Structure And Insight

Structure and Insight

Structures and Algorithms

This book explains exactly what human knowledge is. The key concepts in this book are structures and algorithms, i.e., what the readers “see” and how they make use of what they see. Thus in comparison with some other books on the philosophy (or methodology) of science, which employ a syntactic approach, the author’s approach is model theoretic or structural. Properly understood, it extends the current art and science of mathematical modeling to all fields of knowledge. The link between structure and algorithms is mathematics. But viewing “mathematics” as such a link is not exactly what readers most likely learned in school; thus, the task of this book is to explain what “mathematics” should actually mean. Chapter 1, an introductory essay, presents a general analysis of structures, algorithms and how they are to be linked. Several examples from the natural and social sciences, and from the history of knowledge, are provided in Chapters 2–6. In turn, Chapters 7 and 8 extend the analysis to include language and the mind. Structures are what the readers see. And, as abstract cultural objects, they can almost always be seen in many different ways. But certain structures, such as natural numbers and the basic theory of grammar, seem to have an absolute character. Any theory of knowledge grounded in human culture must explain how this is possible. The author’s analysis of this cultural invariance, combining insights from evolutionary theory and neuroscience, is presented in the book’s closing chapter. The book will be of interest to researchers, students and those outside academia who seek a deeper understanding of knowledge in our present-day society.

Structure and Insight

From the Preface: The goal of this book is to contribute to the improvement of teaching. Yet there is, of necessity, considerable theory in it: an understanding of how the levels can be used in practice requires an understanding of the theory behind it. There are ways to ascend from one level to the next and the teacher can help the pupil to find these ways. To be able to do this we need a theory, and practice follows from it. I was never much interested in the question of how many levels can be identified in a certain topic because it is possible to improve teaching without answering this question. * This text includes: * Structure-based theory of geometric thought development * Context-effected theory of cognitive change * Theory based on the structure of and insight into mathematical thought processes * New theory of mathematics instruction in geometry * New insights into the language of mathematics development * Provides mathematics educators, educational psychologists, test planners, and educational researchers with a forum for instructional thought and development in mathematics, in general, and geometry, in particular * The levels of thinking that are manifested in the ways children learn geometry provide a basis for a new theory of the development of thought and instruction * The teacher-based knowledge, experience, and conception of these levels of thinking are represented in the ways geometric thinking undergoes change as a function of instruction