It's all in the mind
Me: What is 5 add 7?
Sarah: (Hesitantly) 12.
Me: And how did you do that?
Sarah: (Nervously) I did it in my head.
Me: Can you explain how?
Sarah: Just in my head.
Me: Can you explain what came into your head?
Sarah: Well (somewhat embarrassed) I counted the books.
Me: I'm sorry Sarah, but I don't understand.
Sarah: (Going red) I looked up there and saw those five books. So I counted themI together in my headI and then two more.
Me: You counted them together?
Sarah: Well, I counted them over againI Me: Oh. Twice? (that is, 5 + 5) Sarah: Yes, and then two more. (5 + 5 + 2) Me: (Stunned by the originality of the calculation) Can you think of another way of working out 5 add 7?
Sarah: Well (somewhat disparagingly), I used to use my fingers.
Me: But you don't use them any more?
Sarah: No (indignantly), I do it in my head now!
This conversation still fascinates me. Why does "counting on your fingers" seem such a bad idea? And what does "working in your head" really mean to Sarah? Equally important, perhaps, what does "working out in your head" mean to teachers? I remember the advice given to me by a senior member of staff when I was a young teacher: "Get them to sit on their hands. That'll make them work things out in their heads."
I didn't believe this advice had any sound basis, but neither did I understand how mental activity was developed. In some naive way I assumed that children acquired this capacity through capability in, and reflection on, paper and pencil activity. They don't! Yet the issue it raises is an important one. My research has shown that many children use very simple counting strategies when undertaking quite complicated number operations, even at Year 6, and most of these go undetected because pupils seem increasingly reluctant to show their working out as they get older. These children seem to experience difficulties when solving oral questions without recourse to pencil and paper.
Fingers are a powerful counting tool, but are not as flexible and as efficient as the calculating applications of number bonds and place value. So what are some of the stages in becoming competent with calculations? An important step in moving from counting to calculating is gained with a thorough grasp of "doubles" (1 + 1, 2 + 2, etc) and "near doubles" (5 + 6 seen as 5 + 5 + 1, etc). These strategies have a long and successful history of being employed by infant teachers through varied activities and games, supported by discussion. This, combined with a growing understanding of the relationship between addition and subtraction using a number line, can offer flexibility that is denied by a premature introduction to written calculations in traditional vertical format.
For many children, a sustained approach to mental arithmetic at key stage two might be more beneficial than a diet of vertically represented computations worked out by pencil and paper using standard algorithms. But it is important to target which children require support, and agree on its nature. I have worked with some five-year-olds who understand many aspects of number without any formal teaching.
These children seem to pick up the logic of the place value system quite intuitively. Anna is in a reception class.
Anna: Can you work out 10 + 6?
Me: Well, it's 10 (pointing to my head) and 6 (putting up six fingers) so that's 10, 11, 12, 13, 14, 15, 16.
Anna: (Ignoring my actions) I thought so! What's 20 + 6?
Me: Well, that's 20 (again pointing to my head, but with less conviction) and 6 (putting up six fingers) so that's 20, 21, 22, 23, 24, 25, 26.
Anna: It's really easy isn't it?
Me: Is it? How would you do 30 + 6?
Anna: That's 36. It's like the same.
Before I could question her further, Anna's mind had jumped to another question: Anna: What's 5 + 5 + 5 + 5?
Me: That's 20 (flashing two sets of hands).
Anna: I thought so. It's like 10 + 10!
As I was planning what to ask next, Anna rushed over to the construction area leaving me drained and feeling inadequate. Later I discovered from the class teacher that Anna had a thorough grasp of doubles, and a growing awareness of place value, as is clear from our discussion. Anna is having "fun" playing with numbers, and subsequent teaching needs to build upon her developing capabilities. Sadly, many adults have told me of "insensitivity" and "indignation" when discussing calculations.
Claire, a manager for a publishing firm, explained: "I can remember working out a whole page of 'sums' and the teacher marking everyone wrong because I didn't show my working out. He stood in front of me and put a cross next to every one. I did them in my head and I could not understand how to do them any other way. I thought I was good at maths. I've hated that teacher ever since."
You could see from her expression, normally one of enthusiasm, that Claire's attitude to that teacher had never recovered. We bear a huge responsibility. Identifying mental strategies is not easy. Generalising about them for use in our teaching is difficult and open to speculation. But there is a treasure trove to be found by the confident teacher in an open approach. It is a confidence that we need to build. Current pressure on "knowing" can undermine attempts at "discovering".
An important stage in developing calculations is the ability to "transform" numbers - the ability to change to an equivalence that is more suitable for mental manipulation. For example 56 + 29 transformed to 56 + 30 - 1. Transforming numbers builds upon doubles and near doubles, capitalises on the use of number bonds, uses the ability to manipulate tens separately from units, and requires rounding up and down without losing the equivalence. All of this reduced to a number of steps to be performed and held in short-term memory.
The more fluent the children are with each skill, the more adaptable they will be given any set of numbers. Those struggling with this stage often need help in deciding what to do with the "bits" - the parts of numbers used for maintaining equivalence. Pressure on accuracy can lead them back to paper and pencil.
So finally, a brief word about mental methods and written computation. Many schools have successfully used an approach in mental work that is separate from written calculations. For example, children gain skills in "counting on" to work out mental subtraction, but use decomposition for written work. Other schools are now trying to adopt an approach to written calculations that is more related to the approach required of effective mental work. Both approaches, if used successfully, will mean that children are not "all fingers and thumbs" when doing calculations. You can count on it!
John Sharpe, a former Cambridgeshire headteacher, is a High Scope nursery trainer and a freelance primary adviser. He is in charge of the mathematics courses for the county and wrote the television series Wondermaths.