Genesis and development of the theory. The period until 1970: the mechanistic theory It is only recently that changes have appeared in the vision on learning multiplication. Until the1970s ‘drill and practice’ were considered obvious for learning the tables of multiplication. In fact, for centuries, learning to multiply had consisted of little more than learning a number of arithmetic rules which had be acquired through ‘demonstrating, imitating and practicing.’ From the research of Kool (1999) into Dutch arithmetic textbooks from the fifteenth and sixteenth century, we know that at that time teaching multiplication mainly involved giving the students a definition of multiplying and a table with table products. The purpose in laying the foundation for multiplying was memorizing knowledge that could be used for arithmetic algorithms. In fact understanding of the procedures was not required for using algorithms either, as can be seen in the work of Xxxxxx Xxxxxxxx, who is an example of a Dutch schoolmaster from the seventeenth century. He became famous with his textbook ‘Cijfferinge’ (published in 1604; see Xxxxxxxx, 2005). His books reflected the arithmetic habits of that time and would remain influential for centuries to come. The expression ‘according to Xxxxxxxx’ typifies the arithmetic of that period; the schoolmaster had to explain how the rule was ‘according to Xxxxxxxx,’ this is ‘according to the rules of the art of arithmetic.’ This meant instrumental explanation: the arithmetic procedures were first told and then practiced through doing an impressive amount of sums. Teaching understanding was considered unnecessary, and even a waste of time. Students had to learn to do arithmetic as quickly and as well as possible for daily life. Adults needed to train for professions such as clerk, teacher or surveyor; these involved skill and certainty, rather than the underlying explanations. There was no benefit in understanding arithmetic, that was for scientists. Associative psychology and behaviorism have had long and intense influence on that ‘practice.’ Barely thirty years ago, there were still textbooks in use that were based on a mechanistic foundation. One example of a much-used mechanistic method is ‘Naar zelfstandig rekenen (Towards independent reckoning)’ from the 1950s. Even today’s education still contains elements of that approach. The following page from ‘Naar zelfstandig rekenen’ illustrates that textbook’s approach (fig. 2.1). 2 1x2=2 2+2 2x2=4 2+2+2 3x2=6 2+2+2+2 4x2=8 2+2+2+2+2 5x...
