Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
|Introduction | Developing Mathematical Ideas Through Communication | Analyzing and Evaluating the Thinking of Others | Additional Strategies | Your Journal|
We have reviewed examples from geometry that show how communication can make it possible for students to deepen their understanding of mathematical processes. Let's take a further look at the teacher's role in fostering learning through communication. The following are some communication strategies that teachers can utilize.
Just as the selection of problems and tasks affects students' growth in terms of their problem-solving ability, well-chosen mathematical problems and tasks can spark a variety of types of communication. Effective tasks have enough complexity to be seen as engaging, even enticing, challenges, while offering entry points for students with varied levels of problem-solving and linguistic expertise. Such tasks cause students to reflect on what they already know, what they need to figure out, and what strategies they want to try. Good problems have an important mathematical purpose; an accessible, interesting real-world or mathematical context; and several likely opportunities for communication.
"Well-posed questions can simultaneously elicit, extend, and challenge students' thinking and at the same time give the teacher an opportunity to assess the students' understanding" (NCTM, 2000, p. 197).
The teacher's choice and timing of questions is an art form that guides the direction of student thinking and prompts students to communicate in a variety of ways. It takes many teachers years of practice and reflection to develop the ability to decide "when to tell" and "when to ask" and then to decide "what to ask." Using an appropriate mix of question types helps students clarify and extend their thinking, and take ownership of their own learning.
Some questions, such as "What do you understand so far and what are you wondering about?", help students as they strive to solve a problem or make sense of a new concept, establish what is already known or understood, or identify problematic areas. Other questions, such as "What have you observed about the number of edges and vertices in the figures?", are direct, often factual, and specific and may be used to focus attention on a particular idea. Open-ended questions, such as "How do you plan to explore and to show your findings about this problem?", are meant to entice students to decide on their own how to investigate a concept or problem, and sometimes lead to additional questions from the teacher and students.
Good questions engage students and elicit communication, leading to student learning and useful assessment information for the teacher. Gradually, students should be expected to pose questions for themselves and their peers.
Encouraging Student Interactions
The teacher can facilitate communication between students by modeling and teaching the use of statement starters, such as "Can you please explain [ . . . ]? and "I'd like to add to [ . . . ]'s idea . . .". NCTM recommends that teachers periodically, "explicitly discuss students' effective and ineffective communication strategies" (NCTM, 2000, p. 198). Equally important is building a classroom climate of inquiry, where a variety of ideas and solution methods are valued and even sought out, rather than one where interactions are rushed and focused on finding an answer and quickly moving on. Otherwise, it is likely that very little worthwhile communication will occur.
The teacher can monitor students' understanding and encourage and model the rephrasing and summarizing of a classmate's explanation. English-language learners benefit from opportunities to practice the use of English through structured questions related to rich mathematical group experiences, as well as from opportunities to express their own ideas and ask questions in their native language. The teacher should also be aware of cultural patterns of communication and ways of asking and answering questions that may affect different students' involvement.
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