Children\'s Mathematics

• counting on from the larger of the two numbers


• using a number line with points marked on it


• using an ‘empty number line’


• partitioning numbers


• exploring alternative ways of working or checking a calculation


• the use of derived number facts


• some understanding of commutativity (see Figure 10.6 ‘super-zero’).

Examples of subtraction with larger numbers using mental methods

or jottings

The empty number line is also a taught strategy introduced in England by the

National Numeracy Strategy(DfEE, 1999a). It is also used extensively in Holland (see

Chapter 1). In England this is one of several forms of notation that are termed ‘jot-

tings’, although these are not intuitive methods. This term ‘jotting’ is ambiguous.

Jotting down is something most adults would be familiar doing when, for

example, estimating rolls of wallpaper for a room – our jottings aid quick calcula-

tions. Sean, 7:6, (Figure 7.12a) also partitioned 86 and 47 in order to combine them,

then checked this by re-working the calculation in a different order beneath what

he’d first written. At the foot of his page he used a drawing of a number track to

check part of his calculation.

There is potential for children to adapt this number line model in flexible ways

depending on their need, as Miles, 7:5, did (Figure 7.12b). Miles’s class were about to

leave for a residential trip; we used a pack containing three nectarines to calculate

how many packs would be needed for the whole class (see Chapter 9). Using a piece

of A4 paper Miles began by drawing a horizontal line across the width of the page.

Because the way in which he had chosen to orientate his paper restricted the number

of jumps he could make Miles adapted his method of subtraction, changing from

jumps of 3 to jumps of 6 several times. Reading from right to left, he wrote beneath

the jumps the cumulative number of packs of nectarines that he was calculating for

26 children to have one nectarine each, to arrive at his answer. This signifies that

Miles has discovered a more efficient method.

In Figure 7.11a, Darryl, 7:3, decided to work out this addition calculation by

partitioning the numbers before adding (partitioning numbers is a taught

strategy). He noticed that he had made an error to begin with and reworked

what he had done, lower down.Working in this way allowed him to go beyond

the hundred boundary. Although at first Stefan, 7:7, had added 8 and the 4 rather

than 80 and 40, he was soon able to see that this did not make sense and

reworked this part of his calculation (Figure 7.11b). Stefan used his jottings to

help him arrive at an answer which he finally resolved mentally.

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