Seeing the complexity of standing to the side: instructional dialogues. From line 8 until line 17, two students even talked to each other instead of talking to the teacher. Vier Stunden zur analytischen Geometrie wurden sukzessive in Zusammenarbeit mit der Lehrperson entwickelt. Let l be the line given by the following vector equation: We consider the lines \(ax+2y=c\), where \(a\) and \(c\) are constants. In the third step of data analysis, all four transcripts were coded using the developed code manual. In the first lesson, the number of convergent teacher actions was much higher than the number of divergent teacher actions. It may take place between partners, small groups, or as a whole class. Reston: National Council of Teachers of Mathematics. 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. The silent and the vocal: participation and learning in whole-class discussion. https://doi.org/10.1007/s10649-015-9636-9. We conducted an exploratory case study to enable an in-depth investigation of changes in classroom discourse of a secondary school teacher who is new to orchestrating classroom discourse. Like the researcher, Anna was interested in teaching such that students’ different solution methods are discussed in the classroom. 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. Solving mathematical problems and discussing various solution methods is an important part of learning mathematics. 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 classroom discourse further toward productive classroom discourse would require more mathematical work in the sense of anticipating student responses, monitoring student ideas, and selecting students to contribute (Ball 2017; Stein et al. As shown in Fig. Ryve (2011) states that researchers should be clear about their definition of discourse. https://doi.org/10.1007/s10857-005-3849-2. This shared goal, and the iterative nature of development, made the collaboration a joint learning experience, which undoubtedly influenced the changes in the teacher’s practice. These aspects made orchestration of classroom discourse about different solution methods a unique and particularly challenging undertaking in Anna’s context. In total, five different solution methods were discussed: first, three incomplete or incorrect solution methods and subsequently, two correct solution methods. This problem involves calculating the distance between a point and a line. Review of Educational Research, 78(3), 516–551. 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. 3 shows that the percentage of student utterances greatly increased from the first to the second lesson. 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. 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. 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. Effective orchestration of classroom discourse shifts students’ cognitive attention from problem solutions and procedural rules to sense-making and the reasoning that leads to a solution (Yackel and Cobb 1996). 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 Post #19c Orchestrating Mathematical Discourse to Enhance Student Learning Here is the third of five questions and responses from Gladis Kersaint’s blog post, Orchestrating Mathematical Discourse to Enhance Student Learning. https://doi.org/10.1007/s11858-009-0214-4. In this excerpt, the teacher’s actions were mostly closed progress details. Orchestrating Mathematical Discourse to Enhance Student Learning by Dr. Gladis Kersaint Creating successful classroom environments where every student participates in rigorous discussions. The teacher’s role in classroom discourse and specific teacher actions formed another important topic of discussion. Design experiments in educational research. In addition, the teacher’s openness to feedback and “will to learn” (Van Eekelen et al. In E. A. Forman, N. Minick & C. A. The use of less convergent actions and more divergent actions is a strong indication for building the discussion on students’ ideas and reasoning, because our categorization in convergent and divergent teacher actions is based upon the distinction between teacher actions that serve to control the subject of discussion, or those that serve to build further on student ideas. For the Learning of Mathematics, 18(1), 41–51. Oxford: Oxford University Press. The collaboration between Anna and the researcher can be characterized by three central aspects: goal, roles, and topic. The students were presented with a problem which involved calculating the distance between a point and a line in the Cartesian coordinate system. Excerpt 1.1Footnote 3 presents the beginning of the discourse. (2012), as well as the data of this study. (2016). Mercer, N., & Sams, C. (2006). 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. A balancing act: developing a discourse community in a mathematics classroom. The analysis of teacher moves showed that teachers with higher levels of mathematical knowledge for teaching used more open-ended questioning and prompted more student contributions. 2016), and the procedural way in which these textbooks present mathematics (Gravemeijer et al. Journal of Mathematics Teacher Education, 5(3), 205–233. Apr 05, 2021. The development was content-focused on analytic geometry and in particular on how to teach it. The authors specifically address the challenges one might face in implementing the … Google Scholar. Yes. Van Eekelen, I. M., Vermunt, J. D., & Boshuizen, H. P. A. Orchestrating Mathematical Classroom Discourse About Various Solution Methods: Case Study of a Teacher's Development Author(s): Kooloos, C ; Oolbekkink, H. ; Kaenders, R ; Heckman, G Stein et al. https://doi.org/10.3102/0013189x032001009. In the design phase, Anna and the researcher chose an appropriate problem and discussed the lesson plan. 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. 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. This involves more than merely being able to solve routine tasks. New York, NY: Routledge. A study of whole classroom mathematical discourse and teacher change. As displayed in Fig. 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. Video as a catalyst for mathematics teachers’ professional growth. Journal of Mathematics Teacher Education, 14(5), 355–374. 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. The error remained unsolved until the end of the lesson. 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. In such a meta-discussion, the teacher’s role can be different than in the discussion concerning students’ solution methods. 5607. https://scholarsarchive.byu.edu/etd/5607 Studies in Philosophy and Education, 27(4), 283–297. Fig. Previous case studies have investigated various aspects of professional development of mathematics teachers who succeeded in orchestrating productive classroom discourse or were making extraordinary progress in their development (e.g., Hufferd-Ackles et al. Nathan, M. J., & Knuth, E. J. When students were asked to calculate the distance from a point to a line, several possible solution methods were formulated and discussed. https://doi.org/10.1080/10986065.2016.1107821. We have shown that it was possible for Anna to take important steps in developing classroom discourse throughout these four lessons. Thinking as communicating: human development, the growth of discourses, and mathematizing. 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 … We define a discourse community as a class in which productive classroom discourse is a regular course of action. They conclude that once a discourse community has been established, listening carefully without talking may also constitute active participation in classroom discourse. In mathematics education, students should learn to think mathematically. Journal of Mathematics Teacher Education, 20(5), 409–413. 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. https://doi.org/10.2307/3609122. During the 2016–2017 school year, both Anna and the researcher participated in a Teacher Design Team (TDT) at Radboud University Nijmegen. Cognition and Instruction, 35(4), 290–316. Whereas Drageset’s framework focusses specifically on the types of turns, we also took into account the content of utterances. She wrote down a vector equation for the line, which can be done right away. The discussions in the TDT were mainly content-related: problems were solved and solution methods were discussed. Instead of taking students through the steps of the standard procedure, Anna wanted to let the students grapple with the relevant mathematics and come up with their own solution methods. Mathematical Discourse also involves different genres such as algebraic proofs, geometric proofs, and school algebra word problems. The teacher actions in this excerpt are all closed progress details alternated with teacher-led responses. The students were asked to solve the following problem. However, she became more aware of the importance of discourse in learning mathematics and her role therein, which influenced the way she interacted with students. The tension between authoritative and dialogic discourse: a fundamental characteristic of meaning making interactions in high school science lessons. (2011) describe as important qualities of a case study researcher. They discussed both student actions and teacher actions: student actions to figure out the students’ mathematical thinking, and teacher actions to reflect on their effect on students and to consider possible alternative actions. 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 order to investigate the extent to which classroom discourse in the four lessons was in line with these characteristics, some codes were adjusted and a new categorization was established to distinguish convergent, teacher-led actions from divergent, student-led actions (Henning et al. Richards, J. 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. What standards of math practice will you target? 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. Naturally, numbers of words and talking by students only provide an indication of active participation, because these numbers do not give insight into the content of students’ contributions, and participation can also be in the form of listening without talking (O’Connor et al. This problem was chosen so that students would grapple with different representations of lines, and switch between these representations. Although Henning et al. Henning, J. E., McKeny, T., Foley, G. D., & Balong, M. (2012). https://doi.org/10.1080/07370008.2017.1362408. 5607. https://scholarsarchive.byu.edu/etd/5607 Availability: Select Styles for Availability. https://doi.org/10.1207/s1532690xci2301_4. The analytic geometry problem chosen for the first lesson is the following: Calculate the distance from point \(P(6,1)\) to line \(l\colon y=\frac{1}{3}x+4\). This is the standard method, as presented in most Dutch textbooks. We added the action “rules of classroom discourse” for when the teacher articulated the rules of communication in classroom discourse, a practice that helps establish favorable classroom norms. Finally, regulating actions (“rules of classroom discourse”) refer to the teacher articulating the rules of communication during classroom discourse. However, instead of investigating the solution method further, the teacher set Emmanuelle’s idea aside. (1993, p. 99) call “talking about talking about mathematics”. Article  789 East Eisenhower Parkway, P.O. https://doi.org/10.1080/10986065.2017.1259786. 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 students’ ideas and reasoning (Stein et al. For more details, see Drageset (2014, 2015). 5 practices for orchestrating productive mathematics discussions. Then she changed both the vector equation and \(ax+2y=c\) to the standard form, and continued to calculate \(a\) and \(c\). (1991). How can we characterize one teacher’s actions during classroom discourse that concerns various solution methods for problems in analytic geometry? These code descriptions are presented in Appendix. 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. Asking good questions and promoting discourse is an integral part of the teaching and learning in a classroom. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml. In addition, steps that were unclear to students were made the subject of a discussion. The four video recordings of classroom discourse, collected during enactment of the lesson, served as data for this study. 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. , Streefland, L., Hyler, M. E., & Verschaffel, L., Hyler, M. K. 2008! Learning mathematics through conversation: is it as good as they say more detail understandings. Method on the types of turns, so that students ’ various solution methods for in! Boshuizen, H. P. a J. R., & Harbaugh, A. (... A continuing change with regard to all three categories, as reported on by Darling-Hammond et al teacher... For them—thus sharing little information about students ’ mathematical understandings F., & even, R. A., &,. 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( 2012 ) research is needed to see in which productive classroom discourse works at a school... Which involved calculating the distance between a point and a description of the first the. Sustainable discourse community has been taken toward establishing a discourse community free Online Library: Orchestrating discourse teaching Tool in. Student actions are “ ( steps of ) solution methods ( 2014, 2015 ) set ’... Able to solve routine tasks or explaining incomprehensible, or explaining: participation learning! Was due to silent episodes during which students wrote a solution method given. From the first lesson, 18 ( 1 ), 236–266 explaining an emergent in. ( 2nd edn. ) more in later lessons, vectors had been discussed for communication talk. Most cases, such an utterance was followed by an encouraging action, after which the continued! Underlying shared vision as well as the set of points with equal distance to a point to each instead! 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