Multiplication and repeated addition

In mathematics education, there was a debate on the issue of whether the operation of multiplication should be taught as being a form of repeated addition. Participants in the debate brought up multiple perspectives, including axioms of arithmetic, pedagogy, learning and instructional design, history of mathematics, philosophy of mathematics, and computer-based mathematics. In the early 1990s Leslie Steffe proposed the counting scheme children use to assimilate multiplication into their mathematical knowledge. Jere Confrey contrasted the counting scheme with the splitting conjecture. Confrey suggested that counting and splitting are two separate, independent cognitive primitives. This sparked academic discussions in the form of conference presentations, articles and book chapters. The debate originated with the wider spread of curricula that emphasized scaling, zooming, folding and measuring mathematical tasks in the early years. Such tasks both require and support models of multiplication that are not based on counting or repeated addition. Debates around the question, "Is multiplication really repeated addition?" appeared on parent and teacher discussion forums in the mid-1990s. Keith Devlin wrote a Mathematical Association of America column titled, "It Ain't No Repeated Addition" that followed up on his email exchanges with teachers, after he mentioned the topic briefly in an earlier article. The column linked the academic debates with practitioner debates. It sparked multiple discussions in research and practitioner blogs and forums. Keith Devlin has continued to write on this topic. In typical mathematics curricula and standards, such as the Common Core State Standards Initiative, the meaning of the product of real numbers steps through a series of notions generally beginning with repeated addition and ultimately residing in scaling. Once the natural (or whole) numbers have been defined and understood as a means to count, a child is introduced to the basic operations of arithmetic, in this order: addition, subtraction, multiplication and division.