# 國一學生最大公因數與最小公倍數解題困難之研究

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## Date

2006

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## Abstract

本研究的目的是要探討國一學生在解最大公因數與最小公倍數相關試題時所遇到的困難，進而了解學生在解題時，為何分不清楚應該使用最大公因數與最小公倍數的原因。
本研究為質性的探索性研究。研究之初，研究者先以一般的例行性試題對學生進行晤談，但發現學生會使用其記憶中的解題技巧來幫助自己解題，因此在與指導教授及研究小組討論後，決定發展非例行性試題來了解學生的解題情形，此階段為本研究的第一階段──探索期。接著進入第二階段──創造期，主要是在開發新的非例行性試題，藉由訪談不同能力的學生及考量其問題基模，並且透過預試來了解本試題的功效。
本研究的第三階段為實踐期，這階段開始時發現只以單一試題較為薄弱，故將創造期發展的試題與另一結構相同但解題概念不同的試題合併為「最大公因數與最小公倍數之診斷試題」，作為正式的研究工具。進行方式式訪談24位不同能力及不同性別的國一學生，以了解他們解研究工具的試題時所遇到的困難，以及他們無法分清楚應該是求最大公因數與最小公倍數的原因。
透過質性與量化的資料分析，本研究發現國一學生解最大公因數與最小公倍數相關試題的困難在於：一、有了解題意的困難，多數中、低能力學生無法將題目轉譯成數學式，無法將題目中的條件細分，亦無法找出有用的條件；二、有理解整除概念的困難，多數學生認為整除就是「沒有餘數」而忽略「除出來的商是整數」之條件，此外多數學生無法了解「a可以被b整除」概念中的數值間之關係；三、有理解各基本概念的困難，學生會在「公」、「最大」、「最小」等概念上有誤解的情形發生，且學生也會對兩數之間的因數（倍數、公因數、公倍數、最大公因數、最小公倍數）」關係混淆；四、受試學生過於依賴短除法且在計算過程中容易發生錯誤而影響到其解題；五、受試學生較無法判斷題目中有哪些條件是不必要的；六、受試學生較無法分辨題目中有多於一個的答案；七、部分中、低能力學生僅從題目中的「整除」條件判斷題目可能與因數或倍數有關，但並無法正確判斷出該找因數還是倍數。
至於學生無法分清楚應求最大公因數與最小公倍數的原因有三：一、部分學生的問題基模不夠精緻；二、一般學生以數值的大小來判斷該找因數或倍數，但部分學生卻無法正確判斷出題目中未知數的大小；三、不清楚「a可以被b整除」概念中的數值關係為「a b」或「b a」，因而無法進行正確的判斷。
本研究建議教師必須判別哪些學生在了解題意部分有困難，並針對不了解的部分加強學生對試題的理解能力，另必須加強學生對於整除概念中數值關係的理解，並幫助學生將其具有的問題基模精緻化。

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.

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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## Keywords

因數, 倍數, 公因數, 公倍數, 最大公因數, 最小公倍數, factor, multiple, common factor, common multiple, greatest common divisor, least common multiple