Behaviorism posits that learning is a form of conditioning or training. Instructors associate correct answers or other “good” student behavior with rewards, and incorrect answers or “bad” student behavior with punishments. Over time, students internalize the connection between the right behavior and being rewarded. Though mostly considered old-fashioned nowadays, behaviorist methods can be helpful in the short term -- for example, rewarding students for completing boring but necessary tasks such as drills or memorization.
Constructivism views learning as a cumulative process in which new knowledge is built on or connected to prior knowledge. According to this view, learners start with certain innate mathematical knowledge -- the ability to recognize the differences between small integers, for example -- and are active participants in the construction of further knowledge. Problem-based learning, in which students learn about a given subject through the experience of solving problems, is a common constructivist teaching method.
Social development theory, developed by Russian psychologist Lev Vygotsky, claims that social learning precedes individual learning. According to Vygotsky, every learned behavior appears twice: first on the social or interpsychological level, and then on the individual or intrapsychological level. Traditional mathematics teaching tends towards a “transmissionist” model, in which the instructor transmits knowledge to students. In contrast, social development theory argues for a collaborative, reciprocal relationship between instructor and students.
Situated learning is one of the newer theories in education, dating from the 1980s. Situated learning claims that learning cannot be separated from its context, and that authentic learning takes place in authentic situations. Proponents of situated learning argue that most classroom learning only teaches students how to respond to the situation of the classroom; the closer the students’ tasks are to “real life,” the truer the education. Situated learning in mathematics focuses particularly on applied rather than pure math.