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ENACTING PROOF IN MIDDLE SCHOOL MATHEMATICS Kristen N. Bieda Michigan State University, East Lansing, Michigan, USA Although justifying and proving have been gaining in importance in school mathematics in all grade levels and across all content areas over the past decade, little research has examined how tasks designed to develop students’ competencies in justifying and proving are enacted in the classroom. This paper presents findings from a study of the implementation of proof-rich tasks in classrooms utilizing the Connected Mathematics Project (CMP) curriculum. Findings from this work suggest that middle school students’ classroom experiences with justification and proof are insufficient for developing desired conceptions of mathematical proof. Implications of this work for curriculum development and teacher education will be discussed. INTRODUCTION Proof in school mathematics within the United States since the decline of the “New Math” movement in the late 1960’s has typically been instantiated as a topic in a secondary-level geometry course. However, unlike other topics in mathematics education, little attention has been paid to how the process of proving is taught. Students’ experiences with proving at the middle school level is becoming increasingly important, as the research indicates that students struggle with producing mathematical proof within and beyond the high school geometry course (Chazan, 1993, Harel & Sowder, 1998, Hoyles, 1997, Weber, 2003). Given that tasks exist in middle school curricula, particularly the Connected Mathematics Project (CMP), that provide opportunities to engage students in justifying and proving their reasoning (G.J. Stylianides, 2007), it has been unknown, to this point, the process of enacting proof-related tasks. The aim of the work presented here is to shed light upon the process of learning to prove in middle school classrooms from the perspective of understanding a pedagogy of proving as it currently emerges in practice —specifically, to answer the question: How do curricular opportunities intended to engage middle school students in justifying and proving become enacted? CONCEPTUAL FRAMEWORK As the mathematics education community has understood the potential and importance of developing students’ competencies in proof and more research is starting to examine the pedagogy of proving, there is a need for a widely held definition of proof that takes into account the process of proving as it evolves in classrooms. The following framework of proof, which guided the work presented here, aligns conceptions of proof from the discipline with the demands of school mathematics: Proof is a mathematical argument, that is, a connected sequence of assertions for or against a mathematical claim, with the following characteristics: (i) it uses statements accepted by the classroom community (set of accepted statements) that are true and available without further justification; (ii) it employs forms of reasoning (modes of argumentation) that are valid and known to, or within the conceptual reach of, the classroom community; and; (iii) it is communicated with forms of expression (modes of argument representation) that are appropriate and known to, or within the conceptual reach of, the classroom community. (A.J. Stylianides, 2007, p. 6) One critical feature of the definition is the prominence to the role of the classroom community. As proof emerges as a product of discussions about what is required to justify a conjecture for all cases and how various forms of an argument influence a proof’s potential for justification, students learn that proof goes beyond convincing one’s self and convincing a friend, but it must convince the broader classroom community. With Stylianides’ (2007) proof framework in mind, the design of this research draws from a framework of the implementation of a mathematical instructional task (Stein et al., 1996). The process of enacting mathematical tasks begins with the curricular task either developed by a teacher or written in a textbook (task as written), evolves as the teacher conceptualizes the task to be implemented (task as intended), the task is introduced and taken up by the students (task as implemented), and student learning results (task as enacted). Along each stage of the model, there are several factors that influence the success of the task enactment. For example, the teacher’s content knowledge, both pedagogical and mathematical, plays a significant role in how the teacher decides to use the task from its intended form to its implemented form. As this study seeks to understand the pedagogy of enacting proof-related tasks, the focus of the research is on how the task as intended is implemented and becomes enacted in the classroom. METHODOLOGY Data was collected through observing lesson implementations in seven middleschool teachers’ classrooms. Three 6th grade teachers, two 7th grade teachers, and two 8th grade teachers and their students participated in the study. Each lesson observed was identified to contain a task that engaged students in generating conjectures prior to the observation. All of the classrooms in the study utilized the Connected Mathematics Project (CMP) curriculum, a curriculum where 40% of the tasks are proof-related (G.J. Stylianides, 2007). At least six lessons were observed per teacher; there were a total of 49 observations across all seven classrooms. During the lesson observation, the researcher took fieldnotes along with audiotaping to document the most salient classroom events and discussions related to students’ conjecturing and justifying. The fieldnotes were analyzed to distinguish elements of the implementation as proving events. A proving event consisted of an initial response provided by a student to a task eliciting a conjecture, with the teacher’s and students’ feedback to the student’s initial conjecture. Elements of the fieldnotes were coded using the A.J. Stylianides’ (2007) proof framework at the turn level; thus, individual student and teacher moves related to each part of the proving process were coded as separate instances. The product of this analysis provided information as to what parts of establishing proof were enacted in the proving events, as well as how frequent each of these activities was in the proving process for each classroom. Once the parts of the proving events were determined, the data were utterances from the teacher and the students were coded using a constant comparative method of coding (Glaser & Strauss, 1967), focusing on the teacher’s and students’ moves during proving events. RESULTS Analysis of the fieldnote corpus revealed that while proof-related tasks were generally implemented as written (71% of the time), roughly half of the opportunities to prove, students’ generalizations provided in response to the tasks, were never supported with justifications (see Table 1). Grade Level Number of Problems as Implemented Number of Opportunities to Prove1 Number of Opportunities to Prove Never Supported with Justifications 6th 31 65 24 7th 13 28 17 8th 8 16 9 TOTAL 52 109 50 Table 1: Opportunities to prove generated from problems implemented Table 1 shows that there was little difference in the rate at which students in 7th or 8th grades produced justifications compared to 6th graders, suggesting that students have difficulties with justification and proof throughout middle school and teachers fail to address the needs of students as they learn to justify their reasoning. Data analysis revealed several trajectories that proof-related tasks followed during implementation (see Figure 1). Although each task ultimately resulted in either being enacted or not enacted, depending upon whether justifications were provided to generalizations in response to proof-related tasks, there were three outcomes in terms of the teacher’s and students’ responses to the generalization. 1 As each problem was implemented, more than one opportunity to prove could be generated. As such, the totals reflect situations where more than one opportunity to prove emerged during the implementation. Student(s) Make Generalizations (i.e. claims, conjectures) Student(s) Justify Generalizations (Proof-Related Problem Enacted) Student to Student Feedback Teacher Feedback No Feedback Given Student(s) Do Not Justify Generalizations (Proof-Related Problem Not Enacted) Teacher Sanctions Conjecture without Justification Given Teacher Request for Student to Student Feedback No Feedback Given Figure 1: Trajectory of Proof-Related Tasks Of the 50 proof-related problems not enacted, where students did not provide a justification for their generalizations, the most prevalent response was for no feedback to be given to the students’ generalizations (42% or 21 out of 50). In essence, these emergent opportunities to prove were ignored by the classroom community. The following excerpt from Mr. Zeff2’s 8th grade class illustrates this type of response, or lack thereof. In this excerpt, students work on an investigation from CMP based on the classic “wheat and chessboard” problem, using a context of a king providing a reward to a peasant for saving the life of the king’s daughter. The king allows the peasant to stipulate her reward, and the clever peasant suggests that the king place 1 ruba, comparable to a penny, on the first square of a chessboard, followed by 2 rubas on the second square, and 4 rubas on the third square, continuing in the same manner of doubling the previous amount until all of the squares had been covered. An alternative plan presented, Plan 2, reduces the size of the chessboard to 16 squares, but the king has to triple the amount of rubas he places on each subsequent square. In the exchange below, Mr. Zeff leads a discussion about the two plans: Mr. Zeff: Ok, what are the similarities between both plans? Brian: One gets multiplied by 2 and the other is multiplied by 3. Damon: Yeah, Plan 2 is always higher. Mr. Zeff: Let’s have a little caution on that one, just from what we see so far. Lakesha, will you share your equation? 2 All names presented here are gender-appropriate pseudonyms. Lakesha: I got . Mr. Zeff: Usually I start the equation with r equals. You can do it anyway you want. [writes on the overhead projector] Mr. Zeff: What are we doing over and over again? Brian: Multiplying by 3. Mr. Zeff: Right, and just by trial and error you see that it is always one box before it. What is another reason...why do you subtract one from it? Brian: You triple every single square but one. Mr. Zeff: Which one? Brian: The first square. Mr. Zeff: Ok, so we didn’t triple the first time so it’s always one less. Mr. Zeff: Which plan is better? Damon: Plan 1 Mr. Zeff: Which is better for the King? Brian: Plan 2. Mr. Zeff: [puts up graph of exponential equation on overhead projector] OK, now let’s look at your graphs. You should have a very, very straight line at the bottom. Throughout the excerpt, we see few requests from Mr. Zeff for students to provide justification for their generalizations. As a result, he implicitly sanctions their generalizations without insisting upon justification. Further, Mr. Zeff’s decision to have students continue sharing answers, without justification, reinforces a classroom culture where justifying and proving are not essential components of everyday mathematical practice. Although space limitations do not allow for a full presentation of results and examples from the data, there was evidence of missed instructional opportunities during the proving events across the middle school classrooms. Across all grade levels, there were only 28 opportunities to prove supported by justifications consisting of general arguments. When examining how teachers addressed students’ justifications as they were made public, nearly 30% of the justifications, primarily based on general arguments, never received feedback. Further, when a teacher sanctioned a justification with a positive appraisal, it was just as likely that it was a justification based on non-proof arguments as it was a justification based on general arguments. CONCLUSIONS The results in this study reveal that middle school students produced few general arguments in response to proof-related tasks, a finding not surprising considering the literature on students’ difficulties with proving. Stein et al. (1996) defined several factors that contribute to the decline of high-level cognitive activities such as generating proof, including time constraints and lack of accountability. As the excerpt presented illustrated, the relative lack of sufficient critical feedback once a proof-related task was enacted (e.g., pushing students to move beyond examples-based justifications) contributed to relatively few justifications based on general arguments. A confluence of factors, including time constraints, lack of accountability, and lack of curricular support, worked against the development of a community in the classroom where one’s reasoning, once public, was expected to be scrutinized. During the study, I will present additional data to support these conclusions. REFERENCES Chazan, D. (1993). High school students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387. Glaser, B. and Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine. Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education, 7, 234-283 Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. Learning of Mathematics, 17(1), p.7-16. Hoyles, C. & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193-223. Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., & Phillips, E.D. (1998/2002/2005) Connected Mathematics Project. Upper Saddle River, NJ: Prentice Hall. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Stein, M.K., Grover,B. and Hennigsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), p. 455488. Stylianides, A.J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. Stylianides, G.J. (2007). Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics. International Journal of Science and Mathematics Education. 5, DOI: 10.1007/s10763-007-9074-y. Published online: May 10, 2007. Weber, K. (2003, June). Students’ difficulties with proof. Research Sampler, 8. Retrieved March 17, 2006, from http://www.maa.org/t_ and _l/sampler/rs_8.html.