Communication and Discourse
By Jennifer Ririe and Karen Redford
“Our lives begin to end the day we become silent about things that matter.” – Martin Luther King, Jr.
WHY PROMOTE COMMUNICATION IN THE MATHEMATICS CLASSROOM?
Too often in schools, children have little opportunity to talk, draw, and write about mathematical or scientific ideas. Principles and Standards for School Mathematics (NCTM 2000) emphasize the importance of these communication skills. It cautions that because “mathematics is so often conveyed in symbols, oral and written communication about mathematical ideas is not always recognized as an important part of mathematics education” (p. 60). Talking, drawing, and writing, however, can give students the opportunity to justify their thinking, formulate questions, and summarize important insights.
Talking, drawing, and writing provide students with important avenues for explaining and exploring mathematical ideas. Talking helps them explore ways to express general observations and describe patterns, as well as work through puzzling concepts and propose theories. Drawing and writing serve as permanent records of students’ thinking that can be shared with their peers and teacher. Communication is enhanced and understanding is increased when mathematical ideas are represented in different ways, such as through a story, with manipulatives and charts, and through personal metaphors. Talking is an effective way for children to clarify their thinking, discuss new possibilities, extend the thinking of others, and rehearse their ideas for writing. In these ways, children can become proficient and articulate in communicating mathematical ideas (Whitin, 2002).
Students must learn to reason through situations and develop a problem-solving system that is successful for them. Being able to effectively communicate their solutions and defend their reasoning are life skills that students need not only in school but also beyond the classroom walls throughout their lives (Anderson, 2004).
WHAT ARE THE RESPONSIBILITIES OF STUDENTS AND TEACHERS IN MATHEMATICAL COMMUNICATION AND DISCOURSE?
In the revised Principles and Standards for School Mathematics (NCTM, 2000), communication was recommended as its own standard, with the recommendation that mathematics programs at all grades preK–12 enable students to:
- · Organize and consolidate their mathematical thinking through communication;
- · Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
- · Analyze and evaluate the mathematical thinking and strategies of others;
- · Use the language of mathematics to express mathematical ideas precisely. (p. 348)
In the Professional Standards for Teaching Mathematics (NCTM, 1991), one standard focuses on discourse, with the teacher of mathematics having the responsibility to orchestrate discourse by:
- · Posing questions and tasks that elicit, engage, and challenge students’ thinking;
- · Listening carefully to students’ ideas;
- · Asking students to clarify and justify their ideas orally and in writing;
- · Deciding when and how to attach mathematical notation and language to students’ ideas. (p. 35)
MODES OF COMMUNICATION
Speaking, listening, writing, and reading are varied modes of communication that are integral to effective communication in the mathematics classroom and to the development of mathematical literacy. Teachers need to integrate all four aspects of communication to help students build a robust understanding of mathematics and develop fluency with all aspects of language. Just as we expect students to use all four modes of communication in ordinary language, teachers need to expect students to use all four modes of communication in mathematics language.
Speaking and Listening to Mathematics
Perhaps the most natural forms of mathematical communication are speaking and listening. In the classroom, students should regularly engage in talking mathematics, both in terms of student-to-teacher talk and student-to-student talk. Because some students learn best aurally, hearing mathematics is important. Furthermore, when students discuss mathematics concepts aloud with others, they are more likely to demonstrate a deeper understanding of the concepts than when they simply solve a problem. Upon reflection, mathematics teachers may realize that their own mathematical learning was solidified when they had to explain mathematical concepts to others, notably their students. For K–12 students, the same phenomenon is true. Students may often believe they understand a concept until they have to explain it to someone else, at which point they may realize that their understanding of the concept is not as solid as they thought.
Writing Mathematics
Writing in mathematics takes different forms, such as journals, logs, daily diaries, and explanations about problems and processes. By asking students to explain their thinking, write their own problem, or compare and contrast concepts, teachers can pinpoint difficulties students are having with content. They can then adjust instruction to address those misconceptions early, rather than waiting until an assessment to determine what students do not know.
IDEAS FOR IMPLEMENTING COMMUNICATION AND DISCOURSE INTO MATHEMATICS
- Personal Dictionaries—Students write a vocabulary term, give its definition, include any symbolic representation for the term, draw a diagram, provide an example, an provide non-examples.
- Stories
- Manipulatives and Charts
- Personal Metaphors
- Number Talks
- Student-to-Student Talk
- Student-to-Teacher Talk
- Creative writing, poems, or songs that involve mathematics
- Journals, logs, daily diaries, and explanations about problems and processes
Ideas for Starters/Prompts:
· “Write about what you learned.”
· “Write about what you are still confused about.”
· “Write what you are still wondering about.”
· “Show how to solve the following problem in two ways.”
· “Write what you know about___.
Examples of prompts that promote communication and discourse:
CONCLUSION
There is no better way to know what students think about a concept or what connections they are making than by asking questions and engaging them in dialogue. Students and teachers benefit equally from interactions that take place during discourse, which creates opportunities for students to convince their peers, for teachers to help children gain better mathematical understanding, and for students to internalize concepts that they have learned. When discourse is a common event in the classroom, students develop a sense of shared responsibility in the learning.
Teachers must also build in plenty of time for reflection, both before and after discussions. Reflection includes thinking aloud and silently and recording thinking both numerically and through writing. Writing in math is a skill. It entails a special language, representations, explanations, and proofs. It needs to be explicitly taught for students to be successful with it. The process of communication and reflection is essential (Bahr, Deepening understanding through communication and numeration).
REFERENCES
Anderson, M. & Little, D. (2004). On the write path: improving communication in an elementary mathematics classroom. Teaching Children Mathematics, May 2004, Vol. 10 Issue 9, p. 468-472.
Cawley, J. F. & Reines, R. (1996). Mathematics as communication. Teaching Exceptional Children, Winter 1996, Vol. 28 Issue 2, p. 29.
Weibe Berry, R. & Namsook, K. (2008). Exploring teacher talk during mathematics instruction in an inclusion classroom. Journal of Educational Research, July/August 2008, Vol. 101 Issue 6, p. 363-378.
Whitin, P. & Whitin, D. (2002). Promoting communication in the mathematics classroom. Teaching Children Mathematics, December 2002, Vol. 9 Issue 4, p. 205.
WEBSITES
www.criced.tsukuba.ac.jp/math/apec/apec2008/papers/PDF/21.Tran_Vui_Vietnam.pdf, Enhancing classroom communication to develop students’ mathematical thinking, Article by TranVui.
http://www.math.umd.edu/~dac/650/huangpaper.html, The importance of communications in the mathematics classrooms.