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Over the past decade, researchers have established a predictive relationship between students’ fraction knowledge and algebra performance. In middle school, fraction magnitude knowledge is predictive of algebra skills, however, fraction computation knowledge predicts algebra performance for college students. When do different facets of fraction knowledge become more or less important for algebra? The present study examines developmental and individual differences in fraction magnitude knowledge, fraction computation knowledge, algebra feature knowledge, algebra equation-solving, and the relationships between these measures for students (N=700) in grades 4-10 who were tested in two subsequent years as part of an ongoing study. Results will examine developmental changes in student performance on these individual measures and on the relations between the measures both cross-sectionally and longitudinally. Preliminary findings indicate fraction magnitude knowledge develops early and is most important for development of early algebra skills, while fraction computation remains important even for later algebra learning.

Resumen tomado de la publicación Al explicar su razonamiento, los estudiantes deben comunicar su pensamiento matemático con precisión, sin embargo, no está claro si la terminología formal es necesaria o si los estudiantes pueden explicar el uso del lenguaje cotidiano para describir conceptos matemáticos. Este documento informa sobre los resultados de dos estudios. El primero explora la relación entre el uso de los estudiantes del lenguaje de las matemáticas formal e informal y el conocimiento de los procedimientos matemáticos; el segundo replica y extiende estos hallazgos usando un diseño longitudinal a lo largo de todo un año escolar. El estudio 1 utiliza un método de prueba previa, intervención y prueba posterior en una unidad de estudio. El estudio 2 utiliza el mismo método de prueba previa, intervención y posprueba, pero implementa un diseño longitudinal a lo largo de todo un año escolar, utilizando el análisis de la curva de crecimiento. Los resultados muestran que los estudiantes se benefician más cuando intentan describir los conceptos matemáticos objetivos, independientemente del tipo de lenguaje utilizado. Este hallazgo es consistente con el trabajo previo que muestra que hacer que los estudiantes pasen por el proceso de autoexplicación, independientemente de la calidad de esas explicaciones, produce beneficios para el aprendizaje (Chi, 2000). Concluye que aunque los maestros todavía deben usar el lenguaje formal en sus clases, no deben desalentarse si los estudiantes inicialmente no pueden usar el lenguaje formal correctamente. Se sugiere que los maestros permitan a los estudiantes explicar su razonamiento usando términos formales o informales, especialmente mientras los estudiantes están en el medio de desarrollar una comprensión de los conceptos matemáticos. ESP

Science and technology have had enormous impact on many areas of human endeavor but surprisingly little effect on education. Many large-scale field trials of science-based innovations in education have yielded scant evidence of improvement in student learning (1, 2), although a few have reliable positive outcomes (3, 4). Education involves many important issues, such as cultural questions of values, but we focus on instructional decision-making in the context of determined instructional goals and suggests ways to manage instructional complexity.

This study examined whether the quality of first graders' (mean age = 7.2 years) numerical magnitude representations is correlated with, predictive of, and causally related to their arithmetic learning. The children's pretest numerical magnitude representations were found to be correlated with their pretest arithmetic knowledge and to be predictive of their learning of answers to unfamiliar arithmetic problems. The relation to learning of unfamiliar problems remained after controlling for prior arithmetic knowledge, short-term memory for numbers, and math achievement test scores. Moreover, presenting randomly chosen children with accurate visual representations of the magnitudes of addends and sums improved their learning of the answers to the problems. Thus, representations of numerical magnitude are both correlationally and causally related to arithmetic learning.

The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed.

Estimation is an important part of mathematical cognition, one that is pervasively present in the lives of both children and adults. Consider just a few everyday examples. How long will it take you to finish your chores? What will a large pizza cost us? How many people were at the game? About how much is 75 x 31? How fast can that Lamborghini go? Estimation may be used more often in everyday life than any other quantification process.

Two experiments examined kindergartners', first graders', and second graders' numerical estimation, the internal representations that gave rise to the estimates, and the general hypothesis that developmental sequences within a domain tend to repeat themselves in new contexts. Development of estimation in this age range on 0-to-100 number lines followed the pattern observed previously with older children on 0-to-1,000 lines. Between kindergarten and second grade (6 and 8 years), patterns of estimates progressed from consistently logarithmic to a mixture of logarithmic and linear to a primarily linear pattern. Individual differences in number-line estimation correlated strongly with math achievement test scores, improved estimation accuracy proved attributable to increased linearity of estimates, and exposure to relevant experience tended to improve estimation accuracy.