Abstracts - Earli

Solving subtraction problems flexibly by means of indirect additionLieven Verschaffel, Katholieke Universiteit Leuven, BelgiumJoke Torbeyns, Katholieke Universiteit Leuven, BelgiumLien Vanderveken, Katholieke Universiteit Leuven, BelgiumThis paper deals with an aspect of the inverse relation between addition and subtraction that hasreceived little research attention so far, namely the extent to which people are able to solveadaptively subtraction problems of the type a-b=. by means of indirect addition strategies ("howmuch do I have to add to b to get at a?"). After a review of the relevant math educational literaturewherein a strong plea is made for the teaching and learning of this indirect addition strategy forsolving small-difference subtraction problems like 21-18=. or 2012-1988=., we will report threerecent studies done at our centre. In a first study adults were asked to solve three-digit subtractionswith a small difference between the integers (e.g., 812-786=.) using the choice/no-choice method.Many adults were found to spontaneously apply the indirect addition strategy and to use it in arather efficient and adaptive way. In a second study, 2nd to 4th graders solved small-differencesubtractions up to 100 in two conditions: in the first condition, they could use their preferentialstrategy on each item; in the second condition, they were instructed to report at least onealternative strategy for solving each item. Generally speaking, children did not spontaneouslyapply indirect addition, but also could not generate it as an alternative solution method whenexplicitly asked for an alternative method. In a third study we compared the strategic performanceon subtractions with small differences up to 100 of 2nd to 4th graders from regular classes withthat of children from a school wherein the clever use of indirect addition got ample instructionalattention. Although the children from the latter school generated somewhat more indirect additionstrategies, the number of indirect additions remained remarkably low. Finally, we discusstheoretical and educational implications of our work.Causal relations between children’s conceptual and procedural knowledge about a mathematicsproblem: Two experimentsMichel Schneider, Institute for Behavioral Sciences, ETH Zurich, SwitzerlandElsbeth Stern, Institute for Behavioral Sciences, ETH Zurich, SwitzerlandConceptual knowledge gives humans a deep and abstract understanding of general relations in adomain while procedural knowledge enables them to quickly and efficiently solve problems.Cognitive learning theories provide contradicting predictions as to the causal relations betweenconceptual and procedural knowledge in the course of knowledge acquisition: These relations areassumed to be either none-existent, or bi-directional, or uni-directional. To date, the empiricalevidence for each of these hypotheses is quite rare and weak. In two experimental studies, we tried(1) to derive treatments adequate for influencing students’ conceptual and procedural knowledgeabout decimal fractions independently of each other and (2) to investigate whether anexperimentally induced increase in one knowledge kind will subsequently lead to an increase inthe other knowledge kind. The treatment for improving conceptual knowledge was derived fromtheories of conceptual change, while the treatment for improving procedural knowledge was basedon theories of skill acquisition. To evaluate the breadth of the treatment effects, four typicalmeasures of each knowledge kind were used. The samples of the two studies comprised a total ofabout 170 fifth-graders. Contrary to our expectations, the two treatments failed to show an at leastpartly independent influence on conceptual and procedural knowledge. So the causal interrelationsof both knowledge kinds could not be investigated. The findings were consistent over the twostudies as well as over the eight knowledge measures and confirm previous findings obtained withstructural equation models. In sum, the results show that the conceptual and the procedural– 120 –