The purpose of this paper is to document when and how the use of clinical interviews may have impacted a prospective teacher enrolled in a mathematics methods course within an elementary education certification program. In particular, we present changes or shifts in the ways in which she: (1) reflected on the clinical interview videos she viewed (using VITAL software); and, (2) applied the ideas to her own clinical interviews with young children. Preliminary results point to an increased sophistication in the nature of observations made about the children in the VITAL videos. In addition, there is a progression in the ways in which this teacher was able to follow student thinking and ask follow-up/extension questions while conducting her own interviews.
Providing prospective teachers with opportunities to gain insight into children’s 2 mathematical thinking should be an important component of teacher education programs. One way to do that is to expose them to situations in which they can observe and interact with children, thereby providing them with the opportunity to better understand and appreciate the ways in which the children interpret, internalize and represent various mathematical concepts (Schorr & Ginsburg, 2000). In particular, they can become better acquainted with children’s thinking through the use of the clinical interview method. Ginsburg (1997) notes that prospective teachers’ understanding of children’s mathematical thinking can be enhanced through clinical interviews—whether by actually interviewing the child themselves, or critically observing another do so. The clinical interview method was used extensively by Piaget (1952). His use of the flexible style of questioning allowed him to observe children’s problem-solving behaviors as they worked on tasks, and then ask questions that were tailored to the child’s observed behavior. This type of interview has the potential to provide prospective teachers with modes of observation and questioning we believe can be transferred to their actual instruction (Ginsburg, Jacobs, & Lopez, 1998).
While this paper is limited to a case study of one teacher, the larger study involved 17 prospective teachers, all of whom were enrolled in an elementary/middle grade level Mathematics Methods Course at Rutgers University. Two of the main research questions guiding the overall study are: what (if any) types of changes occurred in prospective teachers’ reflections on the archived clinical interviews, and how, if at all, did they revise their ways of understanding young children’s thinking as reflected in their own interviews with children?
The subjects for the study were enrolled in the mathematics methods course during the Fall 2008 semester (September through December). A mixed-methods design was used for the larger study, with quantitative data coming from a survey (White, Way, Perry & Southwell, 2006) measuring attitudes towards mathematics and mathematics teaching and perspectives on what that teaching entails. This 20-question survey was a composite of subscales from the Fennema-Sherman Mathematics Attitudes Scales (Fennema & Sherman, 1976) and parallel scales to measure Attitudes to Teaching Mathematics, as used by Nisbet (1991). The survey was given to the prospective teachers at the beginning and end of the course in order to document if indeed the teachers’ ideas about what it means to know and teach math might have changed, and if those changes reflected an increased or decreased willingness to teach math.
The qualitative data for this study came primarily from the prospective teachers’ written work and audio-recorded classroom discussions. Their written work consisted of lesson plans, field observations, reflections on the archived videos (using VITAL), and their reflections on their own clinical interviews with children. The VITAL reflections were assigned weekly, totaling 11 throughout the semester.
This paper looks closely at the VITAL assignments and clinical interviews of one particular teacher. For the mathematics methods course, the prospective teachers conducted their own clinical interviews at least twice, and often three times, throughout the semester. They were asked to include either audiocassettes or videotapes of the interviews, along with a written account of their experiences, observations, and insights gleaned from the interview(s). The first clinical interview assignment occurred approximately six weeks after the prospective teachers began to work with the VITAL software. This time frame allowed them to view others engaged in the practice of clinical interviewing, read about particular techniques used, and then prepare and conduct their own interviews.
Our preliminary analysis has begun to reveal several findings based upon responses to the VITAL essays and clinical interviews. One major finding relates to the type of assumptions that the teachers made regarding the mathematical thinking of the children they observed. More specifically, early on, with almost no exceptions, most of the teachers felt that children either “knew” or “didn’t know” a whole range of mathematical ideas based upon very little actual evidence. So, for example, if a child could recite several number facts, the teachers often were willing to state that the child “knew 7 addition”. Over the course of the semester, these types of conjectures continued to occur, but with less frequency and the prospective teachers were more careful to support their ideas with evidence (based upon assigned readings/literature). Another major finding related to when and how the students analyzed the VITAL interviews. The teachers began noticing critical aspects of the interview as time went on (to be described in detail below). Further, they were able to apply some of this knowledge in their own interviews.
As noted above, we focus this report on the work of one prospective teacher (Tania1 ) who was chosen because she exhibited a notable change in her own beliefs about what it means to know and teach mathematics (per the survey results) and she also showed marked change in her own understanding of the mathematical ideas of the children that she interviewed and viewed using VITAL. In looking at her VITAL essays over time, Tania’s observations became increasingly sophisticated, revealing a deeper level of analysis and explanation. As an example, we noticed a shift in the ways in which she described what she thought the young children were capable of doing. As time went on, she recognized that many of her initial thoughts were quite inaccurate. For example, in the third VITAL assignment of the course, the teachers were asked to respond to a writing prompt on Numerical Operations (using five associated VITAL videos, as noted in Table 1). We describe her analysis of Rufus, a young boy being interviewed in the VITAL video. Rufus, a 1st grade boy at the time, was responding to the interviewer’s probe: “Emily and Ginger have 12 cookies. And they want to split them all up… so both girls get the very same number… they want to split them between the two of them so that each girl gets the same number of cookies.” In the video, Rufus began by drawing twelve dots on a piece of paper. He then drew two boxes to represent the two girls. Next, he drew a line from each dot (cookie) to each box (girl) and counted as he was doing this, to show that each girl would end up with 6 cookies. The interviewer followed this with an extension question: “Now, let’s pretend that their friend Paul comes over, and he wants some cookies too. So how many would each child have, if these two girls each give some of their cookies to Paul.” Rufus responded by drawing twelve new dots to represent the twelve cookies along with three boxes to represent the three children. Then, using the same strategy he used in the initial question, he started to draw lines from each dot to each box. The lines started to get a bit messy, so Rufus redrew the dots as circles (so the cookies appear larger and can be distributed easier). His strategy to distribute the cookies was essentially the same as before (drawing lines to connect cookies to children), but he added in an additional piece—drawing a ring around each group of three cookies, and then drawing the lines from each cookie to a child. The instructions guiding this writing prompt were: Tying in connections to Van de Walle [the text for the course] & the other readings, talk about the numerical operations expressed in each video, the tasks that students worked on while exploring the operations, and what you can tell about the students’ understanding of these numerical operations through the clinical interviews & group lesson posed.