Tests done by a colleague and myself demonstrate that there are two distinct aspects of what we normally call place value: quantity value and column value.
The former concerns knowing that 73 is "seventy" and "three". The latter involves knowing that 73 is "seven" in the tens column and "three" in the ones column.
Quantity value is learnt at a younger age than column value, and is essential for mental calculation and the informal written strategies listed in the National Numeracy Strategy.
Column value, however, is a more difficult concept that is not needed for mental calculation. It needs to be taught only when children are about to learn standard written algorithms.
To test our hypothesis about quantity value, my colleague and I interviewed 144 children in Years 2 to 4 from eight primary schools.
These are the questions we asked:
* Can you read this number to me? (Show card with 16 written on it).
* Please take 16 cubes out of the box. Can you show me with the cubes what this part (6) of the number means? (Circle the 6 with the back of a pen).
* Can you show me with the cubes what this part (1) of the number means? (Circle the 1).
In our sample, 54 per cent of the Year 2 children showed 10 cubes to the interviewer, and an impressive 77 per cent of the Year 3 children were also successful.
In the United States, researchers carrying out a comparable study found that not a single child in Grade 1 (Year 2) got it right. In another American study 20 per cent of the Grade 2 (Year 3) children obtained the correct answer, and in Australia, 44 per cent of children of the same age were successful on a similar task.
So, how might we account for the large discrepancy in these figures?
There is substantial pressure in American schools to teach children standard written algorithms for the four basic operations as early as possible. A similar but less extreme pressure also exists in Australia. In this country, however, the National Numeracy Strategy has placed great emphasis on the teaching of mental methods of calculation; on the need for children to discuss these methods; and on the importance of delaying the introduction of formal written calculations until Year 4.
It would therefore seem plausible that these aspects have contributed substantially to the success of the children in our study. The emphasis given to place value cards and Gattegno charts, which clearly model the way that a two-digit number such as 53 can be partitioned into 50 and 3, must also have had a beneficial effect.
A different question in our study asked children to calculate 25+23 mentally and to explain how they did it. More than three quarters were correct and 63 per cent partitioned 23 into 20 and 3 in order to make the calculation.
In the "16 cubes" question the ability to partition 16 into 10 and 6 would obviously be very useful for getting the correct answer. Are we therefore to conclude that English children have an excellent grasp of the difficult concept of place value? A brief look at two other questions might help us answer this.
After discussing a picture of a car dashboard showing a mileage of 6,299 the children were asked what the milometer would show after the car had travelled one more mile.
Only 24 per cent were correct. Another question asked the children to say how the value of several cubes had changed after they had been physically moved by the researcher from the "ones" to the "tens" column on a base-10 board. Only 10 per cent were correct (this rose to 28 per cent when they were asked the follow-up question How many times bigger is it now?).
Now, both of these questions address important aspects of place value. So how do we reconcile an average of 70 per cent success on the "16 cubes" question with a mere 10 per cent success on the "base-10 board" question?
We think quantity value is the aspect tested in the "successful" 16-cubes question and the two-digit mental addition. Column value on the other hand is what is tested in the "unsuccessful" milometer and base-10 board questions.
Our conclusion therefore is that teachers should consider postponing work on "tens and units", with columns headed T and U, until much later.
Ian Thompson is an independent maths consultant and researcher If you would like to receive a copy of the full Nuffield-sponsored report An investigation of the Relationship Between Young Children's Understanding of the Concept of Place Value and their Competence at Mental AdditionEmail: ianthompson.pi@btopenworld