Case Study on the Fractional Scheme for enhancing the connection between the arithmetic and the algebraic thinking
We observed the process for solving linear equations of two 5th grade elementary students, who do not have any pre-knowledge about solving linear equation. The way of students' usage of fractional schemes and manipulations are closely observed. The change of their scheme adaptation are carefully analyzed while the coefficients and constants become complicated. The results showed that they used various fractional scheme and manipulations according to the coefficients and constants. Noticeably, they used repeating fractional schemes to establish the equivalence relation between unknowns and the given quantities. After establishing the relationship, equivalent fractions played important role. We expect the results of this study would help shorten the gap between the arithmetic and the algebraic thinking.
We present a parallel algorithm for the problem of computing the transitive closure for an acyclic digraph D with n vertices and m edges. We use the BSP/CGM model of parallel computing. Our algorithm uses O(log p) rounds of communications with p processors, where p n, and each processor has O(mn p ) local memory. The local computation of each processor is equal to the product of the number of edges and vertices of D that are stored in p.