## Electronic Theses and Dissertations

# Learning from errors in the adaptive mathematics tutoring system

Data is provided by the student.

#### Abstract

Errors are considered to play a crucial role in facilitating self-reflection and knowledge acquisition. Nevertheless, nowadays how students benefit from errors in learning is still open to debate. That is, help is superior to practice for learning from errors or not. The goal of this dissertation is to systematically explore how students use help and practice to learn from errors in ALEKS (i.e. Assessment and LEarning in Knowledge Spaces), an adaptive math learning system. Based on the theoretical framework, the learning phase theory, this dissertation defined strategies to learn from errors in three types: help strategy (requesting worked examples in the next two steps after an error); practice strategies (solving problems in the next two steps after an error); mixed strategies (requesting a worked example and solving a problem in the next two steps after an error). Practice strategies were decomposed into four sub-categories: giving two wrong answers after an error; giving a wrong answer and then a correct answer after an error; giving a correct answer and then a wrong answer after an error; giving two correct answers after an error. Mixed strategies were composed of four sub-categories: requesting a worked example then giving a wrong answer after an error; requesting a worked example then giving a correct answer after an error; giving a wrong answer then requesting a worked example after an error; giving a correct answer then requesting a worked example after an error. In addition, the dissertation considered the learning process as three learning phases: the low-skill phase, the medium-skill phase, and the high-skill phase. Specifically, this dissertation examined the likelihoods of strategies occurring after errors, strategy shifts, and the changes of strategies in different learning phases. Additionally, the dissertation investigated the relationships of prior knowledge, error types and topic difficulty with strategies for learning from errors. Furthermore, the dissertation examined the relationships of strategies for learning from errors with learning outcomes (i.e. immediate and delayed learning outcomes) as well as the relationships between different levels of prior knowledge or topic difficulty.The analysis was applied to two datasets: 6th graders (N = 165) and college students (N = 179). The results of the one-way ANOVA suggested that students were most likely to utilize mixed strategies after making errors. The results of comparisons of strategies likelihoods in different learning phases indicated that students were inclined to use the strategy of requesting a worked example and then solving a problem after an error in the low-skill phase, the strategy of solving a problem after an error and then requesting a worked example in the medium-skill phase, and practice strategies in the high-skill phase. Additionally, help strategy was found to be used frequently in the medium-skill phase and the high-skill phase, and practice strategies were used frequently in the low-skill phase and the medium-skill phase. For strategy shifts, it was found that help strategy tended to transition to help strategy. Mixed strategies tended to transition to mixed strategies. Practice strategies tended to transition to practice strategies. In the low-skill phase, the strategies were more likely to transition to the strategy of requesting a worked example and then solving a problem after the next error. In the medium-skill phase, the strategies were more likely to transition to the strategy of solving a problem and then requesting a worked example after the next error. In the high-skill phase, the strategies were more likely to transition to practice strategies. The practice strategies tended to transition to mixed strategies or help strategy in the medium-skill phase and the high-skill phase as well.Only 6th graders with the high prior knowledge were more likely to use the strategy of giving a wrong answer and then requesting a worked example after an error than those with the low prior knowledge. For students with the low prior knowledge, the strategies were inclined to transition to practice strategies whereas students with the high prior knowledge were apt to transition to the strategy of requesting a worked example and then solving a problem after an error. Furthermore, students with the high prior knowledge presented a more disorder pattern of using strategies. When making careless errors, students tended to use practice strategies and presented a more disorder pattern of using strategies. For different levels of topic difficulty, 6th graders were inclined to use practice strategies or the strategy of requesting a worked example and then giving a wrong answer after an error on difficult topics, whereas they tended to utilize the strategy of giving a wrong answer and then requesting a worked example after an error on easy topics. On easy topics, students presented a more disordered pattern of using strategies. As for strategy shifts, students tended to transition to the strategy of requesting a worked example and then solving a problem after an error on difficult topics and preferred to transition to practice strategies on easy topics. The correctness after the strategy of giving two correct answers after an error was the highest among all the strategies. This finding was stable between different levels of prior knowledge and topic difficulty. Help strategy, the strategy of requesting a worked example and then solving a problem after an error, the strategy of giving two wrong answers after an error, and the strategy of giving a wrong answer then a correct answer after an error benefited college students delayed performance. However, the strategy of giving a wrong answer and then requesting a worked example after an error, the strategy of giving a correct answer and then requesting a worked example, and the strategy of giving two correct answers after an error hindered delayed performance. This finding remained stable across different levels of college students prior knowledge as well. In the low-skill phase, college students benefited from shifts to practice strategies while the shifts to strategies that involved requesting worked examples (i.e. help strategies and mixed strategies) did not facilitate their delayed performance. In the medium-skill phase and the high-skill phase, the shifts to strategies that involved requesting worked examples facilitated delayed performance. In addition, college students with both low and high prior knowledge benefited from strategy shifts to practice strategies.The study revealed students strategies to learn from errors in the adaptive learning system and build a foundation for a finer investigation on students strategies to learn from errors in future. Moreover, the findings will hopefully provide insights to understand students learning strategies and improve the effectiveness of intelligent tutoring systems.

*This paper has been withdrawn.*