Children's understanding of quantity and their ability to use graphical information

This investigation concerns the ways in which young children (ages 5 to 8) compare quantities and how they work out the difference between them. The experiments involved children's understanding of mathematical problems and their ability to make use of graphical information in such problems. Each child was shown a series of illustrations, each representing two sets of quantities where the numerical difference was represented either discontinuously or continuously. The children were asked Equalize and Compare questions about each illustration and had to choose the correct answer from the set which represented the choice stimuli. Children's use of strategies was observed. In Experiment 1 (5-to-8-year-olds), only the younger children (5-to-6-year-olds) were observed to perform much more accurately on the Equalize-type question than on the Compare in both discontinuous and continuous conditions. The 7-to-8-year-olds reached a ceiling effect in performance, suggesting that by this age they can already deal with different types of arithmetic problems and with different types of graphical information. Experiment 2 (5-to-6-year-olds) repeated the first experiment presenting the graphical information on a microcomputer, but the discontinuous and continuous conditions were subdivided on the basis of the use of the comparative term "more" or "less". Children are helped significantly by the use of discontinuous material and by the use of "more" in Equalize-type questions only. These results did not support those of Experiment 1 where the Equalize and Compare difference was significant with both discontinuous and continuous material. Experiment 3 introduced part-whole manipulations in order to find out why Compare questions are more difficult to solve than Equalize questions. Five-to-6-year-olds' performance on Compare word problems was not affected by this type of manipulation. Experiment 4 explored the Equalize and Compare difference by presenting the material in a story-telling context. Again, the 5-to-6-year-olds' performance on Compare word problems was not affected by this type of manipulation. However, Equalize questions were helped by the use of the comparative term "more", as in Experiments 2 and 3, and by the presentation of discontinuous material, as in Experiment 2. Experiments 5 and 6 explored children's (5-to-8-year-olds) performance on Equalize- and Compare-type questions using spatial imagery manipulations. Experiment 5 involved manipulations of display in order to examine children's relative ease with Equalize word problems. Again, children's performance was not affected by this type of manipulation. In addition to the display manipulations, Experiment 6 introduced different level manipulations. However, in this experiment, the comparative pair was not represented in the choice stimuli. Children's performance on Compare word problems improved. There was no sign of the Equalize and Compare distinction which may be due to the fact that there was no representation of the comparative pair. The results show that the Equalize and Compare difference is due to a combination of their inherent structural and linguistic factors. Furthermore, the difficulty children have with Compare word problems is non-number-specific, but their relative ease with Equalize word problems is number-specific. Such type results indicate that children represent these two problems very differently.