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With regard to transfer, is it better to provide pupils with ready-made representations or is it more effective to scaffold pupils' thinking in the process of generating their own representations with the help of peers and under the guidance of a teacher in a process of guided co-construction? The sample comprises 10 classes and 239 Grade 5 primary school students, age 10-11 years. A pretest-posttest control group research design was used. In the experimental condition, pupils were taught to construct representations collaboratively as a tool in the learning of percentages and graphs. Children in the experimental condition outperformed control children on the posttest and transfer test. Both high-and low-achieving pupils profited from the intervention. This study shows that children who learn to design are in a better position to understand pictures, graphs, and models. They are more successful in solving new, complex mathematical problems.Keywords: mathematics education; learning; transfer; collaborative learning; providing versus generating; models; primary school Introduction In most research that focuses on the use of strategies in learning and problem solving, ready-made representations or models provided by the teacher or a textbook are regarded as proper objects of study (Hattie, Biggs, & Purdie, 1996). Research findings support the idea that representations are important for the learning of science and mathematics (Ainsworth, 2006;Gilbert & Boulter, 2000;Greer, 1997;Lehtinen & Hannula, 2006;Mayer, 1989;Perkins & Unger, 1999). Moreover, learning to design representations can be expected to promote problem solving in relation to both familiar and new problems. When students are actively involved in the construction and collaborative evaluation of representations, they develop mathematical knowledge that enables them to generate new solution processes. This article reports on a research project aimed at describing and analyzing the effects of an experimental mathematics program that focuses on the construction and use of student representations in primary mathematics education.

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In the attempt to improve mathematical thinking for safeguarding our future societal needs, there is a worldwide tendency in schools to start training mathematical and arithmetical operations at an earlier age in children's development. Recent theoretical developments and empirical research have pointed to alternative ways of approaching early mathematical thinking. In these latter approaches, mathematical development in the early developmental stages is seen as an emerging process in the context of children's own activities that contributes to meaningful learning and stimulation of children's cultural identity (Bildung approach). The discussion between the training approach versus the 'Bildung' approach is still intemperately going on. In this article, some outcomes of a research programme (based at the Free University Amsterdam) are discussed that present empirical studies and their theoretical background (cultural-historical theory, elaborated in an educational concept called 'Developmental Education') that demonstrates the promising potentials of promoting mathematical thinking through supporting young children's appropriation of schematic representations and notations in the context of play.