mathematics in their own ways, we have found that during our visit some children
will spend time on elaborate drawings and colouring-in, viewing the task as a
drawing exercise. This view may originate from classes where children have been
used to using worksheets or adding a picture to their work after they have
completed a piece of writing. In mathematics lessons it is children’s mathematical
understanding and written methods that are important. Whilst some children may
use a pictographic form of representation, teachers need to help children
understand that their thinking (of mathematics) is of far greater importance in this
context than pretty pictures.
Another problem we have met is of children being expected to represent some
mathematics which is insufficiently challenging, for example asking 6-year-olds to
draw a story representing an addition for the number seven. There is a danger that
children’s own recording is only of teacher-suggested number stories. There is an
abundance of literature supporting this kind of ‘story’ calculation, including Whitin,
Mills and O’Keefe (1990) and Hopkins, Gifford and Pepperell (1999). We believe thatUnderstanding children’s developing calculations 115pictures of beans in pots to numerals and using the word ‘is’ to stand for the equals
sign.This example shows that Barney had a very good understanding of the
operation and that he is beginning to explore the use of abstract signs. It may be
logical to use the word ‘is’ in a narrative.
In the same session he explored further written methods (see Figure 7.5). Next
he again used numerals but this time experimented by drawing a hand instead of
arrows, to show his own action of taking away one bean.
The other children in the group used a range of strategies. Alex also drew hands
to show the operation (action) of taking away but Emma used drawings of hands to
different effect. In her first example she has represented ‘5 –1 = 4’ (she forgot to
represent the one bean removed) using arrows pointing to the final amount after she
had removed one. In her second example Emma included the bean that she
removed. Kristian used arrows in a similar way to Barney, but used tallies to
represent the beans. Like Barney he is beginning to explore the use of the equals
sign in his own way.
Matthew used a simple means of showing the amount subtracted that is similar to
Louisa’s (Figure 7.2b). Jennifer has combined iconic and symbolic representation and
used the standard subtraction sign, whilst Francesca used iconic symbols combined
with both the standard plus and equals signs. Finally, Eleanor chose to represent her
calculations in a standard symbolic form, apart from the final calculation.The beans
were all in use so she used some counters, writing ‘4 counters take 1 = 3’.This
variety of written methods was found within one class when children have been used
to choosing how they will represent their mathematical thinking on paper and have
seen a variety of mathematical graphics modelled (see Chapter 10).8657part 2.qxd 04/07/2006 17:16 Page 115