Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## Communication

The goals of the NCTM's communication process standard are that, 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."
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 26)

Video Overview
This video profiles classroom excerpts that focus on mathematics as communication. The excerpts show students representing, discussing, reading, writing, and listening as vital parts of learning and using mathematics. Communication fosters an understanding of mathematical ideas and the language of mathematics. For example, students are observed in the following contexts:

• reaching consensus on estimates of the number of seeds in a pumpkin
• making a group estimate of the number of cranberries in a jar
• working in teams to collect data for a bubble gum contest
• working in small groups to sort cubes
• collaborating to determine the amount of stain needed for their pencil boxes
• identifying relationships between products and sums and writing about the activity
• deciding how to explain their group's answers in writing
• using a variety of materials to solve a problem about vehicles
and wheels
• working in pairs to develop and illustrate story problems
• representing domino patterns with dot stickers and number sentences
• acting out a story about marching ants
• studying ladybugs in a bilingual classroom
• naming geometric shapes in a bilingual classroom
• recording measurements they found when working in teams
• explaining the reasoning behind their computations
• creating patterns in small groups
• presenting their group's results to their whole class
• reading the equations they developed to their whole class

An Exploration
For Teacher Workshops

Describe It
This investigation focuses on communication in mathematics. Each pair will need these items:

• a partition, such as a file folder
• two identical sets of twenty pattern blocks
• an empty box for each pair of teachers

1. Have teachers work in pairs. The partition should be set up so that the partners cannot see each other's work. Give each person a set of twenty pattern blocks. Explain that one person will build a structure with the pattern blocks and then give verbal directions to her or his partner on how to build an identical structure. Then have one teacher build a structure using fifteen to twenty of the pattern blocks, while the partner waits without looking at the structure being built.

2. The teacher who built the structure gives the partner verbal directions on how to construct an identical structure without viewing the partner's attempts to build it. The person building may ask for directions to be repeated or rephrased.

3. On completion of the structure, the partition is lifted and the two structures are compared. Then the partners switch roles and repeat the activity.

4. As a whole group, discuss the success and the difficulty the teachers had when giving and receiving verbal directions. Some questions to consider include the following:
• How successful were teachers?
• How easy or difficult was it for teachers to communicate with their partners?
• What mathematical terms did teachers use to describe the structures?
• What particular aspects of the structures were difficult to describe?
• How would the task have been affected if fewer blocks had been used or if all the blocks had been the same size or the same color?

5. As an extension of this activity, have pairs of teachers work together to build a structure and write the directions on paper using words only, no drawings. The original structures are then covered by placing a box over them and the written directions are set on top of the box. Pairs of teachers rotate from station to station, trying to build the various structures by reading the written directions. When a pair thinks they have successfully built a structure, they lift the box to compare their attempt with the original structure.

Topics for Discussion
The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.

Developing Mathematics As Communication

1. In what ways did communication provide a context for students to reflect on and clarify their thinking about mathematical ideas and situations?
2. Several classroom episodes showed students writing. How did the writing facilitate their understanding of mathematical ideas and promote mathematical reasoning?
3. Communication plays an important role in helping students establish connections among physical materials, pictures, diagrams, and symbols. Cite examples of students making these types of connections. How do connections among these various representations facilitate the students' understanding?
4. What opportunities allowed students to relate their everyday language to mathematical language and symbols? Describe ways to provide these types of opportunities.

Planning for Communication

1. Brainstorm a list of ways that students can communicate mathematically. Which of these were used in the various classroom episodes?
2. Students were given many opportunities to communicate mathematically in the segments in the video. Cite examples of the conditions that allowed for and encouraged this communication.
3. Students are often not accustomed to communicating their thinking and reasoning verbally or with words and pictures in mathematics classes. How can teachers help students who are having trouble?
4. What are ways to assess the students' ability to communicate mathematically and to document their progress?
5. What is the teacher's role in a classroom that emphasizes mathematics as communication? Comment on the interactions of the teachers and their students in the various lesson episodes.
6. Several teachers had their students present the results of their work to the whole class at the end of the lesson. What is the value and importance of these opportunities to communicate?

Extensions
Assessing Students' Written Communication

Give the teachers about ten student work samples to evaluate. They should place the work samples in two piles: (1) students' written work shows acceptable or good understanding, or (2) students' written work is unclear or shows misunderstandings. How did the teachers decide where to place each work sample? From this discussion, generate a rubric that can be used to assess the students' written communication.

Examining Mathematics as Communication
Each classroom excerpt profiled in this video is from a featured lesson in TEACHING MATH: A Video Library, K-4. You may want to watch the full version of these lessons to further examine and explore the role of communication in learning mathematics. The following list provides information about each full video and the page number of the accompanying print unit. Videos are listed in the order in which their excerpts appear in the Communication video.