Mathematical Communication

Mathematical Communication: Effective Teachers Facilitate Classroom Dialogue that Is Focused Towards Mathematical Argumentation

Teaching ways of communicating mathematically demands skilful work on the teacher‘s part (Walshaw & Anthony, 2008). Students need to be taught how to articulate sound mathematical explanations and how to justify their solutions. Encouraging the use of oral, written and concrete representations, effective teachers model the process of explaining and justifying, guiding students into mathematical conventions. They use explicit strategies, such as telling students how they are expected to communicate (Hunter, 2005).

Teachers can also use the technique of revoicing (Forman & Ansell, 2001), repeating, rephrasing, or expanding on student talk. Teachers use revoicing in many ways: (i) to highlight ideas that have come directly from students, (ii) to help the development of students‘ understandings implicit in those ideas, (iii) to negotiate meaning with their students, and (iv) to add new ideas, or move discussion in another direction.

When guiding students into ways of mathematical argumentation, it is important that the classroom learning community allows for disagreements and enables conflicts to be resolved (Chapin & O'Connor, 2007). Teachers‘ support should involve prompts for students to work more effectively together, to give reasons for their views and to offer their ideas and opinions. Students and teacher both need to listen to others‘ ideas and to use debate to establish common understandings. Listening attentively to student ideas helps teachers to determine when to step in and out of the discussion, when to press for understanding, when to resolve competing student claims, and when to address misunderstandings or confusion (Lobato, Clarke, & Ellis, 2005). As students‘ attention shifts from procedural rules to making sense of mathematics, students become less preoccupied with finding the answers and more with the thinking that leads to the answers (Fravillig, Murphy, & Fuson, 1999).

Mathematical Language: The Use of Mathematical Language Is Shaped When the Teacher Models Appropriate Terms and Communicates Their Meaning in a Way that Students Understand

If students are to make sense of mathematical ideas they need an understanding of the mathematical language used in the classroom. A key task for the teacher is to foster the use, as well as the understanding, of appropriate mathematical terms and expressions. Conventional mathematical language needs to be modeled and used so that, over time, it can migrate from the teacher to the students (Runesson, 2005). Explicit language instruction and modeling takes into account students‘ informal understandings of the mathematical language in use. For example, words such as ―less than‖, ―more‖, ―maybe‖, and ―half‖ can have quite different meanings within a family setting. Students can also be helped in grasping the underlying meaning through the use of words or symbols with the same mathematical meaning, for example, ‗x‘, ‗multiply‘, and ‗times‘.

Teachers face particular challenges in multilingual classrooms. Words such as ―absolute value‖, ―standard deviation‖, and ―very likely‖ often lack an equivalent term in the students‘ home language. Students find the syntax of mathematical discourse difficult. Prepositions, word order, logical structures, and conditionals are all particularly problematic for students. Students may also be unfamiliar with the contexts in which problems have been situated. Language (or code) switching, which involves the teacher substituting a home language word for a mathematical word, has been shown to enhance student understanding, especially when teachers are able to use it to capture the specific nuances of mathematical language (Setati & Adler, 2001).

Assessment for learning: Effective teachers use a range of assessment practices to make students‘ thinking visible and support students‘ learning.

Mathematics teachers make use of a wide range of formal and informal assessments to monitor learning progress, to diagnose learning, and to determine what can be done to improve learning. Within the everyday activities of the classroom, teachers collect information about how students learn, what they seem to know and are able to do, and what they are interested in. This information helps teachers determine whether particular activities are successful and informs decisions about what they should be doing to meet the learning needs of their students (Wiliam, 2007).

Effective teachers gather information about students by watching students as they engage in individual or group work and by talking with them. They monitor their students‘ understanding, notice the strategies that they prefer, and listen to the language that they use. The moment-by-moment assessment helps them make decisions about what questions to ask next, when to intervene in student activity, and how to answer questions. Classroom exchanges in the form of careful questioning provide a powerful way to assess students‘ current knowledge and ways of thinking (Steinberg, Empson, & Carpenter, 2004). For example, questions that have a variety of solutions, or that can be solved in more than one way, can help teachers gain insight into students‘ mathematical thinking and reasoning.

As well as informing the teacher, assessment for learning involves providing feedback to students. Helpful feedback explains why something is right or wrong, and describes what to do next, or describes strategies for improvement. Effective teachers also provide opportunities for their students to evaluate and assess their own work. They involve students in designing test questions, writing success criteria, writing mathematical journals, and presenting portfolios as evidence of growth in mathematics.