Peer influences: Subtraction with small numbers
 Barney's beans Barney’s class were (in England), Year 1, comprising children of 5  6 years of age. The teacher had introduced a game in which  using beans and flowerpots – one child counted a quantity of beans as she added them onebyone to a flowerpot, saying the final quantity. Keeping it secret, her partner then removed one or more beans and her partner worked out the quantity of beans that remained in the pot. Their teacher had put some paper and pens on the table, suggesting that they put something down on paper to help their thinking. This was the first time the children had represented subtraction. Barney (5 years, 3 months) began by using a combination of numbers and letters, writing “10 take 1 is 9." He experimented with drawing arcs of arrows to show the action of removing one bean in his game. At the foot of the page he used numerals rather than beans and flowerpots. Barney was the first child in the class to use arrows to denote 'take away'. Finally, he substituted a hand with the numeral '1' written on it to show that his hand had removed one bean. In several of the calculations Barney used the word 'is' to stand for the symbol '='.

As humans (and also mammals and many other animals) our cultural understandings are socially influenced. In respect of speech, drawing and writing, such influences are also known as intertextuality.
Barney had been sitting with the other members of his group, who quickly saw Barney's arc of arrows and, used their own ideas responded in a variety of ways.
Barney: second thoughts This was Barney's second response, combining numerals, a drawing of a hand removing one bean, (signifying 'subtract' or 'takeaway 1'), and he wrote 'is' to signify equals. Finally he wrote the total '5'.
 
Alex Alex also drew a hand to show the operation (action) of taking away.
 
Emma's arrows In her first example, Emma represented '5  1 = 4" (forgetting to show the 'one' she was subtracting). Building on Barney's idea of arrows to show 'take away', Emma drew arrows pointing towards the final amount, writing her answer '4' on the second hand.


Kristian's arrows Using arrows in similar way to Barney,, Kristian's arc of arrows points to '1' (bean). Kristian used tallies to represent the two quantities of beans, 'is' denoting ''equals'.
 
Matthew's beans Matthew rapidly drew five beans, using a line scored though one to signify the one bean that he'd subtracted.  
Francesca's standard abstract symbols Francesca used dots (iconic signs), combined with the standard abstract symbols for 'subtract' and 'equals'.
 
Eleanor Eleanor chose to represent her calculations in the standard symbolic form (with the exception of her final calculation, which can be read as " 4 counters, take1, equals 3' (or '4  1 = 3').
 
Jennifer's ideas To begin with, Jennifer drew circles or stars (both iconic signs), to represent the quantities of beans in each of the two sets, separating them with the sign for subtraction, (the minus symbol). Finally she added the total beneath each, using a standard written numeral.
