本研究旨在從乘法結構觀點瞭解學生學習國中理化中物理公式的學習困難。研究分為三個部份：一為對國中理化常見的單元進行概念分析、二為針對加速度和牛頓第二運動定律兩單元，蒐集民國91年到101年國中基測相關題目、乘法結構試題及翰林版習題中相關題目進行任務分析、三為由研究者自行編製的乘法結構測驗中，測驗學生的表現。乘法結構測驗使用調查法，工具為研究者研發，以a=∆V/∆t、a=F/m此兩公式為例，以研究者自行編製不同結構題目進行施測。
根據Vergnaud（1988）提出的三種乘法結構：ƒ(λx)=λƒ(x)、ƒ(x)=αx、ƒ(λx+λ’x)= λƒ(x)+ λ’ƒ(x)中，此乘法結構為一種解題策略，研究者將此三種方法應用於物理中，稱為比例法、定義法、拆開法，研究者考慮在物理中成反比情況和公式法的拆開算法，另外提出三種乘法結構ƒ(λx)=1/λ ƒ(x)、ƒ(x)= α/x、ƒ(x+x’)= αx+αx’。題目結構會影響學生使用的乘法結構，根據不同題目結構易使用何種乘法結構，研究者將區分為：定義結構、比例解構、拆開結構題目。施測題目包含此三種結構題目及反比題目，經由預試以及專家審查題目後進行施測。
研究對象採用方便取樣，為新北市某國中三年級的學生，以研究者曾經任教過的三個班級進行施測，捨去幾乎完全空白的無效試卷後，採用有效問卷共40份。根據受試者表現以及高低分各取三位學生進行訪談，以了解學生解題想法以及所使用的乘法結構，再據學生作答以及晤談結果分析，除歸納學生學習困難外，盼從數學角度了解學生在物理公式上的困難。
本研究重要的發現如下：(1) 學生會因題目結構不同而使用不同的乘法結構解題。 (2) 使用比例法時，有些學生無法掌握不變量且易忽略物理條件及意義。 (3) 學生對於變化量、牛頓、分析受力感到困難。 (4) 學生對不同乘法結構看法迥異。 (5) 基測題目聯結數越多其答對率越低。

The purpose of this study was to explore junior high school students’ difficulties in learning physics equations when studying Physics and Chemistry with multiplicative structures. This study contains three parts. The first part analyzed the concepts of the common units in junior high school Physics. The second part collected the questions from the Basic Competence Test for Junior High School Students from 2002 to 2012, the multiplicative structure test questions, and exercises published by Han-Lin regarding the two units, acceleration and Newton's second law of motion, for the task of analyses. The third part tested students’ performances using the multiplicative structure test developed by the researcher. The method adopted for the multiplicative structure test was the survey method. The research tool was developed by the researcher. Two examples were used in the test, a=∆V/∆t and a=F/m. The test was performed with questions of different structures designed by the researcher.
Vergnaud (1988) proposed three multiplicative structures:. ƒ(λx)=λƒ(x)、ƒ(x)=αx、ƒ(λx+λ’x)= λƒ(x)+ λ’ƒ(x). Applying a multiplicative structure is a strategy to resolve a question. The researcher applied these three methods in physics and named them the proportion method, the definition method, and the decomposition method. After considering the cases of inverse ratio and calculations through decomposition of formulas in physics, the researcher proposed another three multiplicative structures: ƒ(λx)=1/λ ƒ(x)、ƒ(x)= α/x、ƒ(x+x’)= αx+αx’. Structures of questions might influence students’ choices of multiplicative structures to be applied. The researcher categorized questions based on suitable multiplicative structures as questions of definition structures, questions of proportion structures, and questions of decomposition structures. The test contained questions from all these three categories and inverse questions. After the pre-test and after the questions were reviewed by the experts, the test was performed.
The research subjects were selected using convenience sampling. They were all third-grade students from a junior high school in New Taipei City. The test was performed in the three classes which the researcher had taught before. After the invalid questionnaires which were almost blank were discarded, a total of 40 valid questionnaires were used. Based on the participants’ performances and scores, three students of high scores, three of medium scores, and three of low scores were selected for interviews in order to find out their thoughts and multiplicative structures they used. According to their answers and the interviews, it was hoped that, besides summarizing students’ difficulties in learning, their difficulties regarding physics equations could also be explored from the aspect of mathematics.
The important findings of this study are as below: Students used different multiplicative structures to solve questions of different structures. When using the proportion method, some students had problems with invariants and therefore may have easily overlooked physical conditions and meanings. Students felt frustrated with variables, Newton’s laws, and analyses of force. Students had different views on multiplicative structures. Higher number of associations of the task analysis on the questions from the Basic Competence Test for Junior High School Students led to lower percentage of correct answers.

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