Long division

Abstract: A fault-tolerant convolution algorithm that is an extension of residue-number-system fault-tolerance schemes applied to polynomial rings is described. The algorithm is suitable for implementation on multiprocessor systems and is able to concurrently mask processor failures. A fast algorithm based on long division for detecting and correcting multiple processor failures is presented. Moduli polynomials that yield an efficient and robust fast-Fourier transform (FFT)-based algorithm are selected. For this implementation, a single fault detection and correction is studied, and a generalized-likelihood-ratio test is applied to optimally detect system failures in the presence of computational noise. The coding scheme is capable of protecting over 90% of the computation involved in convolution. Parts not covered by the scheme are assumed to be protected via triple modular redundancy. This hybrid approach can detect and correct any single system failure with as little as 70% overhead, compared with 200% needed for a system fully protected via modular redundancy. >

Abstract: Any delayed version of a linear binary sequence can be obtained from a suitable linear combination of the outputs of the feedback shift register which generates the sequence. A simple method is given for calculating the linear combination required to give a specified delayed version. The method involves polynomial long division, and is not restricted to short sequences.

Abstract: Students' strategies for solving long division problems under a realistic mathematics approach (RME) at Dutch primary schools were categorized in two ways: (a) according to the level of how students created multiples of the divisor (chunking) to be subtracted from the dividend; and (b) according to their use, or nonuse, of schematic notation. These categories could be quantified on two dimensions: use of schematization and use of number relations. Just after the introduction of long division problems, students' strategies varied from no-chunking to high-level chunking. Five months later, this variation of strategies was reduced to mainly high-level chunking using a scheme. However, strategy development depended on students' prerequisite knowledge and the type of textbook used. The results from this study contribute to the efficacy of RME for the advancement of strategies and achievement in the domain of division.

Abstract: In the Netherlands, national assessments at the end of primary school (Grade 6) show a decline of achievement on problems of complex or written arithmetic over the last two decades. The present study aims at contributing to an explanation of the large achievement decrease on complex division, by investigating the strategies students used in solving the division problems in the two most recent assessments carried out in 1997 and in 2004. The students’ strategies were classified into four categories. A data set resulted with two types of repeated observations within students: the nominal strategies and the dichotomous achievement scores (correct/incorrect) on the items administered. It is argued that latent variable modeling methodology is appropriate to analyze these data. First, latent class analyses with year of assessment as a covariate were carried out on the multivariate nominal strategy variables. Results showed a shift from application of the traditional long division algorithm in 1997, to the less accurate strategy of stating an answer without writing down any notes or calculations in 2004, especially for boys. Second, explanatory IRT analyses showed that the three main strategies were significantly less accurate in 2004 than they were in 1997.