So who's cheating whom?
On completion of the allotted time he announced that the work was done and was reluctant to bring it to show me. I was told not to tell him whether it was right or not, as this would be cheating. He sat with his fingers in his ears in case I should be tempted.
Of course I was tempted to make some comment - at the very least to the effect that it wasn't cheating. At a glance it wasn't obvious whether some of the results were right so I asked him to explain how he got the answers. He misunderstood and said he could not do them without a calculator; neither could I without recourse to a considerable amount of pencil and paper calculation. But he was most decidedly of the opinion that to have used a calculator was to have cheated.
Now it was me who was fuming, but not outwardly. I felt it most important that this child who was certainly not a failure in mathematics and who could confidently do mental calculations using his understanding of numbers rather than the algorithms his primary teachers had failed to implant successfully, should not be left believing himself "no good at maths". He must be no good, he says, as he is only in the second set. If he thinks it is cheating to use a calculator what do others at his school (one of the so-called top comprehensives in the country) think?
I had an idea - I looked up equations in my fifth year mathematics book (sorry, Year 11 book). There in Exercise 26a of Ordinary Level Mathematics by L H Clark (no relation) were the following questions: A selection from the questions set for my son's homework are: It doesn't take long to spot the differences. In my Year 11 work we see only integers from 1 to 6 as the numbers used in the equations, whereas the current exercise intended for Year 9 pupils contains a motley mixture of numbers both integer and containing decimal fractions; the numbers in the example ranging from -2.9 to 864.
On glancing at the questions from my text book my son was willing and quick to demonstrate that he could easily do these questions without a calculator. He was impressed to think he could do Year 11 work ... I didn't disappoint him by telling him that Year 11 work is not what it was and he will meet more difficult questions than mine.
This interlude is interesting because of the number of current issues relating to maths education it raises, but the one it has most made me consider has been the role of the calculator in mathematics education.
Society in general is quick to agree with my son (and presumably his mathematics teacher) that using a calculator is cheating. But how many have thought carefully about the possible assistance in learning mathematics that the calculator can give?
The latest version of the mathematics national curriculum helps to clarify this in one of the statutory "opportunities" stated in the Number section of the Key Stage 2 programme of study where it is stated "Pupils should be given opportunities to: . . . use calculators . . . as tools for exploring number structure and to enable work with realistic data". My son's homework equations could easily have been equations modelling some scientific process with the "untidy" sorts of number which come from such situations. With the use of the calculator he could cope with realistic data; without it he could not.
And what about that other use of the calculator "as a tool for exploring number structure"? An interesting example of this turned up in a lesson of mine recently when I was working with a class of Year 7 pupils (mixed ability grouping) and had given them graphic calculators to generate number sequences. They had the graphic calculators so that they could see the sequence of numbers appearing as a list on the large screen. Two girls reached the point where their sequence was following the rule of subtracting a constant number and in their books they had recorded a list of numbers: 4, 1, -2, -5, -8, -11, but then stopped in confusion. One of them said that the numbers should have been getting smaller as they were taking away but they seemed to be getting larger. A golden opportunity for me to explain the labelling of negative numbers on the number line.The calculator enabled them to question and learn in a way that the old text book examples could never have done.
Mary Clark is a curriculum development adviser in Hertfordshire