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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
Citation: URL: http://www.allacademic.com/meta/p117539_index.html Direct Link: HTML Code:

Warner, Lisa. and Schorr, Roberta. "From Primitive Knowing to Formalising: The Role of Student-to-Student Questioning in the Development of Mathematical Understanding" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117539_index.html>

Warner, L. B. and Schorr, R. Y. , 2004-10-21 "From Primitive Knowing to Formalising: The Role of Student-to-Student Questioning in the Development of Mathematical Understanding" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117539_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: OBJECTIVES/THEORETICAL FRAMEWORK Prompting students to talk about mathematics is an important goal of education (NCTM 1989, 2000; Sfard, 2000; Dorfler, 2000; Cobb, Boufi, McClain, and Whiteneck, 1997). Cobb, (2000) notes that student exchanges with others can constitute a significant mechanism by which they modify their mathematical meanings. Carpenter and Lehrer, (1999) state that “the ability to communicate or articulate one’s ideas is an important goal of education, and it also is a benchmark of understanding.” (p. 22). Researchers such as those cited above (and others, see for example, Schorr, 2003; Maher, 2002; Shafer and Romberg, 1999) maintain that it is important to provide students with opportunities to discuss their ideas with each other, defend and justify their thinking both orally and in writing, and reflect upon the mathematical thinking of others. One important component of this involves students’ questioning of the mathematical thinking of their peers. This report focuses on the impact of student questioning on the development of mathematical thinking. We do this within the context of the Pirie/Kieren theory for the growth of mathematical understanding (Pirie and Kieren1994). Our central premise is that when students have the opportunity to question each other about their mathematical ideas, both the questioner and the questioned have the opportunity to move beyond their initial or intermediate conceptualizations about the mathematical ideas involved. As students reflect on their own thinking in response to questions that are posed by their peers they have the opportunity to revise, refine, and extend their ways of thinking about the mathematics. As they do this, their earlier conceptualizations and representations become increasingly refined and linked. We stress the role of representations in this dynamic since “the ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas” (NCTM, p.67). In this paper, we will trace the development of ideas (using the Pirie/Kieren model) and the student-to-student interactions and questions that contribute to this development. Pirie (1988) discussed the idea of using categories in characterizing the growth of understanding, observing understanding as a whole dynamic process and not as a single or multi-valued acquisition, nor as a linear combination of knowledge categories. There are eight potential layers or distinct modes within the growth of understanding for a specific person, on any specific topic (Pirie and Kieren, 1994). The theory is non-linear, repeating itself. In this study, we will illustrate movement through several of the layers as we focus in on how the transitions from one layer to the next occurred in association with student-to-student interactions and questioning. METHODS/DATA SOURCES/ANALYSIS The study took place over a two-month period, which involved two visits a week (50 minutes each session) in a diverse, low performing, high poverty eighth grade inner city classroom with approximately 32 students. The visits were part of a professional development project in which the teacher/researcher (first author), who is a mathematics education researcher at a local university, routinely met with local teachers, planned classroom implementations, and then modeled or co-taught lessons with the teacher. After each lesson, the teacher/researcher would “debrief” with the classroom teacher and a University mathematics education professor (the second author) to discuss key ideas relating to classroom implementation, the development of mathematical ideas, and other relevant issues. During the course of the two months, several different tasks were explored. The teacher/researcher, along with the classroom teacher encouraged the students to exchange, talk about, and represent ideas; conjecture, question, justify and defend solutions; discuss disagreements and differences; revisit ideas over time; and, generalize and extend their ideas. Generally, the students worked in groups of 3-5, and each group discussed, argued, and ultimately presented its solutions. During each class session, two cameras captured different views of the group work, class presentations and associated audience interaction. In addition, careful field notes were taken after each session. This study focuses on 6 of the 35 videotapes generated in this manner, as the students explore variations of a task. The initial problem task was as follows: John is having a Halloween party. Every person shakes hands with each person at the party once. Twenty-eight handshakes take place. How many people are at the party? Convince us. This particular problem entails a context that may suggest a structure that ultimately leads to a solution that is generalizable to a larger class of problems. In this case, such a solution is [n(n-1)]/2. Episodes were transcribed and coded to identify critical events (c.f. Powell et al), which in this case were determined by student-to-student questions and/or interactions. In the sections that follow, we examine the development of a particular student, Aiesha, by identifying student-to-student questions and/or interactions in the context of the Pirie/Kieren model for mathematical understanding. RESULTS 1. Moving to image making Primitive knowing is the starting place for the growth of any particular mathematical understanding, what the student can do initially, with the exception of the knowledge of the topic. In this case, Aiesha begins by shaking hands with a member of her group and then moves to a picture and number representation for her idea. She moves to the image-making layer (doing something to get the idea of what the concept is). She uses a picture representation to construct an idea of multiplying the number of people by one less than the number of people to arrive at the number of handshakes. Every time she multiplies, however, she arrives at double the number of actual handshakes in the correct solution. At first, she doesn’t notice this mistake and becomes frustrated, explaining that there is no answer. After another student shares his solution with her, she questions her own idea, and reorganizes it so that she divides her answer by 2. 2. Moving to image having Two weeks later the students were challenged to begin a new task involving an extension of the original task. In this episode, another group member, Bea, questions Aiesha about her initial representation. Aiesha then restructures her knowledge to generate a representation that is more understandable to her peers. In so doing, she has developed a new and ultimately more useful representation. Aiesha’s explanations indicate that she has moved to the image having layer. At the level of image having a person can use a mental construct about a topic without having to do the particular activities that brought it about (Pirie, and Kieren, 1994). Aiesha now appears to have an “image” of the handshakes, and is no longer tied to the action of showing each handshake. 3. Moving to property noticing In this episode, two students question Aiesha’s idea. This helps her to realize that her drawing shows each person shaking hands twice. Aiesha now begins to consider why division by 2 actually works. In this case, the students’ questioning helped Aiesha to notice properties about her representations, thereby prompting her movement to the property noticing layer. This layer may be characterized as one in which the individual can manipulate or combine aspects of his/her images to construct context specific, relevant properties (Pirie, & Kieren, 1994). In this case, Aiesha noticed that her picture representation had double the amount of handshakes, which prompts her to build on her older representation, thereby constructing a new chart. 4. Moving to formalising Aiesha and the other members of her group spontaneously attempted to generate a more generalized symbolic representation that could work for any number of people or handshakes. For this, they reverted back to the original problem. Aiesha drew a chart and constructed a number sentence along with a formula using both words and standard algebraic notation. Ultimately, Aiesha was able to come up with the formula [n(n-1)]/2, which she presented to the class. We suggest that during this episode, she was working in the formalising layer, creating a “for all” statement, in which a method or common quality from the previous image was abstracted (Pirie, and Kieren, 1994). As Aiesha was presenting her formula, another student, Shaniqua, questioned her, setting up a hypothetical situation based on the existing problem. Aiesha showed that her idea was valid, using multiple representations to solve Shaniqua’s hypothetical situation (words, numbers, symbols, a chart, picture representation and acting it out), and was able to link the representations to each other. NOTE: We are aware that due to space limitations, we have oversimplified the results and the connections to the Pirie/Kieren model. Further, we have not, at this time, provided full documentation for the results that were presented. The full paper will include more detailed descriptions, and the actual questions, discussions, and pictorial and symbolic representations that are needed to support our results and conclusions. CONCLUSIONS We conclude that student-to-student interactions and questions played a key role in Aiesha’s movement from primitive knowing to formalising. We believe that an analysis of this type has the potential to help teachers, teacher educators, and researchers recognize, support, and consider meaningful opportunities for such student-to-student interaction, and the impact of this on the development of mathematical understanding. Further, we believe that by using an existing theory of learning to guide and understand the particulars of actual classroom implementation, we have furthered the connection of research to practice.

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