Last chant for times tables
Heightened concern has led to proposals for mental arithmetic tests in the SATs. Of course, testing will not of itself improve skills, any more than weighing a pig will fatten it. To make gains pupils have to acquire strategies which go beyond simple counting methods, and these strategies have to be taught. Doing even more practice of pencil and paper methods will not help much either, because the methods are different. Written methods were designed to mechanise working in sums with several non-zero digits, whereas mental methods are appropriate to what can be held in the head.
Mechanical arithmetic has been taken over by machines Q employers, for whom time is money, would not allow it to be otherwise. However, the importance of mental arithmetic remains because it promotes understanding and a sense of personal command over figures, useful in everyday situations where it can be rapidly applied, as well as a means of estimating and keeping a check on calculator working. It also provides the essential foundation for algebra.
It is widely recognised that mental methods of calculation are flexible and varied: different people will do the same sum in different ways; any one person may choose a different method if the numbers are changed slightly. Two important questions to ask are:
* Are there a few essential mathematical principles on which most methods are based?
* Are there ways of teaching pupils to use these principles flexibly and confidently?
The most fundamental need is for understanding of number language and place value. Key to the Hindu-Arabic number system, which lent itself to manipulating symbols rather than using an abacus or other physical aid, was the admission of zero as a place holder. Some words used for smaller numbers hide the regularity of the system, but it becomes clearer if large numbers are introduced. Pupils need to understand how numbers are composed, for example, 8463 = 8000 + 400 + 60 + 3. This is read as "eight thousand four hundred and sixty three", partitioning the number from left to right and reading the "6", for example, as "sixty" rather than "6 tens". Working from left to right and preserving the zero place-holder at all times seem to be crucial to mental methods. Traditional written methods usually break these principles by working from right to left and detaching digits by "borrowing" and "carrying". This has been a cause of many problems in classrooms.
Understanding number language enables pupils to order numbers and to count Beyond counting, addition and subtraction requires another fundamental notion, complementation. The human brain has the ability to recognise a number of objects arranged randomly without having to count them individually, provided the number is quite small. For our purposes,the five fingers on one hand is a useful case. Fold down 2 fingers and 3 remain unfolded. It can be seen that 5 is made up from the 2 folded and the 3 unfolded (5 = 2 + 3), or vice versa (5 = 3 + 2), that 3 remain after taking 2 away from 5 (3 = (5 - 2), or that from 5 folded fingers one can unfold 3, leaving 2 folded (2 = 5 - 3). In summary, 3 and 2 are the complements of each other in 5. Viewed in different ways, this image is rich in meaning, including the relationship between addition and subtraction, and the commutativity of the former but not of the latter.
There is more. Raising a finger generates a whole new set of relationships. This can be extended further, particularly to involve the fingers of both hands, so that pupils are able to generate all the relationships associated with complements in numbers up to 10. Also, renaming a finger, for example as "ten" or "hundred", yields results such as 50 = 30 + 20 and 500 = 300 + 200. Rather than a collection of disconnected facts to be memorised separately, there emerges a network of interrelated results, any one of which enables others to be generated immediately.
The awarenesses described facilitate learning by relieving the burden on memory. They need to be cultivated as much as possible. The Japanese would be surprised at our curriculum requiring pupils to learn addition and subtraction facts to 20. They would see this as effort unnecessary to retention because such results can be generated easily when required, for example, 17 P 9 = 10 P 9 + 7.
The "basics" consist not only of facts, but also processes and relationships. To make progress in classrooms requires a broad consensus on a sensible minimal set and a progression which ensures that work is consolidated before moving on.
Taken together, place value and complementation can be used to add and subtract bigger numbers by splitting them down and re-combining in different ways. Here are some examples with 2-digit numbers: 45 + 38 = 45 + 30 + 8 = 75 + 8 = 80 + 3 = 83 45 + 38 = 45 + 40 - 2 = 85 - 2 = 83 64 - 36 = 64 - 30 -6 = 34 - 6 = 30 - 2 = 28 64 - 36 = 64 - 40 + 4 = 24 + 4 = 28 64 - 36 = 4 + 24 = 28 A close examination of several examples illustrates that, although the chosen method will vary, the underlying principles are the same. An approach which allows scope for creativity and choice, but which is also structured, provides an excellent stimulus to learning. Pupils enjoy discussing different strategies and, with practice, are less likely to regress to more primitive counting methods.
Multiplication is founded on knowledge of tables. As a teaching strategy, chanting is limited because it does not get one directly to the result needed. Also, it ignores many important multiplicative relationships. Can a way be found which reduces memorisation and increases awareness? The key new skill seems to be doubling and halving. Discussion and practice of doubling with 2-digit numbers is particularly useful. Types to consider are exemplified by: 34 (double the digits), 73, 27, 87, 69 (= 70 - 1). Repeated doubling should also be practised. Halving can be tackled in a similar manner.
Tables x2, x4 and x8 can be generated by repeated doubling. Since x10 is easy, x5 can be generated by x10 and then halving. This leaves x3, x6 (double x3) and x9 (related to x10, for example 7 x 9 = 7 x 10 - 7) and the slightly tricky x7 is left until last, which makes it easier. Apart from doubling and halving, there are other strategies, such as working from near neighbours. For example, 8 x 7 can be found from 7 x 7 + 7, 9 x 7 - 7, 10 x 7 - 14, 7 x 4 x 2, etc. As with addition and subtraction, facts not yet memorised can be generated from others, leaving pupils with fall-back strategies more sophisticated than addition. Anyone who can use these methods understands the number operations and is more truly fluent than someone who can only chant tables. Appropriately structured, such an approach builds knowledge and confidence.
With numbers having more digits, place value understanding is used to break them down: 38 x 8 = 30 x 8 + 8 x 8 = 240 + 64 = 304 406 x 5 = 400 x 5 + 6 x 5 = 2000 + 30 = 2030.
Again, the most sensible approach is to work from left to right, with all zeros retained.
Division involves trial multiplication and subtraction, to leave progressively smaller remainders. It is most easily displayed in a vertical format, for example 977:
97 10 x 7 70 27 3 x 7 21 6 977 = 13 r 6 In summary, mental arithmetic needs to be taught using methods quite different from traditional pencil and paper methods. Offering only one method is too rigid. Leaving pupils to find their own methods will deprive many of more advanced strategies. A better approach recognises that there are a few big ideas. Teaching should aim to help pupils to understand these ideas and to acquire strategies for adapting them to solve many problems.
Alan Wigley is adviser for mathematics in Wa