In teaching, representations are used as ways to illustrate the concepts underlying a specific topic. For example, use symbols (e.g., 1+203) to express the concept of addition. To compare students' abilities to interpret different representations in mathematics, the symbolic representation (SR) test and the pictorial representation (PR) test were designed, and then administered to 681 sixth graders in Taipei, Taiwan. This study adopts two different modeling perspectives, the testlet perspective and the multi-ability perspective, to analyze this SR and PR test data in the context of item response theory. The main results show that: 1. Students scored on average significantly higher on the SR test than the PR test. 2. The effects of the item stem testlets could be large, but they are statistically nonsignificant; however, the influence of the number of items in the testlet should also be considered. 3. The nature of the option representations, SR and PR, represents two different mathematics abilities. 4. The main factor that influences students' item responses is students' abilities to interpret SR and PR, and the testlet effects generated from the shared item stem can be ignored. 5. Regarding the parameter estimates of the best-fitting model: (a) the person ability variance estimates show that the ability distributions on the SR and PR dimension may not be the same, (b) the correlation estimate between the SR and PR dimension indicates that these two abilities are moderately correlated, and (c) the item difficulty estimates for different models are similar.
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