The Montessori Method, 2nd Edition - Restoration # Chapter 19 - Teaching of numeration, the introduction of arithmetic Children of three years already know how to count as far as two or three when they enter our schools. They therefore ***very easily*** learn numeration, which consists ***in counting objects**.* A dozen different ways may serve toward this end, and daily life presents many opportunities; when the mother says, for instance, "Two buttons are missing from your apron," or "We need three more plates at the table." One of the first means used by me is that of counting with money. I obtain ***new*** money, and if it were possible I should have good reproductions made in cardboard. I have seen such money used in a school for deficients in London. The ***making of change*** is a form of numeration so attractive as to hold the attention of the child. I present the one, two, and four centime pieces, and the children, in this way learn to count to ***ten**.* No form of instruction is more ***practical*** than that tending to make children familiar with the coins in common use, and no exercise is more useful than that of making change. It is so closely related to daily life that it interests all children intensely. Having taught numeration in this empiric mode, I pass to more methodical exercises, having as didactic material one of the sets of blocks already used in the education of the senses; namely, the series of ten rods heretofore used for the teaching of length. The shortest of these rods corresponds to a decimeter, the longest to a meter, while the intervening rods are divided into sections a decimeter in length. The sections are painted alternately red and blue.  Some day, when a child has arranged the rods, placing them in order of length, we have him count the red and blue signs, beginning with the smallest piece; that is, one; one, two; one, two, three, etc., always going back to one in the counting of each rod, and starting from the side A. We then have him name the single rods from the shortest to the longest, according to the total number of sections which each contains, touching the rods at the side B, on which side the stair ascends. This results in the same numeration as when we counted the longest rod 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Wishing to know the number of rods, we count them from side A and the same numeration results; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. This correspondence of the three sides of the triangle causes the child to verify his knowledge and as the exercise interests him he repeats it many times. We now unite the exercises in ***numeration*** with the earlier, sensory exercises in which the child recognized the long and short rods. Having mixed the rods upon a carpet, the directress selects one, and showing it to the child, has him count the sections; for example, 5. She then asks him to give her the one next in length. He selects it ***by his eye***, and the directress has him ***verify*** his choice by ***placing the two pieces side by side and by counting their sections**.* Such exercises may be repeated in great variety and through them, the child learns to assign a ***particular name to each one of the pieces in the long stair**.* We may now call them piece number one; piece number two, etc., and finally, for brevity, may speak of them in the lessons as one, two, three, etc. ## [19.1 Numbers as represented by graphic signs](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+19+-+Teaching+of+numeration%3A+Introduction+to+arithmetic#19.1-numbers-as-represented-by-graphic-signs (Link to Montessori.Zone's Translation Base Text "The Montessori Method")) At this point, if the child already knows how to write, we may present the figures cut in sandpaper and mounted upon cards. In presenting these, the method is the same used in teaching the letters. "This is one." "This is two." "Give me one." "Give me two." "What *number* is this?" The child traces the number with his finger as he did the letters. ***Exercises with Numbers**.* Association of the graphic sign with the quantity. I have designed two trays each divided into five little compartments. At the back of each compartment may be placed a card bearing a figure. The figures in the first tray should be 0, 1, 2, 3, 4, and in the second, 5, 6, 7, 8, and 9. The exercise is obvious; it consists in placing within the compartments some objects corresponding to the figure indicated upon the card at the back of the compartment. We give the children various objects to vary the lesson, but chiefly make use of large wooden pegs so shaped that they will not roll off the desk. We place a number of these before the child whose part is to arrange them in their places, one peg corresponding to the card marked one, etc. When he has finished he takes his tray to the directress so that she may verify his work. ***The Lesson on Zero**.* We wait until the child, pointing to the compartment containing the card marked zero, asks, "And what must I put in here?" We then reply, "Nothing; zero is nothing." But often this is not enough. It is necessary to make the child ***feel*** what we mean by ***nothing**.* To this end, we make use of little games which vastly entertain the children. I stand among them, and turning to one of them who has already used this material, I say, "Come, dear, come to me ***zero*** times." The child almost always comes to me, and then runs back to his place. "But, my boy, you came ***one*** time, and I told you to come ***zero*** times." Then he begins to wonder. "But what must I do, then?" "Nothing; zero is nothing." "But how shall I do nothing?" "Don't do anything. You must sit still. You must not come at all, not any time. Zero times. No times at all." I repeat these exercises until the children understand, and they are then immensely amused at remaining quiet when I call to them to come to me zero times or to throw me zero kisses. They often cry out, "Zero is nothing! Zero is nothing!" ## [19.2 Exercises for the memory of numbers](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+19+-+Teaching+of+numeration%3A+Introduction+to+arithmetic#19.2-exercises-for-the-memory-of-numbers (Link to Montessori.Zone's Translation Base Text "The Montessori Method")) When the children recognize the written figure, and when this figure signifies to them the numerical value, I give them the following exercise: I cut the figures from old calendars and mount them upon slips of paper which are then folded and dropped into a box. The children draw out the slips and carry them still folded, to their seats, where they look at them and refold them, ***conserving the secret**.* Then, one by one, or in groups, these children (who are naturally the oldest ones in the class) go to the large table of the directress where groups of various small objects have been placed. Each one selects the ***quantity*** of objects corresponding to the number he has drawn. The number, meanwhile, has been left ***at the child's place***, a slip of paper mysteriously folded. The child, therefore, must ***remember*** his number not only during the movements which he makes in coming and going but while he collects his pieces, counting them one by one. The directress may here make interesting individual observations upon the number memory. When the child has gathered up his objects he arranges them upon his own table, in columns of two, and if the number is uneven, he places the odd piece at the bottom and between the last two objects. The arrangement of the pieces is therefore as follows: ``` o o o o o o o o o o X XX XX XX XX XX XX XX XX XX X XX XX XX XX XX XX XX X XX XX XX XX XX X XX XX XX X XX ``` The crosses represent the objects, while the circle stands for the folded slip containing the figure. Having arranged his objects, the child awaits the verification. The directress comes, opens the slip, reads the number, and counts the pieces. When we first played this game it often happened that the children took ***more objects*** than were called for upon the card, and this was not always because they did not remember the number, but arose from a mania for having the greatest number of objects. A little of that instinctive greediness, which is common to primitive and uncultured man. The directress seeks to explain to the children that it is useless to have all those things upon the desk and that the point of the game lies in taking the exact number of objects called for. Little by little, they enter into this idea, but not so easy as one might suppose. It is a real effort of self-denial that holds the child within the set limit and makes him take, for example, only two of the objects placed at his disposal, while he sees others taking more. I, therefore, consider this game more an exercise of willpower than of numeration. The child who has the ***zero***, should not move from his place when he sees all his companions rising and taking freely of the objects which are inaccessible to him. Many times zero falls on the lot of a child who knows how to count perfectly, and who would experience great pleasure in accumulating and arranging a fine group of objects in the proper order upon his table, and in awaiting with security the teacher's verification. It is most interesting to study the expressions on the faces of those who possess zero. The individual differences which result are almost a revelation of the "character" of each one. Some remain impassive, assuming a bold front to hide the pain of the disappointment; others show this disappointment by involuntary gestures. Still, others cannot hide the smile which is called forth by the singular situation in which they find themselves, and which will make their friends curious. There are little ones who follow every movement of their companions with a look of desire, almost of envy, while others show instant acceptance of the situation. No less interesting are the expressions with which they confess to the holding of the zero when asked during the verification, "and you, you haven't taken anything?" "I have zero." "It is zero." These are the usual words, but the expressive face, and the tone of the voice, show widely varying sentiments. Rare, indeed, are those who seem to give with pleasure the explanation of an extraordinary fact. The greater number either look unhappy or merely resigned. We, therefore, give lessons on the meaning of the game, saying, "It is hard to keep zero secrets. Fold the paper tightly and don't let it slip away. It is the most difficult of all." Indeed, after a while, the very difficulty of remaining quiet appeals to the children, and when they open the slip marked zero it can be seen that they are content to keep the secret. ## [19.3 Addition and subtraction from one to twenty: multiplication and division](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+19+-+Teaching+of+numeration%3A+Introduction+to+arithmetic#19.3-addition-and-subtraction-from-one-to-twenty%3A-multiplication-and-division (Link to Montessori.Zone's Translation Base Text "The Montessori Method")) The didactic material which we use for the teaching of the first arithmetical operations is the same already used for numeration; that is, the rods graduated as to length which, arranged on the scale of the meter, contain the first idea of the decimal system. The rods, as I have said, have come to be called by the numbers which they represent; one, two, three, etc. They are arranged in order of length, which is also in order of numeration. The first exercise consists in trying to put the shorter pieces together in such a way as to form tens. The most simple way of doing this is to take successively the shortest rods, from one up, and place them at the end of the corresponding long rods from nine down. This may be accompanied by the commands, "Take one and add it to nine; take two and add it to eight; take three and add it to seven; take four and add it to six." In this way, we make four rods equal to ten. There remains the five, but, turning this upon its head (in the long sense), it passes from one end of the ten to the other, and thus makes clear the fact that two times five makes ten. These exercises are repeated and little by little the child is taught the more technical language; nine plus one equals ten, eight plus two equals ten, seven plus three equals ten, six plus four equals ten, and for the five, which remains, two times five equals ten. At last, if he can write, we teach the signs *plus* and *equals* and *times.* Then this is what we see in the neat note-books of our little ones: ``` 9+1=10 8+2=10 5x2=10 7+3=10 6+4=10 ``` When all this is well learned and has been put upon the paper with great pleasure by the children, we call their attention to the work which is done when the pieces grouped together to form tens are taken apart and put back in their original positions. From the ten last formed we take away four and six remains; from the next, we take away three and seven remains; from the next, two and eight remains; from the last, we take away one and nine remains. Speaking of this properly we say, ten less four equals six; ten less three equals seven; ten less two equals eight; ten less one equals nine. Regarding the remaining five, it is the half of ten, and by cutting the long rod in two, that is dividing ten by two, we would have five; ten divided by two equals five. The written record of all this reads: ``` 10-4=6 10-3=7 10 / 2=5 10-2=8 10-1=9 ``` Once the children have mastered this exercise they multiply it spontaneously. Can we make three in two ways? We place the one after the two and then write, so that we may remember what we have done, 2+1=3. Can we make two rods equal to the number four? 3+1=4, and 4-3=1; 4-1=3. Rod number two in its relation to rod number four is treated as was five in relation to ten; that is, we turn it over and show that it is contained in four exactly two times: 4/2=2; 2x2=4. Another problem: let us see with how many rods we can play this same game. We can do it with three and six, and with four and eight; that is, ``` 2x2=4 3x2=6 4x2=8 5x2=10 10/2=5 8/2=4 6/2=3 4/2=2 ``` At this point we find that the cubes with which we played the number memory games are of the help:  From this arrangement, one sees at once which are the numbers that can be divided by two all those which have not an odd cube at the bottom. These are the ***even*** numbers because they can be arranged in pairs, two by two; and the division by two is easy, all that is necessary is to separate the two lines of twos that stand one under the other. Counting the cubes of each file we have the quotient. To recompose the primitive number we need only reassemble the two files thus 2x3=6. All this is not difficult for children of five years. The repetition soon becomes monotonous, but the exercises may be most easily changed, taking again the set of long rods, and instead of placing rod number one after nine, place it after ten. In the same way, place two after nine, and three after eight. In this way we make rods of a greater length than ten; lengths which we must learn to name eleven, twelve, thirteen, etc., as far as twenty. The little cubes, too, may be used to fix these higher numbers Having learned the operations through ten, we proceed with no difficulty to twenty. The one difficulty lies in the ***decimal numbers*** which require certain lessons. ## [19.4 Lessons on decimals: arithmetical calculations beyond ten](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+19+-+Teaching+of+numeration%3A+Introduction+to+arithmetic#19.4-lessons-on-decimals%3A-arithmetical-calculations-beyond-ten (Link to Montessori.Zone's Translation Base Text "The Montessori Method")) The necessary didactic material consists of a number of square cards upon which figure ten is printed in large type, and of other rectangular cards, half the size of the square, and containing the single numbers from one to nine. We place the numbers in a line; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then, having no more numbers, we must begin over again and take the 1 again. This 1 is like that section in the set of rods which, in rod number 10, extends beyond nine. Counting along ***the stair*** as far as nine, there remains this one section which, as there are no more numbers, we again designate as 1; but this is a higher 1 than the first, and to distinguish it from the first we put near it a zero, a sign which means nothing. Here then is 10. Covering the zero with the separate rectangular number cards in the order of their succession we see formed: 11, 12, 13, 14, 15, 16, 17, 18, 19. These numbers are composed by adding to rod number 10, first-rod number 1, then 2, then 3, etc., until we finally add rod number 9 to rod number 10, thus obtaining a very long rod, which, when it alternates red and blue sections are counted, gives us nineteen.  The directress may then show to the child the cards, giving the number 16, and he may place rod 6 after rod 10. She then takes away the card bearing 6, and places over the zero the card bearing the figure 8, whereupon the child takes away rod 6 and replaces it with rod 8, thus making 18. Each of these acts may be recorded thus: 10+6=16; 10+8=18, etc. We proceed in the same way to subtraction. When the number itself begins to have a clear meaning to the child, the combinations are made upon one long card, arranging the rectangular cards bearing the nine figures upon the two columns of numbers shown in figures A and B. Upon card A we superimpose upon the zero of the second 10, the rectangular card bearing the 1: and under this the one bearing two, etc. Thus while one of the ten remains the same the numbers to the right proceed from zero to nine, thus:  In card B the applications are more complex. The cards are superimposed in numerical progression by tens. Almost all our children count to 100, a number which was given to them in response to the curiosity they showed regarding learning it. I do not believe that this phase of the teaching needs further illustrations. Each teacher may multiply the practical exercises in the arithmetical operations, using simple objects that the children can readily handle and divide. > ##### **The License of this page:** > > This page is part of the “**Montessori Restoration and Translation Project**”.\ > Please [support](https://ko-fi.com/montessori) our “**All-Inclusive Montessori Education for All 0-100+ Worldwide**” initiative. We create open, free, and affordable resources available for everybody interested in Montessori Education. We transform people and environments to be authentic Montessori worldwide. Thank You! > > [](http://creativecommons.org/licenses/by-nc-sa/4.0/) > > **License:** This work with all its restoration edits and translations is licensed under a [Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-nc-sa/4.0/). > > Check out the **Page History** of each wiki page in the right column to learn more about all contributors and edits, restorations, and translations done on this page. > > [Contributions](https://ko-fi.com/montessori) and [Sponsors](https://ko-fi.com/montessori) are welcome and very appreciated! * [The Montessori Method, 2nd Edition](https://montessori-international.com/s/the-montessori-method/wiki/+Chapter+Index+-+The+Montessori+Method%2C+2nd+Edition+-+Restoration+-+Open+Library#the-montessori-method%2C-2nd-edition---restoration---open-library "The Montessori Method on Montessori Zone - English Language") - English Restoration - [Archive.Org](https://archive.org/details/montessorimethod00montuoft/ "The Montessori Method on Aechive.Org") - [Open Library](https://openlibrary.org/books/OL7089223M/The_Montessori_method "The Montessori Method on Open Library") * [Chapter Index](https://montessori-international.com/s/the-montessori-method/wiki/+Chapter+Index+-+The+Montessori+Method%2C+2nd+Edition+-+Restoration+-+Open+Library) * [Chapter 00 - Dedication, Acknowledgements, Preface to the American Edition, Introduction](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+00+-+Dedication%2C+Acknowledgements%2C+Preface+to+the+American+Edition%2C+Introduction) * [Chapter 01 - A critical consideration of the new pedagogy in its relation to modern science](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+01+-+A+critical+consideration+of+the+new+pedagogy+in+its+relation+to+modern+science) * [Chapter 02 - History of Methods](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+02+-+History+of+Methods) * [Chapter 03 - Inaugural address delivered on the occasion of the opening of one of the “Children’s Houses”](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+03+-+Inaugural+address+delivered+on+the+occasion+of+the+opening+of+one+of+the+%E2%80%9CChildren%E2%80%99s+Houses%E2%80%9D) * [Chapter 04 - Pedagogical Methods used in the “Children’s Houses”](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+04+-+Pedagogical+Methods+used+in+the+%E2%80%9CChildren%E2%80%99s+Houses%E2%80%9D) * [Chapter 05 - Discipline](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+05+-+Discipline) * [Chapter 06 - How the lesson should be given](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+06+-+How+the+lesson+should+be+given) * [Chapter 07 - Exercises for Practical Life](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+07+-+Exercises+for+Practical+Life) * [Chapter 08 - Reflection the Child’s diet](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+08+-+Reflection+the+Child%E2%80%99s+diet) * [Chapter 09 - Muscular education gymnastics](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+09+-+Muscular+education+gymnastics) * [Chapter 10 - Nature in education agricultural labor: Culture of plants and animals](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+10+-+Nature+in+education+agricultural+labor%3A+Culture+of+plants+and+animals) * [Chapter 11 - Manual labor the potter’s art, and building](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+11+-+Manual+labor+the+potter%E2%80%99s+art%2C+and+building) * [Chapter 12 - Education of the senses](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+12+-+Education+of+the+senses) * [Chapter 13 - Education of the senses and illustrations of the didactic material: General sensibility: The tactile, thermic, basic, and stereo gnostic senses](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+13+-+Education+of+the+senses+and+illustrations+of+the+didactic+material%3A+General+sensibility%3A+The+tactile%2C+thermic%2C+basic%2C+and+stereo+gnostic+senses) * [Chapter 14 - General notes on the education of the senses](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+14+-+General+notes+on+the+education+of+the+senses) * [Chapter 15 - Intellectual education](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+15+-+Intellectual+education) * [Chapter 16 - Method for the teaching of reading and writing](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+16+-+Method+for+the+teaching+of+reading+and+writing) * [Chapter 17 - Description of the method and didactic material used](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+17+-+Description+of+the+method+and+didactic+material+used) * [Chapter 18 - Language in childhood](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+18+-+Language+in+childhood) * [Chapter 19 - Teaching of numeration: Introduction to arithmetic](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+19+-+Teaching+of+numeration%3A+Introduction+to+arithmetic) * [Chapter 20 - Sequence of exercise](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+20+-+Sequence+of+exercise) * [Chapter 21 - General review of discipline](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+21+-+General+review+of+discipline) * [Chapter 22 - Conclusions and impressions](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+22+-+Conclusions+and+impressions) * [Chapter 23 - Illustrations](https://montessori-international.com/s/the-montessori-method/wiki/Chapter+23+-+Illustrations)