. [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.]
Students' answers on tests don't always show their true level of understanding. Sometimes they understand more than their answers indicate, and sometimes, despite their regurgitating the correct words, they don't understand what they write. This article discusses a method to probe what they actually understand.
Background and Purpose
Drawing on and expanding techniques developed by Jean Piaget in his studies of the intellectual development of children, researchers in mathematics education now commonly use content-based interviews to understand students' mathematical understandings. What started as techniques for researchers to assess student understanding has more recently been used by mathematics instructors who are interested in understanding more deeply the nature of their students' mathematical understandings. Although it is possible to use interviews for the purpose of grading, that is not the intention of this article. If the instructor approaches the interview by informing the student that the goal is to understand how students think instead of to assign a grade, students are generally quite ready to share their thinking. Many seem fairly pleased that the instructor is taking this kind of interest in them. The primary purpose of the interviews described in this article is the enhancement of the mathematics instructor's understanding of students' understandings with its subsequent improvement of mathematics teaching.
My first experiences with using interviews to understand student understanding came when I interviewed high school students to learn about what kinds of activities made them think deeply in their high school classes. A few years later, I wanted to explore differences between good and poor students' understandings of calculus, so I began by interviewing an "A" student and a "D" student (as reported by their instructor). One of the questions I asked each of them was "What is a derivative?" The "A" student answered in the following way: "A derivative? I'm not sure I can tell you what one is, but give me one and I'll do it. Like, for x2, it's 2x." Further probing revealed no additional insight on the part of the student. The "D" student gave an explanation of the derivative as the limit of the slope of a sequence of tangent lines, complete with illustrative drawing. This experience first suggested to me the potential for interviews as vehicles to reveal students' mathematical understandings in a way that grades do not.
Since my first experiences with interviews, I have used interviews in a variety of circumstances. The goals of interviews like the ones I have conducted are to find out more about each student's personal understanding of selected mathematical concepts and ideas and to assess the depth and breadth of students' mathematical understandings. Many assessment techniques are designed to inventory students' abilities to execute routine skills or their abilities to demonstrate a pre-identified set of understandings. This type of interview, however, is designed to uncover a network of understandings that may or may not be part of a pre-identified list. The interview is not just an oral quiz or test but rather a way to dig more deeply into the complexities of students' mathematical understandings.
Method
Using interviews to examine student understanding usually involves several steps.
Identifying the goal(s) of the interview. Before beginning to construct an interview schedule, the interviewer needs to clarify his or her goals for the interview. Interviews may be directed toward goals like describing the interviewee's concept image of derivative, determining the extent to which the student sees connections among the concepts of limit, derivative and integral, or determining the depth of a student's knowledge of a particular set of concepts.
Designing an interview schedule of questions to be asked. An interview schedule is a set of directions for the interview, including questions that the interviewer plans to ask, directions for how to follow-up, and tasks to be posed during the course of an interview. The schedule should include a core set of questions or tasks that will be posed to every interviewee and a set of potential follow-up questions or tasks — items whose use would depend on the interviewee's initial set of responses. The schedule should also include a plan for what the interviewer will do under different circumstances. For example, the interviewer will want to plan ahead of time what to do in case the interviewee gets stuck on particular facets of the task, or in case the interviewee asks for assistance, or in case the interviewee brings up an interesting related point. Finally, attached to each copy of the interview schedule should be copies of the core set of questions or problem tasks in large enough type to be visible by interviewer, interviewee, and the video-camera.
Piloting and revising the interview schedule. Just as for any assessment instrument, interview schedules need to be piloted and revised based on the success of those pilots in achieving the goals of the interview. I recommend taping a pilot interview or two to enable more accurate analysis. After conducting a pilot interview, ask yourself questions such as:
Preparing for and conducting the interviews. I have found it useful to prepare, before each round of interviews, an "interview box" which contains a copy of the interview schedule, tapes, batteries, an accordion pocket folder for each interviewee, a pen, blank paper, graph paper, a straightedge, and an appropriate calculator. Knowing that needed supplies are ready allows me to focus my attention on the interview.
Conducting the interviews. I will illustrate my method of conducting an interview by describing a particular set of interviews. (See sample interview schedule at the end of the article.) In a yearlong curriculum development, teaching, and research project, I examined the effects of focusing an applied calculus course on concepts and applications and using a computer algebra system for routine skills. Central to my understanding of the effects of this curricular approach was a series of content-based interviews I conducted both with students from the experimental classes and with students from traditionally taught sections. These interviews were loosely structured around a common set of questions. As catalysts with which to start students talking about their understanding of calculus concepts, these interviews used questions like: "What is a derivative?", "Why is it true that `If f(x)= x2, then f´(x) = 2x.'?", "How could you find the slope of a tangent line like the one shown in the sketch?", "Could you explain in your own words what a second derivative is?", and "How can you estimate the area under a given curve?" In these and other interviews I have conducted, I continued to probe as long as the probing seemed to produce additional information about the interviewee's mathematical understandings. I asked the interviewees to share their rationales for each answer, regardless of the "correctness" of the response. At every perceived opportunity as interviewees responded to these initial questions, I encouraged them to talk openly and freely about their understandings of course concepts. When their answers contained phrasing that appeared to be uniquely personal, I probed so that I could understand its meaning. When they made statements that resembled the language of their textbook or of my classroom explanations, I asked them to explain their ideas in another way. When they made generalizations, I asked them to give instances and to explain them. Perhaps my most useful guideline is when I think I understand what the student is saying, I probe further. Many times in these cases, the student's response turns my initial interpretation on end. Interviews like these were designed to allow each student's personal understandings to emerge.
Analyzing the results of the interviews. When possible and reasonable, I have found it useful to watch interview tapes with a colleague who is willing and able to engage in an in-depth discussion of what the tapes seem to indicate about the interviewee's mathematical understandings. Because we have a videotape, we can see what the student is writing and entering into the calculator or computer while making a particular statement. The question we continually ask ourselves is "How does this student make sense of the mathematics he or she is using?" rather than "Is the student's answer right or wrong?"
Findings
Content-based interviews can reveal a variety of aspects of students' mathematical understandings that are not visible through many other methods of assessment. The interrelatedness of one student's understandings of various mathematical topics was revealed in an interview I designed to tap understanding of limit, derivative, and integral. Early in the interview, I asked the interviewee to use the word limit to describe a few graphical representations of functions which I had provided. In each case, the interviewee's response suggested a view of the limit as the process of approaching and never reaching. When asked to explain what was meant by a derivative, the interviewee said that the derivative approached, but never reached, the slope of a tangent line. And finally, the interviewee spoke of the definite integral as approaching the area under a curve, never exactly equal to that area. Further probing in each of these cases uncovered a strongly held and consistent belief based on the notion of limit as the process of approaching rather than as the number that is being approached. The interview allowed me to discover this aberrant belief and to test its strength and consistency in a way that other assessment techniques could not.
Use of Findings
As mathematics instructors learn more about their students' mathematical understandings through interviews, several benefits arise. First, the experience of designing and conducting content-based interviews can help instructors to listen with a new attention and ability to focus on the student's personal interpretations and ways of thinking. Second, an instructor who is aware of possible pitfalls in reasoning can construct examples that are likely to pose cognitive conflicts for students as they struggle with refining the ways they are thinking about particular aspects of mathematics. These cognitive conflicts are helpful in inducing a more useful and robust way to think about the concept in question. For example, after recognizing that students were viewing the limit as a process of approaching rather than as the number being approached, I have centered class discussion on the difference between thinking of limit as a process and thinking of limit as a number. I have been more attentive to the language I use in describing functions, and I have identified examples that would engage students in thinking about limit as a process and as a number. For example, a discussion of whether is likely to bring this issue to the fore. However, even the best discussion is not enough to move a student who is not ready beyond a procedural understanding of limit.
Success Factors
Key to success in using this method is a deep-seated belief on the part of the interviewer that each student's understanding is unique and that this understanding is best revealed through open-ended questions and related probes. Also key to the success of interviewing as a way to assess student understanding is the belief that each student constructs his or her own mathematical knowledge and that this knowledge cannot be delivered intact to students no matter how well the concepts under consideration are explained.
Perhaps just as important as understanding what interviews are is understanding what they are not. Interviews are not tutoring sessions because the goal is not for the interviewer to teach but to understand what the interviewee is thinking and understanding. A successful interview may reveal not only what the interviewee is thinking about a piece of mathematics but also why what he or she is thinking is reasonable to him or her. Interviews are not oral examinations because the goal is not to see how well the interviewee can perform on some fixed pre-identified set of tasks but rather to be able to characterize the interviewee's thinking. And this sort of interview is not usually a teaching experiment because its goal is not to observe students learning but to access what they think in the absence of additional purposeful instruction.
For further examples of what can be learned about student understanding from content-based interviews, the reader can consult recent issues of the journals, Educational Studies in Mathematics and the Journal of Mathematical Behavior, and David Tall's edited volume on Advanced Mathematical Thinking [1].
The interview schedule on the next page was developed by Pete Johnson (Colby-Sawyer College) and Jean Werner (Mansfield University), under the direction of the author.
Reference
[1] Tall, D. (Ed.). Advanced Mathematical Thinking. Kluwer Academic Publishers, Dordrecht, 1991.
INTERVIEW ON REPRESENTATIONAL UNDERSTANDING OF SLOPES, TANGENTS, AND ASYMPTOTES*
Goal for this interview: To assess interviewee'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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* This interview schedule was developed by Pete Johnson (Colby-Sawyer College) and Jean Werner (Mansfield University), under
the direction of the author.
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