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INTERVIEW ON REPRESENTATIONAL UNDERSTANDING OF SLOPES, TANGENTS, AND ASYMPTOTES *
Goal for this interview: To assess student's numerical, graphical, and symbolic understanding of slopes, tangents, and asymptotes and the connections they see between them.
The interview schedule consisted of four parts: one on slope, one on tangent, one on asymptote, and one on the connections among them. Below are sample questions used in the "tangent" and "connections" parts.
Part II TANGENT
1. DESCRIBE OR DEFINE
In your own words, could you tell me what the word "tangent" means?
[ If applicable, ask] You said earlier that the slope of the function is the slope of the tangent to the function. Now I want you to explain to me what you mean by "tangent." [ If they say something like slope of a (null) function at a point, ask them to sketch a picture.]
I have a graph here [ Show figure.], with several lines which seem to touch the curve here at just one point. Which of these are tangents?
What does the tangent tell you about a function?
2. ROLE OF TANGENT IN CALCULUS
Suppose it was decided to drop the concept of tangent from Calculus. How would the study of Calculus be affected?
3. SYMBOLIC REPRESENTATION
Suppose I gave you a function rule. How would you find the tangent related to that function?
Can you find the tangent in another way?
Discuss the tangent of the function  . [ Then ask,] What is the tangent at the point x = 0? [ Follow up with questions about the limit to the right and left of 0.]
4. GRAPHICAL REPRESENTATION
Can you explain the word "tangent" to me by using a graph? Explain.[ Show a sketch of (null) .] Discuss the tangents of this function.
Part IV - CONNECTIONS
1. Some people say that the ideas of asymptote and slope are related. Do you agree or disagree? Explain.
2. Some people say that the ideas of asymptote and tangent are related. Do you agree or disagree? Explain.
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