| Summary: | 碩士 === 國立臺灣師範大學 === 科學教育研究所 === 94 === The purpose of this study is two-folded. First, it attempts to find out the difficulties of seventh grade students in solving greatest common divisor and least common multiple problems. Second, it explores why students con not discriminate when to use the concept of greatest common divisor and least common multiple.
This study adopted a qualitatively approach and was executed in three stages, each with its specific purpose. At first, this study began with administering to seventh grades general routine problems. Through in-depth interviews, it was found that students had problem schema and could simply use their memories to solve problems. After discussing with an expert and a research group, it was decided that a new non-routine problem was needed understand where lied the students’ problem-solving difficulties. This was the first stage, namely, the explorative stage. In the second stage, the creative stage, a novel problem format was created for the study. A number of seventh graders were interviewed in order to make sure that the item was unseen at their level. Proper wordings were also decided as a result of the interviews. A pilot study on thirty-seven seventh graders was then executed to test out the proper functioning of the item.
The third stage is the formal stage. Early on in this stage, it was decided that only one item was insufficient, and another problem that have the same structure but different problem-solving concept was created. Together they formed the major test instrument of this study. The formal data collection was done by interviewing twenty-four seventh graders with different academic ability and different gender. Data analysis was performed both qualitatively and quantitatively.
The major finding of this study about the difficulties of seventh graders in greatest common divisor and least common multiple problems were as follows. First, they had difficulties in understanding the problems. Some students could not translate the problems to mathematical representations. Some could not clarify the conditions in the problems, and some could not identify what is the useful condition in the items. Second, they had difficulties in understanding the concept of divisibility, some students think divisibility is equivalent to having no remainders. They tended to neglect the condition that “the quotient is an integer”. Besides, most students could not understand the numerical relationship between A and B in the phrase “ A is divisible by B”. Third, some students had difficulties in basic concepts. They misunderstood the concepts of “common ”,“greatest” and “least” , and they were confused with the relationship among the various terms (i.e. multiple, common factor, common multiple, greatest common divisor and least common multiple). Forth, they tended to rely on short division too much and made mistake easily during the process of using short division. Fifth, they had difficulties about identifying what were the unnecessary condition in the problems. Sixth, some students had difficulties about determining whether there was only one answer or numerous answers to the problems. Seventh, some medium and low ability students knew that the concepts of factors and multiples based on are related to the concept of divisibility. However, they could not determine whether they should find factors or multiples.
Three reasons were identified regarding why students could not discriminate between applying the concept of greatest common divisor or least common multiple. First, they only had partial problem schema to solve the problems. Second, they tend to decide whether to find factors or multiples by whether the number to be formed is big or small. Yet, some students could not correctly determine whether the unknown will be bigger or smaller. Third, they could not understand the numerical relationship between A and B in “ A is divisible by B ”.
This study suggested that mathematics teachers should identify who among the students have problems in understanding the question. Efforts should be directed at helping them to understand the question first. Also, teachers should enhance students’ understanding of the concept of divisibility. Moreover, they should help students to build stronger problem schema with respect to the greatest common divisor and the least common multiple problems.
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