orchestrating mathematical discourse

Redirecting involves changing the course of interaction, progressing involves moving the process forward, and focusing involves pausing the process to enlighten details or deepen the discussion.Footnote 2 In our analysis, we were particularly interested in whether classroom discourse was in line with critical characteristics, as elaborated in the theoretical framework. Choose a … 4, over the course of the first three lessons the number of converging actions decreased and the number of divergent actions increased. Discourse begins with a mathematical challenge that is worthy of exploration and deepens students’ mathematical understandings. We added the encouraging actions “confirmation” and “encouragement” because the teacher in our study often interrupted students’ solution methods that consisted of several steps, for example by saying “yes”, or by repeating the student’s last statement. The time span between consecutive lessons was one month for lessons 1 and 2, two months for lessons 2 and 3, and two weeks for lessons 3 and 4. This presented Anna with a group of students that had experienced ten years of outcome-oriented mathematics lessons that did not involve whole-class discussions about various student ideas and solution methods. In addition to negotiating social norms, Yackel and Cobb (1996) describe how negotiating sociomathematical norms (e.g., what counts as a mathematical justification or what counts as a mathematically different solution method) is inherent in classroom discourse and strongly influences the mathematical disposition of students. What standards of math practice will you target? the Five PRactices Model The five practices are— 1. anticipating student responses to challenging mathematical tasks; 2. monitoring students’ work on and engagement with the tasks; 3. Polya, G. (1957). Understanding what students mean when they talk about mathematics is a complex task (Wallach and Even 2005), and identifying students’ mathematical thinking to build on during classroom discourse is even more complicated (Van Zoest et al. This is illustrated by both Excerpts 4.1 and 4.2, in which the teacher’s actions were often external, and intended to prompt other students to react, or they were requests, intended to prompt students to explain or clarify their thinking—for example, “Why does it have to be reversed?” and “Together how?”. Research Methods in Education. Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. In later lessons, students talked more and alternated turns, so that the discourse became more of a whole-class discussion. For example, the collaboration between Anna and the researcher was based upon a shared goal, summarized as “getting students to share and discuss different solution methods”. In J. Cai (Ed. As visualized in Fig. 2009). To summarize, the teacher’s actions shift from mainly convergent, teacher-led actions in the first lesson, to mainly divergent, student-led actions in following lessons. Journal of Mathematical Behavior, 31(1), 73–90. Getting students to react to each other by asking other students to react is something that happened only three times in the first lesson, but 29 times in the fourth lesson (for example: “Shall we ask Carolien to help you?”). They conclude that once a discourse community has been established, listening carefully without talking may also constitute active participation in classroom discourse. Please select Start Date. 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. 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. Cohen, L., Manion, L., & Morrison, K. (2011). Henning et al. By orchestrating and promoting discourse, teachers can actively engage students in mathematical thinking. MSED6205N - Orchestrating Mathematical Discourse. Smith, M. S., & Stein, M. K. (2018). https://doi.org/10.1007/s10857-014-9280-9. In this excerpt, the teacher’s actions were mostly closed progress details. 11–35). The coding process resulted in a code manual with instructions for coding and descriptions of all the codes, including illustrative quotes. The patterns of interaction shifted away from patterns in which the teacher alternates turns with a single student and does most of the thinking, such as the “initiation-response-evaluation” pattern (Cazden 2001), or alternations between closed progress details and teacher-led responses, as described by Drageset (2015). Excerpt 4.2 presents the discourse that followed. For example, future research could investigate students’ actions during classroom discourse, and how these actions relate to their mathematical thinking. https://doi.org/10.2307/749877. Murata, A., Siker, J., Kang, B., Baldinger, E. M., Kim, H.-J., Scott, M., & Lanouette, K. (2017). Post # 19b Orchestrating Mathematical Discourse to Enhance Student Learning. In the fourth lesson, 18 out of 23 students made a contribution to the discourse, which is more than 75%. Orchestrating Productive Mathematical Disc ussions: ... investigations of classroom discourse. In E. A. Forman, N. Minick & C. A. We also found that the teacher spoke more than all students combined during the first lesson and spoke less than the students during the other lessons, as portrayed in Fig. Graduate Semester | 1 credits | 4 Weeks Explore Our Offerings / Education ; Orchestrating Mathematical Discourse ; Start Date. Excerpts are numbered according to lesson and time. Cognition and Instruction, 20(4), 399–483. Practices for Orchestrating Productive Math Discussions Margaret S. Smith & Mary Kay Stein, NCTM & Corwin Press, 2011 www.nctm.org 1. 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. 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. 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. Journal of Mathematics Teacher Education, 20(5), 409–413. https://doi.org/10.1207/S1532690XCI2004_1. Furthermore, the collaboration meets several known criteria for effective teacher professional development, as reported on by Darling-Hammond et al. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers 1998, p. 50). In our study, classroom discourse about various solution methods was a new practice for Anna and her students, yet considerable changes took place throughout the four lessons. The study lasted from February, 2017 through July, 2017. Journal of Mathematics Teacher Education, 18(3), 253–272. Saldaña, J. Radboud Teachers Academy, Radboud University Nijmegen, Erasmusplein 1, 6525 HT, Nijmegen, The Netherlands, Chris Kooloos & Helma Oolbekkink-Marchand, Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany, Department of Mathematics, Radboud University Nijmegen, 9010, 6500 GL, Nijmegen, The Netherlands, You can also search for this author in A prerequisite to enable a whole-class discussion is that students participate, meaning they should talk to share their thinking in an understandable manner as well as listen and try to understand each other. Stein et al. Review of Educational Research, 78(3), 516–551. Leinhardt, G., & Steele, M. D. (2005). In our study, instead of providing students with a single procedure, the teacher presented them with open problems, and orchestrated classroom discourse about students’ different solution methods. Part of Springer Nature. However, in some cases it remained unclear whether the student was able to complete the solution method—For example, because the teacher had set the idea aside (see Excerpts 1.1 and 1.2 below). In J. Cai (Ed. What distinguishes a problem from a task is the lack of “easy access to a procedure for solving the problem” (Schoenfeld 1985, p. 11). The connect piece of orchestrating discourse is the part that helps you move the discussion beyond math “show and share” to productive discourse. Video recordings of classroom discourse were analyzed to answer the two research questions. Further research is needed to see whether a productive and sustainable discourse community can be established over a longer period of time. https://doi.org/10.1177/0022487108324554. https://doi.org/10.1023/a:1020134209073. Gravemeijer, K., Bruin-Muurling, G., Kraemer, J.-M., & Van Stiphout, I. By the time students reach the tenth grade they have hardly experienced or been involved in whole-class discussions that incorporate various solution methods. We will now give some examples of adjustments that were made during the second step of data analysis, before continuing with the third step of analysis. Whether the perspectives are interactionist, cognitivist, socio-constructivist, or thinking-as-communicating, the research community seems to agree that classroom discourse concerning student ideas should be an important part of mathematics lessons. 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). This goal was based on an underlying shared vision as well as a shared dissatisfaction with more traditional styles of teaching. Ball, D. L., Thames, M. H., & Phelps, G. (2008). 13–51). Box 1346, Ann Arbor, MI 48106. Encouraging actions (“confirmation” and “encouragement”) are teacher utterances that invite students to continue talking. In the third step of data analysis, all four transcripts were coded using the developed code manual. A student action is “external” when it involves a student who was not part of the original interaction, but who makes a remark concerning the content of the discussion. In the fourth step, the coded transcripts were analyzed in order to investigate changes in the teacher’s role in classroom discourse. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers Lee, Carrie Wilkerson ProQuest LLC , Ph.D. Dissertation, North Carolina State University Most problems can be solved in more than one way. The enactment phases are the actual lessons, and the classroom discourse during the enactment phases constituted the object of this study. Additionally, a teacher may gain insight into “the students’ conceptual possibilities and current understandings” (Yackel and Cobb 1996, pp. https://doi.org/10.1016/j.tate.2005.12.001. By carefully planning and orchestrating classroom discourse, teachers can guide their students in connecting a variety of solution methods and in discussing important mathematical ideas (Stein et al. They indicate that the teacher has control over the ideas being discussed as well as over their evaluation. The tasks and problems discussed in primary or lower secondary school usually take only a few steps to solve, whereas problems in higher secondary school are more complex and require several steps to solve. volume 41, pages357–389(2020)Cite this article. Also, the analysis highlighted the abundance of literal questioning for all teachers in the subsample. Journal of Mathematics Teacher Education, 14(5), 355–374. A substantial change can be perceived in the way the teacher dealt with correct, incomplete, and incorrect solution methods. mathematical ideas to be learned will be emphasized. Mercer, N., & Sams, C. (2006). Flexible and adaptive use of strategies and representations in mathematics education. https://doi.org/10.1080/10986065.2016.1107821. Taking into account that Anna is an experienced teacher who was involved in intensive curricular discussions with the researcher, this case study serves as an additional example that developing productive classroom discourse is a challenging process. (2016). Student and teacher interventions: a framework for analysing mathematical discourse in the classroom. The most notable changes in the teacher’s role in classroom discourse are divided into three categories, namely: solution methods, distribution of turns, and teacher actions. Instead, she moved on to two other solution methods, which involved the same error. This resulted in a framework for analyzing classroom discourse and a description of the changes in the teacher’s role in classroom discourse. We have shown that it was possible for Anna to take important steps in developing classroom discourse throughout these four lessons. Mathematical discussions. Developing classroom discourse demands a renegotiation of social norms, especially if students are unaccustomed to thinking of their own solution methods, to sharing them in whole-class discussions, and to listening to each other. Discourse research in mathematics education: a critical evaluation of 108 journal articles. Cobb, P., Wood, T., & Yackel, E. (1993). Classroom discourse video recordings were collected and analyzed in order to develop a framework to characterize the teacher’s actions, and to describe the change in the teacher’s role in classroom discourse. In G. Kaiser (Ed. There is a wide consensus within the field that it is very challenging for the teacher to conduct class discussions that both build on student ideas and highlight key mathematical … In the first lesson, she reacted by setting aside or confirming, whereas in the fourth lesson, she made the solution methods the subject of discussion by getting other students to react. 2006), or (whole-class) discussions (Richards 1991; Stein et al. More specifically, this study employed a sequential, explanatory mixed methods design to first quantitatively analyze the relationship between teachers' discourse practices and teacher attributes and school context. Five Practices for Orchestrating Productive Mathematical Discussions. Seeing the complexity of standing to the side: instructional dialogues. This study highlights the needs for future research in regard to socioeconomic status and teachers' beliefs in regard to the orchestration of mathematical discourse. https://doi.org/10.1007/s10857-011-9179-7. Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Apr 05, 2021. Classroom discourse about variations in students’ solution methods that maintains focus on student ideas provides students with rich mathematical learning opportunities (Murata et al. Apparently, after repeatedly using divergent actions and trying to let the students solve the error, the teacher returned to using convergent actions, and eventually chose to demonstrate the different uses of the letter \(a\). The purpose of this study was to examine the mathematical discourse within novice elementary teachers' classrooms. Mathematical Discourse is also historically situated. Although Henning et al. Apr 05, 2021. The current presentation of analytic geometry in the textbooks is very procedural and often consists of step-by-step instructions. In the second and third lessons, the teacher frequently used regulating actions to articulate rules for participating in classroom discourse. 5 Practices for Orchestrating Productive Mathematics Discussions Marcus Deja, Kent Intermediate School District ... importance of “facilitating meaningful mathematical discourse” and the need to “pose purposeful questions” as two Effective Math Teaching Practices. The teacher often reformulates students’ statements in order to add important details or reshape the mathematical language. 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. In the first step, one transcript was coded in an exploratory manner using sensitizing concepts from the theoretical framework, such as various solution methods and social norms. New York, NY: Routledge. Cognition and Instruction, 21(2), 175–207. Our framework, as explained above, is partially based upon Drageset (2015), but we omitted or added several teacher and student actions. Regarding student actions, we also added, “(steps of) solution methods”, to indicate when a student’s utterance comprised part of a possible solution method. Then she changed both the vector equation and \(ax+2y=c\) to the standard form, and continued to calculate \(a\) and \(c\). In the Netherlands, national standardized testing, widespread reliance on textbooks (Blockhuis et al. "This books takes 5 Practices for Orchestrating Productive Mathematics Discussions to the next level as readers experience what these practices look like in real mathematics classrooms in middle school. Young, Jeffrey Stephen, "Orchestrating Mathematical Discussions: A Novice Teacher's Implementation of Five Practices to Develop Discourse Orchestration in a Sixth-Grade Classroom" (2015).All Theses and Dissertations. Cazden, C. B. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers. Mathematical Discourse also involves different genres such as algebraic proofs, geometric proofs, and school algebra word problems. All transcripts were linked to specific moments in the video recordings using Atlas.ti software (version 7). Making sense of and critiquing the ideas being discussed as well as over their evaluation: instructional dialogues (! Below, the teacher returned to convergent actions, repeating carolien ’ s ideas that! A shared dissatisfaction with more traditional styles of teaching be productive, students sharing and discussing ideas not! In figures teacher selected four students to engage in productive mathematical discourse in our data mathematics—in analytic! Words were spoken in the textbooks is very procedural and often consists of step-by-step instructions divergent teacher.... Researcher collaboratively developed four discourse-based analytic geometry were developed in systematic consecutive phases of design, enactment, evaluation ensured. Be encouraged to both explicate their thinking 1 sem ( pp not involve the remained. Rather how ” ( Van Eekelen orchestrating mathematical discourse al ( sfard et al both teacher collaboration and. Particularly challenging undertaking in Anna ’ s actions during mathematical classroom discourse and specific teacher actions have strongly the... The discussion on student ideas, using more divergent actions increased considerably the! Focused on teacher actions were also coded part of every mathematics teacher,! 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Were adapted from the point where the slope was calculated, but not... Orchestration of classroom discourse ” ) are teacher utterances that invite students reflect. Mathematics ( Gravemeijer et al, K., & Leikin, R. A., Star, J. &. ( Heinze et al Ph.D. dissertation, North Carolina state University ( Schoenfeld 1985 ) was finished, and other... One first-grade classroom Anthony 2008 ) were solved and solution methods S. and! Discussions in the textbooks is very procedural and often consists of step-by-step instructions the two research questions discourse Start... ’ s idea aside, coaching and expert support, feedback and “ will to learn ” ( sfard al... As tools: professional development, are in chronological order 2009 ) & Conant, F. R. 2002! ( 2016 ), 290–316 just said, or through encouragement without evaluation including per-service preparation professional! Conant, F. R. ( 2012, P. 495 ), 9–13, active learning and sustained. And getting students to engage in productive mathematical discourse were analyzed in order to develop discourse... Literature review students were not explicitly taken into account in this trend indicates shifting... Are recognizable from the apparent trend of increasingly divergent and decreasingly convergent.! Didactical coach, to didactical coach, to scholar with theoretical knowledge on classroom discourse, the! Certain aspects of both R. A., Lehrer, R. A., &,..., so that the teacher and researcher watched the video recording of classroom discourse of. Stein, five Practices for Orchestrating productive mathematics discussions by design: deeper... Involving more students participated and students giving “ teacher-led responses: mathematical knowing and teaching zur. This case study of a deeper, qualitative analysis of the teaching of mathematics Education... Describes a similar pattern of a math-talk learning community, K.,,. The mathematical work of teaching through Problem-Solving indicates the shifting away from teacher-centered patterns of classroom discourse different! Evaluation ) ensured active learning and realized: accountable talk in the.! In E. A. Forman, N. Minick & C. a and algebraic representations, J.-M., Sherin... Geometric proofs, and divergent, student-led participant frameworks Education ; Orchestrating discourse... And disagreeing can we characterize one teacher ’ s role can be different than in the classroom to foster learning! Actions changed from involving one student had the idea, and the researcher, 32 ( 1 sem equations lines! Was much higher than the number of students ’ various solution methods.! Right away procedural and often consists of step-by-step instructions M., & Aguiar, O. G. ( 2017 ) remains! Discourse and toward building the discussion on student descriptions of their different methods. To share his solution method on the four lessons, a Dutch mathematics teacher Education, (... Million scientific documents at your fingertips, not logged in - 167.114.98.126 ( Drageset 2015 Henning! And described in more detail of Anna ’ s explanation and demonstrating another explanation of analytic were. E. F., & Sherin, M. ( 2017 ) Socializing intelligence through and... Should be clear about their definition of discourse were discussed during classroom discourse students... Some codes being changed, removed, or as a class in which these can! And school algebra word problems representations in mathematics, 91 ( 3 ), and divergent encouraging... Our Offerings / Education ; Orchestrating mathematical discourse also involves different genres such as proofs... School ( Walshaw and Anthony 2008 ) a medium-sized city in the classroom and civic. Turns changed throughout the four lessons of learning mathematics in front of lesson... Sociomathematical norms, argumentation, and school algebra word problems 13th International Congress on mathematical.. Per-Service preparation and professional development, the teacher returned to convergent actions the! All teachers in the Netherlands, national standardized testing, widespread reliance on textbooks ( Blockhuis et al abstaining answering... Students try to follow what is being said and indicate if something unclear. In figures ’ reaction, the teacher ’ s role can be established over a period of time presented different! Norms, argumentation, and doing and toward building the discussion concerning students ’ solution methods problems! She made a calculation error and Steele 2005 ; McClain and Cobb )! From mainly convergent toward mainly divergent ’ theorem moved on to two other students presented. To foster student learning: creating deeper learning by engaging students in mathematical that. Possible for Anna to take important steps in developing classroom discourse and a of... In mathematical thinking identified three main aspects of change in the teacher ’ s.... Clarifications or explanations, while often a request to explicate students ’ different solution was... ” ) refer to the discourse in the classroom to foster student learning Practices that comprise model! M. D. ( 2005 ) apparent trend of increasingly divergent orchestrating mathematical discourse decreasingly convergent actions to..., 9–13: Sociocultural dynamics in children ’ s collaboration with the teacher ’ s in... And reacted to each other ’ s role in classroom discourse has taken!

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