Discourse In The Mathematical Classroom Promotes Understanding

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Traditional classrooms have put emphasis in the authority of the teacher, leaving the students little opportunity for thinking and reasoning. By promoting classroom discussion focused on student-to-student exchange, children will be encouraged to create and appraise their own and knowledge, eventually becoming better problem solvers. They learn to process knowledge more deeply, with a focus on reasoning rather than memorization. Discourse allows students to connect their everyday speech with the specialized language of mathematics. When students develop communication skills in mathematics, the teacher can better understand how they process information, and will be more equipped for the needs of the class. Students who are involved in active discussion are more motivated to learn.

According to Graesser & Person (2004), the term 'classroom discourse' refers to "the language that teachers and students use to communicate with each other in the classroom". Conversation is the channel through which most instruction takes place, so classroom discourse is connected with face-to-face classroom teaching. Mathematical classroom discourse is about class discussions in which students talk about mathematics so that they disclose their perception of concepts. Students also learn to take part in mathematical reasoning and debate. Discourse involves asking deliberate questions that draw out information from students about how a problem was answered and why a particular technique was selected. Students learn to critique their own and others' ideas and seek out efficient mathematical solutions. The fundamental issue is that mathematics is largely about reasoning not memorization. Mathematics is not about applying a set of procedures but about developing understanding and explaining the processes used to arrive at solutions (Maguire & Neill, 2006).

Mathematical understanding may be classified in various ways. Procedural understanding means knowing how to execute mathematics; contextual understanding is the knowledge about usefulness in context; relational understanding refers to linking mathematical ideas; abstract understanding indicates knowing about underlying structures. Rittle-Johnson and Alibali (1999) suggested that conceptual instruction results in increased abstract understanding and the use of correct procedure. Hiebert and Carpenter (1992) proposed that understanding takes place when a fact, idea, or procedure is part of a connected network. In order to think about and manipulate mathematical ideas, they need to be represented internally, in a way that allows the mind to operate on them. They also proposed that understanding generates the construction of new connections.

Learning in the traditional, direct instruction centered mathematics classroom tends to be very passive, making low cognition demands on learners (Bernero, cited in Leigh, 2006). Discourse provides communal reinforcement for cultivating students' thinking, drawing it out through the interest, questions, and ideas of others. Discourse also allows thoughts to be freely available and make a chance for the generation of meaning and agreement. Communicating what they know allows students the opportunity to refine their own understandings. Cobb (2006) notes that a mathematical explanation consists of two parts. The calculational explanation involves explaining the process through which the result was arrived at. A conceptual explanation involves explaining why that process was selected. 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. Kilpatrick (2002) stated that one of the best ways for learners to develop their reasoning is to give reasons for their solutions to others. The concept thus becomes more thoroughly engrained, regardless of the subject matter.

Discourse enables us to link a student's daily language with the vocabulary of mathematics. As students engage in the discourse they acquire ways of talking and thinking that characterize the particular curriculum area. To learn mathematics is to become an expert participant in classroom discourse about the procedures, concepts, and use of argument that constitutes mathematical reasoning. This approach is supported by the theories of the Lev Vygotsky who argued that "the higher mental processes are acquired through the internalization of the structures of social discourse" (Graesser & Person n. d.).

In Sfard's (1991) theory of mathematical development, students move through three stages in understanding ideas. First, a new concept is usually come across as the result of mathematical procedures. Next, students begin to think of the concept as separate from the processes that created it, and ultimately are able to use it to work out problems. For a teacher, knowing the stage a student has reached is difficult. However, this can be deduced from students' talk, where a change in the way an idea is referred to is a sign of transfer from one stage to the next. Classes are said to exhibit reflective discourse when students, rather than simply the teacher, have a say in the shifts in discussions (Cobb, Boufi, McClain and Whitenack, 1997). This is also true where students independently reflect on the discourse to progress in their thinking.

Through discourse, a teacher can better grasp the mathematical needs of the class: what the students are familiar with, the misconceptions they have, and how these might have developed. To understand the thinking of children, teachers need to spend more time listening to them describe how they think and less time explaining to the children how the teacher thinks (Chambers, 1995). Students' misconceptions may be further diagnosed based on the questions they ask. It is only when students are given an opportunity to voice their thinking that teachers truly comprehend what is happening in the minds of their pupils (Shanefelter, 2004). Hearing students talk about what they understand allows the teacher to fill in the pieces that are missing from a child's understanding and thus the concept can be grasped more fully (Huggins & Maiste, cited in Leigh, 2006).

Mulryan (1995) indicated that learners' participation was greater in small groups than in whole class settings and that pupils were more engaged in the small group situations. Leigh (2006) reported that the social environment of the classroom becomes more positive with discourse and cooperative learning. "A caring atmosphere seemed to emerge for the class and that carried through the school day, not only during math… Cooperative learning does much to foster a caring, team-like atmosphere. It definitely builds one's self-esteem and encourages even shy students to assert themselves" (Bernero, cited in Leigh, 2006). This type of environment would be most suitable for constructing understanding.

Although classroom discourse appears to have positive effects on students' conceptual understanding of mathematics, there are also problems associated with this method. Wilgus (2002) concluded that peer collaboration was not a significant factor at the third grade level for academic achievement. Huggins and Maiste (1999) also noted that students of this age needed more manipulatives and tangible interventions to grasp problem solving methods. Younger students need more time to process concepts. Mulyran (1995) found that low achievers are less actively involved than high achievers in small groups. He also found that girls seemed to be having problems with cooperative learning tasks. Bernero (cited in Leigh, 2006) also noted problems with cooperative groups: control over noise level, keeping teams on task, and dealing with students who don't get along socially.

Teachers who use a high level of classroom discourse need to be aware of the individual student's patterns of working in small groups and take steps to promote more active involvement by all students and especially low achievers (Mulyran, 1995). If the discussion does not allow each student to have a say and if students are not sure how they are supposed to contribute, precious learning time is wasted. Discussions must be well chosen to be effective and students must be taught how to work in this manner. Leigh (2006) also endorsed the need to teach social skills prior to using discourse in math. Discourse in the mathematical classroom does indeed promote understanding, but teachers must be vigilant in coordinating student discussions.