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Peer influences: Subtraction with small numbers







  • Taxonomy: Calculations: children's own methods 
  • Subtraction





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

one-by-one 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 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










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



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 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.