Counting On
The Counting On web site has been developed to provide educators with an overview of the Counting On project.
Mental computation
Most calculations we carry out each day we do in our heads. With some calculations we feel the need to reach for paper and pencil or a calculator. Yet when we play a game of darts or cards, or even watch a league game, we rely on carrying out lots of mental calculations. How many points are needed to win, how many converted tries will put us in front and is there enough time?
Social club activities, travelling, sharing a bill, shopping, playing or watching sport and preparing a meal, all involve invisible mathematics ? mental calculations. The methods that we use when we work things out in our heads are often not the same methods that we use when we write down a sum. So how do we learn to do this invisible mathematics?
The empty number line
New teaching methods are being used as part of the Counting On project across the state. For the first time ever, for many students, the invisible mathematics is starting to become visible. Teaching in the Counting On project emphasises using problems and ways of recording that allow students to share what they are thinking.
Think about adding together two numbers in your head, say 157 and 36. When this addition is carried out mentally, we often start from the left rather than from the right. To get the students to show their thinking, the teachers in the Counting On project have been using an empty number line.
Each student can show how he or she thought about the problem by drawing and filling in an empty number line.
A number line showing 30 and 6
One of the interesting things about mental calculations is that we do not all think the same way. Some people start by breaking the 36 into 33 and 3. This turns the question into the problem of adding 33 to 160.
A number line showing adding 3 and then 33
Comments from the teachers
The empty number line is only one of the innovative teaching methods being used in the Counting On project. Some teaching ideas have been developed from international research while others are distinctively home grown. One thing about the new methods is clear — they appear to be working.
Jennie Kidd from Liverpool Boys' High School had this to say about the project: "Counting On has really worked on the students' understanding of place value and improved their confidence in working with large numbers. It is wonderful to hear a student saying excitedly, ’Oh Miss, I can do that now.’"
These comments were echoed by Safija Poricanin, a teacher involved in Counting On at Kogarah High School. "I think Counting On is a terrific program because the students have become so excited and enthusiastic about the work. It shows what they are thinking and they’re achieving success."
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