A place for everything;Maths Year 2000
The National Numeracy Strategy has shifted the focus in early number work, away from sorting ordering and matching - the infamous "pre-number activities" - towards a carefully-structured introduction to and development of important counting sub-skills. It has also made mental calculation (rather than "mental arithmetic") a key underpinning principle.
Detailed suggestions for teaching mental strategies have stressed that mental calculation comprises more than mental recall. But vital as all this is, the strategy falls short in one key area - place value.
In the Numeracy Framework document, "place value" forms an important sub-section in the number strand. For example, children in Year 1 should be able to represent numbers to 20 on a spike abacus - a fairly formal representation of the place value principle. The Mathematical Vocabulary booklet that accompanies the framework contains the phrase "units place, tens place" in the list of words reception and Year 1 children need to understand and use, with "place value" appearing alongside words deemed appropriate for Year 2.
I believe children do not need to learn about place value until the time they are expected to perform standard written algorithms for the basic operations - currently a goal for all children by the end of their primary schooling (Year 6). I also believe attempts to teach the concept as early as we do - when children are already struggling to make sense of the "awkward" teen numbers - are counter-productive, and have probably contributed to our relatively poor performance in the number section of international surveys.
It is important to be clear that I am talking about the standard, traditional interpretation of place value as emphasised in all schemes of work, which involves terminology such as "tens column", "the hundreds place", "four tens and six units", and so on. I am taking "place value" to mean the value assigned to a digit according to its position in a number. For example, in the number 365, the 6 stands for six tens because it is in the "tens" column rather than in the "hundreds".
We need to think in terms of a concept to precede place value in the learning sequence - a concept that is less formal and more readily understood by young children. Unfortunately, we lack the necessary language to describe it. Perhaps we could call it "quantity value", which would mean the actual quantity represented by each digit in a given written number. For example, in 365 the 6 stands for 60 and the 3 for 300. This may appear little more than hair-splitting, but in terms of the development of mental calculation skills and subsequent written algorithmic competence, the difference is crucial.
The two most common mental strategies used for two-digit addition and subtraction are the "split method" (47+36 becomes 40+30=70; 7+6=13; 70+13=83), and the "jump method" (83-47 as 83-40=43; 43-7=36). A key procedure of both strategies is "partitioning" - the splitting of two-digit numbers, not into separate digits as is required for the execution of written standard algorithms, but into the quantities represented by the number names. So, 47 (forty-seven) is partitioned into 40 (forty) and 7 (seven), and not into "4 in the tens and 7 in the units column" or "4 tens and 7 units".
Two other less common mental strategies are "complementary addition" for difference problems (45-27 as 27 to 30 (3); 30 to 40 (10); 40 to 45 (5), so it's 3+10+5 = 18); and "compensation" for adding or subtracting numbers ending in 7, 8 or 9 (36+27 as 36+30 = 66; 66-3 = 63). Neither method involves the place-value concept as described above.
Two specific strategies for mental multiplication and division recommended in the framework are doublinghalving and partitioning. The latter is a generalised version of the former, and makes use of the distributive property: ax (b+c) = axb + axc. So, for example, to double 36, children are expected to partition 36 into 30 and 6, double the 30, double the 6, then add the two answers together. Halving 36 would be done in a similar way. In Year 6, for example, children are expected to be able to mentally calculate 86x7 by adding 80x7 to 6x7. All of these strategies involve "partitioning" - a procedure more heavily dependent on an understanding of "quantity value" than on "place value".
A consideration of the informal written strategies described in the framework for the basic operations at key stage 2 confirms the importance of "quantity value" even at this level. The recommended algorithm for addition is described as "adding the most significant digits first", and the example given for Year 5 is: 587
------- = 900
------- = 1062
This algorithm is an almost exact model of the "split" (partitioning) mental strategy described above - the digits represent quantities; they are added from the left according to their significance, and there is no carrying (a procedure that demands an understanding of column-based place value). In a similar way, no "borrowing" ("exchanging") is involved in the informal subtraction methods recommended in the framework.
For multiplication, the framework recommends an informal "grid method" and a slightly more formal written version for Year 4: 23 x 8 -- = 160 (20x8) + 24 (3x8) -- = 184 As with mental calculation, this procedure uses the quantity value of the numbers - the 2 in 23 is "twenty" not "2 in the tens column". Neither the informal nor even the "standard" algorithm recommended for division involves the traditional place-value concept .
Only when standard algorithms are introduced some time in Years 4 or 5 do column-based place value concepts need to be taught, to enable children to make sense of "carrying" or "exchanging" procedures.
There is almost universal agreement about the need to concentrate on teaching mental calculation methods before written algorithms. The logical consequence of this move is to develop children's understanding of "quantity value" well before the more formal "place value" concept. We have recently reassessed the way in which we teach early number - let us move on to reappraise the teaching of place value.
A free copy of the Nuffield-funded report 'Mental Calculation Strategies for the Addition and Subtraction of 2-digit Numbers' can be obtained from Ian Thompson, Department of Education, University of Newcastle upon Tyne, St Thomas St, Newcastle upon Tyne NE1 7RU