Longer bars for bigger numbers? Children's usage and understanding of graphical representations of algebraic problems

In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students’ understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also examined whether usage and understanding varied with grade. Secondary 2 (N = 68, Mage = 13.9 years) and Primary 5 students (N = 110, Mage = 11.1 years) were administered a production task in which they drew bar diagrams of algebraic word problems with operands of varying magnitude. In the validation task, they were presented with different bar diagrams for the same word problems and were asked to ascertain, and give explanations regarding the accuracy of the diagrams. The Kuchemann algebra test was administered to the Secondary 2 students. Students from both grades drew longer bars to represent larger numbers. In contrast, findings from the validation task showed a more abstract appreciation for how the bar diagrams can be used. Primary 5 students who showed more abstract appreciations in the validation task were less likely to use the bar diagrams in a concrete fashion in the production task. Performance on the Kuchemann algebra test was unrelated to performance on the production task or the validation task. The findings are discussed in terms of a production deficit, with students exhibiting a more sophisticated understanding of bar diagrams than is demonstrated by their usage.