Some years ago I saw one of the charts below (Fig 1) on a classroom wall. So what, I thought, and passed on. More recently, I have come to see the power of its imagery in working with children. All of place value can be found in the layout. It offers some significant mental images similar to the soroban (the Japanese abacus).
At first I had no idea how to make pupils aware of the patterns. I now know a key lies in making the link between the regularity in language as we count and the regularity of the symbols we use to record that count.
One way is to use chorusing. It needs to be rhythmical. It needs to be fun. It must have elements of certainty and elements of surprise. The chart can be used, for different reasons and with different emphasis, with children as young as six and as old as 14.
Start with some simple counting, from 42, for instance. Point at 40 - they chorus "40". Point at 2 - they chorus "2". Point to 40 - chorus "40". Point to 3. So you hear 42, 43, 44, 45, 46 . . .
Then arrange to go up and down over 10 numbers on the chart: 68, 69, 70 71, 72, 73, 72, 71, 70, 69. The teachers' role is that of conductor and rhythm keeper. The group supplies language for the symbols.
Keep speech to a minimum. When one or two offer different spoken answers, go back and repeat that series of numbers until the majority are at home with words for the symbols.
Counting is fundamental and counting is fine in itself. But children in Year 3 need to be confident with adding and subtracting in tens. By Year 5 they should be confident in multiplying and dividing by 10. You can use the chart to work on both these skills.
For the first, extend the group's counting in ones to counting in tens and listen to the language. 123, 133, 143, 153, 163 . . . Which row are we going up in? Tens.
Once this pattern is established, other members of the group can take over the conducting role. This gives wonderful formative assessment information. For instance, counting in tens is easier from say, 40 than from 38. It is easier on the chart too. But we want the children to become at home with counting in tens, up and down, from any number.
So, to work with multiplying by 10, we need to shift their attention from the numbers to the operations on the numbers. Start by pointing at 7 (they say "seven"). Point down two rows to 700 and say "hundred" as you move the pointer. Some surprising but useful insights occur when working in this way. For instance, seventy hundred is seven thousand - two names for the same number. Write this: 70 x l00 = 7,000. Also, seven thousand hundred is seven hundred thousand (7,000 x l00 = 700,000) - and so on.
The regularity and connectivity of the number language is beautifully stressed. We say "nineteen hundred," and "one thousand nine hundred". The names for the rows reading up from the bottom are: hundred thousands; ten thousands; thousands; hundreds; tens (said "ty", although there are irregularities in this row), and units (unsaid). A second chart (Fig 2) extends the pattern to decimals. It reads from hundreds on the bottom row, through "ty", units to tenths (written 0.1, 0.2 . . .) and above that, hundredths (written 0.01, 0.02 . . .).
Work on "times ten" or "ty" for many trips on both charts, with "divide by ten" and "tenth" as trips in the other direction.
* Detailed articles about working with the charts appear in 'Mathematics Teaching', 163, June 1998, a journal of the Association of Teachers of Mathematics.
* A large, double-sided chart is available on tough white plastic sheets. Cost is Pounds 6.50 for one, with discounts for quantity. Contact: Education Initiatives, Cardew Farm, Dalston, Carlisle CAS 7JQ. Tel: 01228 710 661. Fax: 01228 711 090.
* Association of Teachers of Mathematics, 7 Shaftesbury Street Derby DE 23 8YB. Tel: 01332 346599. Fax: 01332 2O4357.
* Geoff Faux is co-ordinator of the ATM's science of education working group.