Reprintedfrom " Mathematics Teaching," the Bulletin of the Association of Teachers of Mathematics No. 50,

Spring, 1970.

Papy's Minicomputer

ttFREDERIQUE PAPY

Centre for the Pedagogy of MathematicsErusse/s

The nrethod we have used to introduce childrento mechanical and mental arithmetic employs thedistinct advantages of the binary system overother positional systems, while at the same timcta'<ir-rq account of the decimal environrnent inwhich we are immersed. It is thanks to the Mini-computer Paplt that we have been able to achievethis.

Inspired by some work o[ Lemaitre, this varietyof two-dimensional abacus uses the binary systemon boards which are arranged according to thedecimal system.

2nd October, 1967Frédérique hangs one ofthe display boards onthe blackboard

Fig.2

- Oh, they're our colours! said Jean-.Jacques,showing a white, red, pink and brown rod.

- Yes, indeed!Ilte will pla-y a game together; yu will use your rodsand I uil! uçe the board and these tounters-

Frédérique puts two black counters on the whitesquare.

- They stay up by then?selves ! one child marvelled.- Couple two white coaches together.

- It's the same as the red rod!- A white-white train is the same length as a red

coach.Frédérique takes two counters from the whitesquare and puts one on the red square.

l;lil [-,;l;;l I;T;ffiEtrtri

Fig. I

The colours recall the red familyCuisenaire rods and help to makeof the maehine easily accessible,

ffiffiffi:6e796

of the set ofthe four rules

H:HFig.3

- Two counters on the white square are equal to one on

th.e red.

The children are quick to adopt the shorthand 'ared' for 'a counter on the red square' and statethe rule 'two whites are equal to a red'.

- Let us go on with the garne !Frédérique puts two counters on the red squareand the children put two red rods together.

- It is the sarne as the pink rod !

Frédérique takes the two counters from the redsquare and puts one on the pink square.

Fig.4

- Two reds are equal to a pink.

Jean-Jacques is on fire with desire to move thecounters and begs:

- Madarne! Can I play?

Frédérique aÉ{rees and.fean-Jacques lifts the blackcounter from the pink square and replaces it withtwo green counters.

- It is beautiful !

- Couple two pink coaches together.

- It is the same as a brown rod.

Frédérique removes tl.re green counters and putsone on the brown square.

Fig.5

Two pinks are equal to a browrr,

5th October, 1967

- A new game !Frédérique places a counter on the white square

-lShe places a second counter on the white square_.)- I can play another way ! says Carine. She lifts

off the 2 counters from the white square andputs one on the red.

Fig.6

Add I to the number 2, Fft.dérique suggests,putting a new counter on the white square.It is 3.Like with rods, red-w'lriteor light green.Add I to the number 3, Frédérique continues,putting a new counter on the white square.It is 4.

- I can play another way, says Sylvie, who takesthe trvo counters frorn the white square and putsone on the red.

- And anothef way, states Jean-Jacques, replacingthe two counters on red by one on the pink.

- Let us go on adding | . . .

The recital continues and the numbers 5, 6, 7. 8, 9appear on the N{inicomputer.

Fig. 7

- Add I to the number 9.A child puts a counter on the white square.

- It is 10.

- I play in another way, says Anita, replacing thetwo counters on white by one counter on thcred.

- It's the red-brown train.- The orange rod.- It's 10.

Frédérique hangs a second Minicomputer boardon the blackboard, to the left ofthe first. She takesthe two counters from the red and brown squaresand puts a counter on the white square of the newboard, saying simply- And this is still lO.

ffiffi:Hm*Fig.8

- Jump ! On to the second board, remarks Jean-Jacques, not in the least surprised.

Thus armed, our pupils are able to representon the Minicomputer the number of elements inany set ofcounters put on square l. The applica-tion (in reverse order) ofthe four fundamental ruleson a small scale lets us keep a concrete link between anumber in its written form and as represented on themachine.

l3th October, 1967Two small Nfinicomputer boards and a box of

counters are on the deslis in front of each child.- How many counters doyou hauc inlouT 6t*7- 29...32... r8... 26... 3sThe children's boxes are not all filled with thesame number !

- Put lhe box of counters on the uhite squarc of the frstboard. Play . . . and then write down the result.T'he first extellcled piecc of inclividu:rl rvork on

the Minicomputer: concentration, precise andquick movements: some children arrive at thercsult rvithout a mistake.

Clunrsy techniques, colrnters spilled, false moves;the others rnust be helped.

The seconcl part of the lesson is played on thewall Nlinicomputer.

We start with the number 25 on the machine.We work all the counters back to the whitesquare on the fir'st board and count them:confirmation !

After a fortnight the demands of the classforced Frédérique to introduce a third board andto accept numbers over 100.

- If I put a counter on each square of the machine,will I make the number 100? asks Didier, whostill has a machine with only two boards.

Frédérique doesproblem to ariseprefers to let theçoursç,

Fig.9

not answer. She expects thelater in a different form andchild's thinking follow its own

24th October, 1967

Didier returns to the attack.- I want to make 100 on the machine.- All right . . . let us haue 2our method !- 100 is twice 50, Didier goes on, showing:

ffiffiFig. l0

- I can play !

FIe replaces the two connters on the white squareby one counter on the red square and the twocounters on the pink square by one on the brown.

ffiHFig. ll

I want to play again! he says, taking thecounters from the red and brorvn squares andputting a counter on the left of the secondboard.

ffiffi? .mffiFig. t2

And Didier demands his rights:- I must have another board, Madame!F'rédérique gives him one. Didier triumphantlyforms the number.

ffiffiffi:10.0

Fig.13

Overexcited, he shouts out the numbers as heshows them on the machine.

ffiffiffi:'*

- In the car park I counted 75 Volkswagens and 49IVIercedts. How manl cars are there altogether?

- Show 75 in red!

0

ffiffiffiFig.15

- And 49 in green!

ffiffiffi:110 ffiffiffiffiffiffi:120

Fig. 16

- Can I play, Madame?

120

ffiffiffi:12s75+4e:ffiffiffi'*=,r* -mffiffi:mffiffi:mffiffi:124

124

50

01',

Impressed, the class has shared in this discovely.

The addition of whole numbers is carried outautomatically by 'playing' the machine. The sameis true lor doubling which is a primitive experiencefor children and fundamental to the Mini-computer.

The part played by arbitrary memorisations, so

often a distasteful part of learning to compute, isreduced to a minimum. The addition of smallnumbers is carried out by means of intelligiblerules: a purely binary system when the sum isless than 9 and a mixed decimal-binary system inother cases. From the beginning, therefore, thechildrcn are initiated into a positional system ofnumefation, Fig. 17

The children make a note of the temperature indegrees centigrade (Celsius) each morning. Atthe start of the school year, in Brussels, these arealways natural numbers, but the winter monthsforce the use of negative numbers. From the sirtfrrnonth, negative integers are part of the commonknowledge of the children, having authenticstatus as numbers 'measuring' a 'size' which hasbeen experienced.

The results of a sequence of two-person gamesare written with red and blue numbers which aremutually destructive, since each point capturedby one of the players 'kills' a point captured by theother. The addition of red and blue numbersfollows. From the given notation the childrenarrive at the additive group ofintegers.

7+4=3

Fig. 18

Eventually, the writing is simplified and all thenumbers are written in black, those originallyblue having a bar across the top. In effeàt thistime we have the gtoup Z, f , except for a veryminor differerr..,5 being used instead of -3. Wehave known the advantages of the T notation forbeginners for a long time since we have used it tosimplily logarithmic calculations.

At the beginning we show the result of eachgame by putting red and blue counters on a flatsurface. A battle to the death gives the final score.The children spontaneously transfer the procedureto the Minicomputer.

- Let us calculate this to fnd how big it is:

100 + 23

Great excitement at all the desks !

- Do it on the machine!

Fig. t9

fl The black dots in Figs. lB, 19 and 20 should be inter-preted as being blue.

ffiffiffiNothirry either way!

HffiffiAttack the blues !:mffiffiAttack the blues !:ffiffiffiAttack the blues !:mffiffiAttack the blues !:mffiffi

Nothing e'ither wa1! Two dead!-ffiffiffia.a o

ffiffiffi:ffiffiffi-Hffiffi

Fig.20 (opposite) loo+23= 77

In our teaching, the function 'a half of ' appearsas the reciprocal of the function 'twice' which is atransformation of Z; that is, a function mappingZ into Z. Brft some numbers exist which are nottwice an integer. They do not have integral halves.

A bar of chocolate can be broken fairly intotwo pieces. A 100 franc note can be changed intotwo 50 franc notes. 100 is certainly an evennumber. But one franc can be changed into two50 centime coins, and fortunately this alwaysappeals to six-year-old children. Consequently itseems very natural to them to try to find a half ofI on the machine.

It was because they wanted to write 100, gotby doubling 50, that the pupils literally obligedme to give them the third board. We can nowput 100 on the board and watch the way in whichwe find a half of it; we can start with l0 in thesame way.

,O*FHffiffi

:Hffiffi

Who will win?Red!Red soldiers ! Attack the blues !

Fig.22

The problem of dividing 100 francs fairlybetween three children introduces an exceptionallyinteresting situation. The first glimpse of a non-terminating decimal comes through in this remarkof one child: 'We will still be here tomorrowmorning. . .'

The interest which six-year-old children showin quite large numbers forces us to respond bygiving them unmotivated calculations which offervarious kinds of challenge. Thanks to the Mini-computer, our pupils can acld and subtractthree-figure numbers and multiply them by simplefractions. The lt'ork is a good foundation becauseit requires considerable concentration to find thebest strategies each time. These exercises help toproduce a harmonious cooperation between theintelligent human being and a true machine,which is what the Minicomputer is in the eyes of'the pupils.

BibliogrophyG. Lemaitre, 'Comment calculer' in Bulletin de I'Academietoyh dc Belgique: Classe des Sciences, Brussels, 1954,G. Lemaitre, 'Pourquoi de nouveaux chiffres?' in Reaua des

Qustions Scicntifques, July, I 955.G. Lemaitre, 'Le calcul élémentaire' in Bulletin de I'Academieroyale dz Belgique: Classe des Sciewes, Brussels, 1956.G. Lemaitre, Calaulons sans fatigue. E, Nauwelaerts, Louvain,1954.Papy, Minicomputer, Ioac, Brussels, 1968. English version:Collier-Macmillan, New York (in press),P apy, Mathématique moderne l. Didier. Brussels-Paris-Montreal,1963. (English version: Collier-Macmillan, New York, 1969.)Frédérique and Papn L'enfant et les graphes. Didier, 1968.English version: Algonquin Publishing, Montreal, Canada,1970.

l--T_l l-];; l-T--l il:ilil'ï;.,..;l 'Hg:,T,:":ii!:K{,"i,1;,2*:'4"'"'.'.i

=l__H [_-H l__|__--] :2 xsg(i" PreParation)'

I I I I lool I I I Theaccountsoflessonsdescribedinthisarticleareel(tractedt I I from the last reference and are reproduced here by kind

permission of the editor.

Fig.2l

How can the same rule be applied to finding ahalf of I ? The children ask for a new board, tothe right of the others, and at once call it the 'tinynumbers board'. How shall we remember thatit is the 'tiny numbers board'? By putting a greenline between the boards which will later becomethe decimal point. So we can calculate a half of Iand we write 0'5.

50