Using Clinical Interviews To Promote Preservice Teachers'

Understanding of Childrens' Mathematical Thinking

Roberta Y. Schorr	Herbert P. Ginsburg
Rutgers University Schorr@email.rci.rutgers.edu	Teachers College, Columbia University

Abstract: The purpose of this research paper is to share results of a “teaching experiment” in which preservice teachers were provided with opportunities to learn to use the clinical interview method with children. Our central hypothesis was that the clinical interview can be a fundamental method for helping teachers—both preservice and in-service -- develop their own personal and instructionally relevant theories of how children interpret mathematics and then devise better ways of teaching it. Results indicate that preservice teachers do develop a deeper understanding of the ways in which children build mathematical ideas as a result of clinical interviewing children.

Theoretical Framework

Teaching mathematics well is a difficult task. In addition to managing the behavior of some 20 or more children, the teacher should understand the mathematics to be taught, have a grasp of useful pedagogical techniques, and have insight into students' minds (Ginsburg, 1998; Schorr and Lesh, 1998). For example, to help first graders understand the notion of "equivalence" as it is involved in a statement such as 2 + 3 = 5, the teacher must understand that equivalence refers to a special mathematical relationship, that certain kinds of manipulatives (for example, a balance) may be used as a "model" of that relationship, and that first graders are likely to provide their own distinctive interpretations of the "equals sign." Children tend to believe that the "equals sign" refers not to a relationship-- the teacher's view-- but to an act of adding. For first graders, the "equals sign" usually does not mean "the same as," but is instead interpreted as "makes" or "get the answer" or, as one child put it, "the end is coming up." The teacher who has insight into children's minds can appreciate the sense in their interpretations-- after all, the "equals sign" can legitimately refer to the outcome of an operation-- and can deal with them constructively. (For example, the teacher can help the child to understand that the "equals sign" has at least two legitimate meanings, both of which can be useful.)

By contrast, the teacher who lacks understanding of children's minds is left in a kind of pedagogical delusional state: the teacher understands equivalence in a certain way, thinks that concept is being taught to the child, but the child is in fact learning an entirely different concept of which the teacher is unaware. In this case, there is a wide gap between the mind of the teacher and the mind of the child. The teacher tends to deal with what is seen as the child's failure to learn equivalence by "shouting louder"-- that is, by redoubling efforts to teach the concept (as interpreted by the teacher)-- and remains unaware that the child is in fact attempting to learn something else entirely. In our view, such gaps between teachers’ minds and students' minds are widespread, and characterize teaching from preschool through university. As Piaget (1976) pointed out, it takes a psychological equivalent of the Copernican revolution for the adult to realize first that the child's thinking does not necessarily revolve around or take a form similar to that of the adult's, and second that children's minds, although often radically different from the adult's, can nevertheless make their own kind of sense.

This research focuses on one technique for reducing these wide mind gaps, namely the "clinical interview" method. The clinical interview, as originally developed by Piaget (1976), is a flexible and deliberately non-standardized method of questioning, which aims at providing insight into children's ways of thinking-- into their personal "constructions"-- which are often different from the adult's. In the clinical interview, the adult poses a specific task to the child, and usually begins with some predetermined questions. However, the adult is free to modify the questions as necessary, depending on the child's apparent understanding of the questions, the child's motivation, and particularly the child's response to the initial question. The interviewer has the freedom to rephrase the questions to ensure that the child understands them, to follow up on interesting remarks, to clarify responses, and even to challenge them so as to establish the child's degree of conviction. The clinical interview method has been used as the basis of a good deal of research on children's understanding of school mathematics for many years, perhaps beginning with Davis & Greenstein (1964), and is now receiving increasing recognition as a major tool for psychological research into cognitive functioning (Ginsburg, 1998).

Our central hypothesis is that the clinical interview can be a fundamental method for helping teachers—both preservice and in-service, achieve the needed Copernican revolution in their thinking about children's thinking. The clinical interview method helps teachers to both develop their own personal and instructionally relevant theories of how children interpret mathematics and then devise better ways of teaching it. This becomes especially important if we intend to move away from teaching techniques that simply enable students to repeat, without understanding, various specific algorithms, rules or procedures. As Cohen points out, “The teaching that reformers seem to envision would require vast changes in what most teachers know and believe” (Cohen & Barnes, 1993, p. 246). “Teachers who take this path must... have unusual knowledge and skills... They must be able to comprehend students’ thinking, their interpretations of problems, their mistakes...they must have the capacity to probe thoughtfully and tactfully. These and other capacities would not be needed if teachers relied on texts and worksheets” (Cohen, 1988, p. 75).

This research builds upon previous work which shows that it is possible for teachers to learn the clinical interview method and to develop useful forms of it for practical implementation in the classroom (Ginsburg, Jacobs, & Lopez, 1998). That research showed that elementary level teachers from very different types of schools—inner city, suburban, and private— were able to become rather good interviewers and develop distinctive styles of interviewing appropriate for their classrooms. For example, some teachers developed forms of interviewing individual students; others developed methods for interviewing groups of students; others integrated interviewing into their teaching; and another teacher taught her students to interview each other. Almost all teachers said that the process of learning and implementing clinical interview methods was an extremely valuable experience and indeed changed their whole approach to understanding children and teaching them. Colleagues have shared similar anecdotal results: the student teachers or in-service teachers with whom they have worked report that conducting clinical interviews can be a transforming educational experience. The goal of the present research is to provide data concerning what preservice teachers learn from clinical interviewing.

Methods and Procedures

The research reported in this paper took place over a 15 week period as part of a Math Methods course for elementary and middle school preservice teachers at Rutgers University. As part of the course, the prospective teachers were provided with opportunities to learn the clinical interview method, interview children, and reflect on the interviews during classroom sessions. They were also provided with opportunities to deepen their own understanding of the mathematics that they were expected to teach, and consider the pedagogical implications of teaching mathematics in a thoughtful manner.

More specifically, during weekly class sessions, the prospective teachers would investigate a particular mathematical idea by solving a problem or series of problems related to the idea, generally in a group setting. They would then reflect on their own solutions and the solutions of others in the class. Next, they would watch an interview involving a child or series of children grappling with the same or similar mathematical ideas. During and after the interview, they would share reflections about the questions posed and the interview techniques used. They would also discuss the child's mathematical thinking, and the pedagogical implications of teaching the ideas in a thoughtful manner. Afterwards, they were encouraged to actually interview a child about the same ideas, and share the results during the next class session. As part of their final written project, they had to interview a child (either the same child, or a different child) about a particular mathematical idea, record significant aspects of the interview, and discuss the overall interview and the implications for teaching.

The following example will illustrate the process. Several class sessions were devoted to the development of ideas relating to numerical operations. In one particular session, preservice students worked in small groups to consider the development of a base 5 number system and then use their system to solve a series of addition and subtraction problems. After sharing their results with other groups, the class spent some time discussing their mathematical thinking, and sharing some of the misconceptions that occurred over the course of the session. Next, they observed a series of videotaped interviews¹ in which they could begin to consider children’s understanding of place value and written calculation. One such interview involved a second grade girl who had not memorized the basic addition combinations, but was quite capable of solving computational problems by using different counting strategies, some of which involved using her fingers. The students then reflected on a series of questions including the following: How did this girl figure out the different combinations? What strategies did she use? What do you think she needs to learn next? A second interview involved a second grade child, who could perform simple subtraction with regrouping easily enough, yet appeared to have a weak notion of the meaning of place value. After watching the interview, the class discussion for the prospective teachers revolved around the following questions: What do you think this student understands about place value? Given this understanding, what is the meaning for her of written calculation procedures involving regrouping? How does she use the chips to represent numbers? What kind of instruction would be useful for her?

After solving the problems, watching the interviews, and having the class discussions, the prospective teachers performed their own interviews.

Results and Conclusions

To document and discuss insights attained by the teachers as a result of clinically interviewing children, this paper focuses on specific examples of interviews conducted by prospective teachers. One preservice teacher wrote the following about a second grade student that she had interviewed (after watching the videotaped interviews described above). As background, it is important to note that the child being interviewed was in a basic skills math program due to overall poor performance in mathematics.

Conducting this interview left me surprised at how much Jake really knew. Because I had been privy to his “academic standing” in school, I had expected his knowledge to be rudimentary at best. At the very least this interview taught me not to accept labels as a defining standard for any child. In the future I will not assign ceilings to a child’s capabilities. They are so often arbitrary, as well as debilitating. Beyond that, I was astounded to see that the war between his outlook of his “life math” and his “school math” was so pervasive. It was as if he had on and off switches that programmed him to either flourish and find solutions or render him rigid; searching his memory bank of useless, incoherent symbols that would somehow mysteriously pop into his head, gifting him with the appropriate response….This interview has done me a greater service than just highlight some of the common issues we face in trying to teach students math. It has blatantly pointed out that the major hurdle in teaching is connecting to the students.

As documentation to support her comments, this teacher provided several excerpts, one of which included the following:

When I asked Jake about school [math], a somewhat blank, confused look came over him. “Uh, I don’t know. Um, oh, I forget. Oh, uh, no, now I remember. We are doing teens.” I then asked what was he doing with the teens. He said they were “like adding and taking away.” I asked a few additions facts. He got the correct answers, but did not seem as sure of himself…. At one point, I asked him what 12 take away 8 was. He tried to unobtrusively look down at his fingers. He had all 10 splayed and then curled down four fingers from each hand, leaving only the 2 thumbs. He looked up at me, smiled triumphantly and said “4”!”. Then he lowered his eyes and sheepishly informed me that his teacher gets mad and hollers at him if he uses his fingers. It broke my heart. I told him I thought it was okay to use his fingers, and in fact sometimes I use my fingers too. I then asked him to explain how his fingers helped him to figure out 12 minus 8. “Well, I have 10 fingers, and 2 more invisible ones is 12. Then I put down 8. Then I had my 2 thumbs and my 2 invisible ones…so I have 4!”

This teacher went on to share her own interpretation of Jake’s informal strategies for doing addition and subtraction. She (and many of the other preservice teachers) noticed that the children that they interviewed often used informal strategies that they already knew in order to solve new problems rather than traditional algorithms. They noted that they would never have know this had they not had the chance to interview actual students who could share their thinking about mathematical tasks. Taking the time to interview students was really important for this preservice teacher. She realized that although Jake did not perform well when asked to solve computational problems in school, he appeared to be quite good at using his own invented strategies. In her journal reflections, she noted that prior to this course “numbers were numbers to me-nothing more. I’ve used them when I had to-for the typical things such as checking accounts, budgeting and of course sales-but even still they were just a necessary evil”. After reflecting on her own thinking, the thinking of her classmates, and most particularly, the thinking of children, she said the following:

I would not be rigid in my approach to teaching. I would use manipulatives freely and frequently; this includes allowing them [the children] to use their fingers if they feel so inclined. I would encourage them in finding their own solutions to the problems, thus giving them the ownership that creates a freedom of exploration over their own work and processes. It does no one any favor to demand a one-way problem and solution strategy useful only in repetitious math drills.

Other teachers also reported that the clinical interviewing process had been helpful. The excerpt below is representative of the comments of others:

After spending a great deal of time on the clinical interviewing process, I was exposed to the reality that many classrooms do not lend to a student enough time to frame solid thoughts, or invite students to use their own logical methods of calculation. I have also come to understand that being a teacher is not solely comprised of facilitating information but rather being a part of teaching that invites reciprocal processing. The reciprocal process allows students and teachers to learn from each other by sharing. During the time I spent with clinical interviewing, I felt that the role of the preservice teacher and student became inter-changeable and communal. Each question that I posed to Joseph, the student, he offered an answer with reasons to his approach. This in turn allowed me to witness his implementation of different cognitive processes …. The constant feedback in the interview had a positive impact on both my becoming a better teacher and my teaching practices. Moreover, the student was able to gain insight into his own mathematical thinking. Through clinical interviewing, I was able to see the challenges Joseph experienced that would have not been seen in a traditional classroom setting. This …gave me a chance to see how a student’s mind works.

The point of sharing these reflections is not merely to confirm that indeed, preservice teachers enjoyed the clinical interview process, but rather, to suggest that as a result of the experience, these prospective teachers will spend more time considering children’s thinking, and how children build mathematical ideas, when they actually become teachers.

Space limitations of this paper do not allow a more complete description of these or other preservice teacher’s comments. The documentation is provided to suggest that using the clinical interview method had an impact on the prospective teachers’ approach to understanding children and on how they intend to teach them.

Note

1. The tapes and corresponding guide are part of a series entitled “Children’s Mathematical Thinking-videotape Workshops for Educators” developed by Herbert Ginsburg, Rochelle Kaplan, and Rebecca Netley. They are distributed by the Everyday Learning Corporation.

References

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Cohen, D. K., & Barnes, C. A. (1993). Pedagogy and policy. In D. K. Cohen, M. W. McLaughlin, & J. E. Talbert (Eds.), Teaching for understanding: Challenges for policy and practice (pp. 207-239). San Francisco: Jossey-Bass Publishers.

Davis, R. B., & Greenstein, R. (1964). Jennifer. Mathematical Teachers Journal, 19, 94-105.

Ginsburg, H. P. (1998). Entering the child’s mind: The clinical interview in psychological research and practice. New York: Cambridge University Press.

Ginsburg, H. P., Jacobs, S. G., & Lopez, L. S. (1998). Flexible Interviewing In the Classroom: Learning What Children Know About Math. Boston: Allyn Bacon.

Piaget, J. (1976). The child's conception of the world (J. and A. Tomlinson, Trans.) . Totowa, NJ: Littlefield, Adams & Co.

Schorr, R.Y. & Lesh, R. (1998). Using thought-revealing activities to stimulate new instructional models for teachers. In S. Berenson, K. Dawkins, M. Blanton, W. Coulcombe, J. Kolb, K. Norwood, & L. Stiff (Eds.), Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.(pp. 723-731). Raleigh, North Carolina, USA.