| Summary: | 碩士 === 國立嘉義大學 === 應用數學系研究所 === 104 === This study is designed to analyze the making errors from the 11th graders in test related to the trigonometric functions of a generalized angle. Samples are selected from five senior high schools and more than three-hundred of the 11th graders included. This study lasted for approximately nine months. Basically, both of quantitative and qualitative data are collected and analyzed. Main results can be summarized as follows:
1. The typical types of making errors in problem solving include:
(1) Students don’t understand some relevant definitions to the trigonometric functions, such as a generalized angle, coterminal angles, radian system, and so on.
(2) Students often commit cognitive errors about the definition of the trigonometric functions involving of a generalized angle.
(3) Students have misconceptions on determining the sign of the functional value, sometimes they forget to determine the critical sign.
(4) How to transform a function with a generalized angle to the corresponding function with an acute angle often brings students much trouble.
(5) Students often commit cognitive errors to the functional values about the special acute angles or quadrant angles.
(6) Students have misconceptions about the points located in four quadrants.
(7) Students have operational errors about number, fraction and square root.
(8) Students make mistakes in reading or interpreting the test questions.
2. The potential reasons of making errors in problem solving include:
(1) Students have difficulty in solving problems because of insufficient backgrounds.
(2) Students are unable to determine the corresponding quadrant of a generalized angle due to incorrect concept.
(3) Students cannot sketch the corresponding right triangle or the proportion of its sides by using the definition of trigonometric functions with a generalized angle.
(4) Students have no memory of special acute angles or quadrant angles.
(5) Students have difficulties in determining the sign of functional values related to the quadrant angles.
(6) Students also have difficulties in finding the functional value by using the concept of coterminal angles. That is, they should find the corresponding smallest positive coterminal angle first.
(7) Students are not proficient in solving methods, computational capabilities and skills.
(8) Students make errors due to carelessness or omission.
Hopefully, these findings can provide some mathematics teachers to understand students’ misconceptions and learning difficulties in this topic. Meanwhile, referring to these highlights, teachers are thus expected to improve and promote their teaching strategies and remedial instruction in classroom.
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