Algebra tile

Abstract: The book gives a thorough introduction to the mathematical underpinnings of computer algebra. The subjects treated range from arithmetic of integers and polynomials to fast factorization methods, Grobner bases, and algorithms in algebraic geometry. The algebraic background for all the algorithms presented in the book is fully described, and most of the algorithms are investigated with respect to their computational complexity. Each chapter closes with a brief survey of the related literature. The book is designed as a textbook for a course in computer algebra for advanced undergraduate or beginning graduate students. Every chapter contains a considerable number of exercises, some of which are solved in the appendix. In bridging the gap between the algebraic theory and computer algebra software, the book should be of interest to both mathematics and computer science students.

Abstract: Approaches To Algebra R. Lins, et al. The Historical Origins Of Algebraic Thinking L.G. Radford. The Production Of Meaning for Algebra: A Perspective Based On A Theoretical Model of Semantic Fields R.C. Lins. A Model For Analysisn Algebraic Processed Of Thinking F. Arzarello, et al. The Structural Algebra Option Revisited D. Kirshner. Transformation And Anticipation As Key Processes In Algebraic Problem Solving P. Boero. Historical-Epistemological Analysis In Mathematics Education: Two Works In Didactics Of Algebra A. Gallardo. Curriculum Reform And Approaches To Algebra K. Stacey, M. MacGregor. Propositions Concerning The Resolution Of Arithmetical-Algebraic Problems E. Filloy, et al. Beyond Unknowns And Variables - Parameters And Dummy Variables In High School Algebra H. Bloedy-Vinner. From Arithmetic To Algebraic Thinking By Using A Spreadsheet G. Dettori, et al. General Methods: A Way Of Entering The World Of Algebra S. Ursini. Reflections On The Role Of The Computer In The Development Of Algebraic Thinking L. Healy, et al. Symbolic Arithmetic vs Algebra The Core of a Didactical Dilemma. Postscript N. Balacheff. References. Index.

Abstract: Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense components, and that several components of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the hope of easing students’ paths in university algebra. The cooperation of the authors in the domain of structure sense originated at a scientific conference where they each presented the results of their research in their own countries: Israel and the Czech Republic. Their findings clearly show that they are dealing with similar situations, concepts, obstacles, and so on, at two different levels—high school and university. In their classrooms, high school teachers are dismayed by students’ inability to apply basic algebraic techniques in contexts different from those they have experienced. Many students who arrive in high school with excellent grades in mathematics from the junior-high school prove to be poor at algebraic manipulations. Even students who succeed well in 10th grade algebra show disappointing results later on, because of the algebra. Specifically, some students drop out of advanced mathematics in 11th grade due to an inability to apply algebraic techniques in different contexts (Hoch & Dreyfus, 2004, 2005, 2006). Similarly, university lecturers involved in training future mathematics teachers often notice their students’ difficulties in developing a deeper understanding of mathematical notions that they meet in their mathematics courses. We refer to experiences from Novotna and Stehlikova’s longitudinal observation of university students—future mathematics teachers—during the course Theoretical Arithmetic and Algebra (Novotna, Stehlikova & Hoch, 2006). Students enter the course having experience with number sets and with linear and polynomial algebra (Novotna, 2000), but they often have problems with basic algebraic concepts.