Analysis of trajectory errors in integrating ordinary differential equations

A method of analyzing and interpreting trajectory errors in the numerical solution of ordinary differential equations by digital computers is discussed. Truncation in integrating a set of differential equations leads to errors in the trajectory of the solution. An explanation is given for the use of diagrams in the complex plane to evaluate errors in the trajectory, with a discussion of the properties of a number of frequently used integration formulas via the diagrams. The diagrams portray the characteristics of an integration method in more detail than do the absolutely stable regions presented by Dahlquist. Based on the diagrams, guidelines are listed as to how to choose a proper integration formula for the given set of differential equations. A method is presented to check whether or not the numerical solution is satisfactory.     

The meaning of play: A cross‐cultural study of American and Japanese teachers’ perspectives on play

Abstract

Early childhood teachers’ understandings of play were examined in a qualitative comparative study of 41 Japanese and 41 American participants. Qualitative analysis of the data yielded five themes: the power of living, the opportunity to learn and to develop, fun activities, child's work, and the opportunity to explore nature. Two themes emerged in regard to the meaning of play for adults: enjoyable and voluntary activities and child's play. Finally, we identified two themes regarding playfulness: state of heart and fun feelings. Teachers in both nations used rhetoric that is congruent with the current Zeitgeist of developmentally appropriate early education. One theme that differentiated the groups was the notion that play is children's work. While several Americans mentioned this slogan, this was not the case for the Japanese participants. In contrast, the theme of play as the power of living was more representative of Japanese teachers’ understandings of play, and this reflects the philosophy set forth by the Japanese Ministry of Education, Culture, Sports, Science, and Technology.     

University Mathematics and Science Faculty Modeling Their Understanding of Reform Based Instruction in a Teacher Preparation Program: Voices of Faculty and Teacher Candidates

This study was conducted in a reform-based mathematics and science teacher education program in the USA, the Maryland Collaborative for Teacher Preparation(MCTP). The goal of the undergraduate program was to prepare upper elementary/middle level specialists in mathematics and science. One significant aspect of the MCTP was the expectation that the program's professors (in mathematics and science) would model a new vision of effective pedagogy based on reform-based recommendations. We determined, in general, that the program's mathematics and science content professors accepted the dual role of modeling effective instruction at the same time they were delivering content. However, this dual responsibility raised in their minds an ‘issue of appropriate balance’ between content and pedagogical foci in their courses. Previously, the professors' had not questioned a focus heavily tilted toward content coverage. We also determined that the program's teacher candidates believed that the mathematics and science professors modeled effective instruction. One of the primary reasons that the teacher candidates believed that their professors were modeling effective instruction was that the focus in the courses was primarily on conceptual understanding, not memorization. A major implication was that the professors' modeling of reform-based instruction prompted the teacher candidates to develop a new vision of mathematics and science teaching shaped by their professors' example.     

Some thoughts on implementing U.S. physics doctoral education in Japanese universities

Recent policy debates surrounding Japan??s graduate education raise a serious skepticism in the validity of its ??research-based?? or ??learning-by-doing?? training approach that these institutions have uniquely established. Proponents of reform suggest drastic restructuring of the graduate programs by replacing the experience-based mentoring aspect with more systematic education that blends an array of coursework and examinations. However, whether such a coursework-based model of graduate training will successfully function without sacrificing the traditional virtue of Japanese practices is an unsubstantiated matter. Japanese graduate schools have produced scientists who have made immense contributions to the advancement in our scientific knowledge. Policy makers and academia in Japan should humbly acknowledge this fact and prudently determine what aspects of their educational content should be reformed and what should be retained. In order to do this, the merits and demerits of alternative models must be clearly understood. The objective of this study is to affirm both the strengths and weaknesses of the traditional approach that Japanese universities need to consider preserving and eliminating, in order to capitalize on their institutional heritage while also building in the educational caliber of the rigorous US graduate training model in the reforming process.