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Instructional
programs from prekindergarten through grade 12 should enable all students
to—
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The ability to read, write, listen, think, and communicate about problems
will develop and deepen students' understanding of mathematics. In grades
3–5, students should use communication as a tool for understanding
and generating solution strategies. Their writing should be more coherent
than in earlier grades, and their increasing mathematical vocabulary can
be used along with everyday language to explain concepts. Depending on
the purpose for writing, such as taking notes or writing to explain an
answer, students' descriptions of problem-solving strategies and reasoning
should become more detailed and coherent.
In grades 3–5, students should become more adept at learning from,
and working with, others. Their communication can consist not only of
conversations between student and teacher or one student and another student
but also of students listening to a number of peers and joining group
discussions in order to clarify, question, and extend conjectures. In
classroom discussions, students should become the audience for one another's
comments. This involves speaking to one another in order to convince or
question peers. The discourse should not be a goal in itself but rather
should be focused on making sense of mathematical ideas and using them
effectively in modeling and solving problems. The value of mathematical
discussions is determined by whether students are learning as they participate
in them (Lampert and Cobb forthcoming).
In a grades 3–5 classroom, communication should include sharing
thinking, asking questions, and explaining and justifying ideas. It should
be well integrated in the classroom environment. Students should be encouraged
to express and write about their mathematical conjectures, questions,
and solutions. For example, after preparatory work in decimals, a fifth-grade
teacher engaged her students in the following problem in order to help
them think about and develop methods for adding decimals (episode adapted
from Schifter, Bastable, and Russell [1999, pp. 114–20]).
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In this activity, the teacher presented the students with a problem-solving situation. Although they had worked with representing decimals, they had not discussed adding them. As was customary in the class, the students were expected to talk with their peers to solve the problem and to share their results and thinking with the class. The students used communication as a natural and essential part of the problem-solving process. As the groups worked, the teacher circulated among the students:
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The teacher moved to another group where the students had represented their problem as shown in figure 5.32.
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Listening carefully to the discussions, the teacher rephrased Ned's suggestion in order to make sure she had accurately captured his thinking, to help him focus on the important mathematical concepts, and to guide him in considering how this problem is related to those more » familiar to him. Ned's response gave her important assessment information about whether he understood his method for adding whole numbers. Although he was able to use an algorithm to add whole numbers, he lacked an understanding of the concepts behind the procedure and therefore was unsure if or how it could be used or adapted for this new purpose.
In talking with Jaron's group, the teacher asked a question that led students to think about the reasonableness of their response by considering it in relation to its real-world context. The realization that their response didn't make sense caused the students to revisit the problem. In this particular instance, the teacher chose to let students work through their confusion. The teacher's decisions about what to say or not say, what to ask or not ask, were based on her observations of the students and their conversations. For example, What strategies were they using? Were misconceptions being challenged? Her goal was to nudge the students to reflect on their answer and to do further mathematical reasoning.
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After the groups finished their work, the class as a whole had a discussion. Rob reported that the students in his group represented the problem as shown in figure 5.33 (p. 116).
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| Malik paid close attention during Ben's explanation, nodding that he understood. Teresa was also interested in the explanation, noting the significance of how zero was represented. After this presentation, the » students wrote in their journals, explaining what they thought was a correct procedure for adding the numbers. Many mentioned that the demonstration had made it clear that tenths had to be added to tenths and hundredths to hundredths for the right answer. Some made up new problems and made drawings of the base-ten model. |
Because discussion of thinking was a regular occurrence in this classroom, students were comfortable describing their thinking, even if their ideas were different from the ideas of their peers. Besides focusing on their own thinking, students also attempted to understand the thinking of others and in some cases to relate it to their own. Ned, who earlier had been unable to articulate why he lined up whole numbers in a particular way when he added, questioned Rob about why his group had lined up the numbers the way they did. Ned was taking responsibility for his learning by asking questions about a concept that wasn't quite clear to him. Ben thought about Malik's dilemma and came up with a solution that became clear to Malik.
The use of models and pictures provides a further opportunity for understanding and conversation. Having a concrete referent helps students develop understandings that are clearer and more easily shared (Hiebert et al. 1997). The talk that preceded, accompanied, and followed Ben's presentation gave meaning to the base-ten model. Malik had been "stuck" by viewing the model in one way until Ben showed him another way to look at it.
Throughout the lesson, the interactions among students were necessary in helping
them make sense of what they were doing. Because there was time to talk,
write, model, and draw pictures, as well as occasions for work in small
groups, large groups, and as individuals, students who worked best in
different ways all had opportunities to learn.
With appropriate support and a classroom environment where communication about mathematics is expected, teachers can work to build the capacity of students to think, reason, solve complex problems, and communicate mathematically. This involves creating classroom environments in which intellectual risks and sense making are expected. Teachers must also routinely provide students with rich problems centered on the important mathematical ideas in the curriculum so that students are working with situations worthy of their conversation and thought. In daily lessons, teachers must make on-the-spot decisions about which points of the mathematical conversation to pick up on and which to let go, and when to let students struggle with an issue and when to give direction. For example, the teacher in the episode above chose to let one group of students struggle with the fact that their answer was unreasonable. Teachers must refine their listening, questioning, and paraphrasing techniques, both to direct the flow of mathematical learning and to provide models for student dialogue. 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. »
Periodically, teachers may need to explicitly discuss students' effective and ineffective communication strategies. Teachers can model questioning and explaining, for example, and then point out and explain those techniques to their students. They can also highlight examples of good communication among students. ("I noticed that Karen and Malia disagreed on an answer. They not only explained their reasoning to each other very carefully, but they listened to one another. Each understood the other's reasoning. It was hard, but eventually, they realized that one way made more sense than the other.")
Teachers need to help students learn to ask questions when they disagree or do not understand a classmate's reasoning. It is important that students understand that the focus is not on who is right or wrong but rather on whether an answer makes sense and can be justified. Students need to learn that mathematical arguments are logical and connected to mathematical relationships. When making a concept or strategy clear to a peer, the student-explainer is forced to reexamine and thus deepen his or her mathematical understanding. In settings where communication strategies are taught, modeled, and expected, students will eventually begin to adopt listening, paraphrasing, and questioning techniques in their own mathematical conversations.
Teachers must help students acquire mathematical language to describe objects
and relationships. For example, as students use informal language such
as "the corner-to-corner lines" to describe the diagonals of a rectangle,
the teacher should point out the mathematical term given to these lines.
Specialized vocabulary is much more meaningful if it is introduced in
an appropriate context. Teachers in grades 3–5 should look for, and
take advantage of, such opportunities to introduce mathematical terms.
In this way, words such as equation, variable, perpendicular, product,
and factor should become part of students' normal vocabulary.
Teachers also need to provide students with assistance in writing about mathematical concepts. They should expect students' writing to be correct, complete, coherent, and clear. Especially in the beginning, » teachers need to send writing back for revision. Students will also need opportunities to check the clarity of their work with peers. Initially, when they have difficulty knowing what to write about in mathematics class, the teacher might ask them to use words, drawings, and symbols to explain a particular mathematical idea. For example, students could write about how they know that 1/2 is greater than 2/5 and show at least three different ways to justify this conclusion. To help students write about their reasoning processes, the teacher can pose a problem-solving activity and later ask, "What have you done so far to solve this problem, what decisions did you make, and why did you make those decisions?" As students respond, the teacher can explain, "This is exactly what I'd like you to tell me in your writing."
Having students compare and analyze different pieces of their work is another way to convey expectations and help them understand what complete and incomplete responses look like. For example, students were asked to use pictures and words to explain their thinking for the following question (Kouba, Zawojewski, and Strutchens 1997, p. 119):
José ate 1/2 of a pizza.
Ella ate 1/2 of another pizza.
José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that José could be right.
Students' responses reflected different levels of understanding (see the examples in fig. 5.35). The first student assumed that each pizza was the same size. Although the student used words and drawings in the response, the answer was correct only if the units were the same, an assumption that cannot be made from the statement of the problem. The second student suggested by the drawing that the size of 1/2 depends on the size of the unit. The teacher might ask this student to explain his or her thinking. The third solution, including written words and drawings, was correct and complete in that it communicated why José could be correct. Discussion of various student responses, especially as mathematical concepts and problems become more complex, is an effective way to help students continue to improve their ability to communicate.
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