本研究的主要目的在於探討國二學生在函數的單元學習前後，其函數概念層次的變化情形，並探討學生的錯誤類型與迷思概念。
本研究的研究對象為台北市與基隆市各一所公立國中，其中的二年級學生各兩班（均為常態分配），共計146人。研究者依據Anna Sfard的概念發展理論，將函數概念分類成「內化」、「壓縮」、「物化」三個層次，自行設計編製測驗卷，並在教學單元「一次函數及其圖形」教學前後分別進行紙筆測驗，然後加以分析，期能瞭解學生在學習前後，其函數概念層次的成長及變化情形，並歸納出學生的學習錯誤類型；再者，從中抽取樣本加以訪談，以求更能深入瞭解學生的想法、作答原因及探討出學生的迷思概念。
本研究的發現如下：
一、 在學生初次學習函數的概念前、後，其函數概念有顯著的不同。
二、 函數單元的學習，最有助於「壓縮」層次的學生進階到「物化」層次。
三、 學生存在著許多錯誤類型。如：自變數與應變數的角色混淆顛倒、將y = ax + b中的b當成x截距、平移時移動方向與加減號的混淆等。
四、 學生存在著許多關於函數的迷思概念。如：可以寫出關係式的就是函數、只有型如y = ax + b者才是函數、數值要有規律的增加，才是函數關係等。
五、 就前、後測中相同的題目進行分析，學生在學習函數單元之後，答對率確有明顯的提升。
最後根據本研究的結果加以討論，提出結論與若干教學建議，衷心希望可供教師在其教學呈現及教材編排上作為參考，而對學生的學習有所助益，並對未來研究者提供一些建議。
關鍵字：函數、層次、錯誤類型、迷思概念

The main purpose of this research is to investigate the change in the phases of junior high students’ conception about the function before and after the unit learning of function, as well as to investigate the students’ types of errors and misconception.
The objects of this research were the second graders of two public junior high schools in Taipei City and Keelung City, each school two classes (where students were normally allotted), totally 146 students. The researcher adopted Anna Sfard’s theory of conceptual development and divided the conception of function into three phases – “interiorization”, “condensation”, and “reification.” The researcher also designed the tests and gave the students the written tests before and after the teaching of “the function of the first degree and its graph.” Then, analysis was made in order to get a better knowledge of the development and change in the phases of students’ conception about function before and after the learning, and to sum up the students’ types of errors in learning. Furthermore, some students were chosen to have an interview, for the purpose of further understanding the students’ way of thinking and reasons for answers and figuring out their misconception.
The findings of this research are shown below:
1. There was a significant difference in the students’ conception about function before and after their learning the concept of function for the first time.
2. The unit learning of function was very helpful for the students to upgrade from the phase of “condensation” to the phase of “reification.”
3. There were many types of errors among the students. For example, independent variables and dependent variables were confused, the b in the y=ax+b was taken as the x distance, and the moving direction at the horizontal displacement was confused with the plus and minus marks, etc.
4. Students had different misconceptions about function. For example, when a relative equation could be acquired, it was a function. Only the one with a pattern like y=ax+b was a function. Function relation should have a regular increase of value.
5. According to the analysis made on the same test questions before and after the learning, it was found out that there was a significant increase in the rate of answering correctly after the students learned the unit of function.
Finally, the results of this research were discussed to present the conclusion and several teaching suggestions, in the hope that teachers could use them for reference in their teaching and preparing the teaching materials. It was also the researcher’s hope that this study could help students in their learning process and offer good ideas for future researchers.
Key words: function, phase, type of errors, misconception.

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