Numbers fall into place
Most primary maths teachers would probably agree that one of the key concepts children must grasp is place value. Without a thorough grounding in the idea children in later junior and early secondary years will struggle with the basic operations.
The Schools Curriculum and Assessment Authority report on the 1995 key stage 1 tests says many children had difficulty with the question: "How many tens in 45?" One common error was to write 40 instead of four.
The national curriculum for mathematics says place value must be introduced in key stage 1. Most children will have spent some time counting objects and reciting, reading, writing and ordering numbers, learning addition and subtraction bonds, performing mental calculations and writing answers to simple single-digit "sums" set out vertically (and sometimes horizontally).
Place value is usually introduced by giving children a variety of familiar materials and then having them form "groupings of groupings". This then leads on to the idea of exchanging within the groupings.
To model our number system, which groups in tens, most commercial mathematics schemes advise the use of Dienes structured base-ten equipment. This teaches young children to add and subtract two-digit numbers and in some cases perform single-digit calculations where the answer exceeds ten.
So, what do we know about the strategies children use to solve two-digit calculations mentally? Research literature contains many examples of children explaining their informal mental calculation methods. Three transcripts from my own research provide examples of what seem the three most common strategies. In these examples the children were calculating 27+28 mentally.
Elspeth: "20 and 20 is 40. Seven and eight is...15. So that's 55."
This was by far the most common strategy. It involves partitioning both numbers into the quantities represented by each digit (not into tens and units) and adding these quantities separately before recombining them. This is the split method, so-called because of the actions on the numbers.
Then there was Joanna: "27...47...and three makes 50. So it's 55."
This method was the least common of the three, although many adults seem to use some variation of it. It involves starting from one of the numbers and adding chunks of the other number in turn. The tens are usually added first (as quantities) followed by the units, to give the total.
This cumulative method can be easily modelled on a number line by jumping forward from one of the numbers. Consequently this is called the jump method.
Mark said: "55. I got the two twenties - that's 40, then I added the eight. That makes 48, and the other seven make 55."
Mark's strategy combines the two previous ones. He split the numbers before adding, and then added the units separately. This is the split-jump method.
None of these strategies considers the 27 as two tens and seven units, but instead they make use of the quantity meaning of the digits, seeing the two as "20". As mental addition and subtraction involve working from left to right, dealing with the larger chunks first to generate a reasonable first approximation to the answer, this would appear a more appropriate interpretation. The language we use to read the number 27 also reinforces this interpretation - we read it as twenty-seven, not two tens seven.
This interpretation of our number notation seems to demand a more sophisticated and formal appreciation of the place value concept. Most of those children who answered 40 to the "How many tens in 45?" question (and no doubt received no marks) would probably answer the follow-up question "And how many tens is that?" correctly.
Teachers and maths educators, who understand the structure of our number system, can see the extent to which Dienes base-ten equipment constitutes a suitable model of the tens and units structure of our place value system. They can appreciate how and why the standard written procedures work, and can see how the equipment appears to model the related algorithms.
But it does not follow that children who work with this apparatus will make the necessary connections between the practical activity they are carrying out and the procedures they are setting out on paper.
Research suggests that evidence of this transfer is tenuous. Most children seem to see nothing more than a set of blocks, and fail to perceive the mathematical relationships "experts" recognise.
Over the past few years several maths educators have argued for a delay in the introduction of formal vertical written algorithms. The National Numeracy Project illustrates the extent of the progress made. Its framework document includes no formal written algorithms before key stage 2.
And because Dienes equipment is used (often unsuccessfully) to model these algorithms, we should delay its introduction too. In fact, we should consign it to the equipment cupboard, and search for alternative models. One that comes immediately to mind is the Gattegno table and its related arrow cards (see Figure 1). These cards provide a model for the "quantities" interpretation of multi-digit calculation that underpins the mental calculation strategies of many children.
Compared with other European countries we appear to have an impoverished and narrow interpretation of what place value means. If we are to make progress in developing children's standards of numeracy we should delay what seems our unsuccessful attempts to have our seven-year-olds operate at the formal level of place value understanding, and work on improving their ability to operate at the quantities level. This is what they appear to do in the Netherlands, and in the recent Third International Maths and Science Study, Dutch nine-year-olds came fifth in the number test, behind the Pacific Rim countries.
Ian Thompson is a lecturer in maths education at the University of Newcastle upon Tyne