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.

Focus

Mathematics can be thought of as a language that must be meaningful if students are to communicate mathematically and apply mathematics productively. Communication plays an important role in helping children construct links between their informal, intuitive notions and the abstract language and symbolism of mathematics; it also plays a key role in helping children make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas. When children see that one representation, such as an equation, can describe many situations, they begin to understand the power of mathematics; when they realize that some ways of representing a problem are more helpful than others, they begin to understand the flexibility and usefulness of mathematics.

Young children learn language through verbal communication; it is important, therefore, to provide opportunities for them to "talk mathematics." Interacting with classmates helps children construct knowledge, learn other ways to think about ideas, and clarify their own thinking. Writing about mathematics, such as describing how a problem was solved, also helps students clarify their thinking and develop deeper understanding. Reading children's literature about mathematics, and eventually text material, also is an important aspect of communication that needs more emphasis in the K-4 curriculum.

This standard highlights the need to involve children in actively doing mathematics. Exploring, investigating, describing, and explaining mathematical ideas promote communication. Teachers facilitate this process when they pose probing questions and invite children to explain their thinking. Teachers also can assess students' knowledge and insight by listening and observing. The idea that children should learn mathematics meaningfully is implicit in this discussion, for meaningful learning is necessary if mathematics is to make sense and if communication is to be possible.

Discussion

Young children are active, social individuals. Much of the sense they make of the world is derived from their communications with other people. Communicating helps children to clarify their thinking and sharpen their understandings.

Representing, talking, listening, writing, and reading are key communication skills and should be viewed as integral parts of the mathematics curriculum. Probing questions that encourage children to think and explain their thinking orally or in writing help them to understand more clearly the ideas they are expressing.

Representing is an important way of communicating mathematical ideas at all levels, but especially so in K-4. Representing involves translating a problem or an idea into a new form. Translations of this type often are used by adults and children as they converse with others. Children might draw diagrams, for example, to express an idea or viewpoint in an alternative format that might be more comprehensible to the listener. The act of representing encourages children to focus on the essential characteristics of a situation. Representing includes the translation of a diagram or physical model into symbols or words. A child should be able to examine a set of two bundles of ten and four units each and match the set with the symbol 24. (See fig. 2.1.) Since representing is central to learning and using mathematics, it is important to provide many such experiences for children. Any of the following materials would be useful: base-ten blocks, straws that can be bundled in sets of ten, connecting cubes, or loose counters that must simply be grouped together to show the tens.

Fig. 2.1

Representing is also used in translating or analyzing a verbal problem to make its meaning clear. The problem in figure 2.2 involves a part-whole comparison, but many children who have not had enough experience in modeling such situations or who do not actively model this situation fail to recognize its structure.

Fig. 2.2

Some students simply use the numbers 1 and 3 to arrive at an answer of 1/3. A concrete representation or diagram will help children to correctly identify that four balls can possibly be drawn.

Communicating by talking and listening is also very important. When small groups of children discuss and solve problems, they are able to connect the language they know with mathematical terms that might be unfamiliar to them. They make sense of those problems. The use of concrete materials is particularly appropriate because they give the children an initial basis for conversation. Such occasions also permit the teacher to observe individual students, to ask probing questions, and to note or attend to any conceptual difficulties individual students might be experiencing. The following discussion activity would help children see how several problems that appear to be different in fact share the same underlying structure: 14 - 5 = [_]. The children would be given counters to model each problem. See fig. 2.2a.

Fig. 2.2a

With your group, use counters of different colors to model each of these problems and then discuss how the problems are alike or different.

Maria had some pencils in her desk. She put 5 more in her desk. Then she had 14. How many pencils did she have in her desk to start with?

Eddie had 14 helium balloons. Several of them floated away. He had 5 left. How many did he lose?

Nina had 14 seashells. That was 5 more than Pedro had. How many seashells did Pedro have?

As children talk about mathematics, it is important to keep in mind that what appears at first to be an incorrect response may be, in fact, an inability to communicate. Of primary importance is the value children derive from reflecting on their responses.

Writing is a communication skill that has been used too infrequently in mathematics. It is particularly useful because it allows a child who is uncomfortable in oral situations to express understanding in a less public forum. After children have solved a problem, they can write their answer in sentence form, which helps them exhibit a knowledge of the problem's place in the real world and clarify their thinking.

Students can write a letter to tell a friend about something they have learned in mathematics class. This type of activity allows the students to consider mathematics for a new purpose. If letters are exchanged, then students learn from the thought processes of their peers. See figure 2.3.

Fig. 2.3

Having students keep journals in mathematics class is another way to facilitate communication and give them an opportunity to reflect on their learning. A journal can be a form of free expression about the mathematics studied, or children can be asked to respond to directions such as these: Tell me what you thought were the hardest and easiest parts of today's lesson and why.

Children can also create their own stories or books about mathematics. Many schools have a "young authors" program that encourages children to develop an idea into a book to be shared with parents or classmates. This activity is within the reach of fourth graders and can include mathematics topics as options for development.

Many children's books present interesting problems and illustrate how other children solve them. Through these books students see mathematics in a different context while they use reading as a form of communication. Some of the books most directly linked to mathematics give children insights into the history of mathematics and the development of mathematical ideas. Materials children write themselves can be part of a reading activity and shared with class members. Mathematics texts have not often been viewed as sources of reading material, but taking this perspective can add a valuable dimension to students' learning. Many schools are making efforts to include expository reading as an important part of reading instruction. Mathematics texts and other mathematics reading materials should certainly be included in these efforts.

Children learn from one another as they communicate. Encouraging them to represent, talk and listen, write, and read facilitates meaningful learning. Attending to students' communications about their thinking also gives teachers a rich information base from which they can make sound instructional decisions.