Children\'s Mathematics

These problems illustrate some of the difficulties children experience when they use

examples– including standard symbols – shown by their teacher and which they do

not understand. They want to comply and interpret the request to ‘put something

down to show’ as meaning ‘do what I have just shown you’. This leads to compli-

ance and conformity without understanding.

It is clear that following the teacher’s example without understanding leads to

confusion: if children continue in this way, even when they sometimes get ‘right’

answers, their difficulties are compounded. They also learn that they should not

attempt to work things out in ways which might make sense to them since the

teacher is looking for them to all use the same written method. If they use the

teacher’s method, formula or layout without understanding, many children come to

learn that mathematics often does not make sense. A chasm has then been created

between their informal mathematical understanding and standard ‘school’ mathe-

matics that will be very difficult to bridge.

Group 2 – children’s ideas following direct teacher modelling

I asked if any of the children had an idea of how they might ‘put their ideas down on

paper’. Responses included suggestions of ‘drawing how many bears’, ‘putting tallies’,

‘numbers’, ‘shapes’, ‘letters’ (words) and ‘numbers’. The child who suggested ‘shapes’

explained she might put a square for each bear (iconic representation). Many of the

suggestions they made drew on features I had previously directly modelled.

Emily had drawn the correct number of bears but had also copied my example of

‘3 + 2 = 5’ (Figure 10.7c). She was not able to explain the calculation and did not

know what they symbols ‘+’ and ‘=’ meant, explaining ‘I saw them on the board’.

She read the string of numbers but had no idea what they related to other than

saying ‘you have to count’.

John read ‘one and’ and then turned to me, looking confused.We counted the

two bears and the four bears in front of him together but he was unable to

relate what he had done to the bears he saw (Figure 10.7b).

Louisa started as I had done in my example, by drawing the bears. She had

written ‘5 + 6 = 10’ and when I asked her to tell me what she had found out, she

quickly crossed out the ‘= 10’ and wrote ‘7’ by the 6’ then looking at the bears

she’d drawn, she counted them all and wrote ‘6’ at the end. Her written

calculation had not matched her drawing so she altered the total to match the

drawing that she trusted (Figure 10.7a).

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