There is much discussion of ‘mastery’ in primary schools currently, so, I was fascinated to read an article by Madeleine Goutard in Mathematics Teaching 18 from 1962. She was pioneering the use of the Cuisenaire-Gattegno method with 8/9 year-old children who had been using the rods for 5 months. She describes how the class were given a test which included examples of division they would not have come across before. The children were told it did not matter if they did not answer these questions as they had not been taught division yet. However, when she collected the test in many of the children had answered the questions correctly. Intrigued, she gave the children blank sheets of paper and asked them to write down how they had carried out the calculations 235 ¸ 5 and 402 ¸ 6. A couple of examples of their responses are:
There are five 5s in 25. So, in 50 there are 10 and in 100 there are 20 and in 200 there are 40. 35 is 7 x 5 so 235 ¸ 5 = 47.
There are 8 6s in 48 so there are 16 in 96 and 32 in 192 and 64 in 384. This leaves 18 and there are 3 6s in 18. So 402 ¸ 6 = 67
The full article contains 12 explanations from pupils about the methods they had used. Madeleine Goutard suggests that they discovered methods of division through their mastery of the distributive law. What goes around?
Mathematics Teaching is the journal of The Association of Teachers of Mathematics and is available at www.atm.org.uk. The full article described above will be available in MT256 and is also in the archives on the ATM website.
mathematicsteachingblog 