從邏輯推理能力探討學生犯錯的成因— 以對數解題錯誤為例

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本研究的目的主要有三,一是探討各對數錯誤類型及其成因,二是探討邏輯推理錯誤類型及其成因,三則是進一步探究邏輯推理能力與對數運算能力之間的關係。本研究共分三個階段進行,且各有不同的研究對象;其中,研究一是透過對四位參加過國際數學比賽的國中數學資優生的長期觀察與施測的結果,而逐漸形成「數學解題能力是與邏輯推理能力有正向的關係」之一個初始假設,並以國中數學資優生在對數運算的錯誤類型為例,進行假設的引證。研究二和三則分別以兩位一般的中學生以及一百九十一位一般國中生為研究對象,進一步引證研究一所形成假設的可能性。 研究一主要是透過研究者自編的「新對數題甲」及「推理問題甲」兩份測驗卷來收集資料。結果顯示四位受試者在兩份測驗卷的表現似乎能支持上述之研究假設。其中,在「推理問題甲」的表現上,發現部份學生會對題目做過度的自我解釋;在「新對數題甲」的表現上,則發現其所犯的主要錯誤類型為分配律與交換律型的錯誤,而經由訪談的結果得知,此兩錯誤類型的產生應與學生過去所學習的分配律有關。 研究二則選取一位國二及一位高一學生,透過「新對數題乙」測驗卷與後續訪談,進一步探討一般學生的對數運算錯誤類型,同時也試圖探究學生是如何詮釋抽象公式並運用之。研究結果顯示他們所產生的錯誤類型較研究一的數學資優生為多,其中更以轉換型錯誤最為明顯,根據訪談結果得知,此錯誤類型的形成主要是因為學生在對抽象對數公式的主動詮釋過程中出現錯誤所致。其中,「新對數題乙」測驗卷的表現上,高中生似乎比國中生較容易產生分配律型錯誤,但在轉換型錯誤上則屬國中生較多。此現象可能的原因是當學生正式學習對數後,轉換型錯誤會因之減少,但分配律型錯誤的增加則似乎意味著學生還是容易受其先備知識影響。在後續的訪談中得知,一般學生在使用公式上的困難似乎與「記憶」有關,顯示學生並無法真正掌握公式;此外,學生也建議運算符號應有意義、越短越好,但又不能有太多意義在內。 研究三則是承續研究一和二的結果,以研究者自編的「新對數題丙」及「推理問題乙」兩份測驗卷對一群未學過對數的國中生進行施測,藉以驗證研究一的初始假設,並廣泛地瞭解有哪些先備知識與經驗會影響學生產生對數錯誤。研究結果發現未學過對數的國中生在「新對數題丙」測驗卷中有三種主要的錯誤類型,包括分配律、交換律以及轉換型的錯誤,此結果與研究一、二的結果頗為吻合。此外,與李芳樂(1997)針對高一生之對數錯誤類型的研究相較之下,發現某些錯誤類型在經正式對數教導後,其發生比率會有所減低;然而,亦有某些錯誤類型顯示其可能是由高中生在學習對數中所習得的。而在「推理問題乙」的表現則顯示,此群國中生有幾個主要的錯誤推論類型,其成因則大多與他們對於題目的詮釋有關,但整題來說,所有國中生在推理問題的表現並不十分理想。再者,透過年齡、數學成就以及邏輯推理能力三自變數對國中生在對數解題的表現進行迴歸分析後發現,此三變數雖可預測學生在對數中之表現,但只能解釋13%的變異數;另外,不同年齡、數學成就以及邏輯推理能力的學生在抽象對數的表現差異並不大,此結果似乎顯示在邏輯推理能力與對數解題能力之間的關係並不如研究者所預期的大。 總括來說,本研究從最初在研究一中觀察到的現象、建立假設、初步驗證假設、提出現象與問題,進展到研究二及三的後續研究,透過這一連串的探索過程,發現國中生的推理能力宜多加訓練。此外,學生在對數題上也有眾多的錯誤類型,部份錯誤原因似乎與先備知識及對公式過度推論有關。而在驗證初始假設方面,雖然發現邏輯推理能力與對數解題能力之間只呈弱相關,但這可能與研究工具設計得太難有關,由於本研究可屬一初探性質,關於這兩種能力之關係及此三研究中所引發出來的問題,還有待未來研究繼續進行之。本研究最後建議教師教導對數時,宜注意學生的先備知識與其對於公式的詮釋能力。
There are three main purposes for this study. The first is to explore the error pattern and factor of logarithm. The second is to explore the error pattern and factor of logical reference. The third is to explore the relationship between the ability of logically inference and logarithm operation. The whole study is divided into three sub-study (named study 1, study 2 and study 3), each has its own subjects. In study 1, researcher produced an initial hypothesis " there is a positive relationship between the mathematical problem solving ability and the logically inference ability " by observation and test of four gifted mathematically students who are seventh grade and were participated in international mathematical contest to be subjects. And then focused to the error pattern of logarithm operation for testing the hypothesis. In study 2 and 3, 2 and 191 students study in a common high school are respectively involved in this study to proving the possibility of the exit of the hypothesis produced in study 1. In study 1, researcher collecting information by " New Log A " and " Logical Inference A" two instruments made by researcher. And the result showed their performance in the above-mentioned instruments seems to suppose the initial hypothesis. In " New Log A ", some subjects tend to over-explain, in " Logical Inference A" their main error pattern are distributive error and commutative error, and by the continuing interview, the cause of this two error pattern is related to distributive law and algebra operation they had ever learned. In study 2, researcher choosing 2 students to be subjects, including one seventh grade and the other tenth grade who study in different common middle schools, and collecting needed information by " New Log B " and instantaneous interview. Not only explore the error pattern of general students in logarithm operation but also attempt to probe how they interpret and use the abstract formula. The result showed the error pattern they made is more than the subjects in study 1, especially the transformative error. By instantaneous interview, this kind of error might form by the initiative interpret of students in abstract logarithm formula. Besides, the tenth grade student who had learned logarithm made more distributive error than the eighth grade student never learned about logarithm. The most possible reason is concerned about that if someone had learned logarithm, transformative error will reduce and distributive error will increase after learning. The above-mentioned condition seems to appear that students' performance in logarithm operation is affected easily by prior knowledge. In addition, the circumstance also demonstrated the difficulty in utilizing formula is related to memory and the impotence in mastering the formula. Besides, students suggested that the operation signal should not only have meaning and be short, but also not with much meaning in it. Study 3 inherited the outcome of study 1 and 2. To inspecting the hypothesis and realizing generally which prior knowledge or experience could influence the logarithm error student made, researcher chosen 191 junior school students who haven't learned about logarithm to be subjects and collected material by " New Log C " and " Logical Inference B". There were three main finding in this sub-study. The first is concerned subjects' performance in " New Log C ". Subjects never learned logarithm produced many expected error pattern including distributive, commutative and transformative error, moreover, comparing with another logarithm error pattern research were involved senior high school students, the outcome appeared the emerge rate of some error patterns will decrease by instructing, but increase by giving them guidance. The second is their performances in " Logical Inference B". The result displayed that they aren’t good at it and the cause they made error is concerned about their explain. The third is connected with the factor of subjects’ performance in logarithm. By the regress analysis with age, mathematical achievement and logical inference ability this three independent variables, they could forecast 13% performance in logarithm. Furthermore, the three independent variables have no big difference in abstract logarithm. This finding seems to appear that there doesn’t exit the original relationship between two ability. In a word, researcher built and tested the hypothesis by observing phenomenon in study 1, and then proposed the problem progress to the following study 2 and 3. In the series of explored process, it is found that regarding the inference junior student made should be trained, in addition, students have various error patterns in logarithm, and the reason why they made error is related to prior knowledge and over-inference of formula. Although there is a weak relationship between the logical inference ability and logarithm problem solving ability in proving the hypothesis, but it might be concerned with the difficulty of instrument. As a result of the whole study is with primitive explore, the future research could continue the problem induced from the above three sub-study and the relationship between this two ability. Finally, researcher suggest that teacher should notice the prior knowledge ability of formula of student when they instruct the logarithm.

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邏輯推理能力, 學生犯錯的成因, 對數解題

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