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Changing Students` Conceptions of Mathematics through the Introduction of Variation
( Ngai Ying Wong ),( Chit Kwong Kong ),( Chi Chung Lam ),( Ka Ming Patrick Wong ) 한국수학교육학회 2010 수학교육연구 Vol.14 No.4
Some 400 Secondary One (i.e. seventh-grade) students from 10 schools were provided with non-routine mathematical problems in their normal mathematics classes as exercises for one academic year. Their attitudes toward mathematics, their conceptions of mathematics and their problem-solving performance were measured both in the beginning and at the end of the year. Hierarchical regression analyses revealed that the introduction of an appropriate dose of non-routine problems would generate some effects on the students` conceptions of mathematics. A medium dose of non-routine problems (as reported by the teachers) would result in a change of the students` conception of mathematics to perceiving mathematics as less of "a subject of calculables." On the other hand, a high dose would lead students to perceive mathematics as more useful and more as a discipline involving thinking. However, with a low dose of non-routine problems, students found mathematics more "friendly" (free from fear). It is therefore proposed that the use of non-routine mathematical problems to an appropriate extent can induce changes in students` "lived space" of mathematics learning and broaden their conceptions of mathematics and mathematics learning.
Residual empirical processes for nearly unstable long-memory time series
Ngai Hang Chan,Wei Wei Liu 한국통계학회 2010 Journal of the Korean Statistical Society Vol.39 No.3
This paper studies the goodness-of-fit test of the residual empirical process of a nearly unstable long-memory time series. Chan and Ling (2008) showed that the usual limit distribution of the KolmogorovSmirnov test statistics does not hold for an unstable autoregressive model. A key question of interest is what happens when this model has a near unit root, that is, when it is nearly unstable. In this paper, it is established that the statistics proposed by Chan and Ling can be generalized to encompass nearly unstable long-memory models. In particular, the limit distribution is expressed as a functional of an OrnsteinUhlenbeck process that is driven by a fractional Brownian motion. Simulation studies demonstrate that the limit distribution of the statistic possesses desirable finite sample properties and power.