Thde Development of the Concept Mapping Tool and the Evolution of a New Model for Education: Implications for Mathematics Education

Introduction When I began my graduate studies in 1952 at University of Minnesota, the only psychology of learning presented was behavioral psychology, based largely on research with rats, cats and other animals. The only philosophy of knowledge, or epistemology, I was taught was logical positivism, for which the Philosophy Department at Minnesota was world famous. I did not see much value in behavioral psychology as a theory to guide research on human problem solving and ways to enhance this ability, which was the subject of my PhD thesis. Nor did I see value in a view of knowledge creation that centered on proving axioms and logically deriving new knowledge from basic premises, a view that did not appear to apply to the work I was doing in laboratory research in the Botany Department. Although there was the work of Barlett (1932) theorizing on how cognitive learning takes place, and the extensive work of Piaget beginning in 1926 describing how children's cognitive operations advance over time, I was taught none of this. I did, however discover the writing of Conant (1947) and his ideas on how the sciences create new knowledge. Later his protege, Kuhn (1962) would expand Conant's ideas in his enormously popular, The Structure of Scientific Revolutions. Lacking a psychology of learning that made sense to me, I chose to base my research on Wiener's (1948) cybernetic ideas, and we continued with these ideas until our research data failed to fit the theory. Most fortunately for us, Ausubel's (1963) cognitive psychology of learning was published about this time, and we embraced this as a foundation from 1963 onward. Today cognitive learning theories have essentially replaced behavioral theories, although much school learning still proceeds on behavioral learning principles, such as repetition and reinforcement. This is also evident in the common drill and practice observed in mathematics classrooms. One of the issues debated in the early 1960s was the extent to which children could profit from instruction on abstract, basic science concepts such as the nature of matter and energy. The dominant thinking in science education and development psychology was centered on the work of Jean Piaget (1926), particularly his ideas about cognitive operational stages. Piaget had devised some ingenious interviews administered to children, the results of which could be interpreted to support his theory of stages of cognitive operational development. It was widely assumed that children could not profit from instruction in such abstract concepts, such as the nature of matter and energy, before they reached the formal operational stage of thinking at ages 11 or older. Similar misperceptions are common with math educators who do not think young children can understand the basic concepts behind math procedures, or they may not even be aware of these concepts. The fundamental questions that concerned me and my research group were: 1. Are these claimed cognitive operational limitations of children the result of brain development, or are they at least partly an artifact of the kind of schooling and socialization characteristic of Piaget's subjects, and those commonly tested in US and other schools? 2. With appropriate instruction in basic science concepts such as the nature of matter and energy, can six to eight year-old children develop sufficient understanding to influence later learning? 3. Can the development of children's understanding of science concepts be observed as specific changes in their concepts and propositions resulting from the early instruction and from later science instruction? 4. Will the findings in a longitudinal study support the fundamental ideas in Ausubel's (1963) assimilation theory of learning? Answers to these questions could only be obtained by first designing systematic instruction in basic science concepts for 6-8 year-old children, and then following the same children's understanding of these concepts as they progressed through school, including later grades when formal science courses were taken. …