| Errors are valuable learning resources.Learning from errors has attracted increasing interest and attention from researchers.A significant portion of high school students’ math learning time is spent solving various math problems and reading math solutions,both of which contain rich opportunities to learn from errors.A student may read a solution that implicitly contains an error,or solve a mathematical problem whose conditions are contradictory.In practice,there are cases where students find answers wrong or even questions unrealistic.However,few empirical studies have explored the thinking process of students in detecting and correcting these errors.The main purpose of this study is to explore the cognitive mechanism of mathematical error detection,to reveal the cognitive characteristics and differences of senior high school students with different mathematical abilities in solving mathematical error-detecting problems,and to understand students’ mathematical beliefs and learning habits that affect the performance of error detection,as well as students’ views on the value of mathematical error-detecting problems and their suggestions on their use.Two types of mathematical error-detecting problems are used in this study: type 1 error-detecting problems which require the subjects to judge the correctness of the solution and then correct the error,and conditional error-detecting problems(type 2)which require the subjects to reveal the contradiction of conditions and modify conditions to make the problem correct.Research questions are as follows:Question 1.How do high school students solve mathematical error-detecting problem? Namely,what are the general cognitive processes for solving error-detecting problems?Question 2.How do average ability students and high ability students perform in solving mathematical error-detecting problems? What’s the difference? Specifically,it includes the following two sub-questions:Question 2.1 How do average ability students and high ability students perform in solving type 1 mathematical error-detecting problems? What’s the difference?Question 2.2 How do average ability students and high ability students perform in solving type 2 mathematical error-detecting problems? What’s the difference?Question 3 What are the mathematical beliefs and learning habits that affect the performance of high school students in solving error-detecting problems? What are the views of high school students on the value of mathematical error-detecting problem?Any suggestions on its use?In this study,using think-aloud method and case study method,and referring to the research paradigm of expert-novice comparison,8 students with high mathematical ability and 8 students with average mathematical ability in the second grade of a key middle school were recruited as the subjects.Each subject solved error-detecting task through thinking aloud and accepted the follow-up interview.For research question 1,based on the think aloud protocols and interview data,five types of error-detecting episodes are extracted: read,analyze,inspect,correct,and judge.The cognitive processes of error detection are combination of different kinds of error-detecting episodes which begins with reading question.Due to the differences in task difficulty,task characteristics and subjects’ personal characteristics,error-detecting episodes have different ways of combination.On the whole,the two groups present two different debugging styles.High ability students are more active in debugging,while average ability students are more passive and more dependent on the debugging text.For research question 2.1,the results show that average ability students pay more attention to the surface features of the solution process,and mainly check the calculation.They pay less attention to the meaning of concepts and the validity of reasoning and have difficulty in finding and explaining conceptual errors and logical errors.High ability students pay more attention to the logical structure of the solution process,and mainly check reasoning process.They have more knowledge related to errors,and can quickly find out the conceptual and logical error.High ability students are all able to write correct solutions,with more varied solutions and more flexible use of correct solutions.They can not only write correct solutions after detecting errors,but also independently write correct solutions and compare them with the solutions in the text after reading the questions or solutions.Average ability students are not willing to write solutions by themselves,and their ability is insufficient.When writing solution independently,they are easy to make mistakes and difficult to self-correct.For research question 2.2,the results show that high ability students focus more on the deep structure of the problem and the relational understanding of knowledge,have a more profound perception of the conditions of the questions and the connections and restraints between the conditions,are more active in the analysis of the conditions,can reveal the contradictions and modify the conditions more quickly,and propose more abundant modification approaches.Average students’ analysis of the conditions are more superficial,and spend more time in the analysis and verification of the solution.Because they do not carefully analyze the implicit restrictive relationship between the conditions,average ability students are slow to reveal the contradictions,and they give a relatively simple modification approach or make mistakes because of neglecting the constraint relationship between conditions.For research question 3,the interview results show that the performance and its differences between the two groups of subjects are related to their daily error-detecting experience and learning habits.The main results include the following.More subjects are most impressed by type 2 task.Mathematical error-detecting problem is relatively novel to them.Daily problems are all done directly to get answers.Compared with type 1 task,type 2 task is more challenging.Students need to infer some new conclusions according to conditions.The process of derivation of contradiction is exploratory,and the modification of conditions is open to some extent.There is no significant difference in the number of subjects who find wrong answers.There is a significant difference in the number of subjects who find unrealistic problems.There are significant differences in the number of students who have the habit of solving one problem with multiple solutions and verification and reflection.Without being told beforehand,high ability students are more likely to find that conditions of type 2 task are contradictory.The subjects generally held the following beliefs: questions are generally solvable,their goal is to solve them directly and get right answer,and that if questions are wrong in itself,they can get points anyway.There are three ways for students to deal with mistakes: sorting out using wrong problem book,sorting out using notebook and correcting directly.High ability students all correct error directly and remember them,ordinary students may use three ways.Usually,the progress of mathematics teaching is very fast,and teachers seldom comment on students’ mistakes and usually provide correct answers directly.Their teachers would make use of type 1 error-detecting problems in teaching,but rarely make use of type 2 error-detecting problems.All the subjects agree on the value of mathematical error-detecting problems,and the main values include developing mathematical thinking,deepening impression,enhancing understanding,cultivating reading error detection ability,and being useful for mathematics teachers’ teaching and propositions.Students suggest that mathematical error-detecting tasks can be used in classroom teaching,mathematical practice,review and exercise comment,but there is no agreement on whether mathematical error-detecting tasks should be used in the examination.Finally,the implications and limitations of this study and suggestions for future research are discussed. |
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