Program Algebra over an Algebra

We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Faculty of Computer Science, Białystok Technical University, Wiejska 45A, 15-351 Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.Grzegorz Bancerek. K¨onig’s theorem. 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Effect of Computer-Aided Instruction on Grades in Middle School Algebra

Computers and targeted software are increasingly common in the middle school algebra classroom, but the efficacy of computer-aided instruction on classroom achievement in middle school algebra remains unclear. Participants were 1452 middle-school algebra students who were seldom, occasionally, or often exposed to the algebra modules of Destination Math, an interactive mathematics software designed to be appropriate for in-school use. Data were collected over three years (2002–04) and included the cooperation of the algebra teachers, who were subsequently interviewed. Data were analyzed with ANOVA, ANCOVA, and multiple regression. Statistically significant effects were localized using pairwise tests with Bonferroni correction for multiple comparisons. Findings were considered statistically significant at p \u3c .05. Students who received the greatest exposure to Destination Math achieved higher first and second semester algebra grades than participant groups who received less treatment, even when parent education, Gifted and Talented Education (GATE) program status, subsidized lunch status, teacher, and grade in which the student took algebra (7th or 8th) were included as covariates. No significant differences were seen in the California Achievement Test (CAT4SS), a standardized statewide test of general mathematics, possibly because of content validity in that the statewide test was not algebra specific. Multiple regression revealed that parent education, gender, exposure to Destination Math, and GATE program status may be predictive of higher first and second semester grades for middle school students in algebra. Interviews with algebra teachers (n=6) generally revealed support, and suggested that improved performance came from improved compliance, specifically because interactive software captures and holds student attention and thereby increases focused time on task. Increased exposure to Destination Math computer software was associated with increased first and second semester classroom achievement in middle school algebra students. These findings provide empirical evidence supporting the efficacy of developmentally appropriate computer-aided instruction on achievement in the middle school algebra classroom