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* Standard symbolic calculations with small numbers

Standard symbolic calculations arise directly out of the preceding ones, children's previous knowledge and understandings combining to support simple calculations with small numbers and quantities.


Jax (5 years, 2 months) implies numerals

or quantities by using dots (an iconic form) and

reinforced by the numerals '6' and '4'. This can

be read as "6 and 4 = ten", Jax having written

the initial 't' of  ten.  


  • Taxonomy: children's own methods, symbolic operations with small numbers    
  •   Addition          

Peter (5 years, 9 months) has used his own

abbreviated calculation, which, provided he  

reads it to us, can be understood as "4 - 3 = 1".


  • Taxonomy: children's own methods, symbolic operations with small numbers    
  • Subtraction


Mary (5 years, 4 months) used a combination

of iconic and symbolic responses, with the

minus and equals signs implied, denoting "6,

minus/subtract 3 equals 3".


  • Taxonomy: children's own methods, symbolic operations with small numbers  
  • Subtraction

















Barney’s beans

     Barney’s 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. 

     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 '='. 

     The second page includes examples from other

children in Barney's group, after they had seen

what Barney had done, and is an example of


     Barney's example here shows that he then

abandoned his drawings of flowerpots, beans and

arcs of arrows, moving to a more concise form of representation [known as 'successive shorthand'].

The hands he drew indicate the 

removal of one bean, and the word ‘is’, is Barney’s personal 

means of representing ‘=’ or the final total of his calculation.


Note: We term this use of arrows and hands as "narrative action".


  • Taxonomy: Calculations: children's own methods - 

         Symbolic operations with small numbers; separating 

         sets; explorations with signs and symbols

  • Subtraction