Showing papers by "Michael Schneider published in 2015"

Developing Conceptual and Procedural Knowledge of Mathematics

Abstract: Mathematical competence rests on developing knowledge of concepts and of procedures (i.e. conceptual and procedural knowledge). Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative. The chapter reviews recent studies on the relations between conceptual and procedural knowledge in mathematics and highlights examples of instructional methods for supporting both types of knowledge. It concludes with important issues to address in future research, including gathering evidence for the validity of measures of conceptual and procedural knowledge and specifying more comprehensive models for how conceptual and procedural knowledge develop over time.

Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics

Abstract: There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional. However, alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another. We consider possible reasons for why mathematics education researchers often believe that a conceptual-to-procedural ordering of instruction is optimal and why so little research has evaluated this claim. Future empirical research on the effectiveness of different ways to sequence instruction on concepts and procedures is greatly needed.

Bridging the Gap: Fraction Understanding Is Central to Mathematics Achievement in Students from Three Different Continents.

Abstract: Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The integrated theory of numerical development posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of the magnitudes to which they refer, and this magnitude understanding is central to general mathematical competence. We investigated relations among fraction magnitude understanding, arithmetic and general mathematical abilities in countries differing in educational practices: U.S., China and Belgium. Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders’ fraction magnitude understanding was positively related to their general mathematical achievement in all countries, and this relation remained significant after controlling for fraction arithmetic knowledge in almost all combinations of country and age group. These findings suggest that instructional interventions should target learners’ interpretation of fractions as magnitudes, e.g., by practicing translating fractions into positions on number lines.

Hochschuldidaktik als quantitativ-empirische Wissenschaft

Abstract: Fragen der Gestaltung effektiver Hochschullehre werden mehr und mehr mit den Mitteln der empirischen Lehr-Lern-Forschung untersucht und in internationalen Fachzeitschriften mit Peer Review publiziert. Diese Forschung zeigt, dass es Gestaltungsprinzipien fur Lehre gibt, die ihre Wirkung unabhangig von Hochschultypen, Studiengangen und Studienphasen entfalten. Das vorliegende Buch gibt einen anwendungsorientierten Uberblick uber die empirische Forschung zu diesen Gestaltungsprinzipien und ihren Wirkmechanismen. Eine komplexe Kompetenz wie die Fahigkeit zu effektivem Lehren erfordert sog. deliberate practice, also Jahre der praktischen Ubung und des gezielten Arbeitens an den eigenen Schwachen. Die Teilnahme an hochschuldidaktischen Weiterbildungen erhoht die Lehrkompetenz von Dozierenden umfassend, wobei insbesondere Weiterbildungsmasnahmen mit sog. microteaching-Methoden effektiv sind, also mit Video- und Kollegenfeedback zu den Details der Durchfuhrung einer Unterrichtseinheit.