Pip's adventures with nine thieves

Chapter 7 of Great Expectations gives us an insight into the pedagogical problems that face our primary education system. 

Pip is trying to learn to read and write with Biddy.  He has been attempting to decipher alphabet and numbers.  Some letters and combinations seem almost impossible for him.  The nine thieves are nine numbere.  He imagines them as troublesome little thieves. 

Dickens is making a joke out of Pip's struggle with literacy. To an adult reader, the letters are perfectly ordinary; to Pip they are mysterious marks that refuse to behave.

There is also an important contrast here: Pip can observe the world extremely well -- people, objects, voices and gestures.  Reading and writing are artificial skills that he has not yet mastered.  Dickens presents literacy from the child's point of view.

Pip is beginning to move from an oral

experiential world into the world of books, schooling, and social advancement.  Pip can understand complicated human situations and remember vivid details.  In his childish understanding the numerical symbols were a mysterious system.

Pip's education is elementary.  Biddy teaches Pip reading, writing and arithmetic.  What he learns he passes on to Joe.  Joe in turn begins his trouble.  Pip lives in a concrete world. Counting numbers are abstract representation of concrete objects.  The teaching method is divorced from the child's concrete experience.

If the teacher had put seeds, stones, or similar objects before Pip the nine figures could have been connected to things he could see and handle.  Instead the child is confronted with abstract marks on a page.  Pip's world is overwhelmingly material and concrete.  Yet his education suddenly ask him to deal with abstraction.

Here we are coming to a well established principle in modern mathematics education.  It is often called Concrete -- Representational --

Abstract (CRA) approach, or Concrete --Pictorial -- Abstract (CPA) approach. 

The U.S. Institute of Education Sciences describes CRA as a method in which children first manipulate concrete objects, then work with visual representation, and finally solve problems using abstract notions. 

Pip was being introduced to the symbol before the concept had been grounded in experience.  Pip lives in a world of objects --- stones, tools, food, fire etc.  Modern pedagogy would normally try to build a bridge:

Objects = representation = symbols.

There was a traditional play in Malabar using the seeds of bullet wood tree:

1. One child takes an unknown handful of seeds into his closed fist.

2. He aks his peer Odd or even?

3. His peer makes a prediction.

4. The fist is opened and the seeds are counted.

5. The prediction is checked against the actual quantity.

6. If the prediction is correct the handful of seeds go to the winner.

What is particularly interesting is that the children are not being taught odd and even through abstract numerals. They are experiencing the difference through physical quantities. And there is a second mathematical idea hidden in the game: the child is making a prediction under uncertainty. He doesn't see the seeds, but he knows that the hidden quantity must be either odd or even.

Here concrete experience precede abstract mathematical notation.  A child who goes through the play acquires an intuitive understanding of the concept of odd, even, counting and quantity.

But this type of traditional game is not unique to Malabar.  Remarkably close to this odd even game of Malabar was played in England.  Alice Bertha Gome's The Traditional Games of England, Scotland, and Ireland records "Odd or Even" as boys' game played with buttons, marbles and halfpence.  A boy concealed a few objects in one hand and asked his opponent to guess whether the number was odd or even.

Joseph Strutt's Earlier Sports and Pastimes of the People of England gives essentially the same procedure   According to Strutt the game was known to Greeks and Romans as well.  Greek children used beans, nuts, almonds and money.  The Malabar game has striking parallel.  These children were doing mathematics without writing mathematics.

Edgar Allan Poe in his detective story

The Purloined Letter presents this story this game l.

Pip's circumstances were entirely different.  He was an orphan  brought up by hand.