Louise was using a number grid to plot the results of repeating a function (for example +3). She was using the calculator constant ([+][+][=]. ..[=]...[=]) to generate a pattern.
I had chosen the calculator because I wanted her to work with the symbols on the calculator and the grid. This meant she had some support "counting on" on the grid and through the patterns generated, but was working with number symbols rather than counting objects such as cubes.
Louise realised she was filling in a pattern on the grid. I asked her to suggest a number that might be coloured in soon. She pointed to 63 and confirmed this with the use of the constant. She then asked if she could colour in the pattern without using the calculator. I said she could as long as she checked with the calculator afterwards. She did this and was pleased with the result.
Finishing the task slightly before the other children in the group, Louise wanted to know how to work "times sums" on the calculator. I asked her for a times sum she knew so she could check the answer for herself. She offered 3 x 2. I showed her which keys to press and she was delighted when 6 appeared on the display. She tried several multiplications and became increasingly confident with the [x] key. I knew Louise had been introduced to multiplication through grouping and that she had learned some tables. So I took this as an informal opportunity to link this knowledge to notation.
She was doing so well that I decided to introduce her to the inverse. As it turned out she made a connection between [#214;] and sharing, rather than 6 #214; 2 = 3 being the inverse of 2 x 3 = 6. This was new mathematical territory for Louise, but she was confident and involved. I asked if she knew what [#214;] meant? She said she didn't know. I told her it meant divide, or share, and suggested she put 6 on the display and then press [divide sign] to divideshare by  makesequals [=]; 3 appeared. Louise was familiar with simple sharing of objects, but not this abstract symbolisation, where the calculation disappears before you have finished the sum.
Taking a step back into the slightly less abstract world of mental images,I suggested we repeat the sum but this time she should picture 6 buns shared between 2 bears. How many would each bear receive? She was sure this was 3, and the 3 duly appeared. I set her further buns and bears problems, taking care that the numbers were small enough for her to imagine and had whole number answers.
Louise was finding it hard to trust the notation and accept the pure mathematics of inverse but offered an elementary understanding of sharing. I followed that lead and because the notation was not secure, introduced mental imaging to support the formal calculation.
At this point I decided to go a step further. I hoped to move Louise into territory which might be beyond her, but as I was with her and she was on a roll I took the chance. I suggested there were 11 buns and 2 bears. How many would they each get? Louise decided 51#218;2 (fortunately the image chosen, the bun, could be divided as well as remaindered). She divided 11 by 2 on the calculator and read out 55. I said that seemed rather a lot as we had only 11 to start with. She thought about this and said it should be 51#218;2, using her confidence in mental calculatio n to support her argument. I showed her the decimal point and explained that .5 was the calculator's way of saying half. We did two more bear and half-bun calculations before finishing.
I would not anticipate Louise being able to do decimal division independently because it was a first experience, carefully structured, and lying at the periphery of her mathematical knowledge. She had already moved into the unfamiliar with the calculator keys [x] and [#214;], into multiplication and simple division. Why take her this far then? We must never underestimate pupils' abilities. Sometimes mathematics is presented like a narrow country lane, with the route carefully prescribed and terribly safe. Louise was playing in the field alongside and glimpsing the road ahead. Children need challenge at the limits of their knowledge but in this case the teacher could ensure a confident and successful experience.
The calculator made the exercise possible. If it were not for the rapid calculation, Louise's thinking would have been fragmented and lost in the laborious procedure of recording. Here, mental imaging, previous knowledge and abstract symbol were being confidently linked.
Within this session the calculator was used in two ways: the first carefully planned to promote the use of notation, the second as a medium for a spontaneous exchange of mathematics. In both situations the children had an appropriate level of prior knowledge to allow purposeful use of the calculator.
The move to trusting notation was important. Louise's responses offer a good example of a child using the calculator to consolidate prior knowledge in an increasingly abstract situation as well as advance into new mathematical territory. Louise had to move between the abstract and the concrete - in this case mental images - to confirm the results shown on the calculator. It also shows the need for teacher support and decision-making when a child is working at the outer limits of his or her knowledge.
Margaret Sangster is senior lecturer in the school of education at Sheffield Hallam University