高中生对切线的错误理解

Senior High School Students’ Misconceptions of the Tangent Lines

切线问题是导致微积分诞生的基本问题，是学生学习导数概念的认知起点．对苏、沪、皖三地高二和高三224名学生的调查表明，尽管学过“切线是割线的极限位置”这一近代分析定义，但绝大多数学生仍然持有“圆的切线”或“与曲线只有一个公共点的直线”的表象，与该定义完全分离；绝大多数高中生对切线的理解只达到古典几何阶段，他们只是根据公共点个数来判别切线，与古希腊数学家的理解具有相似性；绝大多数高中生在从圆和圆锥曲线的切线过渡到一般曲线的切线，从切线的静态的直观定义过渡到动态的分析定义时存在困难，表现出高度的历史相似性．

The tangent line is one of the basic problems leading to the birth of the calculus and is the starting point of its learning. A questionnaire survey was conducted to 224 senior high school students from Jiangsu, Shanghai and Anhui. It is found that, in spite of the modem analytic definition of the tangent line, most of students hold the images of ＂the tangent line of circle＂ and ＂the straight line sharing only one point with the curve＂; most of students only attain the stage of classic geometry, similar to the understanding of ancient Greek mathematicians; most students have difficulties in passing from the static definition to the analytic one, showing evident historical parallelism.