Communication

It is important that students are given the opportunity to communicate both orally and in writing. Writing can be words, pictures, graphs, tables, etc., any means the student uses that explains their thinking. As students communicate about problems, they are better able to sort out concepts, discover relationships and deepen their understanding of mathematics. They are also better able to attach meaning to symbols. They become aware of the connection between the symbols (abstract) and concepts.

Students begin to assume responsibility for their own learning when given the chance to communicate. They validate their thinking and see where modifications are needed.

We, the classroom teachers, need to help students become better communicators, especially in mathematics. Students need many opportunities to share ideas and justify solutions. Asking reasoning questions and posing problems that can actively engage students in discourse are imperative.

Key Concepts Related to the NCTM Standards

Mathematics as Communication

In grades K-4, the study of mathematics should include numerous opportunities for communication so that students can:

• relate physical materials, pictures, and diagrams to mathematical ideas;
• reflect on and clarify their thinking about mathematical ideas and situations;
• relate their everyday language to mathematical language and symbols;
• realize that representing, discussing, reading, writing, and listening to mathematics are a vital part of learning and using mathematics.

In grade 5-8, the mathematics curriculum should include opportunities to communicate so that students can:

• model situations using oral, written, concrete, pictorial, graphical, and algebraic methods;
• reflect on and clarify their own thinking about mathematical ideas and situations;
• develop common understandings of mathematical ideas, including the role of definitions;
• use the skills of reading, listening , and viewing to interpret and evaluate mathematical ideas;
• discuss mathematical ideas and make conjectures and convincing arguments;
• appreciate the value of mathematical notation and its role in the development of mathematical ideas.

The ability to read, write, listen, think creatively, and communicate about problems will develop and deepen students' understandings of mathematics.

This standard highlights the need to involve children in actively doing mathematics. Exploring, investigating, describing, and explaining mathematical ideas promote communication.

(NCTM. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: The Council,1989, p. 26)

Questioning Techniques

What questions we ask students and what questions we stimulate students to ask are important for a deeper understanding of mathematical concepts. We want students to explore, think, discover, and reason, and it is up to the teacher to stimulate and facilitate this.

The following is an excerpt from Professional Standards For Teaching Mathematics, National Council of Teachers of Mathematics, March 1991, pp. 3-4. It is a good guide to help teachers develop their questioning techniques.

"One would expect to see teachers asking, and stimulating students to ask, questions like the following:

• Helping students work together to make sense of mathematics

- What do others think about what Janine said?

- Do you agree? Disagree?

- Does anyone have the same answer but a different way to explain it?

- Would you ask the rest of the class that question?

- Do you understand what they are saying?

- Can you convince the rest of us that that makes sense?

• Helping students to rely more on themselves to determine whether something is mathematically correct

- Why do you think that?

- Why is that true?

- How did you reach that conclusion?

- Does that make sense?

- Can you make a model to show that?

• Helping students learn to reason mathematically

- Does that always work?

- Is that true for all cases?

- Can you think of a counter example?

- How could you prove that?

- What assumptions are you making?

• Helping students learn to conjecture, invent, and solve problems

- What would happen if...? What if not?

- Do you see a pattern?

- What are some possibilities here?

- Can you predict the next one? What about the last one?

- How did you think about the problem?

- What decision do you think he should make?

- What is alike and what is different about your method of solution and hers?

• Helping students to connect mathematics, its ideas, and its applications

- How does this relate to...?

- What ideas that we have learned before were useful in solving this problem?

- Have we ever solved a problem like this one before?

- What uses of mathematics did you find in the newspaper last night?

- Can you give me an example of...?"

Article

Nesbitt Vacc, Nancy. "Questioning in the Mathematics Classroom." Aritmetic Teacher (October, 1993): 88-91.