Summary of the characteristics. As described before, the history of the development of the theory of learning and teaching multiplication shows a drastic evolution. The content of the theory varies from a collection of simple concepts, rules and views, dominated by associative psychology and behaviorism, to a refined, content knowledge based system of concepts, suggestions and guidelines. In all cases there is a strong connection between the theory and ‘the practice of education.’ The relationship with that practice becomes a solid and specific one only after the 1970s. Xxxxxxxxxxx’x mental legacy, specifically his didactic phenomenology of mathematical structures, stimulates the experts involved (developers, researchers, supervisors, teachers) to change their views. Thinking in terms of increasing complexity of the material offered to students changes to thinking about progress in students’ learning processes. Development of theory on that largely occurs as reflection on practical experiences during developmental research. In that respect the theory can be seen as beginning with reflection on practice, which then evolves into a theoretical foundation and justification of that practice. The reciprocal relationship between theory and practice is determined to a large extent by a theory-led, cyclical process of brainstorming, designing, trying, analyzing and reflecting (from the theory); the designers themselves, as an exalted kind of ‘bricoleurs’ keep this process going, while positioning themselves as learners. The description in the TAL brochure, even more than for Ter Xxxxx and in ‘De Proeve,’ is characterized as a practical theory; that character is partly determined by the practice- oriented goals, as represented in activities, and the immediately included theoretical justification (called ‘theory in activities’). It is a characteristic that is associated with the description Xxxxxxx and Xxxx (1997) give of practical theory19. The evolution of the theory for learning to multiply is influenced by its relationship with practice from another point of view as well, as debate and looking for consensus among those involved play an important part in the relationship between theory and practice. In the course of the process of the development of the theory there has been an increase in the consensus about the theory being developed among experts. From the 1970s onwards a close network of experts and interaction with practice develops, which leads to theoretical statements becoming gradually ...
