Probability in a Primary School
P. SHERWOOD
 
At what age can children begin to grasp principles of probability?  In Philip Sherwood's experience a great deal can be done before the age of 11.
 
In the current atmosphere of 'back to basics' it may seem that introducing probability is better left until the secondary school.  The Hungarian Tamas Varga has never doubted that combinatorics and probability are important in a primary mathematics scheme.  He has tried out his ideas with young children and developed them over the years.  Working with children he usually pretends that he is as puzzled by the phenomena as they are, and that, when looking at it together, their suggestions and strategies are more valuable than his, consequently he offers very few.
Twelve years ago when I first watched him working with children in my school, he produced three discs marked as follows; 
 

These he held up to the children and asked them to guess what was on the reverse, ,spot or blank'.  After a spell of random guessing he asked them to devise and write down a strategy for guessing, which they would apply each time.  Some tried repeating the last situation each time, others used blank and spot alternately.  None chose the best strategy (whatever is on the face is most likely to be the same on the back).  He then let one child use this and the results showed that he consistently scored best over a range of fifty tries.  The children began to think and to suggest reasons as to why this might be.  The thinking was intuitive, no one came up with a numerical solution but their answers showed that they had begun to grasp some of the ideas inherent in probability.  From all this came a Probability Kit for use with children and two books.

Hasard ou Strategie, Engel, Varga, Walser, O.C.D.L. Paris and Combinatoire, Statistiques et Probabilites Varga, Dumont, O.C.D.L. Paris.

Most of the materials in the kit are familiar, for example the interlocking cubes are very like Unifix.  Generally the work starts with a game or some constructional activity.  These may involve building houses according to certain constraints, e.g. using any three colours you can build 3 storey houses.  No two houses can be the same.  You may use any one colour as many times as you wish (red, red, red) or (red, red, yellow).  How many houses can you build?  If children have had some experience of making their own attribute material the task will seem familiar.  They will adopt some strategy of their own, e.g. "start with all the ones with a red cube on the first floor." Children without any previous experience of this sort attempt random solutions which are usually non productive.  For them we can suggest that a 'tree' might help;

 

It is of course better to use an 'unfinished' tree and to leave the children to develop it.  I have used this activity with the less able element of a third year junior class and they have worked at it with total involvement for a considerable time.  Having completed the task I have asked them to arrange the pieces of a three colour set of 'trimath' to match each house (Trimath, E.S.A., is a 6 colour by 3 shape by 3 hole difference, 0 hole, 1 hole or 2 hole attribute material).  The children have been able to do this by changing the labels on the tree.  Then without looking at their lay out, given any piece of tri-math, they have been able to determine to which house it corresponds.  More able children have gone on to match the base 3 numerals 000, 00 1, - - - 222, with both their houses and the trimath pieces.

Not all of Varga's combinatorial activities are quite so simple.  Other activities require children to make up as many numbers as possible using the digits greater than seven but with the condition that 9 cannot be next to another nine.  The children make up a table showing their results;
 
 
  The digits are greater than 7 
9 cannot appear next to another 9		  
No. of  digits  Numbers which we can make    How many  can we make? 
1  8, 9  2 
2  88, 89, 98  3 
3  888, 889 etc.   
A parallel card asks them to work out in how many different ways they can go up a flight of stairs if they can take either one step at a time or two.  We start with a flight of one stair, then move on to flights of two stairs, three stairs, etc.  There are other similar activities each generating the same number series.  Ideally children should see the structure common to all these problems. It does not come in a flash and children do need prompting.

Probability is approached through games.  Here is a typical board game played using a die with three green and three white faces.

The die is thrown by each player in turn, who then moves one of his stock of counters, to the left if the green face shows, to the right if a white face.  They record each counter's ultimate destination;
 
 
 
            *          * * *  * * * * * * *  * 
* 
* 
* 
* 
* 
*		       * * * *          * * * 
King  fairy   dragon  giant  sorcerer   crocodile
 
Why do so many children (counters) end up eaten by the dragon or the giant, so few reach the friendly king or fairy?  From that question we can go on to build up Pascal's triangle, or at least give some hint of its existence.
 

The games generate interest and involvement from which comes the learning.  Varga's Thieves of Bagdad game ended up as the Colditz Escape in our hands.  The thieves are thrown into the darkest of dungeons.  There are three tunnels leading out from the prison.  One leads to freedom and takes an hour to traverse.  The other two lead back into the dungeon and take two and three days respectively.  In the tunnels there is only sufficient life support (air and water) to last for 5 days and 1 hour.  One day twenty seven robbers are thrown into the dungeon.  How many are likely to survive?  We find out by playing it as a game.


Three players each have nine thieves (counters).  The die, thrown in turns, shows which tunnel each thief takes.  How many thieves survive?

Players can record the progress of their thieves.
 
 Thief No.  route taken  length of journey   Free dead
1  black, red,  black   8 days    +
After playing the game they can then build up a tree which will show the outcome of the various attempts

This is incomplete, on principle, we never give too much help or guidance.

Many of the games the children play are on race tracks or football fields.  They are usually, but not obviously, 'bent' games.

Two dice are thrown, I move on a total of 1, 2, 3, 4, 10, 11, 12, you move on 5, 6, 7, 8, 9. To children the game seems bent from the start. I have more chances of moving and yet I never win.  It suddenly occurs to me that you can never score 1 with two dice.  Elementary I know, but it is rare for children to object until quite late in the game.

The game is adjusted. I now move on 2, 3, 4, 5, 9, 10, 11, 12.  My chances of winning are now improved and obviously I have the better prospects, or do I ? Finally we try to make the game really fair and, settle for, I move on odd, you move on even.' That seems to give us a fair game, but why did the other rule, so obviously bent, prove to be bent in a different way.  We can now investigate the problem numerically.

What have the children learned from it all?  Nothing that you could measure with a standardised test.  There is one item in the kit which we have not generally used-a box with two compartments in to which we can put coloured beads.  The box is clear plastic.  The lower set of beads cannot change places, the others are free.  We shake the box, what are the chances of coincidence?

Recently I handed the problem to two children who had worked through most of the probability cards.  What are the chances of the red matching the red, the blue the blue etc.,? I didn't expect factorial 3 in the reply. I did however get what I wanted; a quick pencil and paper record of all the possible outcomes, the answer 1/6 and ten minutes of experimental shaking to verify the answer.  Experimental results, drawings, trees, are the substance of our use of Varga's work.  Why do it when all that will be required of our children at secondary level will be facility with four rules?

Measuring on the 3P scale-Pleasure, Purity and Profitability-it scores well on all three.  It is very high in pleasure, the purity is unimpeachable and its profitability?  Well, most of our pupils will eventually indulge in gambling on football pools etc., they will now at least do it with awareness.  Our children now approach probability situations with confidence and competence.  The insight they have developed is very real.  These are worthwhile and lasting rewards.

We are not a Mathematics school, we dance, sing, paint, swim, camp etc. with the best and probability is only a minor element in our mathematics work.  We do enjoy probability, as we do dance, gymnastics and swimming.  For that good reason it will stay with us.

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