In Theories of Mathematics Education: Seeking New Frontiers (Advances in Mathematics Education), Paul Ernest gave a brief synthesis of four major learning theories in mathematics education in his article Reflections on Theories of learning. He calls them theories of learning but he also acknowledges that they are more appropriately called philosophies of math education. I’m sticking to learning theories. It’s less scary and makes me conscious that they are for framing teaching and learning studies and practice. In summarizing the ‘learning theories’ Paul Ernest in fact described each in the context of practice and research.
The four theories are Simple Constructionism, Radical Constructivism (Piaget, von Glasersfeld), Enactivism (Varela), and Social Constructivism (Vygotsky).
Simple constructivism suggests the need and value for:
(1) sensitivity towards and attentiveness to the learner’s previous learning and constructions,
(2) identification of learner errors and misconceptions and the use of diagnostic teaching and cognitive conflict techniques in attempting to overcome them.
Radical constructivism suggests attention to:
(3) learner perceptions as a whole, i.e., of their overall experiential world,
(4) the problematic nature of mathematical knowledge as a whole, not just the learner’s subjective knowledge, as well as the fragility of all research methodologies.
Enactivism suggests that we attend to:
(5) bodily movements and learning, including the gestures that people make,
(6) the role of root metaphors as the basal grounds of learners’ meanings and understanding.
Social constructivism places emphasis on:
(7) the importance of all aspects of the social context and of interpersonal relations, especially teacher-learner and learner-learner interactions in learning situations including negotiation, collaboration and discussion,
(8) the role of language, texts and semiosis in the teaching and learning of mathematics.
Each one of these eight focuses in the teaching and learning of mathematics could legitimately be attended to by teachers drawing on any of the learning theories for their pedagogy, or by researchers employing one of the learning theories as their underlying structuring framework.
I agree with Campbell that there are as many theories as theorists: Theories are like toothbrushes… everyone has their own and no one wants to use anyone else’s.