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Emmanuelle had an idea for a solution method which uses Pythagoras’ theorem. Since the end of the twentieth century, a growing emphasis has been placed on classroom discourse in mathematics education research, as shown in several review studies (e.g., Herbel-Eisenmann et al. 2012; Nathan and Knuth 2003; Sherin 2002). Convergent actions are teacher-led. Engle and Conant (2002) describe that providing students with opportunities for disciplinary engagement involves letting them engage in genuine problem-solving, giving them authority in addressing such problems, and holding them accountable to others and to disciplinary norms. ORCHESTRATING PRODUCTIVE MATHEMATICAL DISCUSSIONS 315 promoting productive disciplinary engagement to explain how the practices work ... ics in part by helping students learn mathematical discourse practices (e.g., Chapin, O’Connor, & Anderson, 2003; Michaels et al., 2002), making (1993) describe how a teacher who involved students in negotiation of meaning when talking about mathematics took a more directive role when talking about talking about mathematics. Furthermore, Leikin and Levav-Waynberg (2012) argue that solving geometry problems in a variety of ways fosters students’ knowledge and creativity. Journal for Research in Mathematics Education, 35(2), 81–116. Mathematical discourse includes ways of representing, thinking, talking, agreeing, and disagreeing. This involves more than merely being able to solve routine tasks. This study highlights the needs for future research in regard to socioeconomic status and teachers' beliefs in regard to the orchestration of mathematical discourse. When done in a collaborative and supportive learning environment, this can support achievement of higher order thinking skills, as required by the Common Core Standards for Mathematical Practice. We added the action “reformulate” to indicate when the teacher reformulated a previous statement. A key challenge mathematics teachers face in enacting current reforms is to orchestrate. How can we characterize one teacher’s actions during classroom discourse that concerns various solution methods for problems in analytic geometry? The results of this study are described in three steps: First, the developed framework is presented in tables. Various studies have described the difficulties that teachers may experience in finding balance between being open to student ideas and achieving certain mathematical goals (e.g., Cengiz et al. In order to investigate the extent to which classroom discourse in the four lessons was in line with these characteristics, some codes were adjusted and a new categorization was established to distinguish convergent, teacher-led actions from divergent, student-led actions (Henning et al. Second, the distribution of turns changed throughout the four lessons: in the first lesson, the teacher did most of the talking, and the discourse consisted mainly of sequences of a single student and the teacher alternating turns until the teacher turned to a new student. For most mathematical problems and tasks, several different solution methods are possible; whenever a group of students is confronted with a mathematical problem, students have differing ideas and find various solution methods. However, we discovered many differing ways of calculating the required distance. The importance of engaging students in meaningful mathematical discussion has long been identified as anessential component of students’ mathematics learning(National Council of Teachers of Mathematics, 1991, 2000, 2007, 2014). cr.) Michaels, S., O’Connor, C., & Resnick, L. B. [The dissertation citations contained here are published with the permission of ProQuest LLC. 5607. https://scholarsarchive.byu.edu/etd/5607 MSED6205N - Orchestrating Mathematical Discourse. Leermiddelenmonitor (report on teaching materials) 15/16. What makes the orchestration of productive classroom discourse particularly complex is finding balance between building on students’ ideas and guiding the students toward disciplinary ideas (Stein et al. Discrepancies were discussed until a consensus was reached, resulting in adjustments to the code descriptions and framework. Certain aspects of Anna’s particular context make this finding a worthy addition to research in the field. 3, fewer words were spoken in the third lesson. What happened next is illustrated in the following excerpt. 2008). This is the standard method, as presented in most Dutch textbooks. Since our framework was developed by analyzing a teacher’s first steps in orchestrating classroom discourse, and since the collaboration with Anna was focused especially on “getting students to talk”, the framework may be especially useful for studies that focus on teachers’ beginning process of developing a discourse community and establishing favorable norms within their lessons. Orchestrating Mathematical Classroom Discourse About Various Solution Methods: Case Study of a Teacher’s Development. The results reveal three main changes in the teacher’s role: First, the way the teacher reacted to correct or incorrect solution methods shifted from confirming or setting aside suggestions, toward making the solution methods the subject of discussion; second, the distribution of turns changed such that more students were involved in the discourse and in reacting to each other’s solution methods; third, the teacher’s actions shifted from convergent, teacher-led actions toward divergent, student-led actions. When the problem is not the question and the solution is not the answer: mathematical knowing and teaching. Students’ participation, building the discourse on students’ ideas, and logical argumentation are important criteria of classroom discourse but do not ensure students’ learning of mathematics. The relevant parts (each with a duration between 21 and 24 min) containing classroom discourse were transcribed. Cognition and Instruction, 20(4), 399–483. In the presentation of qualitative results below, the changes are made visible and described in more detail. The analysis of teacher moves showed that teachers with higher levels of mathematical knowledge for teaching used more open-ended questioning and prompted more student contributions. Working through these practices involves considerable domain-specific work. Van Eekelen, I. M., Vermunt, J. D., & Boshuizen, H. P. A. Saldaña, J. The cases described in previous studies usually involved a teacher highly skilled in orchestrating classroom discourse, or involved a teacher who had already been involved in an intensive professional development program. Sfard, A. Journal of Mathematics Teacher Education, 8(5), 393–417. https://doi.org/10.1007/978-3-319-62597-3. Third, we observed a strong shift in the teacher’s actions during classroom discourse. In all three excerpts from the first lesson (Excerpt 1.3 follows below), the teacher interacted about a solution method with only one student. First, the teacher’s reactions to solution methods changed considerably throughout the lessons we studied. 3. (SPONSORED WEB SEMINAR DIGEST) by "District Administration"; Education Students Study and teaching Journal for Research in Mathematics Education, 27(4), 458–477. In conclusion, the teacher has made important steps toward the development of classroom discourse, but establishing a productive discourse community would require additional changes and development. My learners often get frustrated with me because I am always striving to great this type of communication both verbally in the face-to-face classroom but also written in my online classes. We added the action “rules of classroom discourse” for when the teacher articulated the rules of communication in classroom discourse, a practice that helps establish favorable classroom norms. The development was content-focused on analytic geometry and in particular on how to teach it. Classroom discourse about variations in students’ solution methods that maintains focus on student ideas provides students with rich mathematical learning opportunities (Murata et al. Journal of Mathematics Teacher Education, 5(3), 205–233. A study of whole classroom mathematical discourse and teacher change. 5 practices for orchestrating productive mathematics discussions. 1998, p. 50). To answer our first research question, a framework was developed to characterize teacher actions during mathematical classroom discourse. MSED 6205 - Orchestrating Mathematical Discourse♦ (1 sem. Also, the analysis highlighted the abundance of literal questioning for all teachers in the subsample. A balancing act: developing a discourse community in a mathematics classroom. Mathematical Discourse Joe Bysiek-MacArthur. Five Practices for Orchestrating Productive Mathematical Discussions. The teacher’s name and all students’ names are pseudonyms. A substantial change can be perceived in the way the teacher dealt with correct, incomplete, and incorrect solution methods. In the third step of data analysis, all four transcripts were coded using the developed code manual. Orchestrating Productive Mathematical Disc ussions: ... investigations of classroom discourse. This second step of our data analysis is a hybrid form of deductive coding, based on the two existing frameworks and inductive coding that emerged from the data (Saldaña 2016). Sfard, A., Nesher, P., Streefland, L., Cobb, P., & Mason, J. In our quantitative analysis, we found that the number of students involved in the discourse increased from nine in the first lesson to 18 in the fourth (see Fig. Vol. Polya, G. (1957). Students are held accountable to the community of learners by sharing and discussing their ideas, accountable to standards of logical argumentation, and accountable to disciplinary knowledge insofar as their work and discussions relate to the rules of mathematics as a discipline (Michaels et al. In addition, teachers can explicitly state and discuss rules for communication. Educational Studies in Mathematics, 91(3), 375–393. https://doi.org/10.1007/s11858-009-0214-4. volume 41, pages357–389(2020)Cite this article. These code descriptions are presented in Appendix. These patterns do not meet the criteria for classroom discourse elaborated above, because they challenge students to try to guess what their teacher is thinking instead of building on and deepening students’ thinking. The five practices that comprise the model will be the focus of the remainder of this article. In G. Kaiser (Ed. Orchestrating classroom discourse that builds on students’ thinking and also advances students mathematically is extremely challenging (e.g., Nathan and Knuth 2003; Sherin 2002). Box 1346, Ann Arbor, MI 48106. Excerpts are numbered according to lesson and time. After Inez was done explaining her method, the teacher asked another student, Thom, to react. Practices for Orchestrating Productive Math Discussions Margaret S. Smith & Mary Kay Stein, NCTM & Corwin Press, 2011 www.nctm.org 1. Orchestrating Mathematical Classroom Discourse About Various Solution Methods: Case Study of a Teacher’s Development November 2019 Journal für Mathematik-Didaktik 41(5):1-33 Kooloos, C., Oolbekkink-Marchand, H., Kaenders, R. et al. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers. I received this email, "Orchestrating mathematical discourse to enhance student learning", and wondered how many of my colleagues utilize mathematical discourse in the class? How mathematics teachers can develop and orchestrate classroom discourse remains an important question for research, especially regarding various solution methods for mathematical problems in higher secondary school. Graduate Semester | 1 credits | 4 Weeks Explore Our Offerings / Education ; Orchestrating Mathematical Discourse ; Start Date. We join this field of research and choose to employ as our definition of (mathematical) classroom discourse: verbal interaction among teacher and students as a community, in which students’ ideas about mathematical problems or tasks are discussed. An additional challenge was the Dutch teachers’ reliance on textbooks (Blockhuis et al. These different roles can be regarded as a particular strength of this case study, as the researcher was able to adapt to changing situations, to have a sense of realism about the classroom situation, and to build access, empathy, and trust, which Cohen et al. Stanford: Stanford University. In preceding lessons, vectors had been defined and vector equations for lines had been discussed. Anna and the researcher took this lesson as the starting point for their collaboration, and together they developed three more analytic geometry lessons for the same tenth-grade class (ages 15–16) of the pre-university science track at Anna’s school. 2008). For example, in Excerpt 1.1, line 7: “Yes. 2006) may have strongly influenced the changes in her role during classroom discourse. Fig. Like the researcher, Anna was interested in teaching such that students’ different solution methods are discussed in the classroom. Additionally, a teacher may gain insight into “the students’ conceptual possibilities and current understandings” (Yackel and Cobb 1996, pp. O’Connor et al. https://doi.org/10.1007/s10857-011-9179-7. 11–35). (2006). Theses and Dissertations. Redirecting, progressing, and focusing actions‑a framework for describing how teachers use students’ comments to work with mathematics. Based on the quantitative findings, fourteen teachers were selected for the qualitative phase and their classroom discussions were coded to reveal patterns in the teachers' orchestration of discussions. For example, in Excerpt 1.2, the teacher did not ask how Emmanuelle wants to use Pythagoras’ theorem, instead starting by choosing a triangle. mathematical ideas to be learned will be emphasized. The teacher demonstrates or reformulates something, sets an idea aside, or asks for closed progress details: for instance a teacher may ask a closed, simple question for which only one correct answer is possible. - 167.114.98.126. In Socializing intelligence through talk and dialogue (pp. Developing and orchestrating classroom discourse about students’ different solution methods is an essential yet complex task for mathematics teachers. Excerpts 4.1, 4.2, and 4.3 are from the fourth lesson, and they, too, are in chronological order. ], ProQuest LLC. We define a discourse community as a class in which productive classroom discourse is a regular course of action. To engage students in productive mathematical conversations, teachers can orchestrate discourse and structure learning environments to deepen engagement and support learning.Using effective strategies will support students as they learn to participate in mathematical discourse. 2009). 4, over the course of the first three lessons the number of converging actions decreased and the number of divergent actions increased. In designing the lesson, main focus was on orchestrating classroom discourse about students’ solution methods. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers, The purpose of this study was to examine the mathematical discourse within novice elementary teachers' classrooms. This research comes from the book 5 practices for Orchestrating Productive ... mathematical ideas that they want their students to learn and what students The most common form of classroom discourse is referred to as the “initiation-response-evaluation” pattern: the teacher initiates a question, a student responds, and the teacher evaluates the response (Cazden 2001; Mehan 1979). Whereas Drageset’s framework focusses specifically on the types of turns, we also took into account the content of utterances. Orchestrating Mathematical Classroom Discourse About Various Solution Methods: Case Study of a Teacher’s Development, $$\left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} 3\\ 2 \end{array}\right)+\lambda \left(\begin{array}{c} 4\\ -1 \end{array}\right)$$, https://doi.org/10.1007/978-3-319-62597-3, https://doi.org/10.1007/s10857-011-9179-7, https://doi.org/10.3102/0013189x032001009, https://doi.org/10.1007/s10649-013-9515-1, https://doi.org/10.1007/s10857-014-9280-9, https://doi.org/10.1207/S1532690XCI2004_1, https://doi.org/10.1080/10986065.2016.1107821, https://doi.org/10.1007/s10649-015-9636-9, https://doi.org/10.1007/s11858-009-0214-4, https://doi.org/10.1007/s10857-012-9224-1, https://doi.org/10.1007/s10857-017-9387-x, https://doi.org/10.1207/s1532690xci2301_4, https://doi.org/10.1016/j.jmathb.2011.11.001, https://doi.org/10.3102/978-0-935302-43-1_27, https://doi.org/10.1007/s11217-007-9071-1, https://doi.org/10.1080/07370008.2017.1362408, https://doi.org/10.1207/S1532690XCI2102_03, https://doi.org/10.1016/j.learninstruc.2016.11.003, https://doi.org/10.1080/10986060802229675, https://doi.org/10.1016/j.tate.2005.12.001, https://doi.org/10.1080/10986065.2017.1259786, https://doi.org/10.1007/s10857-005-3849-2, http://creativecommons.org/licenses/by/4.0/, https://doi.org/10.1007/s13138-019-00150-2. Based on the four transcripts and the tables, several aspects of change were identified and classified into three categories, as elaborated below. For more details, see Drageset (2014, 2015). https://doi.org/10.1007/s13138-019-00150-2, DOI: https://doi.org/10.1007/s13138-019-00150-2, Over 10 million scientific documents at your fingertips, Not logged in 2008, p. 315). (2016). Cambridge: Harvard University Press. The students were asked to solve the following problem. 3 shows that the percentage of student utterances greatly increased from the first to the second lesson. Both the framework that we developed and our method of development could be used as a basis for further investigations seeking to answer this question. Divergent actions are student-led in the sense that they are teacher actions intended to make students’ thinking public and to build the discussion on student ideas. By asking for explanations and getting students to respond to each other, teachers implicitly negotiate favorable norms. Nathan, M. J., & Knuth, E. J. Attributes of instances of student mathematical thinking that are worth building on in whole- class discussion. Here lies an essential yet challenging task for the teacher, namely orchestrating classroom discourse such that the students are supported in making important mathematical connections and guided toward disciplinary ideas, while maintaining the focus on … Cobb, P., Wood, T., & Yackel, E. (1993). Ryve (2011) states that researchers should be clear about their definition of discourse. We found that these categories roughly matched the categories of Drageset’s framework (2015), which was developed to analyze classroom discourse on a turn-by-turn basis through the categorization of student and teacher actions during their interactions. 4 above, the teacher’s actions changed from mainly convergent toward mainly divergent. Excerpt 1.3 is from the first lesson and regards the fifth solution method (see 3.4.1.). Learning mathematics through conversation: is it as good as they say? Download your free copy here. Anna had enacted one analytic geometry lesson, based on discussions in the TDT, in which she orchestrated classroom discourse about a variety of students’ solution methods for the first time. Additional research is necessary to see in which ways the students benefit from changes in the teacher’s role. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Second, quantitative outcomes of analysis are presented in figures. By making each solution method a subject of discussion, instead of evaluating and moving on, the teaching moves away from practices in which the teacher holds the authority over the right answer, toward collectively establishing truth based on logical argumentation. In each of the four lessons, a variety of students’ solution methods was discussed during classroom discourse. During several steps of data analysis, the second author also coded part of a transcript. Leinhardt, G., & Steele, M. D. (2005). https://doi.org/10.1002/sce.20131. Classroom discourse is initially described as a way to align classroom mathematics with mathematics as a discipline in which mathematical meaning is negotiated. More recently, Güçler (2016) used Sfard’s concept of metadiscursive rules (Sfard 2008) to show that making these rules explicit in discussion fosters students’ mathematical learning. However, the teacher, as a representative of the mathematics community, has the responsibility to make decisions about the ideas students share and to advance the mathematical learning of the whole group toward certain disciplinary mathematical ideas (Yackel and Cobb 1996). Implications for teacher educators, including per-service preparation and professional development, are outlined. It may take place between partners, small groups, or as a whole class. In most cases, such an utterance was followed by an encouraging action, after which the student continued and finished the solution method. , 458–477 deviate slightly from the first in a variety of students working on a mathematical challenge that worthy! Coordinate system 108 journal articles einen Diskurs über verschiedene Schülerlösungen führten hand to react the! Version 7 ) done right away implicit metalevel rules of communication during classroom discourse about various solution is! Not whether to teach it leg ( line 11 ) the video recording of classroom discourse details appear., Pierson Bishop, J. E., & Heyd-Metzuyanim, E. ( 1993.! Kaenders, R. ( 2012 ) mathematics lessons, an important part of every mathematics teacher Education 90! 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