# orchestrating mathematical discourse

19 January 2021For example, to calculate the distance from a point to a line, only one solution method is given. 2 shows that the number of students who contributed to classroom discourse increased during the course of the four lessons. It may take place between partners, small groups, or as a whole class. The silent and the vocal: participation and learning in whole-class discussion. In this study, the researcher collaborated with one teacher, Anna, and developed classroom discourse about various solution methods. They indicate that the teacher has control over the ideas being discussed as well as over their evaluation. 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). We developed a framework for describing the teacher’s actions during classroom discourse about variations in solution methods (Tables 1 and 2). In the third step of data analysis, all four transcripts were coded using the developed code manual. Journal for Research in Mathematics Education, 27(4), 458–477. Second, Drageset’s categorization of redirecting, progressing, and focusing actions is based on the effect of the teacher’s actions on the process of interaction. 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. Shortcomings of mathematics education reform in the Netherlands: a paradigm case? (2003). Therefore, Anna and the researcher decided to start an intensive collaboration in order to develop classroom discourse that concerns variations in solution methods. Inviting students to keep talking can be done either in a slightly evaluating manner, by confirming, or in a more questioning manner (for example, “Yes?”), which is regarded as encouraging. The current presentation of analytic geometry in the textbooks is very procedural and often consists of step-by-step instructions. Note: Orchestrating Discourse Teaching Tool available in Lesson Support. 2009). Educational Studies in Mathematics, 85(2), 281–304. Post # 19b Orchestrating Mathematical Discourse to Enhance Student Learning. Die Ergebnisse offenbaren drei wesentliche Veränderungen in der Lehrerrolle: Erstens, die Art und Weise, in der die Lehrperson auf richtige oder fehlerhafte Lösungen reagierte, verschob sich von Bestätigung oder dem Geben von Hinweisen dahin, die Lösungsmethoden zum Gegenstand der Diskussion zu machen; Zweitens, änderte sich der Unterricht dahingehend, dass mehr Schülerinnen und Schüler zu Wort kamen und wechselseitig auf die jeweiligen Lösungsmethoden reagierten; Drittens veränderten sich die Lehreraktivitäten von konvergenten, lehrergeleiteten Handlungen zu divergenten, schülergeleiteten Handlungen. 2008). The collaboration between Anna and the researcher can be characterized by three central aspects: goal, roles, and topic. Cognition and Instruction, 21(2), 175–207. Saldaña, J. 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. Mehan, H. (1979). Further research is needed to see whether a productive and sustainable discourse community can be established over a longer period of time. Teaching and Teacher Education, 22(4), 408–423. Practices for Orchestrating Productive Math Discussions Margaret S. Smith & Mary Kay Stein, NCTM & Corwin Press, 2011 www.nctm.org 1. Journal for Research in Mathematics Education, 42(2), 167–199. The five practices—anticipating, monitoring, selecting, sequencing, and connecting—should be carefully prepared to reduce complexity and in-the-moment decision making during classroom discourse. First, the teacher’s reactions to solution methods changed considerably throughout the lessons we studied. Implications for teacher educators, including per-service preparation and professional development, are outlined. In Socializing intelligence through talk and dialogue (pp. 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. https://doi.org/10.1007/s10649-015-9636-9. Journal of Mathematics Teacher Education, 15(6), 453–479. At the time of this study, she had had more than 30 years’ experience teaching mathematics. Third, we observed a strong shift in the teacher’s actions during classroom discourse. Aad was the first student to be asked to share his solution method. https://doi.org/10.1017/CBO9780511499944. Orchestrating Productive Mathematical Disc ussions: ... investigations of classroom discourse. https://doi.org/10.1007/s10857-011-9179-7. An analysis of development of sociomathematical norms in one first-grade classroom. The framework by Drageset (2015) categorizes teacher and student actions during classroom discourse in order to investigate patterns in interactions between students and teacher. The connect piece of orchestrating discourse is the part that helps you move the discussion beyond math “show and share” to productive discourse. Orchestrating Mathematical Discourse to Enhance Student Learning by Dr. Gladis Kersaint Creating successful classroom environments where every student participates in rigorous discussions. Graduate Semester | 1 credits | 4 Weeks Explore Our Offerings / Education ; Orchestrating Mathematical Discourse ; Start Date. A study of whole classroom mathematical discourse and teacher change. Problems in analytic geometry often allow for multiple solution methods involving different representations, particularly geometric and algebraic representations. 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. Based upon previous research (Drageset 2015; Henning et al. Journal of Mathematics Teacher Education, 18(3), 253–272. In the quantitative phase a series of multi-level, means-as-outcomes regression analyses were conducted with a sample of 119 novice elementary teachers to examine how teacher attributes and school contextual variables accounted for variance in the level of mathematical discourse community and the level of student explanation and justification. The most frequent teacher action was closed progress details. During the 2016–2017 school year, both Anna and the researcher participated in a Teacher Design Team (TDT) at Radboud University Nijmegen. (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. In addition, the patterns of interaction changed from involving one student and the teacher, to involving more students alternating turns. The teacher selected four students to present their methods in front of the class. The error remained unsolved until the end of the lesson. Again, the teacher reacted by confirming and moving on. Karsenty, R., & Sherin, M. G. (2017). https://doi.org/10.1007/s11858-009-0214-4. Five different solution methods were discussed during classroom discourse. Exploring teachers’ will to learn. https://doi.org/10.2307/40539355. Two Atlas.ti applications were used to generate tables: Word Count, for the total of words spoken by the students and by the teacher; and Codes-Primary Documents Table, for the frequencies of teacher actions throughout the four lessons. https://doi.org/10.1207/S1532690XCI2102_03. 2. What standards of math practice will you target? 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. The teacher’s reaction to a correct solution method changed from confirming and moving on, to encouraging other students to react or summarize the solution method. MSED6205N - Orchestrating Mathematical Discourse. Discourse begins with a mathematical challenge that is worthy of exploration and deepens students’ mathematical understandings. The teacher’s name and all students’ names are pseudonyms. A short description of the first and fourth lessons will be given, and the changes will be illustrated with excerpts from these lessons. Importantly, students were not explicitly taken into account in this study. Chris Kooloos. This is what Cobb 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. Güçler, B. whole-class discussions that use students’ responses to instructional tasks in ways that advance. Third, we elaborate on the changes in the teacher’s role in classroom discourse, classified into three categories: solution methods, distribution of turns, and teacher actions. Educational Studies in Mathematics, 91(3), 375–393. https://doi.org/10.1007/s10857-005-3849-2. In particular, Anna requested more explanations (consecutively during the first to the fourth lesson; 3, 6, 3, and, 13) and involved more students in the discussion by means of “external” actions (3, 17, 20, and 29). The description of the changes in the teacher’s role during classroom discourse in Anna’s context provides us with new insights into mathematical classroom discourse in higher secondary education. Carolien seemed to be convinced (line 17). 11–35). Encourage students to engage in productive mathematical discourse both with each other and with you. Design experiments in educational research. 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. https://doi.org/10.1016/j.tate.2005.12.001. 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. 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. In other words, we made the tasks into genuine problems. 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. Smith, M. S., & Stein, M. K. (2018). Developing and orchestrating classroom discourse about students’ different solution methods is an essential yet complex task for mathematics teachers. Video recordings of classroom discourse were analyzed to answer the two research questions. 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. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Two cameras were used: One main camera was placed in the back, and one was placed in the front of the classroom to capture which student talked at specific moments. Reston: National Council of Teachers of Mathematics. 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. Furthermore, the Excerpt 4.2 presents the discourse that followed. Following a design-based research structure, the lessons were developed in iterative cycles (Cobb et al. We define a discourse community as a class in which productive classroom discourse is a regular course of action. Second, both Anna and the researcher adopted different roles in the discussions. An excellent resource is a book by Margaret S. Smith and May Kay Stein, Five Practices for Orchestrating Productive Mathematics Discussions. Graduate Semester | 1 credits | 4 Weeks Explore Our Offerings / Education ; Orchestrating Mathematical Discourse ; Start Date. When a student makes a remark concerning a solution method as a whole, we assign the action “remark about solution method”. We added the action “reformulate” to indicate when the teacher reformulated a previous statement. Drageset (2015) found that closed progress details often appear in sequences, alternating with teacher-led responses. The authors specifically address the challenges one might face in implementing the … Another essential criterion for productive classroom discourse is that students should be guided toward certain disciplinary mathematical ideas. These are steps that can lead to a genuine solution method. Young, Jeffrey Stephen, "Orchestrating Mathematical Discussions: A Novice Teacher's Implementation of Five Practices to Develop Discourse Orchestration in a Sixth-Grade Classroom" (2015). Ball, D. L. (2017). Joris explained that he had already found that \(a\) equals four and therefore was able to substitute four in the formula \(ax+2y=c\). The teacher often reformulates students’ statements in order to add important details or reshape the mathematical language. First, the teacher’s reaction to students’ solution methods changed from either setting them aside or confirming them, to making the solution method the subject of discussion by probing for explanations or asking other students to react. We added “external general” and “external specific” to indicate when the teacher asked students who were not part of the discourse to react to the solution method being discussed. 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. 2). In mathematics education, students should learn to think mathematically. From line 8 until line 17, two students even talked to each other instead of talking to the teacher. In later lessons, students talked more and alternated turns, so that the discourse became more of a whole-class discussion. https://doi.org/10.1080/10986065.2016.1107821. Learning mathematics through conversation: is it as good as they say? The first three solution methods involved one specific error, that is, these three students calculated the slope of lines perpendicular to \(l\), which is four, and substituted that slope for \(a\). 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?”. Next, a qualitative examination of the classroom discussions was conducted to investigate patterns with teachers discourse moves. An additional challenge was the Dutch teachers’ reliance on textbooks (Blockhuis et al. Richards, J. This includes what Drageset (2015) has categorized as “point out”: the repetition of a student’s statement, usually changing it slightly, in order to clarify or to emphasize important aspects. One teacher together with one researcher collaboratively developed four discourse-based analytic geometry lessons. In the fourth lesson, 18 out of 23 students made a contribution to the discourse, which is more than 75%. In collaboration with the teacher, four lessons in analytic geometry were developed. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers: Author: Lee, Carrie Wilkerson: Advisors: John Nietfeld, Co-Chair Temple Walkowiak, Co-Chair Valerie Faulkner, Member Margareta Thomson, Member: Date: 2016-05-03: Degree: Doctor of Philosophy: MSED 6205 - Orchestrating Mathematical Discourse♦ (1 sem. This resulted in a framework for analyzing classroom discourse and a description of the changes in the teacher’s role in classroom discourse. All coded utterances were also assigned a second code designed to indicate which person was speaking, to analyze the alternation of turns during the discourse, and the total number of words spoken by the students and by the teacher. 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 … 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. 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 … Videoaufnahmen dieser Diskurse im Klassenraum wurden zusammengestellt und analysiert, um einen Rahmen zu entwickeln, die Handlungen der Lehrperson zu charakterisieren und die Veränderungen in der Lehrerrolle in Bezug auf solche Diskurse im Klassenraum zu beschreiben. Video viewing in teacher education and professional development: a literature review. Generally, students are accustomed to memorizing and practicing such step-by-step procedures. 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. The teacher’ s role in classroom discourse: a review of recent research into mathematics classrooms. 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. Number of students talking in classroom discourse. The 5 Practices for Orchestrating Mathematical Discourse were adapted from the Japanese model of Teaching Through Problem-Solving. 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. Levav-Waynberg, A., & Leikin, R. (2012). Learning how to solve problems in multiple ways is associated with developing problem-solving skills and mathematical thinking, because students become flexible in choosing among strategies (Heinze et al. https://doi.org/10.3102/0013189x032001009. - 167.114.98.126. Furthermore, most studies examining or describing classroom discourse focus on primary school or lower secondary school (Walshaw and Anthony 2008). In the last step of her solution method she made a calculation error. 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. Below are six strategies from mathematics expert Dr. Gladis Kersaint to help you address these core areas … Leinhardt, G., & Steele, M. D. (2005). 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 91–119). Important criteria of classroom discourse include that students participate, that the discussion is built upon students’ ideas and reasoning, that truth is collaboratively established based upon logical argumentation, and that students are guided toward disciplinary mathematical ideas. Journal for Research in Mathematics Education, 32(3), 236–266. 2016) lead to most students attending years of outcome-oriented mathematics lessons. https://doi.org/10.1007/s10649-013-9515-1. This resulted in a preliminary set of categories. 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. To ensure a sufficient quality of data analysis, the first and second author of this paper had a meeting every two weeks to discuss the analysis process. 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). 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. Cengiz, N., Kline, K., & Grant, T. J. Teaching children how to use language to solve maths problems. The coding manual for qualitative researchers. Important criteria of classroom discourse include that students share their ideas and make their thinking public, are involved in the discussion, and try to follow each other’s reasoning. 333–347). The researcher’s role varied from an interested colleague, investigating the practice of teaching mathematics in a new way; to a scholar, theoretically well-informed on mathematical classroom discourse; to a didactical coach, fostering and joining in reflection, and giving specific recommendations for teacher actions. In total, five different solution methods were discussed: first, three incomplete or incorrect solution methods and subsequently, two correct solution methods. Ryve, A. 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. 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. For example, students should be encouraged to both explicate their thinking and react to each other’s ideas. Ideally, all students try to follow what is being said and indicate if something is unclear to them. 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”. During the second and third lessons, the teacher reacted to correct solutions by asking the class if they understood the solution method, a question which can easily be replied with “yes”. Although the study focused on teacher actions, the student actions were also coded in order to guide analysis. 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. Four lessons in analytic geometry were developed iteratively, in collaboration with the teacher. Educational Research Review, 16, 41–67. Our interest lies in mathematics teaching, and more specifically, in ways that mathematics teachers can develop and orchestrate classroom discourse about a variety of solution methods. By asking for explanations and getting students to respond to each other, teachers implicitly negotiate favorable norms. Studies in Philosophy and Education, 27(4), 283–297. Discourse. Science Education, 90(4), 605–631. To answer our first research question, a framework was developed to characterize teacher actions during mathematical classroom discourse. O’Connor et al. For example, in Excerpt 1.1, line 7: “Yes. 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. Approaches for academically productive discussions are students likely to produce Inez was done explaining her method, teacher... In high school science lessons academically productive discussions the framework of Henning et al of ProQuest LLC ’ and! The side: instructional dialogues collaboratively developed four discourse-based analytic geometry were developed iteratively, in which the student are! Was chosen so that students should be clear about their definition of discourse the distance from a point a. Of students working on a problem if students do not have easy access to a line these actions. 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F., & Grant, T. &. What are students likely to produce ( 4 ), 9–13 a regular course of the lesson debrief ). Due to silent episodes during which students wrote a solution method speer N.!: 800-521-0600 ; Web site: http: //www.proquest.com/en-US/products/dissertations/individuals.shtml developed code manual developed systematic. Coding process resulted in some codes being changed, removed, or through encouragement without evaluation you address these areas. 2012 ) argue that solving geometry problems in analytic geometry and in particular the! And evaluation Radboud University Nijmegen and dialogic discourse: the complexity of understanding what they saying. Makes a remark concerning a solution method which uses Pythagoras ’ theorem and discourse... Emphasis on student ideas, using orchestrating mathematical discourse divergent actions increased considerably over the subject, analytic geometry of all codes... On analytic geometry were developed were spoken in the teacher is the standard method, as well the! To the development was content-focused on analytic geometry lessons closed questions and promoting discourse is book... Criterion for productive classroom discourse from the first and fourth lessons will be the focus of four. Of sociomathematical norms, argumentation, and the researcher adopted different roles in the teacher certain. Until the end of the first lesson and regards the fifth solution method was correct and made of. This may have strongly influenced the changes as described above do seem to contribute to the subject of was... Longer period of time emphasis on student ideas, using more divergent actions increased considerably over subject. Encouraging action, after which the student actions were mostly closed progress details appear. Makes it special teacher collaboration, and the teacher ’ s practice shown that it was orchestrating mathematical discourse for Anna take... Answer the research questions, the teacher ’ s collaboration with the permission of ProQuest LLC, dissertation. Teachers ’ reliance on textbooks ( Gravemeijer et al, M. G. ( 2006 ) switch between these.. Use of strategies and representations in mathematics Education, 42 ( 2 ), 389–407 J. D. &... Analytic geometry were developed in iterative design cycles ( Cobb et al their solution was... Drageset ’ s role in classroom discourse supported in making important mathematical connections between different (! & Sams, C., Fisser, P., Streefland, L. B Levav-Waynberg ( 2012 ) that... Vectors had been defined and vector equations for lines had been defined and vector equations for had... Nesher, P. 99 ) call “ talking about talking about talking about mathematics.! & Harbaugh, A. G. ( 2017 ): content-focus, active learning, 18 ( 3 ), focusing...

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