從比例問題的表面結構和深層結構探究國一學生的解題表現及解題策略情形
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2007
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本研究的主要目的是從比例問題的表面結構和深層結構探究國一學生的解題表現及解題策略情形，並瞭解表面結構解題和深層結構解題的關聯性。
本研究主要採取量化與質性取向的調查研究方法，包含紙筆測驗及個別訪談兩個部分。第一部分：比例問題的紙筆測驗乃是依據三個表面結構－數字型式、語意類型及量的性質設計而成，共計15題，主要以量的方式分析學生在這三個表面結構下的解題表現與解題策略是否有差異，資料收集的對象為彰化縣立某縣立國民中學460名國一學生；第二部分：個別訪談的主要目的則是在瞭解學生是否具備三個深層結構概念－共變原則、不變原則及相對改變原則，並分析深層結構是否與學生的解題表現及解題策略有關，訪談對象共計24名，依照高、中、低數學學業成就各選取8名學生。
本研究的結果顯示：
(一)不同數學學業成就學生在不同數字型式的解題表現及解題策略有差異，數字型式的認知學習由易至難為第三式→第一式→第二式→第四式；
(二)不同數學學業成就學生在不同語意類型的解題表現及解題策略有差異，語意類型的認知學習由易至難為熟知的量數問題→關係集合問題→放大-縮小問題→部分-部分問題；
(三)不同數學學業成就學生在不同量的性質的解題表現及解題策略有差異，「離散量」的解題表現優於「連續量」，「外比」的解題表現優於「內比」，量的性質的認知學習由易至難為離散量-離散量-外比或離散量-連續量-外比→連續量-連續量-外比→離散量-離散量-內比或連續量-連續量-內比；
(四)不同數學學業成就學生會因題目結構的不同而出現不同的解題策略，高數學學業成就學生傾向使用「公式法」，低數學學業成就學生傾向使用「單價法」；
(五)高數學學業成就學生受表面結構影響程度最低且較瞭解比例深層結構概念，低數學學業成就學生受表面結構影響程度最高且較不瞭解比例深層結構概念，可用深層結構的表現預測學生的數學學業成就及在比例問題的答對題數；
(六)學生受表面結構影響的先後順序可能為數字型式→語意類型→量的性質，深層結構的認知順序由易至難為共變原則→不變原則→相對改變原則；
(七)比例概念的相對改變原則與題目的數字型式關聯度最高，比例概念的相對改變原則是掌控學生是否能正確解題的關鍵因素；
(八)能成功解比例問題不代表瞭解比例的深層結構概念，但瞭解比例深層結構概念的學生卻一定能成功解比例問題。
The purpose of this study is to investigate the problem solving performance as well as strategies used by seventh grade students on proportion problems. Special attention will be directed towards their performance with respect to the surface and deep structure of the problems. In addition, the relationship between performances under the two structures will also be investigated. Towards this end, both qualitative and quantitative methods will be used in this study. It can basically be divided into two stages, the paper and pencil test stage and the interview stage. The items used in the paper and pencil test are based on three features of surface structure, namely, the integer type, the semantic type, and the quantitative properties of the ratios. The test instruments were administered to 460 seventh grade students from a junior high school located in the central region of Taiwan. The data thus collected was analyzed by using descriptive statistics, multiple regressions together with multivariate repeated measures method. The interview stage was focused on the problem solving performance of 24 students in relation to the deep structure of the proportion problems. They were selected based on their high, medium or low performance at school, with eight students coming from each group. The purpose of this stage is to find out how the problem solving performance of the students were related to the three deep structure features of the items, namely, the concept of covariance, invariance and relative change. The results of this study are as follows: 1.Students with different abilities performed differently and used different strategies with respect to the specific kind of integer type items that they encountered. 2.Students with different abilities performed differently and used different strategies with respect to the specific kind of semantic type items that they encountered. The kind of semantic type items in increasing difficulty as perceived by students are, well-chunked measures, associated sets, stretchers and shrinkers, and part-part items, in that order. 3.Likewise, students with different abilities performed differently and used different strategies with respect to the specific kind of quantitative properties of the ratios items. In general, performances on the discrete quantity items are better than those on the continuous quantity items. Moreover, performances on the external ratio items are better than on the internal ratio items. 4.Students with different abilities performed differently and used different strategies with respect to the surface structure of the items. Generally speaking, high ability students tended to use formula to solve problems while low ability students tended to use unit amount method to solve problems. 5.High ability students were not that much affected by the surface structure of the items as the low ability students. Moreover, they understand the deep structure of the items better than the low ability students. Furthermore, one can predict to a certain extent the overall performance of students on proportion items based on their extent of understanding the deep structure of the items. 6.There seemed to be difference in terms of student perception of the feature of surface structure of the items. There are some evidence that students were most easily affected by the feature of integer type, the feature of semantic type and then the feature of quantitative properties of the ratios that underlaid the items. So far as the deep structure is concerned, students were most easily affected by the feature of covariance, the feature of invariance and then the feature of relative change that underlaid the items. 7.The gamma coefficient was biggest between students’ performance in relation to the feature of relative change and their performance in relation to the feature of integer type. Students’performance with respect to the feature of relative change was the best predictor of whether they could solve proportion items correctly. 8.Students who could solve proportion items did not necessarily imply that they could handle the deep structure that underlaid the items. However, for those students who could handle the deep structure of the items, they could all solve the proportion items successfully.
The purpose of this study is to investigate the problem solving performance as well as strategies used by seventh grade students on proportion problems. Special attention will be directed towards their performance with respect to the surface and deep structure of the problems. In addition, the relationship between performances under the two structures will also be investigated. Towards this end, both qualitative and quantitative methods will be used in this study. It can basically be divided into two stages, the paper and pencil test stage and the interview stage. The items used in the paper and pencil test are based on three features of surface structure, namely, the integer type, the semantic type, and the quantitative properties of the ratios. The test instruments were administered to 460 seventh grade students from a junior high school located in the central region of Taiwan. The data thus collected was analyzed by using descriptive statistics, multiple regressions together with multivariate repeated measures method. The interview stage was focused on the problem solving performance of 24 students in relation to the deep structure of the proportion problems. They were selected based on their high, medium or low performance at school, with eight students coming from each group. The purpose of this stage is to find out how the problem solving performance of the students were related to the three deep structure features of the items, namely, the concept of covariance, invariance and relative change. The results of this study are as follows: 1.Students with different abilities performed differently and used different strategies with respect to the specific kind of integer type items that they encountered. 2.Students with different abilities performed differently and used different strategies with respect to the specific kind of semantic type items that they encountered. The kind of semantic type items in increasing difficulty as perceived by students are, well-chunked measures, associated sets, stretchers and shrinkers, and part-part items, in that order. 3.Likewise, students with different abilities performed differently and used different strategies with respect to the specific kind of quantitative properties of the ratios items. In general, performances on the discrete quantity items are better than those on the continuous quantity items. Moreover, performances on the external ratio items are better than on the internal ratio items. 4.Students with different abilities performed differently and used different strategies with respect to the surface structure of the items. Generally speaking, high ability students tended to use formula to solve problems while low ability students tended to use unit amount method to solve problems. 5.High ability students were not that much affected by the surface structure of the items as the low ability students. Moreover, they understand the deep structure of the items better than the low ability students. Furthermore, one can predict to a certain extent the overall performance of students on proportion items based on their extent of understanding the deep structure of the items. 6.There seemed to be difference in terms of student perception of the feature of surface structure of the items. There are some evidence that students were most easily affected by the feature of integer type, the feature of semantic type and then the feature of quantitative properties of the ratios that underlaid the items. So far as the deep structure is concerned, students were most easily affected by the feature of covariance, the feature of invariance and then the feature of relative change that underlaid the items. 7.The gamma coefficient was biggest between students’ performance in relation to the feature of relative change and their performance in relation to the feature of integer type. Students’performance with respect to the feature of relative change was the best predictor of whether they could solve proportion items correctly. 8.Students who could solve proportion items did not necessarily imply that they could handle the deep structure that underlaid the items. However, for those students who could handle the deep structure of the items, they could all solve the proportion items successfully.
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表面結構, 深層結構, 絕對改變, 相對改變, 加法策略, 比例推理, surface structure, deep structure, absolute change, relative change, additive strategy, proportional reasoning