A search for understanding

Abstract

We discuss an important breakthrough for us in our search for a technical meaning of “understanding” in mathematics education. In this article, we describe the background to this discovery, the catalyst for the breakthrough, and a concise explanation of our new definition. In discussing the consequences of this definition, both in terms of the theoretical implications for the internal characteristics and external manifestations of understanding with some initial practical consequences for a teacher's attempts to model a learner's understanding, we begin to describe our ongoing search for a more comprehensive theory. Much of our work during the past 5 years has been devoted to developing a theory of learning which accounts for our own experiences as learners, teachers, and researchers. The theory we have developed in which we identify learning experiences as natural, conflicting, or alien, and consider the ways in which learners might respond to them (Duffin & Simpson, 1993) has been used to analyze a number of learning incidents we have encountered (Duffin & Simpson, 1995). In the course of such analyses, we found that the word “understanding” often entered our discussions and eventually, we felt compelled to seek a definition of the word, which would fit within the basic framework of our theory and might enable us to make more sense of the incidents we encountered. The quest for such a definition has occupied us for more than 3 years and we will use this article to explore how we came to our current position. The article is not focussed on understanding, its focus is on the journey we undertook to make sense of a particular term (which happens to be “understanding”) in the context of our on-going development of a personal theory of learning. As such, the article contains almost as much about what we later came to see as mistaken (or to replace) as it does about the terminus of our journey: our current definition of understanding.