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In fact, as in many recent publications in mathematics educa- tion, much of what is described . . . reflects two movements, “sit- uated learning” and “constructivism”, which have been gaining influence on thinking about education and educational research. In our view, some of the central educational recommendations of these movements have questionable psychological foundations. We wish to compare these recommendations with current empir- ical knowledge about effective and ineffective ways to facilitate learning in mathematics and to reach some conclusions about what are the effective ways. A number of the claims that have been advanced as insights from cognitive psychology are at best highly controversial and at worst directly contradict. | An Episodic History of Mathematics Mathematical Culture through Problem Solving by Steven G. Krantz September 23 2006 To Marvin J. Greenberg an inspiring teacher. iii Preface Together with philosophy mathematics is the oldest academic discipline known to mankind. Today mathematics is a huge and complex enterprise far beyond the ken of any one individual. Those of us who choose to study the subject can only choose a piece of it and in the end must specialize rather drastically in order to make any contribution to the evolution of ideas. An important development of twenty-first century life is that mathematical and analytical thinking have permeated all aspects of our world. We all need to understand the spread of diseases the likelihood that we will contract SARS or hepatitis. We all must deal with financial matters. Finally we all must deal with computers and databases and the Internet. Mathematics is an integral part of the theory and the operating systems that make all these computer systems work. Theoretical mathematics is used to design automobile bodies to plan reconstructive surgery procedures and to analyze prison riots. The modern citizen who is unaware of mathematical thought is lacking a large part of the equipment of life. Thus it is worthwhile to have a book that will introduce the student to some of the genesis of mathematical ideas. While we cannot get into the nuts and bolts of Andrew Wiles s solution of Fermat s Last Theorem we can instead describe some of the stream of thought that created the problem and led to its solution. While we cannot describe all the sophisticated mathematics that goes into the theory behind black holes and modern cosmology we can instead indicate some of Bernhard Riemann s ideas about the geometry of space. While we cannot describe in specific detail the mathematical research that professors at the University of Paris are performing today we can instead indicate the development of ideas that has led to that work. Certainly
