| The problem of the tangent line is one of the most important problems which lead to the birth of the calculus. The slope of a tangent line acts as an introductory example or as an example of the application of the derivative in any calculus book or middle school mathematics textbook involving the calculus. It can be said that the concept of the tangent line is a bridge connecting the middle school mathematics and college mathematics.An questionnaire survey is conducted to 332 students from four senior middle schools in four provinces. Through analysis of the results, the following conclusions are obtained:(1) The common image of the tangent line is the tangent line of a circle or the line which shares one and the same point with a curve. The tangent line of a circle is just the starting point of cognition of the tangent line.(2) Senior middle students'misconceptions of the tangent line are as follows:Taking 'the straight line shares one and the same point with the curve'as the sufficient or necessary condition of the tangent line;Taking 'the straight line doesn't cross the curve'or'the curve is situated on one side of the straight line' as the necessary condition of the tangent line;Believing that the tangent line of a hyperbola can share two points with the hyperbola;Believing that at the vertex or extremity of a curve two different tangent lines can be constructed;Believing that there may be infinite number of tangent lines at the extremity of a curve.(3) Most students' understanding of the tangent line only remain at the stage of the classic geometry, and historical parallelism is identified between the students' understanding and that of the ancient Greek mathematicians, such as Euclid, Apollonius and Archimedes. Most students have difficulties in passing from the tangent line of a special curve to that of a general one, which also shows the historical parallelism.(4) Students'image of the tangent line divorces from its definition.
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