Caleb Gattegno - η-Τάξη ΕΚΠΑ
Caleb Gattegno
The Common Senseof TeachingMathematicsCaleb Gattegno Educational Solutions W orldwide Inc.
First published in the United States of America in 1974. Reprinted in 2010Copyright 1974-2010 Educational Solutions Worldwide Inc.Author: Caleb GattegnoAll rights reservedISBN 978-0-87825-220-6Educational Solutions Worldwide Inc.2nd Floor 99 University Place, New York, N.Y. 10003-4555www.EducationalS olutions com
AcknowledgmentsI wish to express my deep gratitude to my colleagues forimproving the language which in the original text was oftenobscure. First I must thank David Wheeler whose acquaintancewith my work goes back over twenty years and who has beenresponsible for editing the whole book; second, Ms. CarolineChinlund for working on the original of some of the chapters,and Ms. Zulie Catir for re-reading the typed texts as often as wasneeded. My secretary Ms. Yolanda Maranga typed the text anumber of times over the last four years. She deserves mythanks for her skill and patience.
Table of ContentsF o re w o rd . 1P a rt I: A ctions W hich G en erate N u m e ra ls . 51 The Set of One’s Fingers. 72 Reading and Writing Numerals in any Base. 253 Exploiting Complementan· Numerals. 394 Introducing One Aspect of Division.53P a rt II: A ctions W hich G en erate A lg e b ra .595 A Model for the Algebra of Arithmetic.61P a rt III: N u m b ers as N u m erals P rovided w ith anA lg e b ra .776 Integers and Fractions. 79The Generation of Integers and Fractions.797 The Mathematics of Numbers. 911 Prime Factors. 942 Common Factors and Common Multiples. 963 Towers and Sets. 1004 Powers and Roots. 1011 Polynomials. 1032 Exponents. I l lFractional Exponents. 113
Negative Exponents.1143 Logarithms.115P a rt IV: T eaching M a th e m a tic s.1218 Teaching Mathematics to Teachers. 1239 Teaching Mathematics to Children.14110 Generating a Mathematics Curriculum. 155The Characteristics ofGattegno Mathematics, Books 1—7. 156Fractions. 158Fractions as Operators.158Ordered Pairs. 160Classes of Equivalence and Operations. 161Addition. 162Multiplication and Equivalence-M. 165Decimals. 169Percentages. 172Summing Up. 172P o s ts c rip t. 175A ppendix A: A P relu d e to The Science o f E d u c a tio n .177A ppendix B: A M ap o f E lem en tary M athem aticsD erived F rom T ables o f P a r titio n s . 191Books an d M aterials199
Forew ordFor some time I have had it in mind to write a book which wouldsubstantiate my claim that it is now possible to make the studyof education properly scientific. It seems to me at this momentthat my contribution to the study can best be put in the form ofseveral short books, each devoted to a particular part of theeducational field, rather than, as I originally thought, into asingle comprehensive volume. I offer this book as the first insuch a series; in it I restrict myself to elementan’ mathematics,and mainly to the algebra and theory of numbers.Perhaps this book will seem to others to be mainly another bookabout mathematics teaching continuing a line of development Ihave already exemplified in other writings; for me, it representsa radical departure. Now I know that only awareness iseducable, I have found it possible to come up with originalanswers appropriate to many of the areas of educationalendeavor — answers of a universality which ensure that they are,indeed, solutions to problems and not merely bright ideas. Whatcharacterizes this book is that it is concerned with people1
The Common Sense o f Teaching Mathematicsbecoming aware of how to use their own manifestations, asperceivers, actors, verbalizers and thinkers, in order to gathermathematics on the way. It must therefore sen e any learner ofmathematics, whoever he is.The study of the education of awareness has yielded tools whichcan be used to grasp unequivocally the whole universe ofeducation: the method of investigation coincides with the fieldof application, and knowing replaces knowledge as the cardinalnotion. Since knowing produces knowledge, but not the otherway round, this book shows how everyone can be a producerrather than a consumer of mathematical knowledge.Mathematics can be owned as a means of mathematizing theuniverse, just as the power of verbalizing molds itself to all themanifold demands of experience.In this book I show mathematization in action, giving only justenough detail to display what it is, and leaving the elaboration ofthese sketches to the reader. In this way he will find how muchhe has learned through the extent to which he can add to itscontent himself.It is obvious that learning anything always exacts a price — thelearner must always give some of his attention, his effort and histime in exchange for learning. It is also obvious that wherespontaneous self-generated learning is concerned the learnerwillingly pays what is required, whereas in school he often doesnot. My contention is that he cheerfully pays up when thelearning he wants to acquire dictates the price, but that he willrefuse if the price is higher than it need be, or if he is offered the2
Forewordwrong goods in exchange. One criterion, I suggest, for thevalidity of the existence of a science of education is in its abilityto make accurate estimates of the cost of learning.Making the cost fit the learning requires that we know, in detail,what elements must be offered to the learner of a topic becausehe does not already have them and cannot invent them, andmust therefore pay for. In an article, “A Prelude to the Science ofEducation," reprinted as an Appendix to this volume, I discussthis matter fully and show how ‘‘units of learning" can beprecisely calculated. A consequence of the analysis is that wealso know exactly what does not have to be offered by anyonebecause the learner is able to invent it for himself.A second aspect of the cost of learning is what is required of anindividual learner for him to become the master of what helearns. Here it is not possible to predict exactly what eachlearner will need to pay, since much will depend on him, butwhere the matter involved is a skill, as in most of themathematics discussed in this book, we need to know thatmasteiy demands practice, and that awareness must precedepractice. Teachers can minimize this feature of the cost oflearning by finding those situations which carry the correctawarenesses and by suggesting exercises which provide thefacility that shows that the awarenesses are functional. The bookgives many examples to show how this can be done.In restricting the scope of this first volume to elementarymathematics it is my hope that many teachers of elementaryschool can be helped to abandon their belief that mathematics is3
The Common Sense o f Teaching Mathematicsnot for them, and achieve a new confidence through thediscovery that they can indeed function as mathematicians,whatever the lessons of their previous experience. If thishappens they may want to test their growth by transferring theirinsights to mathematical areas that may require additionalfimctionings. Two of these will be the subject of further books inthis series: on geometry and on analysis.Caleb GattegnoNew York CityAugust 19734
Part IActions Which GenerateNumerals
1 The Set o f One’s FingersThe content of this chapter is discussed from the point of view ofthe activity of a teacher and some children, but it would be aswell for the reader to take some time looking at his own handsheld in front of him, palms towards himself, following some ofthe activities described.Since for most of us our fingers will obey instructions from ourwill, a preliminary game goes as follows.T eacher:*Hold your hands in front o f you withall the fingers sketched out. Now folddown your right index finger (or yourleft thumb, or both thumbs, etc.)*We will subsequently use the convenient verb forms 'fold' and unfold."7
P a r tiActions Which Generate NumeralsEach time the teacher gives an instruction he folds the samefingers himself.T eacher: Do with y o w fingers exactly what Ido with mine.He watches the students fold fingers corresponding to his. Itdoes not matter whether they choose to make their left and righthands correspond to his, or whether they reverse them, providedthe exact correspondence of fingers can be seen. Thispreliminary game is quickly mastered as the will can usually actimmediately on the muscle tone of the fingers.*The teacher then folds one or two fingers on one or both handsand asks the students to do what he has done. If they do thissuccessfully, he folds an additional finger or fingers and asksthem to do the same.It may be interesting to readers at this stage to note that thereare ten choices of showing only one finger, forty-five of showingonly two fingers, one hundred and twenty of showing three, twohundred and ten of four, and two hundred and fifty-two of five.The order of these numbers is reversed for the number of waysNevertheless the game can lead to som e som atic aw arenesses. Not all our fingers respondequally readily to our will; som e people find some configurations extremely difficult to produce.Besides the possibility th a t this could lead to a useful physio-psychological test, die discovery ofhow little one owns one's som a m ay have consequences for one s self-education. Practicing thesegames may m ake some people less rigid, m ore supple in th eir som a and in th eir m ind. W hereconfigurations need a considerable effort for th eir production, the teacher may drop them for thepurpose of moving ahead with the m athem atical investigation. Not all configurations arerequired for this end; only the knowing th a t is produced by the activity.8
1 The Set o f One s Fingersof showing six, seven, eight or nine fingers. There is one way ofshowing no fingers and one way of showing all ten fingers. Thefollowing table demonstrates, at this stage of the game, theenormous variety of showings that are available.number of fingers012number of choices110453120421056789252 210 120 4510101The table with its obvious richness may suggest to teachers howlittle they have exploited the set of fingers that eveiyone carriesaround with him every day. In particular it may suggest howthey can show the classes of equivalent sets of fingers whichexemplify what we call cardinal numbers. As well as thisimportant awareness there is another still more important forour purpose. It concerns complementary subsets. Since eachfinger can be characterized as folded or unfolded, and the twostates are mutually exclusive, any subset of unfolded fingers canbe matched with another subset made by changing the state ofeach finger on both hands. The union of the two subsets isalways the whole set of fingers.To bring this out we may change the game.T eacher: Look at my hands. You can see someo f my fingers but not all o f them. (Heturns his hands momentarily to showthe folded fingers.) Will you unfoldyour fingers corresponding to myfolded ones, and fold thosecorr esponding to my unfolded ones?9
P a r tiActions Which Generate NumeralsIf the class clearly understands the instruction they can producea very large number of configurations.Three pairs of complementary subsets.The game with the hands is fun as long as its demands on thestudents do not become too great. As we have noticed, someconfigurations are difficult to make, and the teacher can create asense of relief at a certain moment by proposing a shift to averbal system which is more restricted and easier to deal with.He can ask the students to call out the numeral describing thecount of their unfolded fingers after he has told them his.10
1 The Set o f One s FingersClearly the 252 configurations of five unfolded fingers will nowall be associated with the same sound. This is the restriction ofthe verbal system referred to above. The students experience thefact that a large variety of subsets will only trigger one sound,and if the teacher is sensitive to the importance of theassociation he will not hurry to stop his class from becomingdeeply aware — by acting in the situation, not by being toldabout it — that the sounds for 1, 2, 3, 4, 5, 6, 7, 8, 9 describemany configurations. Each therefore applies to several differentsets of fingers, although zero and ten in the present context eachapply to only one set.But the ease of uttering as against the difficulty of showing alsopermits awareness to be focused on the constancy of the pairingof the sounds — tiuo with eight and eight with tivo, for example.A new opportunity has arisen for a new awareness.Now the exercise becomes the recognition that it is possible toshift from one pair of sounds to another. Bv reversal, forexample, (2,8) becomes (8,2). By folding or unfolding one ormore fingers a given pair of sounds can be changed into another;the passage from (2,8) to (1,9) or (3,7) for example, comes fromallowing one finger to pass from the subset of folded fingers intothe subset of unfolded fingers, or vice-versa.Students soon realize that the teacher is not necessary to thegame and by looking at their own hands they are ablesimultaneously to see a pair of complementan' subsets and toact upon it to produce either the same pair of numerals or arelated pair.11
Part IActions Which Generate NumeralsThough there are so many configurations there are only a smallnumber of pairs of sounds needed to describe them: (zero, ten),(one, nine), (tiuo, eight), (three, seven), (four, six), (five, five).Exactly six if we are not interested in the order of the sounds;eleven if we are.In the students' minds is now a wealth of experience of subsetsof the set of fingers together with a move towards organizing it.*In a similar way it is easy to find the subsets of the set of fingerswhen one of them, a thumb say, is taken out of circulation byfolding it down and regarding it as immovable during theexercise. But it may not be necessary to do this. We shall see thatan alternative route to the same results exists when we come tointroduce the conventional numerals and replace a pair ofsounds with a pair of signs.If the students do not as yet own either the sounds or the signsof these numerals, a preliminary exercise will yield them. Forthis it is useful to obtain from the students a number of timesthe soimd of each numeral simultaneously with showing anappropriate subset of fingers.jt#For instance it can now be brought out th a t since the folding of some fingers produces a t thesam e time a configuration of unfolded ones, there m ust be the same num ber of subsets for twocom plem entary cardinals. Indeed, to show two fin gel's requires th a t eight have been folded, soto each choice of two particular fingers corresponds a particular choice of eight, and vice-veisa. Ift10we use the notationto indicate the n um ber of different choices of two fingers from the wholeset of ten fingers, then we can express this awareness by w riting the following equivalences:f1 Vl J(101 ,10 i 10U )[ioUJ10,8 ;10 I1010 10,3 J l 7 ) U J U )12
1 The Set o f One s FingersAlternative ways of showing one and four.Not too far in the future this will become an instance of animportant theorem in combinatorics, thatinifnVP; v n - pwhose foundation can now be laid at the beginning of amathematical education.After the appropriate sound (one, two, etc.) has been utteredcorrectly several times for various subsets, the sign ( 1 ,2, etc.) is13
Part IActions Which Generate Numeralswritten lip. As the ten numerals are produced in any order tomatch the correct showing of subsets of fingers, the followingtable of signs is built up:1 2 3 4 5 6 7 8 9(10)The sequence can be read from 1 to 9, or from 9 to 1 , or in anyorder.It must be insisted upon that the whole purpose of the exerciseis to give students the experience that some of what they can dowith their fingers can also be done with the set of sounds andthe set of signs — that is, they can present any of the possiblesubsets on their fingers, and therefore can also utter or show anyone of the numerals which correspond to the subsets.To return to the main thread of the game, we may now write(7,) on the chalkboard and ask the students if anyone knowswhat must be written in the blank space for the whole set offingers to be accounted for. Since this is an exercise they havealready practiced, there is little doubt that one or more studentswill answer three and that one or more will be able to enter 3 inthe blank space. From this pair, (7,3) all the other pairs can bededuced and afterwards written up and uttered. With their eyesshut the students can now be asked to give the appropriatemember of a pair when the teacher gives the other.The crucial awareness here is that the folding of any one fingertakes it out of the set represented by the numeral on one side ofthe comma and puts it into the set represented by the numeralon the other side. This transformation is so simple for most14
1 The Set o f One s Fingerspeople to master and to become aware of that we can expect thestudents to learn that any pair can generate all others so thatonly one pair needs to be remembered bv name. This is ameasure of the cost to the memory of this mastery. One pairmust be memorized but all pairs will be retained since none ismore striking than another and, in fact, each one is capable ofgenerating all of the others.Once this material has been explored it is another very easygame to link the complementan- pairs of numerals in nine to thecomplementary pairs in ten. The link can be made in two ways,first by finding the two sets of reversed pairs corresponding toany given pair of complements in ten. For instance, given thepair (4,6) one finger can be removed from either subset to give(3,6) or (4,5) and these pairs reversed to give (6,3) and (5,4).This can be done for each pair of complements in ten. Secondly,taking any pair of complements in nine, say (3,6) it can be usedto deduce all the other pairs by folding or unfolding fingers inorder to shift them from one subset to the other.Of the eleven possible ways of writing complements in ten:(0,10), (1,9), (2,8), (3,7), (4,6),(5,5), (6,4), (7,3), (8,2), (9,1), (10,0),we know that our indifference to the order of the two numeralsreduces the list to only six:(0,10), (1,9), (2,8), (3,7), (4
The Common Sense of Teaching Mathematics becoming aware of how to use their own manifestations, as perceivers, actors, verbalizers and thinkers, in order to gather mathematics on the way. It must therefore sen e any learner of mathematics, whoever he is. The study of the education of awareness has yielded tools which
3 Caleb: Well, do you have any better explanations? Melissa: Caleb, you probably just lost your socks. That's all there is to it. Narrator 2:But Caleb had enough of everyone making fun of him. Caleb: I'm going to prove it to you! Then we'll see who has the last laugh. Melissa: Yeah, we'll see! Just be careful that the
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The Project Gutenberg EBook of Caleb Williams, by William Godwin . This eBook is for the use of anyone anywhere at no cost and with . almost no restrictions whatsoever. You may copy it, give it away or . re-use it under the terms of the Project Gutenberg License included . with this eBook or online at www.gutenberg.net . Title: Caleb Williams
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