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# orchestrating mathematical discourse

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. Enschede: SLO. These student actions are “(steps of) solution methods”. An excellent resource is a book by Margaret S. Smith and May Kay Stein, Five Practices for Orchestrating Productive Mathematics Discussions. 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. The teacher’s role in classroom discourse and specific teacher actions formed another important topic of discussion. Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. (2012) categorize reformulations as divergent, we categorized them as convergent, because in reformulating, the teacher decides what to point out, what to name, or what to clarify. It is worth your time! https://doi.org/10.1207/S1532690XCI2102_03. responses that the teacher almost provided for them—thus sharing little information about students’ thinking. In J. Cai (Ed. https://doi.org/10.1007/s11858-009-0214-4. In each of the four lessons, a variety of students’ solution methods was discussed during classroom discourse. In addition, the teacher’s openness to feedback and “will to learn” (Van Eekelen et al. Availability: Select Styles for Availability. Vier Stunden zur analytischen Geometrie wurden sukzessive in Zusammenarbeit mit der Lehrperson entwickelt. Teachers who develop student-centered classroom discourse may feel the need to refrain from all content-related responses or evaluative statements, in order to build the discussion on student ideas. Field notes were made to guide transcription and analysis. Learning and Instruction, 48, 5–13. Students’ participation, building the discourse on students’ ideas, and logical argumentation are important criteria of classroom discourse but do not ensure students’ learning of mathematics. 3 shows that the percentage of student utterances greatly increased from the first to the second lesson. Facilitating mathematical discourse has been consistently identified as a high-leverage instructional strategy. Excerpt 1.3 is from the first lesson and regards the fifth solution method (see 3.4.1.). Joris explained that he had already found that $$a$$ equals four and therefore was able to substitute four in the formula $$ax+2y=c$$. In designing the lesson, main focus was on orchestrating classroom discourse about students’ solution methods. The topic of the TDT was analytic geometry, a new subject in the Dutch curriculum for higher secondary school. Polya, G. (1957). Lampert, M. (1990). Each lesson consisted of two parts: students’ work on a problem, and classroom discourse about their various solution methods. Third, as visualized in Fig. Having their answers evaluated by peers encourages students to think about things from … 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. 2009). However, there seems to be a consensus that orchestrating classroom discourse should be part of every mathematics teacher’s practice. Based on the four transcripts and the tables, several aspects of change were identified and classified into three categories, as elaborated below. Whereas Drageset’s framework focusses specifically on the types of turns, we also took into account the content of utterances. Classroom discourse is initially described as a way to align classroom mathematics with mathematics as a discipline in which mathematical meaning is negotiated. Moreover, feedback and reflection were an important part of the discussions between Anna and the researcher. New York, NY: Routledge. 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?”). Asking for explanations is generally considered a productive teacher practice in mathematics education research as well as in cognitive science (Star and Verschaffel 2017). These are steps that can lead to a genuine solution method. The convergent actions show a decreasing trend. The results of this study are described in three steps: First, the developed framework is presented in tables. 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. This problem was chosen so that students would grapple with different representations of lines, and switch between these representations. Only Excerpt 4.3, from the end of the lesson, which follows below, clearly shows interaction between the teacher and a single student. Uncovering the special mathematical work of teaching. 2017; Ryve 2011). Leermiddelenmonitor (report on teaching materials) 15/16. In the Netherlands, national standardized testing, widespread reliance on textbooks (Blockhuis et al. This suggests a growing participation of students. As displayed in Fig. In the first lesson, the number of convergent teacher actions was much higher than the number of divergent teacher actions. ), Proceedings of the 13th International Congress on Mathematical Education. The silent and the vocal: participation and learning in whole-class discussion. The teacher still refrained from evaluation, instead asking whether other students agreed. ICME-13 Monographs. Orchestrating Mathematical Discourse: Affordances and Hindrances for Novice Elementary Teachers, The purpose of this study was to examine the mathematical discourse within novice elementary teachers' classrooms. This trend indicates the shifting away from teacher-centered patterns of classroom discourse and toward building the discussion on students’ thinking. ), Compendium for research in mathematics education (pp. The analytic geometry problem chosen for the first lesson is the following: Calculate the distance from point $$P(6,1)$$ to line $$l\colon y=\frac{1}{3}x+4$$. https://doi.org/10.1016/j.learninstruc.2016.11.003. This study highlights the needs for future research in regard to socioeconomic status and teachers' beliefs in regard to the orchestration of mathematical discourse. 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. May take place between these representations not complete, not explained, incomprehensible, through... At a secondary school in a variety of students who contributed to discourse! That she did not plan whole-class discussions concerning students ’ thinking would have been appropriate this ensure! Algebra word problems together with one researcher collaboratively developed four discourse-based analytic geometry was new. 2011 ) the dissertation citations contained here are published with the permission of ProQuest LLC, dissertation... Coded part of a whole-class discussion teacher selected four students to present their methods in four consecutive steps may. Indicate that the teacher ’ s role is crucial 11 ) are from the point ( 6,6 which... Two research questions discourse idealized and realized sustained duration, but rather how ” ( sfard et al important in! The lesson educational researcher, Anna attempted to orchestrate classroom discourse were adapted from first... Ideas being discussed as well as a way to align classroom mathematics mathematics... 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