working through this diskette will gain a deep insight into transformations: their need for solving problems, their role in making one become swift at finding answers and feel sure of the result — their capacity to be telescoped (several being replaced by a single one), the guidance they provide for finding which steps to take and which criteria relate to which initiative. There will be plenty of opportunity for practice and for becoming good at transforming the given to meet some criteria; here, the easiest forms of additions and subtractions so as to lead to answers by sight or by ear. The tests are included to ensure that all operations are done in one’s mind and written down correctly as swiftly as possible. *** The last two diskettes (#9 and #10) coalesce all the skills acquired so far to make possible a full mastery of addition and subtraction by students whose age only needs to be sufficient to make sense of all the games offered in the previous diskettes and that led to the mastery of these skills. This age is therefore an individual variable. In #9 there is a thorough study of all the cases of addition of three digit numbers which require transformations as well as of these transformations. There are seven cases which are surveyed systematically while for subtraction there are only two cases — also studied systematically. #10, extends the algorithms developed in #9 to the case of numbers of any size (limited to 9 digits in this program) and in any base ≤ 10. Diskette #10 represents the first application of all the mathematics taught in this courseware, freeing the students once and for all of any doubt about why these two arithmetical operations on integers (addition and subtraction) are what they are in substance and form.


Visible And Tangible Math

To achieve this mastery we must free the students of all their experience of mathematics being arbitrary; of solutions appearing as if they were all only one process and of the restrictions which can be seen as going beyond the convenience and reasonableness required by the arithmetic operations. For that we work out all possible cases, first in the restricted field of three digit operations and then transferring these gains beyond this restriction. We work out the problems in the common system of numeration — merely because it is the common one — and then show that the algebra remains the same whatever the base of numeration. This is only imposes one additional restrictive structure due to the choice of the base. In diskette #10, we continue to carry out the algorithms from left to rights (in the way we read and write numerals) but we give students alternative” choices: •

the classical, from right to left algorithm, and

the possibility of carrying out any partial additions or subtractions which seem legitimate or resulting from obvious transformations.

For example, in a subtraction using zeros to replace two digits that are equal and hold the same place in each of the two numbers. In the treatment of elementary arithmetic as it unfolds in this courseware, zeros have gained a special prominence because of complementarity and the use of the hands. The algorithms tend to increase the number of o’ s in the transforms” of the given which we select to work on finally. In the classical treatments of the same topics, zeros sometimes presented major problems. Now they are welcome and a special bias leads to select transformations which increase their happening. In fact, that selection of changes — which maintain as equivalent the given and the transformed — is guided by the discovery that the Safest way of getting the sum or the difference between two given numbers, is the one in which the answer is read in the final form;


Two Of Our Breakthroughs

for example in the addition pattern

the reading is immediate of

1823. Practice in getting such forms is provided in diskette #9 The modules of this diskette aim at giving insights and practice on the hierarchy of awarenesses which link the form of the given and the special transformations which are best suited to that given. Since the final stage of the sequence of transformations is one in which one reads the answer and puts it down in the proper manner (and also the only one we made acceptable to the computer) any addition or subtraction, which is immediately obvious, is not made to appear among the challenges asking to be worked out in this diskette. But because of the systematic study of the possible forms of addition and subtraction of three digit numbers, some of the forms only require one step to be converted into the easiest one. These are — in the case of addition — when only the last two digits on the left and on top of each other (in the two numbers) add up to more than the base. The seven cases studied are (calling ABC and abc the two 3-digit numbers) •

only A + a ≥ 10

only B + b ≥ 10

only C + c ≥ 10

C + c and B + b ≥ 10

C + c and A + a ≥ 10

B + b and A + a ≥ 10

A + a and B + b and C + c are ≥ 10.

The four transformations presented on the menu (and which appear on the screen) are:


Visible And Tangible Math


an amount to be added or subtracted is suggested by the computer — because of the structure of the given — but to be entered by the student so that the transform includes O’s in the proper places and so as to lead to a form more easily handled by the student. What is asked for here, is that the student recognize that the choice made by the program (also called the computer) does lead to an easier problem. The signs on the menu are asked for on two separate steps, the computer offering twice the same number to be added (or subtracted) from the top number, and subtracted (or added) from the bottom one. The student enters the presumed answers and gets confirmation from the computer if it is acceptable, otherwise the student tries again;

2 simultaneous addition and subtraction of the same amount, or subtraction and addition of same amount, to be chosen by students so as to shift that amount from one addend to the other, hoping for, or aiming at a transform which can either be handled immediately or treated again in the same way; 3 the choice of the amount is also left to the student who, now, begins with his choice of the transform before the amount shifted from one to the other of the two numbers is determined and he makes it appear on the screen; 4 any student who can see all the shifts in his mind, can ask for a scratchpad — by touching the rectangle on the menu marked “equal” — and print in it the number that is the sum of the given numbers. Whenever an answer is correct, the computer accepts it and does something to let the student know he has succeeded, mainly proposes another problem. But errors are not accepted and students must take steps to put things right, This includes erasing one or more digits from the scratchpad. A choice is now left to the student between either doing the systematic study of subtraction of numbers of three digits or to test himself on addition. Each test on addition has seven problems — corresponding to the seven cases above and offered at random — but only the “equal” sign on the menu is allowed to be used. A device will measure the time taken as


Two Of Our Breakthroughs

well as the number of mistakes. Another test can be taken immediately by just requesting it. The study of subtraction of three digit numbers only includes two cases of solution by equi-addition or by equi-subtraction. The student has the choice, according to the given, to select to enter, as equi-addition or equi-subtraction, the amount which will generate the transform believed to be immediately answerable. Tests follow. *** This courseware shows that the subtraction algorithm is conceptually easier to present and to master than the addition algorithm while it also makes clear that either can be used to solve the other. Indeed, since addition and subtraction are inverse operations, they are always present together but not necessarily seen that way — as students of the traditional curriculum know. To sum up the gains that will follow from work with these two diskettes we can say that: •

no addition or subtraction in any base can now be conceived as harder than any other;

looking at a given pair of numbers to be added or subtracted, will first trigger the mental process of transformation, which will replace that given pair by another one leaving for the last step the getting of the answer;

the answer will present itself as a number that can be written (and then read) from left to right, like the given ones;

this will be done swiftly and

accurately with

a confidence that belongs to the level of the challenge (now recognized not to be too high and not exceeding the powers of primary school children).


Visible And Tangible Math

*** The testing done with this courseware confirms that: 1

very young children (5 years old or so) take to it, comprehend its commands and manage to master the processes of execution and neither get baffled nor bored. On the contrary, they seem stimulated to stay with it and to go on with the games they are asked to play;

2 so-called remedial cases and learning disabled students find it capable of clearing their doubts and of stimulating their mental powers while restoring their confidence in themselves. In addition, they master the chapters covered by the course and may feel able to work better in their classroom set-up; 3 teachers and other adults who experience math anxiety find it can diminish considerably as things begin to make sense and even seem to be exciting. *** Our conclusion is that this computer courseware represents a multiple breakthrough in the field of education. It does all the above to the people exposed to it. It displays many criteria needed to define proper uses of the computer for education. It shows in concrete terms that major educational challenges can be solved and at costs in time and money ridiculously low compared with present-day costs for instruction or/and remediation. It opens the way to enlarging the curriculum, modernizing it, individualizing it, so as to serve students with every kind of gifts or impediments. It puts education technically within the modern industrial revolution — also called electronic, since students in their brains as well as on the screen, manipulate electrons.


A Few Notes On Trials By Students


Intermediate School 84 in the Bronx, 6 students of grades 7 specially selected as part of the field testing connected with the National Science Foundation project. In 2½ hours of intense concentration, working in turn with the computer using the joystick, learned to interact with the programs of Disk #1-4 and found it easy, engrossing and yielding knowledge as a by-product.

2 Public School 3 in Manhattan, students of 4th grade rated learning disabled, concentrated in an unusual manner (said their teacher) enjoying learning numeration through a few modules of the program. They requested to be allowed to continue using this approach. They will form a second group of older students testing the program in school hours and school setting. 3 We are videotaping kindergarten and 1st graders working with the programs as a way of catching learning as it takes place rather than refer to printed matter that hides the actual learning and displays figures and verbal references.


The Mathematics Mini-Tests

Most students in the United States and in some other countries have their progress evaluated through standardized achievement tests. Achievement Tests are organized around what was found statistically from national samples — for each grade, for both reading and mathematics and sometimes other fields — to give some idea of what “ordinary students” retain after one year of study of some prescribed curricula taught in certain ways. The seven most widely U.S. nationally distributed tests differ in many ways. School districts (and perhaps individual schools) usually choose to give their students one of these tests once a year. Students and schools are judged on the scores produced by these tests. Generally, these tests have multiple-choice formats. In 1972, we proposed to use the time of preparation for the test as an opportunity to give future test-takers whatever awareness about reading the tests made accessible. By 1974 and 1975 we had systematized our approaches to the reading tests to the point that we could offer them as a product to be used by schools. They were immediately welcome and our reading Mini-Tests became the first of our numerous products to be accepted by all at the national level. Soon, the satisfied customers asked us whether we had a similar product for mathematics. The study of the contents of the math standardized achievement tests showed us that there was a useful and


Two Of Our Breakthroughs

sizable job to do there to transform the preparation for the test into an education of the test-takers. A team of Educational Solutions’ staff members was formed which put together the necessary competence required in order to blend the demands of the subject (math) with the needs of the students and of the people who prepare them for the tests. *** The first breakthrough was reached when we found that the myriads of questions in all those tests were actually a much smaller number of challenges that could each be met as a set of alternative appearances generated by the transformations of one perception. For example, many questions which are seen as mathematical challenges — because they are found in mathematics achievement tests — became reading and language questions. This lighting made sense at once and unified the approach to a number of topics as varied as the clock, coins, graphs, numeration, decimals and percentages. These can all legitimately be considered as “other ways of saying something,” hence primarily concerned with language. Also, because of this awareness, the notion of equivalence imposed its elf in preference to the usual “equality.” Equality can be reserved to the link between two items that can perceptively replace each other (for example 2 and 2). Equivalence of items, while allowing items to replace one another, no longer requires the perceptive similarity (for example, 1+2 can replace 3). To help students, we introduce the expressions such as: “is another way of saying,” “can be said as,” “is another name for,” “is equivalent to,” and the use of the sign ~ . All of this satisfies several useful conditions: •

perceptive differences do not need to disappear for the relationship to be true, 3 ~ 2+1


The Mathematics Mini-Tests

according to the challenge, one or the other of the items on either side of ~ may be the one made use of in meeting the challenge. For example the item on the right yields the answer at once: 9+7 ~ 10+6

other forms, looking very different, may suggest themselves as possible additional equivalences. For example, 4 ~ 3+1 ~ 1+3 ~ 2+2 ~ 1+2+1 ~ 1+1+2 ~ 2+1+1 ~ 1+1+1+1

no statement, although true, needs to have only one form,

any one of these forms is capable of triggering all the others — the one we see and any of those we may need for given problems — since they all belong to the store of material in one’s mind and are all equally accessible.

In all this the dynamics of evocation is used profitably. *** Of course, the curriculum most generally adopted by the thousands of school districts in the country, concerns itself not only with computation — considered as the basic skill schools are supposed to impart — but with concepts and applications. Under concepts are placed those items which differ from computation but imply some computation in order to solve a problem. Since they are expressed not only in numerical terms, they need to be understood as such (as concepts) before involving them in calculations. For example, •

a mixed number is made of an integral part and a fraction;

a number-line as capable of representing some properties of numbers (e.g. order, magnitude.);

coordinates as used to locate a position in a plane.

These are some of the concepts met in all of the achievement tests. 29

Two Of Our Breakthroughs

Applications have been for a long time the social justification for including arithmetic in the curriculum of the public schools which aim at universal education. The word-problems that are part of all tests, are generally expressed in social terms. When the relationships behind them are formalized, this generally leads to a computation finally interpreted in terms talked about in the problem: that is, money, time, distance, shares etc. Mathematicians when working on their own mathematical challenges do not consider the notions just named, as mathematics, therefore they are included in the category of applications. But for school students and some of their teachers, they are the greatest challenge in their “mathematics” studies. Our contributions — organized in terms of the above categories as in the standardized achievement tests — can be looked upon as trying to improve students’ capabilities to compute, to be clear-minded about what they read (concepts) and to interpret the word-problems found in the applications sections. *** To obtain the greatest improvement in test-taking — for the computation part of the test — we did not hesitate to present new approaches to the mental challenges required by the various computation levels taught in grades 1-12.


Addition And Subtraction

In the case of addition and subtraction we propose one common source for both — namely complementarity — and introduce it asking students (1st or 2nd graders) to use their hands and fold fingers. Then we make them read the situation either as “putting together” the folded and the unfolded subsets (which generates addition) or as “taking away” from the whole set the folded subset (which generates subtraction). We also force awareness that there is a choice in the case of addition in starting with the folded or with the unfolded fingers (this generates commutativity of addition) and that there are two subtractions from the whole according to which of the subsets is related to the whole. We also show that it is possible to obtain at the same time, from the statement — x + y ~ 10 (or 10-y~ x) the statements x0+y0 ~ 100 (or 100-y0 ~ x0) x00+y00 ~ 1000 (or 1000-x00 ~ y00) where x and y are one digit numbers. We can transfer this awareness by giving enough practice with some computation, and then rewriting the problems with zeros as done above. In this way we not only increase factual yields but make students aware that they can extend their knowledge at no cost to their memory. Thus, instead of the traditional 99 facts of addition and 99 facts of subtraction (that people were made to memorize) we ask for the five complementary pairs in 10 (1 and 9; 2 and 8; 3 and 7; 4 and 6;


Two Of Our Breakthroughs

5 and 5) from which four complementary pairs are deduced for 9 (1 and 8; 2 and 7; 3 and 6; 4 and 5). It became possible then to know how to obtain immediately all the complements in 100 without asking for any memorization. Indeed (100 - (a-t y+b) ) is equivalent to (9-a)-t y + (10b) for all a and b ≤ 9. It is also easy to generate all the pairs which are complements in 2, 3, 4, 5, 6, 7, 8 by finding that if only one is known all others are obtainable by transfers of units. Thus if one knows for example 8 ~ 4+4 one gets 5+3 or 3+5 by shifting one unit from one of the 4’s to the other. Shifting again one gets 6+2 or 2+6 and then 7+1 or 1+7. Which one of these is held in one’s mind is not of great importance. So as an alternative we give practice in shifting numbers with enough examples to work on so that we induce students while obtaining accuracy and retention more easily to prefer this use of transformation to the more usual drill and repetition. Because addition and subtraction are studied at the same time and because we can now avoid separating the socalled easy additions and subtractions from those requiring “carrying” or “borrowing,” the one approach which makes sense is called equivalence by transformation. In this way every given problem is made into an easier one. In the case of subtraction for example, because complementarity in numbers that end in 0’s is easy, each subtraction problem is transformed into finding such a complement followed by an addition. Let us illustrate this here as follows: find 1021 — 876 (traditionally requiring two borrowings, which are also transformations although not acknowledged as such). The complement of 876 in 1000 is 124. If we now add to it the 21 we ignored we get the wanted answer of 145. Equivalent subtractions are so easily done and so easily lead to forms answerable by sight, that it is possible to forget now that subtraction has been a painful process to so many students for so many years and now make people take it in their stride. As to addition, by shifting the proper units from one number to the other, it is possible to obtain an equivalent pair in which the answer is directly readable.


Addition And Subtraction

Example: 684+768 becomes 1052+400 or 1452 for showing the transformation in the vertical notation

Another thing to bring to students’ notice is that once additions or subtractions are properly transformed they can see or hear the answer at once, giving themselves criteria to know whether they are right. Later, this power will serve them well to decide at a glance either which of the multiple-choice answers can be discarded as most likely unsuitable, or whether a “not given” answer must be selected.


Multiplication And Division

In the case of the other two operations of arithmetic, known as multiplication and division — which are also two inverse readings of the same situation — it is also possible to operate in a manner that bypasses memorization. Indeed, we already know from numeration that a product by 10, or 100 or 1000 etc. is a mere act of writing 0’s. Similarly dividing by ten or a hundred etc. numbers whose numerals end in 0’s is a mere matter of ignoring one or more 0’s. We can lump together multiplication by 2, 4, 8 simply by learning to double and then double again once (x 4) or twice (x 8). We can learn to multiply by 5 when we can reverse doubling as halving, and see 5 as ½ x 10. Thus even numbers can be easily halved and then have a 0 “tacked onto” the result (i.e. be multiplied by 10). Odd numbers become even if 1 is subtracted from them. Once this is done they can be handled by halving, “tacking one” a 0, and then adding 5 to this result. If we have the multiples of 3 we get those of 6 by doubling. We get the multiples of 3 out of the sequence of integers by ignoring two successive ones (after 3) and retaining the others. We can get the same result by adding to the double of a number that number; ex: 3×7 ~ 2×7 + 7 ~ 14 + 7 ~ 21. The multiples of 9 are obtained from the multiples of 10 (already known) by subtracting the factor other than 9: 8 × 9 ~ (8 × 10) - 8 ~ 72


Two Of Our Breakthroughs

that is a × 9 ~ (a × 10) - a ~ or it can be found that:

It is therefore possible to obtain the multiples of 9 by one figure numbers, even before one tries to obtain the double of the integers. By classifying the products obtained we find that all of the multiples of 7 but one, (7 × 7), have been encountered To find it, since we know 6 × 7 we add 7 to 42. In fact, we can look at the products obtained, in a variety of ways which allow us to become aware of different kinds of questions possible on these newly found relationships. These awarenesses are the ones that prepare for the tests. Indeed, if we work on a very simple example: 2×3 ~ 6, we can see a host of possibilities which appear in the test —


Multiplication And Division

Hence not only division can be met at the same time as multiplication, but the various notations, the corresponding equations, and the fractions, result from our stress on language — that is, on “is another way of saying.” What we actually bring to our students’ notice is that all computation is first algebraic and then arithmetical. That is, to say that the processes remain the same when the values change. Without using the word algebra we make students aware of what they have to do and then do it, by referring to knowledge already met in the Mini-Tests as numeration. When for example we have to multiply numbers of two figures (or more) by a one-digit number, we must “transform” the given to be able to find the answer. Such transformation can be said to replace a number by a sum of numbers and the given multiplication by the sum of the products of every one of these numbers by the one-digit number. This “distribution” of the factor over the terms of the sum is in fact the well-known algorithm of multiplication which looks like this: 37 × 4 ~ (30 + 7) × 4 ~ 30 × 4 + 7 × 4 ~ 120+28 ~ 148 or in the vertical notation

The case of several digit numbers multiplied together (e.g. 238 × 5, 749) is a simple iteration of the distribution to the point that only onedigit numbers are multiplied one by the other, and their equivalent forms are already known. For example, if we can distribute (a + b) over 37

Two Of Our Breakthroughs

(x + y) as a × x + b × x + a × y + b × y we can multiply any two-digit numbers: e.g. 37 × 43 ~ (30 + 7) × (40 + 3) ~ 30 × 40 + 30 × 3 + 7 × 40 + 7 × 3 and write all the answers that need to be added: 1200 + 90 + 280 + 21 whose equivalent is 1591. To make students good at computation we make them look at the given to find out what it tells them to do so as to remain on top of the challenge. Whenever we find that a special attack is more beneficial than the routine approach, we suggest the special one that saves time and gives students the alertness that goes with confidence in oneself. This in turn makes one able to look at tests as a welcome hurdle which is there to confirm one’s mastery on the matters in question.


Fractions, Decimals And Percentages

Fractions in mathematics are not fragments of objects as the name suggests, they are operators that transform numbers and quantities, into other numbers and quantities. We already know that doubling and halving go together, one canceling the effect of the other — so do tripling and taking thirds or quadrupling and taking quarters. Indeed, multiplying any number (or quantity) by n will be canceled if we divide the result by n. Dividing by n or taking 1/nth, are two different ways of naming the same transaction

Operators form classes of equivalence and students look at any fraction as being one among an infinity of others which can replace the first set. Thus in which either the bottom number (or denominator) is the double of the top number or this (or numerator) is one-half the bottom number. Another way of looking at the terms of the infinite class of equivalence is to say that


Two Of Our Breakthroughs

where n can be taken to be 1, then 2 then 3…etc. This is true of all fractions even if the class has been only encountered on the examples of Reading the equivalence

backwards leads to the notion of simplifying fractions, since any factor common to the numerator and the denominator can be ignored. That we can see a well-known process results from our treatments of sections of the classical curriculum in terms of awareness. These treatments make students understand how to handle the challenges in the tests which are considered to be reflecting their educational growth. They give the Starters in the Mini-Tests more value than would result from the capacity of the students to do their best on the achievements tests. From that understanding they can see that it is correct for them to say: “At this level, on these matters I can behave like a mathematician.” If this is reachable by every child, we feel we owe it to them to make them aware of it. In particular, the treatment of fractions in the Math Mini-Tests does precisely this. A fraction is conceived as a relationship between two numbers or two quantities which can be found again when those two numbers are replaced by an infinite number of special other pairs. Thus, since even numbers can be obtained by multiplying any number by two, every even number has a half. Any pair thus found is representative of the relationships “double” and “half,” That is why the pairs (1, 2), (2, 4), (3, 6) etc. can as easily be used to represent one half. Hence, when we come to adding or subtracting fractions we see at once that for finding the answer we need to select from a series of equivalent functions those pairs which let us do the adding or subtracting i.e. have the same name (or denominator). A mystery has disappeared and


Fractions, Decimals And Percentages

students will avoid making mistakes which result from misunderstanding of this search for a common denominator before proceeding with the operation. We bring clarity in this area found difficult by students and teachers alike by letting students see that the sign × between fractions can be read “of.” We make them see that multiplication (and division) of fractions must not be treated as repeated addition (or subtraction) — which does not make sense here — but as the application successively of two operators on a certain quantity (in the case of multiplication) and its inverse (in the case of division) and the finding of a single operator that does the same job. In particular, we alert students to the fact that division of fractions is much more easily understood if each problem is worked out as a “how many ___ go into _____” question. For instance: “how many s go into 7 ?” is easily answered by “14” and as readily “how many s go into ?” is answered by” ”. The classical rule can now be understood, and its mystery is gone. Another mystery vanishes as soon as we recognize that decimals are the “script” of a new language into which we translate fractions whose denominators are powers of ten — or can be given that form through classes of equivalence. By giving enough work to do on the translation and the new “script,” all that one knows about fractions can now be said in terms of decimals. If the converse is also true it becomes possible to see where the new notation makes some problems on fractions much easier and this because the new notation takes us back to integers with a “floating point” and makes available for computation with fractions all that has been learned in the field of integers. Since our aim is to make students experience mastery in mathematics and to know that an increase in awareness and facility, once achieved, helps in the conquest of new ground, we offer students applications — in the format they have been given in the tests. Hence while the education gained in this chapter of knowledge called “mathematics” is measured by the ease with which students call in the proper tools to 41

Two Of Our Breakthroughs

solve problems, the tools acquired are mental tools, now second nature, i.e. coming to one spontaneously, being adequate and routinely used. The students have know-hows rather than memorized knowledge which is in danger of being forgotten. We can claim that it is this fundamental difference between know-how and knowledge that gives our Math Mini-Tests a significance and importance in education that goes far beyond improved competence on tests — however valuable this may be in one’s school career. In fact permanent improvement of the students results so that they can take charge of how best to answer these very simple test questions which on the whole are only designed to test retention. One more language awareness will make “percentages,” another way of speaking of the numerators of fractions having 100 as their denominators. Hence, the computational requirements of problems on percentages are all studied in previous computational Starters. By translating these problems into forms which use the known languages, the answers to percentage problems become as simple as reading a transformation of the given.


Directed Numbers

Since we look at number lines and graphs as new “scripts” for that which has been mastered, the extension of arithmetic to negative numbers results from increased reading abilities. That is, we simply translate addition and subtraction into movements on the number line where right and left (or up and down) are clearly distinguished by the notation that demands that a small minus sign be attached to the numeral (–3) — and sometimes a small plus sign (+3) on it. The Starters — which are instrumental in educating awareness — will make students “know in their flesh” the place of the notation and the role of the operations and thus become clear in matters like products of like or unlike terms — for instance, that the products of like ones in even numbers are positive and those of unlike ones, negative. *** In the traditional curriculum, arithmetic comes before algebra and algebraic problems appear late in the test, when, at that stage, many of the questions are rather trivial. Our Starters and Basic Starters, without using the word algebra, make students aware of processes of operations, and of operators. This is equivalent, in pedagogical terms, to presenting algebra informally, instead of as a mathematical term. Indeed, the only ways available to make students aware that the myriads of questions in the tests are aspects of a very small number of lightings, are found in the unifying role of both algebra and language. That is why both play a prominent part in the Math Mini-Tests.


Two Of Our Breakthroughs

By illuminating the mind of test-takers through the Basic Starters and Starters, we give them a higher viewpoint from which to contemplate all those items which are felt as needed in the tests to inform the public what students have achieved in their studies. We therefore are making the tests a more correct instrument for the measure of precisely this aspect of their school education. If the Math Mini-Tests do no more than this we would not complain.


Notes On The Math Mini-Tests

Once it was decided to embark upon the production of the Math MiniTests, it was clear that Dr. Dorothea E. Hinman would have to be the overall administrator and director of the project. Her long experience with the Reading Mini-Tests gave her a background no other member of the team could claim. Ten years of working with staff, with printers and other suppliers, and with all the achievement tests as well as twenty years of work with reading supervisors, administrators and teachers, with the problems of reading, writing and spelling, and with the market nationwide and locally, gave the project a strong head start proving how lucky it was to have her at its head. To use all this energy, talent and experience for the Math-Mini Test Project more people were needed. Four can be mentioned by name because of their share in the finished product. Dr. Gattegno, as President of Educational Solutions had to provide the financial support for two years of investment before returns could begin to be realized by the final product. As the mathematician who has recast the teaching of elementary mathematics by using it on a long study of human learning, he had to recast the type of content found in the standardized achievement tests in terms that could help test-takers while they were also learning the mathematics expected at school. As a member of the team he read every question on every student sheet once his basic suggestion for the content of the various sections had been provided for by the staff. After the student sheets were completed he proposed the essential content for each page of the teachers’ notes


Two Of Our Breakthroughs

to insure a correct relationship both to the demands of the subordination of teaching to learning and of the field of mathematics — reading every page of the completed manuscript to ensure accuracy and clarity in working out these relationships. Stephen Shuller, as a mathematics teacher, as a student of statistics and testing in education, as someone who played a few roles in the production of the Reading Mini-Tests, brought his attention to detail, his facility in translating Dr. Gattegno’s suggestions into actual sheets of test-like questions, and actually produced the contents for a considerable number of Starters in the various kits up through the Intermediate level. He worked steadily and long hours and did all he could to meet deadlines. His was the responsibility to find the words and the examples which would best take care of the unification of the numerous and varied test questions in order to produce the Starters while avoiding scrupulously any use of the material of the actual tests. This tedious and long job, which required many special qualities, was done very well and on time. As a result test-takers this year can benefit from the help our Mini-Tests can give them. Behind the actual product in people’s hands there is a huge material operation which was mainly under Steve DeGiulio’s supervision. A dozen assistants worked with him making each sheet ready to go to the printers looking as attractive as possible, error-free and satisfying the many varied criteria for producing each which were pedagogical, mathematical, aesthetic and economic. If the product is as functional as we claim it is, a good portion of that merit goes to Steve who relieved Dr. Hinman and the other members of the team by being as efficient and diligent as this big job required. Marie Antoinette Auguste typed every sheet of all the kits with her usual precision, care and speed. Her contribution is visible everywhere. A remarkable feature of this project is that the delays in production have been almost insignificant. The production team and the printers have worked overtime, to make us meet our unbelievable schedule. A feat many educational projects found it impossible to achieve.


Notes On The Math Mini-Tests

Looking at the finished product we are proud of it, of its contents and what it can do for the millions of students who have to score their best in the standardized achievement tests. Now they may do it easily and joyfully. Summary of MATH MINI-TESTS available Grade Level

# of Spirit Masters





2 3 4-5 6-7 8-9

P2 P3 E I A

92 120 147 120 120

Extracted from the above, there are 3 remedial kits X, ER & IR.


News Items

On March 8th, 1982, we received the final report on the experiment of teaching English through the video series which took place a year ago on our premises. We reported about it a number of times in the last two years. This time, we can refer to the official report from New York University — its American Language Institute and the Department of Communication Arts and Sciences — to let our readers know something more about that contribution to education we completed just five years ago. Professor Miriam Eisenstein is the author of the report. Associated with the project from the start, she is thoroughly acquainted with its strengths and weaknesses, as an educational research effort. She therefore does not claim more than others would. Her caution is found everywhere. The few lines of her conclusion only say: “English Language Institutes such as the American Language Institute, whose populations are analogous to individuals from working, middle and professional classes, as represented in this study, might do well to consider utilizing the ‘English The Silent Way’ approach as a viable teaching alternative for individuals who are not able to attend traditional classes. While a sizable initial outlay of funds would be required to purchase the video tapes, this initial expenditure might be offset by enabling the Institute to retain students who might otherwise be lost to them.” Being an academic document, this report gives the data in numerical form although the population was small and not easily handled


Two Of Our Breakthroughs

statistically. The following extracts from the data may help readers make up their minds about whether we now know more about the efficacy of the approach, hailed as a breakthrough by Educational Solutions at the time of its production in 1977. Twenty students were admitted to the experimental class. At the end there were only eleven, whose pre-tests and post-tests could be used to draw some conclusion about teaching and learning with video tapes. Of these eleven, seven were males and four females, three were from the working class, five from the lower middle-class and three were professionals; four were Hispanics, three Italians, one Chinese, one Haitian, one Hungarian and one Polish. All were adults of ages between late teens and the forties. The pre-test (Ilyin Oral Interview) determined that seven were unable to answer the basic questions in it and were deemed not to have adequate English proficiency for that interview, four were able to take the test but scored low (only nine or less). Those who scored more than ten were not accepted for this class, so that all the students accepted were absolute beginners or nearly so, when the course started. As posttest both the Ilyin Oral Interview and the Comprehensive English Language Test (CELT) were administered. Students were also asked to write their own impressions of their experience in their own languages. Since the Ilyin has been used for placement at Alemany Community College in San Francisco it became possible to say where these slower students would place themselves in terms of proficiency. Each level at Alemany represents one term of study or 160 hours of instruction as against 100 for the whole of the video tape course. Since the average for these eleven students was 41 — corresponding to the “low intermediate” level at Alemany — it can be said that they made good progress using the tapes, and increased their English proficiency measurably. According to their actual scores 5 students would have been placed at the second level (advanced beginners) 4 at the 3rd (low intermediate) and 1 at the 5th (advanced). The students own reports in their own language can add another dimension. 50

News Items

“Silent Way is one kind of more or less better teaching method in that it tries to use the students’ independent concentration efforts by using students’ previous knowledge to find the actual answers. This method seems to call for the student to prepare, prior to the first class, somewhat of a heightened concentration effort because other than the shown words there are no other materials that he can make reference to. This helps the students to absorb the English language sounds at a faster rate. If you would pull the four together simultaneously: listening, speaking, reading, and writing, it would even enhance the method.” (Chinese student) “The video tape system is totally new for me and I found it very good. The excellence of the method is proven by the fact that I arrived in New York practically without any knowledge of the (English) language and within four months I am able to speak and understand it. In my opinion the speed of this course is a little slow — but others may have a different opinion. In my experience this course makes it possible to learn good pronunciation, but it could be improved as far as vocabulary is concerned. Grammatically it gives you more than any other course. According to my friends it is easier to understand me with my limited vocabulary, than others who possess a larger one. This is made possible by the well exercised formula. The excellence of this method (technique of colored writing of the words) is enhanced by the instructor’s knowledge of psychology in teaching and directing us in a firm manner.” (Hungarian student) “My opinions of the course are very positive. I find the visual method very useful if one follows the course diligently. It is a course that was 51

Two Of Our Breakthroughs

useful in helping me to begin speaking some English. I found the help of the professor to be very good. I would happily continue with the course in order to learn more.” (Italian student) “…… the three month cycle which I have just completed has permitted me to learn many things, nevertheless, there are comments to be made: 1

was not told that this was a class involving an experimental method.

2 Regarding the method employed Solutions, there is a great deal to say:



The video system is very “pedagogical,” but, as far as we were concerned, we should not have been confronted with other students because when they said something incorrect we, who did not yet know anything, had a tendency to follow them, and certain ones did not have good English pronunciation.

More exercises should have been given, be it in class or at home. Thus, we could have studied more and made use of the dictionary in employing new words or expressions.

It is necessary to make the students talk more often among themselves in class and to allow only English even at the beginning of the course.

The colors created many problems at the beginning, but everything is going well now. Yet, we spent too much time at the beginning on the colors.

3 My last remark is a suggestion: The hope that we may be able to take, to the extent that it is possible, these kinds of courses until our English is perfected….” (Haitian/French student)


News Items

“Although I am not very good in English, I now state my opinion about the following evening course. Phonetics is the method by which the professor establishes what he wants the student to learn. Competence is indispensable in the English language for those who wish to use it to attain other goals.” (Italian student) “I can truly say I learned a lot. I congratulate and admire our teacher for her patience and perseverance. The courses were marvelous and made a great impact on us all.” (Hispanic student) “My modest impressions of the recently finished English course can only be positive with regard to all points of view. It is a system based on direct contact, that is not based on previous knowledge of English but on a much more productive emersion. And so, I consider all of this a fairly easy method of learning the most elementary and basic principles of the English language. It is understood from this that even for those without a great desire to learn, learning comes easily. Though the method is more or less geared to a common level, in my judgment it brings the student through various stages of learning. The emersion, as well as the other mentioned system based on previous knowledge of English based on a phonetic deepening, brings the student to a much slower phase which encourages the learning and memorization of new vowel sounds. As to myself, this system helped me a great deal in communicating with the others.” (Italian student) “I have been taking an English course at Educational Solutions for the past three months. In my opinion, it has been a delightful experience, since I enjoyed the method and the system used in teaching. The success of a course depends on its teacher, and in this case I would say 53

Two Of Our Breakthroughs

our professor was wonderful because of her determination, her dedication and her patience in making us understand the language through any means available.” (Hispanic student)                            


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.

Two of Our Breakthroughs  
Two of Our Breakthroughs  

Two of Our Breakthroughs