Main Article Content
Although several Hungarian studies have been carried out on combinatorial reasoning as a reasoning skill (e.g., Csapó, 2001; Nagy, 2004; Hajduné Holló, 2004; Gál-Szabó & Korom, 2018; Zentai, Hajduné Holló, & Józsa, 2018), we are not aware of any longitudinal research. While some of the cross-sectional studies mentioned above are concerned with the development of combinatorial reasoning, changes in the factors affecting performance have not yet been analyzed. Combinatorial reasoning may be looked at from a number of different perspectives; our study uses Csapó’s (1988) theoretical model as a starting point. With respect to combinatorial reasoning, the aims of our study are to look at (1) students’ progress over time in terms of performance as well as (2) the changes in students’ understanding of combinatorial problems observed between Grades four and five and between Grades six and seven. We used a time-series design with two waves of data collection one year apart. The measurement instrument was the same online test with eight enumeration combinatorial problems in both waves. Data were analysed from 183 students from the younger cohort and from 172 students from the older cohort. Performance was characterised by Csapó’s (2018) j-index based on the number of correct and incorrect answers. Task comprehension was measured using three criteria: (1) element number, i.e., the number of elements corresponding to the task condition, (2) repetition, i.e., the occurrence of repetitive elements, and (3) reversibility, i.e., the order of selection (Gál-Szabó & Korom, 2018). The results reveal a significant overall increase in test performance (p<.01) for both cohorts, which suggests an improvement in combinatorial reasoning. Looking at individual problems, however, shows that this improvement did not apply to all problems (two tasks showed improvement in the younger cohort and five tasks in the older cohort) indicating that spontaneous development does not affect all areas of combinatorial reasoning. With respect to the comprehension of problems, looking at the three criteria separately, no change could be observed for most test items (55.2–87.8%). Looking at the three criteria separately for each task, we found that several solutions met the same number of criteria at the two time points. For the test, the proportion of solutions with no change in the number of criteria met was around 15%, 30% of the solutions met fewer criteria and 40–45% of the solutions met a greater number of criteria. In conclusion, although a minor improvement could be observed in students’ comprehension of the principles of combinatorial problems, there was no substantial progress over a year in either age group. Our results point to the significance of assisting students in understanding the principles of combinatorial problems and encourage researchers to carry out further studies concerning the factors underlying the comprehension of these principles (e.g., reading comprehension and problem-solving strategies) and individual differences in development.