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tTŐYP P!YMasterMaster's Thesis(Algebra is a type of systematical language that uses symbols in idea-developing and problem-solving. The aspect of systematical use of language in Algebra starts from the substitution of expressions consisting of symbols; and algebra can be explained by manipulation of those symbol expressions. The level of students' understanding can be evaluated based upon the errors that they make in their manipulations, for the errors do not occur in random but they are results of systematical correlations that they believe in. In that sense, their errors not only degrade their problem-solving ability but also bring about attainment of incorrect concepts. Considering that such errors provide useful information on causes of failure in learning and offer remedy to correct them, efforts must be made to figure out cognitive causes and to correct them rather by putting more emphasis on such errors that come out of problem-solving process rather than on the result of manipulation.
This study finds and evaluates the types of errors that occur when first, second, and third-year middle school students manipulate symbols. The purpose of this research is to study and analyze characteristics of errors according to students' accomplishment levels, and to ultimately help teacher's curriculum to enhance students' problem-solving ability.
In order to achieve this purpose, we have conceived the following three questions.
1. Can errors that middle school students make in their use of symbols be classified by different types?
2. How is student's level of understanding on the use of symbols related to his accomplishment level (high, middle, low) and year?
3. Can the error-correcting process in the use of symbols be clarified?
We have selected a middle school in Kwacheon City in Kyungki-do to carry out this research and also selected a total of 258 students (88 first-year, 83 second-year, and 87 third-year students) grouped in upper, middle, and low-achiving groups based on their grades in their first-semester's mathematics class. The research paper for error types consisted of 15 questions. We analyzed the types and causes of errors that students make in their use of symbols as shown in their solving process.
With the help of Pippig error model and error models that Kim Ok-kyung and Cha Eun-joo suggested, we classified errors into concept-related and procedure-related errors.
Conceptual knowledge-related errors fall into three groups: error of conceptual understanding, error of letter-manipulation law, and error of technical skills; procedure-related errors fall into two groups: error of manipulation and inappropriate answer for the question.
We presents students' ratio of correct and incorrect answers in percentage according to their years and accomplishment levels. In addition, we selected two first-year students that showed a typical error and carried out the correction process.
The analysis of the results revealed the following facts.
Firstly, errors that the first-year students made had high percentage of conceptual knowledge-related errors . Upper level students had relatively high percentage of interruption errors. This kind of errors result from confusion between the already-learned knowledge and just-learned contents and can be deemed as a result of lack of conceptual knowledge. Lower-level students correctly solved problems in simplifying the equations, but their final answers were incorrect probably because their lack of skill in symbol manipulation.
Secondly, errors t< hat the second-year students made had higher percentage of formal operation errors than students in other years. They wrote down only the answers and skipped the writing part of the solving process, making mistakes as a result. This shows that they lack in skills in systematical narration. Like the first-year students, upper-level students had higher percentage of interruption errors than other types of errors.
Thridly, the third-year students made small number of errors but the distribution of types of errors that they made was similar with students in other years.
Upper-level students had a small number of process knowledge-related errors and a high number of interruption errors. Lower-level students made more frequent mistakes than the first-year and the second-year students; this is because while the lower-level of first and second-year students did not answer at all many times, third-year students mader meaningless answers or gave up on the problem in the middle of solving. Moreover, since incorrect concepts in already-learned knowledge result in errors, we made up an error-correction sheet based on principle-discovery model and congnitive-complication study model with the purpose to change the concepts. Students could succesfully solve a problem with teacher's guide when they were guided to approach the solving process with an active attitude. Interviews with the students reveal that they think that solving symbolic equations is hard and that not only do they not fully understand the definitions and principles that they learned, but also lack in the ability to apply. Students must receive help in establishing a step-by-step systematical algorithm to deal with this problem.
In other words, teachers should present a learning activity so students can fully understand a new concept or definition and help students grow a habit to examine to see if their solving-process is correct and if their answer is what the problem is asking for. In addition, the research should be done to find out where students easily make mistakes, and the result should be included in study materials so that they can present counter-examples so as to minimize students' errors. Study materials should emphasize the concept-learning so that new ideas will be well-established within students and students can apply these. The use of symbols in mathematics simplfy mathematical sentences and helps smooth communication. The symbolic meaning of letters makes meaningful representation of an idea possible.
Therefore, in mathematical communication and problem-solving, substitution of letters and the use of symbolic equations is essential and fundamental in mathematics. In other words, it is critical in middle school education to cultivate the ability to fluently use symbols and letters so to mathematically approach and solve problems students can face in their daily lives. Therefore, teachers are highly recommended to analyze the causes of students' errors because the understanding of the types of errors will help them to understand students better and to plan out their curriculum.
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