LEARNING THEORIES

Applying Learning Theories

Abstract

We do have research reports that encourage us but, in the end, teaching, like parenting, is an activity in which we can never really know how effective were our efforts. We can only try as hard as we can to determine and implement what seems to us to be the most effective approaches, and then hope for the best.

Applying Learning Theories

Theoretical Framework

Our theory begins with an hypothesis on the nature of mathematical knowledge and how it develops. An individual's mathematical knowledge is her or his tendency to respond to perceived mathematical problem situations by reflecting on them in a social context and constructing or reconstructing mathematical actions, processes and objects and organizing these in schemas to use in dealing with the situations. There are a number of important issues raised by this statement, many relating to assessment.

For example, the fact that one only has a "tendency" rather than a certainty to respond in various ways brings into question the meaning of written answers in a timed exam(Caffarella, Merriam, 2007). Another issue is that often the student perceives a very different problem from what the test-maker intended and it is unclear how we should evaluate a thoughtful solution to a different problem. The position that learning occurs in response to situations leaves very much open the sequence of topics which a student will learn.

Learning Strategies

In fact, different students learn different pieces of the material at different times, so the timing of specific assessments becomes important. Finally, the position that learning takes place in a social context raises questions about how to assess individual knowledge.

The last part of our hypothesis relates directly to how the learning might actually take place(Daley, 2006 ). It is the role of our research to try to develop theoretical and operational understandings of the complex constructions we call actions, processes, objects and schemas (these technical terms are fully described in our publications) and then to relate those understandings to specific mathematical topics.

Given our understandings of the mental constructions involved in learning mathematics, it is the role of pedagogy to develop strategies for getting students to make them and apply them to the problem situations. Following is a list of the major strategies used in courses that we develop. * Students construct mathematical concepts on the computer to foster direct mental constructions and provide an experiential base for reflection.

* Students work in ...