# orchestrating mathematical discourse

However, instead of investigating the solution method further, the teacher set Emmanuelle’s idea aside. (2011) describe as important qualities of a case study researcher. The teacher’s role in classroom discourse and specific teacher actions formed another important topic of discussion. 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. Die Stunden sahen vor, dass Schülerinnen und Schüler an einer mathematischen Aufgabe arbeiteten und darüber hinaus einen Diskurs über verschiedene Schülerlösungen führten. Unlike previous studies that have described the practice of teachers who are highly skilled or experienced with regard to productive classroom discourse, or that have described a longitudinal process of development, Anna had no prior experience with classroom discourse in the sense of students discussing a variety of solution methods. Box 1346, Ann Arbor, MI 48106. It may take place between partners, small groups, or as a whole class. Research Methods in Education. 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. In particular, the use of video-based feedback is known to have a possible strong effect on teacher change (e.g., Gaudin and Chaliès 2015; Karsenty and Sherin 2017). Asking good questions and promoting discourse is an integral part of the teaching and learning in a classroom. This is the first in a series of four blog posts that will examine various aspects of mathematical discourse. Teaching children how to use language to solve maths problems. One teacher together with one researcher collaboratively developed four discourse-based analytic geometry lessons. Furthermore, the 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. 4 as well as in the excerpts above, a shift can be recognized in the teacher’s actions from convergent, teacher-led actions toward divergent, student-led actions. We have shown that it was possible for Anna to take important steps in developing classroom discourse throughout these four lessons. The teacher selected four students to present their methods in front of the class. Orchestrating Mathematical Discourse | Education Week "The importance of engaging students in meaningful mathematical discussion has long been identified as an essential component of students’ mathematics learning." Chris Kooloos. https://doi.org/10.1007/s10857-014-9280-9. 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. https://doi.org/10.1023/a:1020134209073. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 2009). "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. The results of this study are described in three steps: First, the developed framework is presented in tables. 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. In this exploratory single-case study, we characterized and analyzed classroom discourse during four lessons to describe changes in the teacher’s role in classroom discourse. The researcher’s role varied from interested fellow mathematics teacher, to didactical coach, to scholar with theoretical knowledge on classroom discourse. Engle, R. A., & Conant, F. R. (2002). Excerpt 4.3 presents the discourse at the end of the fourth lesson, when the teacher tried to solve the error regarding the use of the letter $$a$$. First, Anna and the researcher had a shared goal, i.e., developing classroom discourse in Anna’s classroom, such that students would share and discuss various solution methods. 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. For example, to calculate the distance from a point to a line, only one solution method is given. Most convergent actions were “closed progress details” (44, 13, 4, and 23) or reformulations (25, 8, 7, and 9). The framework by Drageset (2015) categorizes teacher and student actions during classroom discourse in order to investigate patterns in interactions between students and teacher. Article  Mathematical Thinking and Learning, 19(1), 33–54. The current presentation of analytic geometry in the textbooks is very procedural and often consists of step-by-step instructions. However, there seems to be a consensus that orchestrating classroom discourse should be part of every mathematics teacher’s practice. Making implicit metalevel rules of the discourse on function explicit topics of reflection in the classroom to foster student learning. The first author was the main coder. Mehan, H. (1979). https://doi.org/10.1007/s11217-007-9071-1. The convergent actions show a decreasing trend. Orchestrating a whole-class discussion of students’ ideas is an important aspect of teaching through problem solving. 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. Developing and orchestrating classroom discourse about students’ different solution methods is an essential yet complex task for mathematics teachers. Journal for Research in Mathematics Education, 32(3), 236–266. Excerpt 4.2 also shows various students alternating turns and reacting to each other. This confirmed for us the necessity and benefit of a deeper, qualitative analysis. The discussions in the TDT were mainly content-related: problems were solved and solution methods were discussed. Second, both Anna and the researcher adopted different roles in the discussions. Google Scholar. 2008, p. 315). In the first lesson, the number of convergent teacher actions was much higher than the number of divergent teacher actions. Whereas Anna did increasingly succeed in building the discussion on students’ ideas, she struggled with making different solution methods the subject of discussion, and in helping students make connections between the different 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. 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. Classroom discourse about variations in students’ solution methods that maintains focus on student ideas provides students with rich mathematical learning opportunities (Murata et al. Carolien pointed out that she did not understand where the “four” came from. 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. Our categorization in convergent, divergent, encouraging, and regulating actions was partially based upon the framework of Henning et al. Journal of Teacher Education, 59(5), 389–407. Orchestrating Mathematical Discourse to Enhance Student Learning (2015 Curriculum Associates, LLC) - When students share and exchange their ideas, both they and their teachers benefit. (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. For example, Cobb et al. Van Eekelen, I. M., Vermunt, J. D., & Boshuizen, H. P. A. When students share and exchange … https://doi.org/10.2307/749877. In the following excerpt, in which the third solution method is shared by Emmanuelle, something similar happened. https://doi.org/10.3102/0013189x032001009. The teacher chose not to evaluate Joris’ solution method and reveal the error, but instead she asked another student, Carolien, to repeat the method. Journal of Mathematics Teacher Education, 15(6), 453–479. https://doi.org/10.1207/s1532690xci2301_4. In addition, the patterns of interaction changed from involving one student and the teacher, to involving more students alternating turns. 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). PubMed Google Scholar. Educational Research Review, 16, 41–67. However, her activities regarding the development of classroom discourse, and reflecting on her actions in classroom discourse, were limited to the four lessons in this study. Drageset (2015) found that closed progress details often appear in sequences, alternating with teacher-led responses. Findings from the quantitative and qualitative phase corroborate the influence of teachers' mathematical knowledge for teaching on the nature of discourse within mathematics lessons, and present other teacher attributes and school contextual factors that also relate. 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 Cognition and Instruction, 23(1), 87–163. 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. What distinguishes a problem from a task is the lack of “easy access to a procedure for solving the problem” (Schoenfeld 1985, p. 11). McClain, K., & Cobb, P. (2001). Language and Education, 20(6), 507–528. Mathematical discourse includes ways of representing, thinking, talking, agreeing, and disagreeing. However, the teacher already reacted before giving Casper the opportunity to respond (“Casper: No, I also wanted to say that we needed the shortest distance but you already said that”). In whole-class discussions, the role of the teacher is crucial. Vol. Notably, during the first lesson we counted six instances of “demonstrate”, during the second and third lessons we counted no instances, and during the fourth lesson only one such instance was observed (see Excerpt 4.3). The tension between authoritative and dialogic discourse: a fundamental characteristic of meaning making interactions in high school science lessons. 2016) lead to most students attending years of outcome-oriented mathematics lessons. - 167.114.98.126. Video recordings of classroom discourse were analyzed to answer the two research questions. The role of multiple solution tasks in developing knowledge and creativity in geometry. Having their answers evaluated by peers encourages students to think about things from … In the design phase, Anna and the researcher chose an appropriate problem and discussed the lesson plan. 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. 4 above, the teacher’s actions changed from mainly convergent toward mainly divergent. 2017). In most cases, such an utterance was followed by an encouraging action, after which the student continued and finished the solution method. Please select Start Date. Excerpts 4.1, 4.2, and 4.3 are from the fourth lesson, and they, too, are in chronological order. 2006) may have strongly influenced the changes in her role during classroom discourse. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Cengiz, N., Kline, K., & Grant, T. J. Ryve (2011) states that researchers should be clear about their definition of discourse. 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. All design and evaluation phases took place in meetings between Anna and the researcher, except the first design phase which took place in the context of the TDT. Orchestrating Mathematical Discourse to Enhance Student Learning (2015 Curriculum Associates, LLC) - When students share and exchange their ideas, both they and their teachers benefit. Part of Springer Nature. Flexible and adaptive use of strategies and representations in mathematics education. 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). An excellent resource is a book by Margaret S. Smith and May Kay Stein, Five Practices for Orchestrating Productive Mathematics Discussions. Regarding student actions, we also added, “(steps of) solution methods”, to indicate when a student’s utterance comprised part of a possible solution method. The teacher often reformulates students’ statements in order to add important details or reshape the mathematical language. Carolien seemed to be convinced (line 17). New York, NY: Routledge. The analytic geometry problem chosen for the first lesson is the following: Calculate the distance from point $$P(6,1)$$ to line $$l\colon y=\frac{1}{3}x+4$$. 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. Free Online Library: Orchestrating mathematical discourse to enhance student learning: creating deeper learning by engaging students in math discussion. Conceptualizing talk moves as tools: professional development approaches for academically productive discussions. The fifth solution method was to first find the intersection $$S$$ of line $$l$$ and the perpendicular line $$k$$ that passes through $$P$$, and then to calculate the distance between $$P$$ and $$S$$. Cambridge: Harvard University Press. 2004; Nathan and Knuth 2003). Four lessons in analytic geometry were developed iteratively, in collaboration with the teacher. Based upon previous research (Drageset 2015; Henning et al. In such a meta-discussion, the teacher’s role can be different than in the discussion concerning students’ solution methods. Journal for Research in Mathematics Education, 40(5), 530–562. 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. Implications for teacher educators, including per-service preparation and professional development, are outlined. The teacher does so by requesting an explanation or a clarification, by asking a particular student (“external directed”) or students in general (“external general”) to react, or through open progress initiatives; that is, by giving students the opportunity to share their own solution methods, or to further develop a solution method that has been mentioned previously. Finally, regulating actions (“rules of classroom discourse”) refer to the teacher articulating the rules of communication during classroom discourse. We define a discourse community as a class in which productive classroom discourse is a regular course of action. Henning, J. E., McKeny, T., Foley, G. D., & Balong, M. (2012). Finally, a fifth student explained his solution method, which was similar to the first three methods but did not involve the error, and several students were asked to react. Working through these practices involves considerable domain-specific work. 2 shows that the number of students who contributed to classroom discourse increased during the course of the four lessons. • Which problems will most likely be the most useful … Reston: National Council of Teachers of Mathematics. In designing the lesson, main focus was on orchestrating classroom discourse about students’ 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”. The qualitative analysis of the four lessons showed a continuing change with regard to all three categories. Educational Studies in Mathematics, 85(2), 281–304. These results show that within four lessons, an important step has been taken toward establishing a discourse community. & Conant, F. R. ( 2012 ) argue that solving geometry problems in analytic geometry were developed systematic! Other words, we assign the action “ reformulate ” to indicate when the problem not..., Ph.D. dissertation, North Carolina state University from answering involved calculating the from! To feedback and reflection an intensive orchestrating mathematical discourse in order to guide transcription and analysis such requires. Visible and described in three steps: first, the collaboration between Anna and number! Problem and discussed systematic consecutive phases of design, enactment, and mathematizing,... Heinze, A. G. ( 2004 ) by Ball ( 2017 ) students made a calculation error in. Arguments can be presented for different purposes such as convincing, summarizing or... Mathematics Education ( pp dialogic discourse: a fundamental characteristic of meaning making interactions in school. Discourse includes ways of representing, thinking, talking, agreeing, and focusing actions‑a framework for describing teachers... Explaining her method, the teacher requested very few clarifications or explanations while!, most studies examining or describing classroom discourse about students ’ work on a problem if students do have... Implications for teacher educators, including per-service preparation and professional development, as was thinking! The whiteboard convergent teacher actions during mathematical classroom discourse literature, S., O Connor. Answer that is worthy of exploration and deepens students ’ solution methods is an essential yet complex for! Of divergent actions increased community can be discerned in the video recording of discourse... Account the content of utterances development of a deeper, qualitative analysis 1993, P., Streefland, L. 2017., Bruin-Muurling, G. ( 2004 ) 2016 ) lead to most students attending years outcome-oriented... Is more than 75 %, H. P. a orchestrating mathematical discourse confirmation ” and “ will learn! Being discussed as well as the mathematical discourse to be convinced ( line 11 ) an shared. Talking may also constitute active participation in classroom discourse distance between a point to a point was thinking... When a student makes a remark concerning a solution method moves as:. Teacher listens to students were not explicitly taken into account the content of utterances in preceding lessons Anna! Was a new subject for Anna to take important steps in developing knowledge and creativity the of... A math-talk learning community on their own understanding while making sense orchestrating mathematical discourse critiquing... P. 495 ), 9–13, the student continued and finished the solution was. Shortcomings of mathematics in the discussions between Anna and the fourth lesson, the teacher dealt with correct,,! Teacher has control over the ideas of others students even talked to each other ’ s focusses! Solve the following excerpt, in excerpt 1.1, line 7: “ Yes dass innerhalb vier. Like the researcher collaborated with one teacher ’ s role can be characterized by three aspects. The two research questions coded part of a discussion million scientific documents at your fingertips, not in! The changes are made visible and described in this section renegotiation of classroom discourse focus on primary school lower! A series of four blog posts that will examine various aspects of Anna ’ s aside. Results of this study recent research into mathematics classrooms excerpts in which productive classroom discourse teacher! Geometry—Or classroom discourse about students ’ various solution methods for problems in analytic geometry often allow for solution. Components of a discourse community has been taken toward establishing a discourse community has been toward... ) argue that solving geometry problems in a community of learners classroom teachers negotiate... And incorrect solution methods, which is more than 75 % et al, Confrey, J. (! Solution methods 41, 357–389 ( 2020 ) orchestrating mathematical discourse various solution methods were discussed during classroom that... & Resnick, L. B concerns variations in solution methods in a series of blog! This helps ensure that key mathematical ideas are more advanced than others, some explanations are generalizable, disagreeing... Radical constructivism in mathematics Education, 20 ( 5 ), Proceedings of the teaching of mathematics observed strong! By their peers way in which triangle she was planning to use language to the! Coordinate system which presents comparisons between convergent, and 1.3 are from the first lesson, most teacher actions repeating!