Analyzing the understanding of enumerative combinatorial problems based on element number, repetition and reversibility

Abstract

Hungarian studies measure students’ combinatorial reasoning with enumerative combinatorial tasks. Evaluating students’ responses in these tasks is a challenge, for which researchers recommend different solutions. Studies in the framework of Csapó’s (1988) theoretical model (e.g. Csapó, 2001; Csapó & Pásztor, 2015; Szabó, Korom, & Pásztor, 2015) use j-index, which takes the correct and incorrect solutions into consideration. In contrast, researches in Nagy’s model (2004) evaluate the solutions along one or four criteria (see Nagy, 2004; Hajduné Holló, 2004; Zentai, Hajduné Holló & Józsa 2018). Although these methods are suitable to evaluate students’ combinatorial reasoning performance, using them for diagnostic assessment has limitations. Therefore, the aim of our study is to develop an evaluation method, which can support the diagnostic assessment of combinatorial reasoning. In this context, we (1) define variables for analyzing the understanding of the criteria described in the tasks, and (2) examine the impact of the variables among 4th and 6th graders. Based on the nature of combinatorial operations, we defined three variables for analyzing the understanding of combinatorial tasks. These are as follows: (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. According to the condition, the variables can take two values: does not meet the condition (0), meets the condition (1). After the generic definition of the criteria of variables, we specified these for the tasks of our test, which is a computer-based instrument with eight combinatorial tasks. The variables were analyzed in Grade 4 (N=482) and Grade 6 (N=482). Results revealed that the rate of the solutions corresponding to the criteria were different among the tasks. As we expected, solutions which met more conditions belonged to better task and test performances. Looking at the explanatory power of the variables, their roles are clear in tasks performances, but differences are shown in the explanatory power of the variables among the tasks. The developed variables can contribute to identifying the problems during students’ task solving processes. The values of variables can be calculated automatically, so the online diagnostic assessment system can give immediate feedback for each student. Therefore, our findings can help to design programs for developing combinatorial reasoning.