Deborah Mulroney is a Primary Mathematics Adviser for Herts for Learning
In this blog the resource that we are focusing on is the counting stick. It usually has ten intervals but the type with four faces and divided in several ways are most useful. The main use is that it can be thought of as representing a three-dimensional empty number line.
Primarily this resource is a very useful tool for a teacher to promote confidence and flexibility in children’s counting skills as they begin to assimilate the knowledge and understanding they can bring to their calculation as well as the new learning that will enhance their adaptivity. But it is also brilliant when used to explore concepts such as position, order and magnitude of numbers.
All children of a primary age will benefit greatly from counting and exploring position on the counting stick daily. This causes increased number sense through reasoning allowing them to clarify and test their knowledge and understanding for themselves.
You will need to begin by making the counting stick a familiar resource that you use to scaffold and challenge children’s counting and sense of magnitude by using it every day to identify numbers in a sequence from consecutive numbers starting at zero throughout the range seen in the fluency range of the year group and in steps that reflect the year group’s identified multiples or as an empty number line with benchmarks made clear. The expectation is that the children in each year group will work to increase known facts confidently and flexibility when exploring these concepts in varied starting points and continuing to explore the commutativity and divisibility beyond the known facts.
Counting and matching:
At the very beginning of its journey, the counting stick can be fully labelled, using post-its (or card squares stuck on with blu-tac) to indicate the value of each interval. As you repeat its use, the post-its are gradually removed to reinforce the progress in the children’s learning in a sequence. As the children’s confidence improves further, they can be asked to use the post-its to “re-label” the counting stick in a random order when it begins as an empty number line. This provides a further opportunity for the teacher to indicate crucial landmarks on the stick – e.g. the starting and finishing numbers. Use the middle interval as the fulcrum by which you hold the counting stick as this allows it to become a crucial landmark on the stick, upon which they can rely.
The children become increasingly less reliant upon the post-its as their immersion continues.
Once the children are confident with any of the activities above, start to use the in-between points to extend understanding. For example, if the counting stick is being used for the 9 times tables, where would 5, 50 or 62 be?
It is essential that you always hold the counting stick in the centre, touching the middle interval. This fulcrum on the stick becomes a useful calculation strategy since most of the time we are partitioning numbers flexibly in order to calculate in our heads. Some of the “benchmarks” we use are found in the columnar methods of written calculations but many are not.
Think of a number such as 55 for example and add 8. What are you doing in your head? Partitioning the 8 into 5 and 3? The children need to be immersed in rich and adaptive counting to be sufficiently equipped to achieve the fluency required.
So when the children are able to work imaginatively with the counting stick, it is at its most flexible for learning. It is effectively a 3-D empty number line and supports the children through the transition from counting to calculating and recording mathematical statements.
Change the starting numbers three or four times a day and remember to quiz the children when pointing to an interval on the stick. What do you know about this number? Don’t forget to model the high quality standard of answer you are expecting every day and try not to start the habit of: ‘teacher asks the questions and children answer’.
You are counting in threes from zero for example. Point to the fulcrum and ask:
“What do I know?”
“I know that 15 is half of 30 and that there are five threes in 15 because there are ten threes in 30.”
Then point to another interval and ask the children “What do you know?”
After significant immersion in this problem solving, the children are ready to use their own counting sticks to solve the unfamiliar. The counting stick becomes another manipulative for children to consider in linear representations of mathematical solutions. When the children are beginning to use calculation strategies more efficiently, ask the children to use the counting stick to demonstrate the chosen strategy and to justify its selection as the most efficient approach. For example:
Without doing the calculations, decide whether each answer lies between 300 and 350. In each case, explain how you decided.
150 x 2
Half of 720
90 multiplied by 4
1000 ÷ 3
When the children are asked to work independently and reason daily they are now ready to “prove it” with their counting stick!
This time hold the counting stick vertically. If the bottom is zero and the top is 1 litre (1 000ml), how many millilitres is this…and this…? How many millilitres to fill to the top? Where would 730 ml be? Use as a spring balance showing weight – this time the bottom is 0g, the top is 1 000g or 1 Kg. How many grams is this…and this…? Where would 270g be? Do the same for length. Use the stick to convert between grams and kilograms, millilitres and litres, centimetres and metres. Do the same for temperature. If this is 0, where is…? Specify Centigrade or Fahrenheit for a real challenge. For time, set a start time at the beginning of the stick and specify the time interval of each block. So if the beginning of the stick is 10 o’clock, where is ten past 11? Use digital time too.