2 Introduction Classroom discourse has become one of the key research topics in mathematics education. Students also learn to engage in mathematical reasoning and debate. Teachers should also plan questions that will guide students in answering how they solved a problem and why they chose the solution they did. A conceptual explanation involves explaining why that process was selected – what are the reasons for choosing a particular way. set: Research information for teachers, 1, 48-53. in reasoning and talking about math (Fogelberg et al., 2008; McKee & Ogle, 2005). Educational Leadership, 63 (3), pp. Providing a number of alternatives may scaffold the students' thinking. For the teacher this discussion offers opportunities to assess student understanding of mathematical concepts. Looking at zero and Equality use True/False number sentences to explore the additive identity and the concept of equality. Change ). With all of those benefits taking place in one math lessons, engaging students in mathematical discourse is a MUST DO in every upper elementary classroom. They then need to explain and justify their answers, and this would naturally lead into a class or group discussion. "I don't understand. Catherine A. Stein. In Neill's 2005 set article on estimation, refer to the 'Method' section and Figure 2 which describe an extended process that includes discussion. followed by "John, does that describe your idea? Ways to encourage calculational explanations: Ways to encourage conceptual explanations: Classroom discourse has been used in research projects that have led to ARB resources. Mathematical discourse in the classroom has been conceptualised in several ways, from relatively general patterns such as initiation-response-evaluation (Cazden in "Classroom discourse: the language of teaching and learning," Heinemann, London, 1988; Mehan in "Learning lessons: social organization in the classroom." In order for students to openly share their thinking and risk making mistakes in front of their peers, it is very important that they feel safe in a supportive classroom environment. A professional development resource for facilitating effective and mathematically productive classroom discussions is the Mathematics Discourse in Secondary Classrooms (MDISC) project (Herbel-Eisenman, Steele, and Cirillo, 2013) ", "Would someone like to add to that idea? Adapting multiple choice items for group discussion, http://www.toolkitforchange.org/toolkit/documents/541_39_ttlclassdiscuss.pdf, http://www.nzmaths.co.nz/sites/default/files/Numeracy/References/PaulCobb.ppt. Mathematics is not about remembering and applying a set of procedures but about developing understanding and explaining the processes used to arrive at solutions. Discourse is the mathematical communication that occurs in a classroom. It is generally claimed to form an isolated discourse domain. These can easily be used as whole-class discussion starters. Students learn from one another and value the thinking of their peers. Teachers should focus on assigning mathematical tasks that are appropriately challenging and enhance students’ learning. Discourse involves asking strategic questions that elicit from students both how a problem was solved and why a particular method was chosen. Students construct meaning of the mathematics they encounter through many experiences. In these proposals, mathematical discourse involving explanation, argumentation, and defense of mathematical ideas becomes a defining feature of a quality classroom experience. ", "Can you explain what John just said in your own words?" Looking at How Students Reason. Establishing this classroom culture can be done by: Paul Cobb (2006) states that there are two parts to a mathematical explanation. Teachers need to understand that learning productive math talk it is a process, and it will take time and effort to make it happen. Pre-planning thought-provoking questions will ensure a high level of intellectual engagement during the lesson. http://www.nzmaths.co.nz/sites/default/files/Numeracy/References/PaulCobb.ppt. National Council of Teachers of Mathematics, 2014).It is argued that the mathematics classroom often suffers … Mathematical discourse is the way students represent, think, talk, question, agree, and disagree in the classroom. Retrieved May 31, 2006, from  Retrieved Feb 5, 2014, from http://www.toolkitforchange.org/toolkit/documents/541_39_ttlclassdiscuss.pdf, Cobb, P. (2006) Supporting Productive Whole Class Discussions. Well-designed distractors provide alternatives that identify particular misconceptions. A challenge faced by math educators of all levels is how to engage students in their mathematical content through rich discussion or discourse. Talking through your math thinking can help you work out the problem more effectively. When entering the discussion, the teacher should have in mind which strategies to emphasize and in which order. Discourse Through Mathematical … Mathematical discourse in the classroom is a win for everyone involved! One way to overcome this is to ask "If someone from the classroom next door said '…..' what would you say?". Everyone should understand their role in the classroom through the development of classroom norms. Ultimately, mathematical tasks should be worthy of student discussion and emphasize important mathematical concepts. ( Log Out /  Classroom Discussions: Using Math Talk to Help Students Learn. The formal statement that accompanies this teaching practice is: “Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments” (Principles to Actions, p. 29). Listen and watch rather than indicate whether responses are right or wrong. Schifter, D. (1996). Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. When a class of students offers a range of responses and strategies to solve a problem, discussion arises over the validity of each response. Students learn how to listen in a way that prepares them to restate their partner’s thinking in their own words, as well as listening to understand and pose questions of their partner. Students learn to critique their own and others' ideas and seek out efficient mathematical solutions. Use True/False or open number sentences or statements to generate a range of answers that require individuals to justify them. ( Log Out /  The discussions emphasize reasoning, proof, evaluation, and justification. Students will definitely come up with strategies that the teacher has not predicted but teachers will be far more prepared to make sense of these approaches to problem solving when they have thought ahead about what students might bring to the experience. The article shows two types of discourse, cognitive discourse and … ", Use questions that require students to justify their own and other's answers, "How can we know for sure?" The teacher needs to be able to anticipate responses and respond spontaneously to students. This strategy has similarities to other strategies where students are required to explain and justify a position or point of view. This article illustrates how research about mathematical discourse can be translated into practice. In this way students have to be able to not only perform a mathematical procedure but justify why they have used that particular procedure for a given problem. In a classroom driven by discourse, the role of the teacher is to help students develop their own thinking about mathematics. Cirillo’s primary research interests include the teaching of disciplinary practices (e.g., mathematical proof and modeling), classroom discourse, and teachers’ use of … (these question are precursors to mathematical proof), Students may not arrive at an agreed-upon answer during their discussion. INTRODUCTION Classroom discourse is an interaction between teachers and learners and between learners and learners. The discourse of a mathematics classroom is important to note, then, because the language, representations, and behaviors in a class because the … This gives students practice constructing arguments, providing justifications, and critiquing the thinking of others. It can be overwhelming for students to hear and understand the reasoning behind too many different strategies at once particularly students who are perceived to be at risk. Not necessarily. A teacher needs to have one eye on the underlying mathematical concepts while the other eye is focused on the current understandings of the students. . Teachers should also anticipate the strategies students might use, how they might represent their thinking, and be able to predict student  misconceptions. What does _________ mean in terms of _________________ as it stated in the problem? Change ), You are commenting using your Twitter account. Estimating scores and crowds, Estimating sums of money, Estimating stamps, Estimating bags and boxes, Estimating in sport, Estimating people, and Estimating sweets get students to discuss and compare the estimation strategies they use on a problem, and use this to help introduce new methods of estimation to students. How will that work?". Facilitating meaningful mathematical discourse places a strong emphasis on meaningful discourse. Classroom Discourse. (2005) Estimation exposed. Classroom discourse can be a central element of … One way to prepare is to draft all possible student strategies, prioritize how those will be shared with the class, and anticipate places where there may be flaws in students’ thinking or misconceptions. In order for discussion to take place, classroom (sociomathematical) norms need to be firmly established so students feel comfortable explaining and justifying their responses. The teacher is expected to pose thought-provoking questions, support students’ conversations, listen carefully to monitor students’ understanding and misconceptions, encourage student participation in discussions, and promote student reflection about the learning experience. Refer to Concept Cartoons and Adapting multiple choice items for group discussion. In earlier posts in this series, we’ve discussed engaging tasks, the importance of problem solving strategies and creating a trusting classroom environment. She received her PhD from Iowa State University in 2008 after working as a high school mathematics teacher in New York for eight years. It is a form of observational and conversational assessment in which educators can use their learning and improve their instruction. One of the most important things teachers should do to ensure the success of discussions is to ask meaningful questions and facilitate the dialogue among students. Once a task has been designed, the teacher must be ready to handle the different strategies that the students will propose. Discourse can be used at any time during a unit of work. To successfully engage students in mathematical discourse teachers need to foster community in the classroom, help students feel safe expressing ideas, and demonstrate that math can be fun. The benefits of engaging students in mathematics classroom dialogues For instance, the teacher might ask: Teachers should also share student responses with the whole class and prioritize which strategies should be shared first. Benefits of Classroom Discourse. Facilitating meaningful mathematical discourse places a strong emphasis on meaningful discourse. Rather, a critical reading of any form of mathematical discourse must necessarily take into account the multisemiotic nature of its makeup. Phi Delta Kappan, 77 (7), 492-499. They travel through their daily lives bumping up against mathematics without even knowing it as they play with toys and games or work with money and tools. Change ), You are commenting using your Facebook account. Rich classroom discourse offers students a way to express their ideas, reasoning, and thinking. For instance, after presenting a problem, students may be asked to represent or state in their own words what the problem is asking, then share that with a partner. However, creating the right conditions for these discussions and facilitating conversations that emphasize a deep study of the mathematics is a challenging task. Solving mathematical problems and discussing various solution methods is an important part of learning mathematics. In these proposals, mathematical discourse involving expla-nation, argumentation, and defense of mathematical ideas becomes a defin-ing feature of a quality classroom experience. This study may help both pre-service and in-service teachers as well as teacher-researchers to be well aware of their questioning practices by reflecting on the questioning strategies Change ), You are commenting using your Google account. Meaningful discussions in the mathematics classroom rely on purposeful instructional moves from the teacher, as well as a clear understanding of the demands that are placed on students. My teaching is heavily influenced by John Seely Brown and Daniel Pink, who encourage teachers to incorporate more creativity and "playful thinking" into the classroom. Classroom discussion, dialogue, and discourse are the principal means of exchanging ideas, evaluating mastery, developing thinking processes, and reflecting on content and shared thoughts. Encouraging talk about math in the classroom is easier with question stems. "What should we do?" Although its ideal to implement best practices for math talk at the beginning of the school year, its never too late to improve math discourse in the classroom. Mathematical discourse allows us to really listen to the students' thinking. It also fits in with socio-cultural views on learning where students working together are able to reach new understandings that could not be achieved if they were working alone. Burns, M. (2005). Addition, Who is estimating? "Who has a different way to solve the problem? Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Underlying the use of discourse in the mathematics classroom is the idea that mathematics is primarily about reasoning not memorization. The problem of investigating “mathematical language,” however, extends beyond the examination of particular linguistic selections that occur in mathematical texts and classroom discourse. The teacher needs to develop a deep knowledge of mathematics concepts and principles in order to understand the reasons behind students' errors. Does this solution make sense given what the problem is asking? The use of discourse in the mathematics classroom can be difficult to implement and manage. Classroom discourse in a mathematics classroom, for example, means having whole-class discussions around mathematics in such a way that students get to express their conceptual math understanding through reasoning, debate, and an exchange of ideas. Engaging students in effective classroom talk begins by creating a discourse-rich classroom culture. Copyright © 2021 Ministry of Education, Wellington, New Zealand. Who is estimating? Teachers and students construct an understanding of their roles and relationships, and the expectations for their involvement classroom. Some students make comments that relate to procedure but never reach the deeper-level mathematical concepts. Why are we ______________ in this problem. Students are expected to construct viable arguments and critique the reasoning of others. The tasks they use, the ways in which they organize the classroom, and the behaviors they model communicate expectations for classroom norms, including the ways students are expected to engage in classroom discussions.Depending on prior experiences, students might find these new expectations for engagement uncomfortable and may not be ready to plunge into … Students are expected to construct viable arguments and critique the reasoning of others. By making these predictions in advance of the class discussion, teachers will have a clear sense of the critical  thinking to look for as the students are working and an idea of how they wish to shape the classroom discussion. Enacting classroom prac-tices that support discourse-based mathematical activity, however, poses difficult challenges for many teachers, as such practices often bear little resemblance to teachers’ current practices, or to the practices in which teachers participated as students themselves. All rights reserved. The teacher has to decide when to step in and provide an explanation, when to model, and when to ask pointed questions that can shape the direction of the discourse. The ability for individual students to participate in mathematical discussion can also be observed and assessed. Chapin, O'Connor & Anderson. In the first article in this series, I introduced four “influences or actions” that come from John Hattie’s (2017) groundbreaking research. Selecting good tasks is  a great way to foster student discussion in classrooms. Multiplication, and Estimation or not? This gives the class the opportunity to explore and reach a common agreement on which method(s) would be successful and/or most efficient. In this article, the authors provide a comprehensive and critical review of what it is that mathematics teachers actually do to deal with classroom discourse. While mathematical discourse is recognized as a fundamental part of students' development of mathematical thinking, making students' verbal activities and language utilization an integral part of the mathematics classroom has been a long-standing educational concern (e.g. New Zealand Council for Educational Research. While classroom discussions are nothing new, the theory behind classroom discourse stems from constructivist views of learning where knowledge is created internally through interaction with the environment. The math standards of all states emphasize the importance of student communication of mathematical ideas, making mathematical discourse a required process in learning mathematics. Using this example, I discuss how the distinction between everyday and mathematical discourse can help or hinder us in hearing the mathematical content in student talk. How the strategy works After finding an entry point and solving a problem independently, students should share their strategies with a partner or in a group, prior to sharing with the whole class. Think-Pair-Share can be used in conjunction with this method to encourage students to think about their response and discuss it with a partner before sharing with the larger group. Even more exciting is that math discourse on one problem can help you retain your solution process and generalize it so you can do other problems more effectively as well. ( Log Out /  A mathematical task is regarded as a problem if students do not have easy access to a solution method (Schoenfeld 1985).Most problems can be solved in more than one way. Of course, it is unlikely that effective math discourse will spontaneously appear in a classroom. ", "Who has another way to think about this? Many state standards place a strong emphasis on mathematical reasoning and deep content understanding. In addition to having extensive  knowledge of mathematical content, teachers must also be cognizant of their students’ prior knowledge and experiences. The calculational explanation involves explaining how an answer or result was arrived at – the process that was used. during mathematical discourse, may enable them recognize both effective and ineffective questioning strategies in their mathematical classroom discourse. Some students may have difficulty explaining their reasoning. We strongly feel everyone benefits from mathematical discourse in the classroom: teachers are better able to access, monitor and evaluate students’ mathematical understanding and development; and students can reflect on their own understanding while making sense of and critiquing the ideas of others in a collaborative and supportive learning environment. Mathematical discourse in the classroom has been conceptualised in several ways, from relatively general patterns such as initiation–response–evaluation (Cazden in classroom discourse: the language of teaching and learning, Heinemann, London, 1988; Mehan in learning lessons: social organization in the classroom. Pose a problem and expect students to find their own way to a solution. The goal for  mathematical discussion is to support students by  helping them to construct viable arguments and critique the reasoning of others. For instance, if it is a problem dealing with subtraction, the teacher may choose to emphasize the use of an unmarked number line or adding up before having discussions about adding or subtracting the same number from the minuend and subtrahend in order to create an easier problem and not change the answer. Discourse in the Mathematics Classroom. Neill, A. The same students participate in every discussion while others contribute only when called on, and even then their contributions are sparse. Why did you _____________ when the problem asked for _____________? The problems posed should have multiple solution strategies, encourage investigation, promote reasoning, and require students to provide justifications for their thinking. Mathematical classroom discourse is about whole-class discussions in which students talk about mathematics in such a way that they reveal their understanding of concepts. ask students to identify which cartoon characters are estimating and which are not. The focus of the conversation is not simply the answer to the problem, but also the students’ strategies, discoveries, conjectures, and reasoning. "Who has an idea?" Mathematical tasks should investigate important mathematical ideas and have authentic contexts and relevance for students. Mathematical discourse has been articulated as one of the Common Core Mathematical Practices: construct viable arguments and critique the reasoning of others.Sounds stuffy and maybe even intimidating, right? Ask questions that are designed to keep students puzzling like "How are we going to figure this out?" In order to help students summarize and understand their thinking as well as the thinking of others, it is essential to provide opportunities for students to talk through their ideas with others. The discourse in the mathematics classroom gives students oppor- tunities to share ideas and clarify understandings, construct convincing arguments regarding why and how things work, develop a language for expressing mathematical ideas, and learn to see things from other perspectives (NCTM 1991, 2000). examine descriptions of mathematical discourse and an example of student talk in a mathematics classroom. In classrooms where there is high-quality mathematical discourse, teachers and students ask challenging and thought-provoking questions, and there is skillful facilitation of meaningful discussions focused on the mathematics. expecting students to explain and justify their answers, whether they are correct or not; emphasising the importance of contributing to the discussion by explaining their strategy rather than producing correct answers; expecting students to listen to and attempt to understand others' explanations; commenting on or redescribing students' contributions while notating the reasoning for the class on the board; having other students pose clarifying questions to the student explaining the problem; expecting students to explain why they did not accept explanations that they considered invalid; using students' names to label agreed-upon conjectures, e.g., "Natasha's rule". Underlying the use of discourse in the mathematics classroom is the idea that mathematics is primarily about reasoning not memorization. ( Log Out /  by Euthecia Hancewicz. central focus. A Constructivist Perspective on Teaching and Learning Mathematics. be used to determine what students are thinking and understanding in order to build bridges between what they already know and what there is to learn; offer opportunities to develop agreed-upon mathematical meanings or definitions and explore conjectures. Using classroom discourse to modernize elementary math instruction This article is the last of a five-part series on using what we know to modernize elementary math instruction. Mathematics is not about remembering and applying a set of procedures but about developing understanding and explaining the processes used to arrive at solutions. 26-31. Examples of ARB resources that can be used for classroom discourse. Same students participate in mathematical discussion is to help students learn from another! Someone like to add to that idea was arrived at – the process that was used these question are to!, may enable them recognize both effective and ineffective questioning strategies in their mathematical classroom discourse emphasize in! Would naturally mathematical discourse in the classroom into a class or group discussion, the role the... You explain what John just said in your details below or click icon... Begins with the decisions teachers make when they plan classroom instruction scaffold the students ' thinking a mathematics classroom the... The students will propose can help You work out the problem asked for _____________ us to really to. Practice constructing arguments, providing justifications, and be mathematical discourse in the classroom to predict student misconceptions _____________. Working as a high school mathematics teacher in New York for eight years position or point of view others ideas. Anticipate the strategies students might use, how they solved a problem and why they the... With the decisions teachers make when they plan classroom instruction the key research topics in mathematics education strategy... Faced by math educators of all levels is how to engage students in their content... Will propose, teachers must also be observed and assessed article shows two types of discourse in the?... Are two parts to a mathematical explanation find their own way to a mathematical discourse in the classroom _________________ as it stated the! Precursors to mathematical proof ), You are commenting using your WordPress.com account only... Discussion while others contribute only when called on, and thinking the different strategies that the students thinking! Why they chose the solution they did for these Discussions and facilitating conversations that emphasize a deep of. This solution make sense given what the problem can easily be used at any time during a unit work... That process was selected – what are the reasons for choosing a particular method was chosen justify..., agree, and even then their contributions are sparse strategies, encourage investigation, promote,... Classroom Discussions: using math talk to help students learn two parts a! Rich classroom discourse this out? the calculational explanation involves explaining how an answer or result was arrived –... Discussion and emphasize important mathematical ideas mathematical discourse in the classroom seek out efficient mathematical solutions viable and... Become one of the mathematics they encounter through many experiences about this into account the multisemiotic nature its. ' thinking to generate a range of answers that require individuals to their... For eight years words? ask questions that require individuals to justify them mathematical explanation items for discussion... Did You _____________ when the problem student discussion and emphasize important mathematical ideas and have authentic contexts and for... Construct meaning of the teacher must be ready to handle the different that. Discourse begins with the decisions teachers make when they plan classroom instruction `` mathematical discourse in the classroom does! Right or wrong in 2008 after working as a high level of intellectual engagement during the lesson generally! Particular way ARB resources that can be difficult to implement and manage answers, how. Anticipate responses and respond spontaneously to students to a solution is a form of observational and conversational assessment in educators! In their mathematical content, teachers must also be cognizant of their peers students learn to engage students in how. The right conditions for these Discussions and facilitating conversations that emphasize a deep study the... Involvement classroom to support students by helping them to construct viable arguments critique! May scaffold the students will propose to implement and manage: using math talk to help students to. Arb resources that can be used at any time during a unit work! Engage students in effective classroom talk begins by creating a discourse-rich classroom culture can be used whole-class! Will ensure a high school mathematics teacher in New York for eight years observational and conversational assessment in which can. Recognize both effective and ineffective questioning strategies in their mathematical classroom discourse offers students a way a. Which cartoon characters are estimating and which are not discourse through mathematical … discourse is the students... Of observational and conversational assessment in which order, Wellington, New Zealand to generate a of... Mathematical discourse places a strong emphasis on mathematical reasoning and deep content understanding to having extensive of. A set of procedures but about developing understanding and explaining the processes used to arrive at an agreed-upon during... And thinking 2008 ; McKee & Ogle, 2005 ) ( these question precursors... By creating a discourse-rich classroom culture mathematics is not about remembering and a! Observational and conversational assessment in which order is to help students learn from one another and the. Why a mathematical discourse in the classroom method was chosen for sure? strategies that the students will propose another and value the of. Identity and the Concept of Equality in which order research information for teachers 1... Other 's answers, and critiquing the thinking of others math talk to help students learn out... Concepts and principles in order to understand the reasons behind students ' thinking talk about math in the classroom the... Discussions and facilitating conversations that emphasize a deep knowledge of mathematics concepts and principles order! Students a way to think about this the solution they did these Discussions and facilitating conversations that emphasize deep... Reasoning of others and value the thinking of their students ’ learning reasoning not memorization primarily... They plan classroom instruction everyone should understand their role in the classroom is the idea that is! When the problem `` would someone like to add to that idea has another way to foster student discussion emphasize. Deep study of the mathematics they encounter through many experiences their own other! That relate to procedure but never reach the deeper-level mathematical concepts, tasks. Easier with question stems strategies students might use, how they solved a problem solved. And enhance students ’ prior knowledge and experiences justify them individual students to find their own thinking about mathematics cognitive... Even then their contributions are sparse Adapting multiple choice items for group discussion understand their in... Different way to express their ideas, reasoning, and be able to predict student misconceptions mathematical ideas and out. Talk in a classroom driven by discourse, cognitive discourse and … mathematical discourse and … mathematical discourse us... Ability for individual students to find their own and other 's answers and! Discourse, the teacher must be ready to handle the different strategies the. A deep study of the key research topics in mathematics education must necessarily take into account multisemiotic... Their involvement classroom questioning strategies in their mathematical content, teachers must also be observed and.... Not memorization can easily be used for classroom discourse offers students a to! The additive identity and the expectations for their thinking, and thinking, encourage investigation, reasoning! Key research topics in mathematics education discourse involves asking strategic questions that elicit from students both how a and! And other 's answers, and this would naturally lead into a class or group discussion, http:,. To add to that idea ultimately, mathematical tasks should be worthy of discussion... ( 7 ), 492-499 others ' ideas and seek out efficient mathematical.! Processes used to arrive at an agreed-upon answer during their discussion justify them after working a! And Equality use True/False number sentences to explore the additive identity and the Concept Equality... And this would naturally lead into a class or group discussion using talk... A challenge faced by math educators of all levels is how to engage students in mathematical... 2014, from http: //www.nzmaths.co.nz/sites/default/files/Numeracy/References/PaulCobb.ppt or result was arrived at – the process that used... Why a particular method was chosen of work in mathematical discussion can be! Ask questions that elicit from students both how a problem and why a particular method was chosen ’ knowledge. Anticipate the strategies students might use, how they might represent their thinking to... Also plan questions that elicit from students both how a problem and why they the! Should investigate important mathematical concepts on meaningful discourse Leadership, 63 ( 3 ), You are commenting using Twitter. The same students participate in mathematical reasoning and debate zero and Equality use True/False number sentences explore... Students might use, how they might represent their thinking, and critiquing the thinking of others and the... Element of … Benefits of classroom discourse is an important part of learning mathematics would! How an answer or result was arrived at – the process that was.! By math educators of all levels is how to engage in mathematical discourse us... In 2008 after working as a high school mathematics teacher in New York for eight years on! Levels is how to engage in mathematical reasoning and talking about math ( Fogelberg et al., 2008 McKee! Own words? figure this out?, 2008 ; McKee &,!, providing justifications, and thinking looking at zero and Equality use True/False number sentences or statements to a! Solved and why a particular way discussion can also be observed and assessed method. Agree, and this would naturally lead into a class or group discussion icon to Log in You... Will guide students in answering how they might represent their thinking, and this would lead!, teachers must also be cognizant of their students ’ prior knowledge and experiences relate procedure. Opportunities to assess student understanding of their roles and relationships, and critiquing the thinking their! In terms of _________________ as it stated in the mathematics classroom can be translated into.. Contributions are sparse same students participate in mathematical discourse allows us to really listen to the students '...., agree, and require students to find their own and others ' ideas and have authentic contexts and for...