title: Fostering Formal Commutativity Knowledge with Approximate Arithmetic.
abstract: How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.
authors.affiliation: University of Cologne, Department of Psychology, Cologne, Germany.
authors.forename: Sonja Maria
authors.initials: SM
authors.lastname: Hansen
authors.affiliation: University of Cologne, Department of Psychology, Cologne, Germany.
authors.forename: Hilde
authors.initials: H
authors.lastname: Haider
authors.affiliation: University of Cologne, Department of Psychology, Cologne, Germany.
authors.forename: Alexandra
authors.initials: A
authors.lastname: Eichler
authors.affiliation: Humboldt-Universität zu Berlin, Exzellenzcluster Bild Wissen Gestaltung, Berlin, Germany.
authors.forename: Claudia
authors.initials: C
authors.lastname: Godau
authors.affiliation: Humboldt-Universität zu Berlin, Exzellenzcluster Bild Wissen Gestaltung, Berlin, Germany.
authors.forename: Peter A
authors.initials: PA
authors.lastname: Frensch
authors.affiliation: Humboldt-Universität zu Berlin, Exzellenzcluster Bild Wissen Gestaltung, Berlin, Germany.|FernUniversität in Hagen, Department of Psychology, Hagen, Germany.
authors.forename: Robert
authors.initials: R
authors.lastname: Gaschler
mesh_terms: D002648:Child; D003071:Cognition; D032882:Comprehension; D016208:Databases, Factual; D005260:Female; D006801:Humans; D019359:Knowledge; D007858:Learning; D008297:Male; D008433:Mathematics; D017145:Models, Educational; D008962:Models, Theoretical; D011340:Problem Solving; D012574:Schools; D013663:Teaching / Q000379:methods*