During the 2016–2017 school year, both Anna and the researcher participated in a Teacher Design Team (TDT) at Radboud University Nijmegen. 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. As shown in Fig. The second most frequent teacher action in the first lesson was reformulate, which mostly occurred when the teacher tried to repeat the steps of a solution method on the whiteboard. Teaching and Teacher Education, 22(4), 408–423. After Inez was done explaining her method, the teacher asked another student, Thom, to react. Mathematics discussions by design: creating opportunities for purposeful participation. Extending students’ mathematical thinking during whole-group discussions. The topic of the TDT was analytic geometry, a new subject in the Dutch curriculum for higher secondary school. AnnaFootnote 1, the teacher, works at a secondary school in a medium-sized city in the Netherlands. This problem involves calculating the distance between a point and a line. First, the teacher did not give her the opportunity to clarify in which triangle she was planning to use the theorem. To answer our first research question, a framework was developed to characterize teacher actions during mathematical classroom discourse. When the solution method had been completed, the teacher reacted by confirming the method, and then moved on to the next solution method. The teacher’ s role in classroom discourse: a review of recent research into mathematics classrooms. A reformulation can also be used to rectify and model the use of mathematical language or to facilitate communication, for example by naming a geometric object involved in a solution method. The discussions in the TDT were mainly content-related: problems were solved and solution methods were discussed. 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, the relative frequency of divergent actions increased considerably over the course of the four lessons. We define a discourse community as a class in which productive classroom discourse is a regular course of action. Alice said she had something different from the point where the slope was calculated, but could not find the words to describe her method. Even though both scholars and policymakers advocate the inclusion of classroom discourse in mathematics educational practice, many mathematics teachers do not orchestrate classroom discourse at all, or if they do, they do not transcend patterns such as “initiation-response-evaluation” (Cazden 2001) or “show and tell” (Stein et al. ), Compendium for research in mathematics education (pp. Yes. It may take place between partners, small groups, or as a whole class. © 2021 Springer Nature Switzerland AG. 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. A partial answer is an answer that is not complete, not explained, incomprehensible, or not entirely correct. Today’s headlines emphasize the need to prepare students for science, technology, engineering, and mathematics (STEM) careers; yet preparing students to be mathematically literate in today’s world is a heavy charge. 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. 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. Cognition and Instruction, 23(1), 87–163. Google Scholar. This was due to silent episodes during which students wrote a solution method on the whiteboard. 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 \(a\). Walshaw, M., & Anthony, G. (2008). In addition, steps that were unclear to students were made the subject of a discussion. In particular, the closed progress details increased again, and a demonstration was given by the teacher. Uncovering the special mathematical work of teaching. Additionally, Anna and the researcher discussed the lesson plan and considered adjustments. Although the study focused on teacher actions, the student actions were also coded in order to guide analysis. Mathematical problem solving. Henning, J. E., McKeny, T., Foley, G. D., & Balong, M. (2012). Mathematical Discourse Joe Bysiek-MacArthur. Discrepancies were discussed until a consensus was reached, resulting in adjustments to the code descriptions and framework. Therefore, we have investigated the distribution of turns among the teacher and students, and the ways in which the teacher regulated the distribution of turns. The teacher still refrained from evaluation, instead asking whether other students agreed. She wrote down a vector equation for the line, which can be done right away. Güçler, B. Cohen, L., Manion, L., & Morrison, K. (2011). Moreover, feedback and reflection were an important part of the discussions between Anna and the researcher. How can we characterize one teacher’s actions during classroom discourse that concerns various solution methods for problems in analytic geometry? https://doi.org/10.2167/le678.0. The categorization was particularly useful for our analysis, as our aim was also to categorize teacher and student actions with regard to their discourse contribution. At the time of this study, she had had more than 30 years’ experience teaching mathematics. Young, Jeffrey Stephen, "Orchestrating Mathematical Discussions: A Novice Teacher's Implementation of Five Practices to Develop Discourse Orchestration in a Sixth-Grade Classroom" (2015). Yackel, E., & Cobb, P. (1996). 2). (2008) describe a five practices model for the design of classroom discourse that both builds on student ideas, and also guides students to mathematical goals. 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. In Excerpt 1.2, the teacher and Emmanuelle alternated turns and no one else was asked to react. (2011). Parents gave their passive consent for videotaping the lessons. Implications for teacher educators, including per-service preparation and professional development, are outlined. 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?”. As described above, more students participated and students spoke more in later lessons. Feb 08, 2021. 2017). PubMed Google Scholar. Apr 05, 2021. https://doi.org/10.1007/s11217-007-9071-1. A balancing act: developing a discourse community in a mathematics classroom. Our categorization in convergent, divergent, encouraging, and regulating actions was partially based upon the framework of Henning et al. Reston: National Council of Teachers of Mathematics. The four lessons described in this study were enacted over a period of almost four months. (1993, p. 99) call “talking about talking about mathematics”. Video recordings of classroom discourse were analyzed to answer the two research questions. Learning mathematics through conversation: is it as good as they say? In most cases, such an utterance was followed by an encouraging action, after which the student continued and finished the solution method. Los Angeles: SAGE. (2008). 2017). 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. 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. Sfard, A. 2008). Chris Kooloos. 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. This helps ensure that key mathematical ideas remain the focus of the lesson debrief. This study reports on the first stages of classroom discourse development of one Dutch higher secondary school mathematics teacher who had no prior experience in including classroom discourse in her teaching practice. Herbel-Eisenmann, B., Meany, T., Pierson Bishop, J., & Heyd-Metzuyanim, E. (2017). Orchestrating Mathematical Discourse to Enhance Student Learning (2015 Curriculum Associates, LLC) - When students share and exchange their ideas, both they and their teachers benefit. Educational Research Review, 16, 41–67. In G. Kaiser (Ed. "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. 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). Based upon previous research (Drageset 2015; Henning et al. In the fourth lesson, students were encouraged to react to each other (for example: “Well, just ask him”). (2011) describe as important qualities of a case study researcher. 2012; Nathan and Knuth 2003; Sherin 2002). Calculate for which values of \(a\) and \(c\), the line goes through point \(P\left(1,5\frac{1}{2}\right)\) and perpendicular to \(l\). Several divergent teacher actions that we have identified can be regarded as “talk moves” that promote student thinking, such as orienting to the thinking of others, and clarifying and sharing their own thoughts (Michaels and O’Connor 2015). 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. Teachers can foster student explanations and logical argumentation by asking questions and pressing for reasoning. In J. Cai (Ed. For more details, see Drageset (2014, 2015). Those who understand: knowledge growth in teaching lee. Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Finally, regulating actions (“rules of classroom discourse”) refer to the teacher articulating the rules of communication during classroom discourse. 2008). (2016). Several characteristics of the collaborative development of these four lessons can be identified as contributing to the changes in the teacher’s role in classroom discourse. 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. 2008, p. 315). Regarding correct solution methods, we perceive a similar change: during the first lesson, when a student finished giving a solution method, the teacher confirmed the method and moved on. Richards, J. An excellent resource is a book by Margaret S. Smith and May Kay Stein, Five Practices for Orchestrating Productive Mathematics Discussions. Anna’s collaboration with the researcher included aspects of both teacher collaboration, and coaching and expert support. 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 \(a\). The fourth and last lesson of this study took place almost four months later, while the teacher and students were working on a chapter regarding vectors. McClain, K., & Cobb, P. (2001). Journal for Research in Mathematics Education, 32(3), 236–266. 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. Furthermore, Leikin and Levav-Waynberg (2012) argue that solving geometry problems in a variety of ways fosters students’ knowledge and creativity. 333–347). https://doi.org/10.1023/a:1020134209073. Excerpts 1.1, 1.2, and 1.3 are from the first lesson and are in chronological order. https://doi.org/10.1007/s10857-014-9280-9. “Evade answer” refers to students abstaining from answering. This resulted in a framework for analyzing classroom discourse and a description of the changes in the teacher’s role in classroom discourse. 11–35). The study lasted from February, 2017 through July, 2017. In Socializing intelligence through talk and dialogue (pp. Let l be the line given by the following vector equation: We consider the lines \(ax+2y=c\), where \(a\) and \(c\) are constants. 292–307). 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. MSED6205N - Orchestrating Mathematical Discourse. In each of the four lessons, a variety of students’ solution methods was discussed during classroom discourse. Our framework is a useful tool that can be applied by researchers, teachers, and teacher-educators as they analyze or develop classroom discourse, in particular classroom discourse concerning various solution methods. Five Practices for Orchestrating Productive Mathematical Discussions. Language and Education, 20(6), 507–528. 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. This resulted in some codes being changed, removed, or added to fit the discourse in our data. Diese Ergebnisse zeigen, dass innerhalb von vier Stunden wichtige Schritte in Richtung der Etablierung einer Diskursgemeinschaft gegangen werden können. 2011; Nathan and Knuth 2003; Sherin 2002). When a student makes a remark concerning a solution method as a whole, we assign the action “remark about solution method”. 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. Depending on each individual context, an important question remains concerning how to support teachers in developing classroom discourse about different solution methods. This requires that teachers shape their lessons such that students “use each other as resources for working through those problems, and then share their strategies and solutions in whole-class discussions” (Stein et al. Emmanuelle had an idea for a solution method which uses Pythagoras’ theorem. Sociomathematical norms, argumentation, and autonomy in mathematics. https://doi.org/10.1080/10986065.2016.1107821. In E. von Glasersfeld (Ed. 2012). The 5 Practices for Orchestrating Mathematical Discourse were adapted from the Japanese model of Teaching Through Problem-Solving. (2003). 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. (2008). 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. 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. Carolien pointed out that she did not understand where the “four” came from. Portsmouth: Heinemann. 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. (2011). https://doi.org/10.3102/978-0-935302-43-1_27. A mathematical task is regarded as a problem if students do not have easy access to a solution method (Schoenfeld 1985). 789 East Eisenhower Parkway, P.O. 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. During the coding process, Drageset’s framework was adjusted for two main reasons: First, the framework as developed by Drageset (2015) is based upon discourse in higher primary school that deals with elementary mathematics, namely fractions, whereas the classroom discourse in our data concerned more advanced mathematics in higher secondary school, and more specifically, various solution methods for problems that require several steps to solve. Fig. Aad was the first student to be asked to share his solution method. 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). Fig. Vol. Speer, N. M., & Wagner, J. F. (2009). Orlando: Academic Press. Cazden, C. B. In the Netherlands, national standardized testing, widespread reliance on textbooks (Blockhuis et al. 2004; Leinhardt and Steele 2005; McClain and Cobb 2001). 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 2008). In the first lesson, most teacher actions were convergent, and the teacher had control over the subject of discussion. 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. In addition, further analysis showed that the patterns of interaction changed from one-on-one dialogues between the teacher and a single student, toward patterns that involved more students alternating turns. Making implicit metalevel rules of the discourse on function explicit topics of reflection in the classroom to foster student learning. Video viewing in teacher education and professional development: a literature review. To summarize, a substantial change can be perceived in the way the teacher dealt with both correct and incorrect solution methods. Palo Alto: Learning Policy Institute. 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. Die Stunden sahen vor, dass Schülerinnen und Schüler an einer mathematischen Aufgabe arbeiteten und darüber hinaus einen Diskurs über verschiedene Schülerlösungen führten. 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. 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). The quantitative results of the coding process appear to show that the main change with regard to teacher actions seemed to be between the first and second lessons. The complexity, as well as the mathematical work of teaching student-centered mathematics lessons, was recently described by Ball (2017). 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. O’Connor, C., Michaels, S., Chapin, S., & Harbaugh, A. G. (2017). In the textbook, rather than presenting this as a problem, students are provided with a step-by-step procedure to calculate the distance between a point and a line. The students were presented with a problem which involved calculating the distance between a point and a line in the Cartesian coordinate system. Mathematical Discourse also involves different genres such as algebraic proofs, geometric proofs, and school algebra word problems. Found that closed progress details alternated with teacher-led responses presents the beginning the... Particular context make this finding a worthy addition to research in mathematics reform... Regulating actions to articulate rules for participating in classroom discourse is a book Margaret. Of ways fosters students ’ thinking would have been related to the subject a! 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