/ 3 April 2011

# Gymnastics of the mind

In my previous column I looked at the role of investigations and the recognition of patterns in helping children to learn mathematics in general and their multiplication tables in particular.

I made the point that if children can see the underlying patterns and relationships in the multiplication facts they will know their tables in ways that do not rely as much on memory as they do on the interrelationships between the numbers. Teachers can help children develop knowledge and skills through the deliberate and careful creation of learning situations.

So-called mental maths is another example of how teachers can help children to develop their skills through the ways in which they structure the learning situations. That children should be able to do some calculations in their heads with ease is undisputed. How we help them to reach this point is where we may differ.

In some classrooms teachers conduct daily mental maths sessions, calling out a random sequence of sums and expecting children to be able to remember the answers. After many weeks of doing this and seeing little improvement in the children’s performance they give up, often complaining that the children are lazy and/or incapable of learning.

But I know that if we are more thoughtful about how we present the daily mental maths session we can help children to develop more robust insights and understanding and in so doing develop strong mental maths and mental arithmetic skills. Here is a sequence of question types a teacher should follow.

Single-digit arithmetic: At first we start with single-digit arithmetic. We start out by asking questions such as:

• What is 5 plus 1? What must I add to 5 to get 6?
• What is 6 plus 1? What must I add to 6 to get 7?
• What is 7 plus 1? What must I add to 7 to get 8?
• What is 8 minus 1? What must I take away from 8 to get 7?
• What is 7 minus 1? What must I take away from 7 to get 6?
• What is 6 minus 1? What must I take away from 6 to get 5? And so on.

As the children begin to answer these questions without having to use their fingers and can “see” the numbers near each other on their imaginary number lines, it is time to increase the amount being added and subtracted.

• What is 5 plus 2? What must I add to 5 to get 7?
• What is 5 plus 3? What must I add to 5 to get 8?
• What is 5 plus 4? What must I add to 5 to get 9?
• What is 8 minus 2? What must I take away from 8 to get 6?
• What is 8 minus 3? What must I take away from 8 to get 5?
• What is 8 minus 4? What must I take away from 8 to get 4? And so on.

With frequent daily repetition of these questions children will gain confidence and will begin to “know” the answers. As their confidence increases it is time to expand the questions. But do so in a deliberate way that helps them develop awareness of the “patterns”. The kinds of questions now include:

• What is 5 plus 2? What must I add to 5 to get 7?
• What is 15 plus 2? What must I add to 15 to get 17?
• What is 25 plus 2? What must I add to 25 to get 27?
• What is 95 plus 2? What must I add to 95 to get 97?
• What is 8 minus 2? What must I take away from 8 to get 6?
• What is 18 minus 2? What must I take away from 18 to get 16?
• What is 28 minus 2? What must I take away from 28 to get 26?
• What is 78 minus 2? What must I take away from 78 to get 76? And so on.

Children should realise that if they know what 7 minus 2 is, then they know what any number ending in 7 minus 2 is. An important caution: it is tempting to tell children about this “rule” or pattern and then expect them to both remember and use it. Sadly, this does not work. Children need to recognise and see the pattern themselves. The role of the teacher is to provide the opportunity for the children to observe the pattern by asking these questions, to ask them frequently and to ask children to describe any patterns that they observe in their own words.

As children gain confidence with so-called single-digit arithmetic it is time to introduce arithmetic using multiples of 10 (and 100 with the older children) into the daily mental maths session.

Arithmetic with multiples of 10: This is an extension of single digit arithmetic. We start by asking the questions in pairs (or threes if we are working with 100s as well):

• What is 5 plus 1? What is 50 plus 10? What is 500 plus 100?
• What is 5 plus 2? What is 50 plus 20? What is 500 plus 200?
• What is 8 minus 1? What is 80 minus 10? What is 800 minus 100?
• What is 7 minus 2? What is 70 minus 20? What is 700 minus 200?

Arithmetic with 10s (and 100s) is an extension of the single-digit arithmetic pattern and yet many children never realise this, largely because their teachers never expose them to the pattern. Again it is not enough for teachers to tell children about the pattern. Teachers need to expose children to the pattern daily and in different ways.

Doubling and halving is another powerful tool that underpins mental arithmetic in general and mental calculations involving multiplication and division in particular. In my previous column I showed how doubling and halving helped children find relationships and patterns in multiplication facts and in so doing helped them to know their multiplication tables in more robust and meaningful ways.

As children gain confidence with so-called single-digit arithmetic and arithmetic involving multiples of 10 (and 100), teachers can include the completing of tens into the daily routine.

Completing of 10s and 100s:

• What is 8 plus 2? What must I add to 8 to get 10?
• What is 18 plus 2? What must I add to 18 to get 20?
• What is 28 plus 2? What must I add to 28 to get 30?
• What is 12 minus 2? What must I take away from 12 to get 10?
• What is 22 minus 2? What must I take away from 22 to get 20?
• What is 32 minus 2? What must I take away from 32 to get 30? And so on.

We should expect intermediate and senior-phase children to answer questions such as the following with confidence:

• What is 76 plus 24? What must I add to 76 to get 100?
• What is 83 plus 17? What must I add to 83 to get 100?
• What is 28 plus 72? What must I add to 28 to get 100? And so on.

Bridging 100s is a natural progression from the completing of 10s. If a child knows that 8 plus 2 is 10 then they find it easy to calculate 8 plus 5 because they know that it is the same as 8 plus 2 plus 3, which is the same as 10 plus 3, which is 13. They learn not to add all of the five at once but rather to break it up in ways that enable them to complete the 10 first and then to add what is left.

• What is 8 plus 2?
• What is 8 plus 5?
• Tell me how you did that?
• What is 28 plus 2?
• What is 28 minus 5?
• Tell me how you did that?

The development of strong mental arithmetic skills in children requires that teachers structure situations in very deliberate ways to enable child­ren to observe the underlying patterns. Remember that children need frequent exposure over an extended period of time before they begin to internalise and use the patterns without consciously having to think about them.

Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za