Theories of Learning Mathematics

Abstract

Definition According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20 th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. So, for healthy theories on development to occur, four Darwinian functions should function: (a) variation – avoid orthodoxy and encourage divergent thinking, (b) selection – submit all assumptions and innovations to rigorous testing, (c) diffusion – encourage the shareability of new and/or viable ways of thinking, and (d) accumulation – encourage the reuseability of viable aspects of productive innovations. Characteristics The history and nature of theory development To describe the nature of theories and theory development in mathematics education, it is useful to keep in mind the preceding four functions, and to focus on two books that have been produced as key points during the development of mathematics education as a research community: Critical Variables in Mathematics Education (Begle, 1979) and Theories of Mathematics Education (Sriraman & English, 2010). Begle was one of the foremost founding fathers of mathematics education as a field of scientific inquiry; and, his book reviews the literature and characterizes the field when it was in its infancy. For example, before 1978, the USA's National Science Foundation had funding programs to support curriculum development, teacher development, and student development; but, it had no comparable program to support knowledge development (i.e., research). Similarly, before 1970, there was no professional organization focusing on mathematics education research or theory development; there was no journal for mathematics education research; and, in the USA, just as in most other countries, there existed no commonly recognized curriculum standards for school mathematics. Furthermore, most mathematics educators thought of themselves as being curriculum developers, program developers, teacher developers, or student developers (i.e., teachers) – and only secondarily as researchers. And, if any theories were invoked to guide their research or development activities, these theories were mainly borrowed from educational psychology such as: Bloom's taxonomy of educational objectives, Gagne's behavioral objectives and learning hierarchies, Piaget's stage theory, Ausabel's advanced organizers and meaningful verbal learning -and later Vygotsky's socially-mediated learning, and Simon's artificial intelligence models for cognition. However, the practitioners' side of these mathematics education researchers made it difficult for them to ignore the fact that very few of their most important day-to-day decision-making issues were informed in any way by these borrowed theories.