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

Beginning of article

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. This was the instructional development and research project we set out to do.

To avoid problems associated with elementary school teacher's limited knowledge of science and limited time for instruction, we developed audio-tutorial instructional materials in which children were guided by audiotapes that we had developed and that were supplemented with pictures, film clips and equipment. The audio-tutorial lessons were based on ideas in the National Science Teachers Association report, Importance of conceptual schemes for science teaching (Novak, 1964), and an elementary science textbook series, The World of Science (Novak, Meister, Knox, & Sullivan, 1966). Twenty-eight lessons were developed that dealt with the particulate nature of matter, energy types and energy transformations, energy utilization in living things, and other related ideas. For the most part, these kinds of concepts are rarely presented to elementary school children, especially to 6-8 year olds in grades one and two. Figure 1 shows an example of an early lesson on energy transformations. All lessons provided audio-guidance through manipulation of materials in the carrel and other observations, including occasional "loop films" showing animations or time-lapse photography.

[FIGURE 1 OMITTED]

The key principle of the Ausubelian learning theory we considered in the design of our lessons is stated in the epigraph to his 1968 book:

  If I had to reduce all of educational psychology to just one
  principle, I would say this: The most important single factor
  influencing learning is what the learner already knows. Ascertain this
  and teach him accordingly.

As my graduate students and I developed an idea for a new lesson, we would interview 6 to 8 primary grade children in an open ended interview, usually using some of the "props" we were planning to use to teach the central concepts of the lesson, such as pictures, materials to be manipulated, loop films or apparatus we were considering. These interviews gave us some idea of what anchoring concepts most of the children already had, and also gave some preliminary feedback on how they were interpreting or using the props. This process was often repeated several times, and again after lesson prototypes were developed. On average, each lesson underwent 6 to 8 revisions before it was deemed ready to use in classrooms. We also considered Ausubel's ideas of progressive differentiation and integrative reconciliation in designing the lessons and lesson sequences (see the section on concept mapping for further discussion of these ideas). The idea of progressive differentiation requires that students build upon their prior relevant concepts, and elaborate concepts in earlier audio-tutorial lessons in a sequence as they study later related lessons. This required that some students needed to experience earlier lessons in a sequence before we could use these students to help develop later lessons. Furthermore, many concepts were revisited in later lessons, but with different examples or props to effect greater differentiation of concepts introduced earlier, and thus also to achieve integrative reconciliation of concepts that may have been initially confusing to a child or where meanings acquired may have been somewhat distorted. Photos and loop films were selected or constructed in many cases to serve as advance organizers. That is, we would use things that were familiar to the students, and we would build on the familiar to point them to see new aspects or dimensions of the new materials observed, much of this through the audio guidance.

Methodology of the 12-year Study

Ithaca Public Schools had 13 elementary schools, and for logistic reasons we chose to work with first grade teachers in five schools that were representative of the school district. A carrel unit was set up in the corner of the classroom of each of the participating teachers and 191 students in all took turns doing the lessons. These were our experimental or "Instructed" students, so called since very little science is taught in primary grades in Ithaca schools. In the second year of the study, we began to interview 48 students in the same classrooms and with the same teachers as the previous year, but these students did not receive the lessons. This was our control or "Uninstructed" sample.

The lessons were placed in carrel units, usually in a corner of the classroom. The class teacher determined the time provided for student involvement with the lessons, but most often this was during "seat-work" times, or when the teacher was working with small reading groups. Students, one at a time, could take turns doing the audio-tutorial lesson. Some students observed others doing the lessons, and many students repeated lessons one or more times, often during recess, lunchtime, or other free time. Each lesson required approximately 20 minutes for a student to complete; thus the 28 lessons provided some 10-20 hours in carefully designed instruction over the two-year span of the instruction. Those teachers who included science in their instruction (a minority) usually dealt with topics such as seasons, clouds, and plant growth, but only in a descriptive manner and not including the basic science concepts such as energy transformations and the particulate nature of matter.

Each teacher we worked with reported excellent student response to the audio-tutorial lessons, and some of the teachers also noted their value for their own learning. None asked to be dropped from the study and most wanted to continue to use the lessons in future years.

Early in the study we developed various forms of paper and pencil tests, including tests with pictures that students …