A Study On The Learning Mode Of Explaining Erroneous Examples

The new curriculum reform emphasizes the importance of the student-oriented education, gives priority to the training and development of students’ subject consciousness, and promotes the autonomous, collaborative and inquiry learning mode. However, traditional teaching is still teacher-centered and adopts the learning mode of "teaching and practice". Only when carrying out remediation instruction, teachers would analyze the students’ errors. This mode leads students to do lots of practice, but produce little effect.Researchers proposed that a student could use his/her errors to develop a deeper understanding of a concept as long as the error is recognized and appropriate and informative feedback is available. Errors, by its nature, may provide the stimulus and starting point for inquiry. National Council of Teachers of Mathematics suggested that students may enhance mathematics reasoning by analyzing errors. Inspired by this thought, learning from erroneous examples (EE) was developed as a new learning mode based on worked examples. The positive effect of EE on knowledge acquisition and transfer has been well investigated. However, a couple of questions remain to be fully explored:most studies (1) lack of systematic study on learning from EE; (2) lack of study investigating the effect of EE design on learning outcomes, leading students to have difficulties in identifying errors; (3) inefficiency to guide students to explain errors correctly and deeply, leaving students have difficulties to find the principle and rationale underlying errors; and (4) lack of support and help for students with low prior knowledge, leaving them unable to correctly reflect on errors by themselves.Aiming at solving these problems, this thesis focused on the construction of learning model of explaining EE, explored its theoretical basis, and studied the design method of EE, the model of explaining errors, and feedback mechanism by using design-based research methodology. The main research work and results are summarized below.(1) A design method of EE based on variation theory was proposed and investigated. This method suggested that teachers or researchers should determine the key features of worked examples based on deep analysis of learning contents, design the error types based on the frequently-made errors, and then designed the specific EE according to the similarities and differences among the features of EE. Using three-digit subtraction as the learning content, the current study designed 6 error types,36 subtraction EEs and conducted corresponding experiment on third grade students. The students were randomly assigned to two conditions:1) learning from EE (experimental group); 2) learning from correct examples (control group). The experimental group was required to identify and correct errors, and make structured self-explanation as to the reasons of the errors. The results showed that learning from EE helped students acquire conceptual and procedural knowledge, promoted the transfer and retention of knowledge and encouraged students to face errors more positively.(2) The mode of collaborative explaining EE was designed and studied based on the collaborative learning theory and cognitive conflict resolution strategy. To investigate the effect of this mode, this study used a 2×2 factorial design varying in example types (correct vs. erroneous examples) and learning settings (individuals vs. collaboration). An experimental study of a two-week course on subtraction for third grade students (N=109) was conducted. The results demonstrated the effectiveness of the learning model of explaining EE, and showed that collaborative learning from worked examples may improve the transfer knowledge of subtraction. The improvements were maintained over two weeks. In addition, collaboration also enhanced students’ confidence in handling errors. Moreover, these results were interpreted from a cognitive load perspective.(3) A personalized multilevel feedback mechanism was designed and investigated to facilitate the learning mode of explaining EE. This mechanism included minimal feedback, lightweight and heavyweight comparable meta-level feedback and elaborate feedback. These four levels of feedback were interdependent, and the amount of information and invasiveness of thought process were increased gradually. Only when students encountered a problem, these feedbacks were provided according to the students’answers. To investigate the effectiveness of this feedback mechanism, an experimental study was conducted on students with weak prior knowledge, and the results were analyzed using qualitative and quantitative methods. The results indicated that with the help of feedback, the learning mode of explaining EE may help students with weak prior knowledge deepen their understanding of knowledge, therefore reducing procedural errors and misconceptions. In addition, the feedback mechanism may help students learn how to reflect on errors and foster self-explanation. The comparable meta-level feedback may help students correctly determine the type of errors and conjecture the calculation steps of incorrect solutions.(4) Based on the results above and the content-first design approach, an online learning system of explaining EE was designed and developed. This system established a game-like learning situation to arouse and preserve learning motivation, created a database of EE under guidance of variation theory, and formulated an adaptive example distribution mechanism according to students’ achievement. This system could help students acquire and transfer knowledge by identifying, correcting and explaining EE in collaborative settings. In addition, the system also provided teachers with management function to monitor the students’ learning progress.