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Implications of a law of a decelerating logarithmic growth in mathematics and reading achievement
The study of growth in educational achievement has relied largely on correlational and descriptive modelling, methods typical in the social sciences. Using these methods, researchers have concluded that the rate of growth in achievement in mathematics and reading is greater in earlier than later years of schooling, and that achievements in the two periods are related. However, there is no single mathematical equation that distils the essence of the relationship between the variables, the kind of relationship found in the natural sciences. Instead, there are multiple descriptive models used to summarise the different rates of growth at different times of schooling. Using educational achievements placed on a single quantitative scale of modern test theory, this paper first demonstrates that growth in means in both mathematics and reading achievement of different large cohorts of students from different countries and different jurisdictions within countries, decelerates virtually perfectly logarithmically. It then establishes that as a result, the rate of growth on a quantitative scale is inversely proportional to the time spent in formal schooling; and as shown in compelling graphical depictions, it reinforces the centrality of beginning a trajectory of achievement during the early years of rapid growth. The paper suggests that a logarithmic characterisation of growth on a quantitative scale as a function of time in schooling unifies the understanding of the development in school mathematics and reading achievement. In particular, it solves the paradox that when comparisons are made in terms of grade equivalents, cohorts initially disadvantaged appear to become relatively even more disadvantaged over time, even though when comparisons are made in terms of rate of growth on a quantitative scale, the disadvantaged groups may have an equal or higher rate of growth.
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