programs from prekindergarten through grade 12 should enable all students
In classrooms where students are challenged to think and reason about mathematics, communication is an essential feature as students express the results of their thinking orally and in writing. This type of environment is desirable at all grade levels, but there are a few distinctive features of such classrooms in the middle grades. For example, the mathematics under discussion in grades 68 is generally more complex and perhaps more abstract than the mathematics in the lower grades.
A second distinctive feature relates to the norms for evaluating the thinking of members of a classroom learning community. When students in grades 68 explain their thinking, they can be held to standards that are more stringent than would likely be applied to younger students, though not as demanding as might be applied in high school. Each student should be expected not only to present and explain the strategy he or she used to solve a problem but also to analyze, compare, and contrast the meaningfulness, efficiency, and elegance of a variety of strategies. Explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries (Yackel and Cobb 1996).
A third distinguishing feature
pertains to the social norms in a middle-grades classroom rather than
to the content of the students' discussions. During adolescence, students
are often reluctant to do anything that causes them to stand out from
the group, and many middle-grades students are self-conscious and hesitant
to expose their thinking to others. Peer pressure is powerful, and a desire
to fit in is paramount. Teachers should build a sense of community in
middle-grades classrooms so students feel free to express their ideas
honestly and openly, without fear of ridicule.
Consider an extended
example (adapted from Silver and Smith ) of mathematical communication
in a middle-grades classroom.
The students began by working collaboratively in pairs to solve
the following problem, adapted from Bennett, Maier, and Nelson (199899):
As the students worked on the problem, the teacher circulated around the room, monitoring the work of the pairs and responding to their questions. She also noted different approaches that were used by the students and made decisions about which students she would ask to present solutions.
After most students had a chance to solve the problem, the teacher asked Lee and Randy to present their method. They proceeded to the overhead projector to explain their work. After briefly restating the problem, Lee indicated that 3 times 4 is equal to 12 and that they needed "a number that both 3 and 4 would go into." The teacher asked why they had multiplied 3 by 4. Randy replied that the ratio of the length to the width was given as "4 to 3" in the problem. Lee went on to say that they had determined that "3 goes into 15 five times and that 4 goes into 20 five times." Since 15 times 20 is equal to 300, the area of the given rectangle, they concluded that 15 inches and 20 inches were the width and length of the rectangle.
The teacher asked if there were questions for Lee or Randy. Echoing the teacher's
query during the presentation of the solution, Tyronne said that
he did not understand their solution, particularly where the 12
had come from and how they knew it would help solve the problem.
Neither Lee nor Randy was able to explain why they had multiplied
3 by 4 or how the result was connected to their solution. The teacher
then indicated that she also wondered how they had obtained the
15 and the 20. The boys reiterated that they had been looking for
a number "that both 3 and 4 went into." In reply, Darryl asked how
the boys had obtained the number 5. Lee and Randy responded that
5 was what "3 and 4 go into." At this point, Keisha said "Did you
guys just guess and check?" Lee and Randy responded in unison, "Yeah!"
Although Lee and Randy's final answer was correct and although it
contained a kernel of good mathematical insight, their explanation
of their solution method left other students confused.
To address the confusion generated by Lee and Randy, the teacher decided to solicit another solution. Because the teacher had seen Rachel and Keisha use a different method, she asked them to explain their approach. Keisha made a sketch of a rectangle, labeling the length 4 and the width 3. She explained that the 4 and 3 were not really the length and width of the rectangle but that the numbers helped remind her about the ratio. Then Rachel explained that she could imagine 12 squares inside the rectangle because 3 times 4 is equal to 12, and she drew lines to subdivide the rectangle accordingly. Next she explained that the area of the rectangle must be equally distributed in the 12 "inside" squares. Therefore, they divided 300 by 12 to determine that each square contains 25 square inches. At the teacher's suggestion, Rachel wrote a 25 in each square in the diagram to make this point clear. Keisha then explained that in order to find the length and width of the rectangle, they had to determine the length of the side of each small square. She argued that since the area of each square was 25 square inches, the side of each square was 5 inches. Then, referring to the diagram in figure » 6.37, she explained that the length of the rectangle was 20 inches, since it consisted of the sides of four squares. Similarly, the width was found to be 15 inches. To clarify their understanding of the solution, a few students asked questions, which were answered well by Keisha and Rachel.
At this point the teacher might ask the girls if they think their approach would work for similar problems: What if the ratio were not 4 to 3? What if the area were not 300? Other students might be invited to ask questions: What would happen if the product of the length-width ratio numbers does not divide evenly into the area of the rectangle? Such questions could generate lively exchange that would include several students and could invite comparison to methods used by other students. The teacher could encourage students to consider generalizations and work to engage the entire class in this kind of thinking. For homework, students might be asked to come up with some possible generalizations. In the final few minutes of the class, the students could record in their journals their observations about what they had learned during the lesson along with any lingering questions they might have.
The previous example illustrates several important facets of the teacher's role in supporting communication, particularly a whole-class discussion, which was portrayed in the example. One is establishing norms within a classroom learning community that support the learning of all students. Another is selecting and using worthwhile mathematical tasks that allow significant communication to occur. And a third is guiding classroom discussion on the basis of what is learned by monitoring students' learning. »
The middle-grades mathematics teacher should strive to establish a communication-rich classroom in which students are encouraged to share their ideas and to seek clarification until they understand. In such a classroom community, communication is central to teaching and learning mathematics and to assessing students' knowledge. The focus in such classrooms is trying to make sense of mathematics together. Explaining, questioning, debating, and sense making are thus natural and expected activities. To achieve this kind of classroom, teachers need to establish an atmosphere of mutual trust and respect, which can be gained by supporting students as they assume substantial responsibility for their own mathematics learning and that of their peers. When teachers build such an environment, students understand that it is acceptable to struggle with ideas, to make mistakes, and to be unsure. This attitude encourages them to participate actively in trying to understand what they are asked to learn because they know that they will not be criticized personally, even if their mathematical thinking is critiqued.
Communication should be focused on worthwhile mathematical tasks. Teachers should identify and use tasks that
Although the task in the example was in many ways quite simple, it provided
students with an opportunity to use their understanding of area and ratioimportant
ideas in the middle grades. The task was simple enough that all students
could do it, difficult enough to challenge students to think and reason,
and rich enough to allow students to engage at different levels.
Teachers also need to monitor students' learning in order to direct classroom discourse appropriately. Facilitating students' mathematics learning through classroom discussion requires skill and good judgment. In the example, to be sure her objectives were met, the teacher skillfully steered the "mathematical direction" of the conversation by calling on particular students to present a different solution. Teachers must consider numerous issues in orchestrating a classroom conversation. Who speaks? When? Why? For how long? Who doesn't? Why not? It is important that everyone have opportunities to contribute, although it is not necessary to give equal speaking time to all students.
Clearly, the students in the example were accustomed to being asked regularly, not only by the teacher but also by other students, to explain their mathematical thinking and reasoning. The teacher and several students pressed for justification and explanation as each solution was presented. Because not all students regularly participate in whole-class discussions, teachers need to monitor their participation to ensure that some are not left entirely out of the discussion for long periods. But encouraging all students to speak can sometimes conflict with advancing the mathematical goals of a lesson because students' contributions may occasionally be irrelevant or lack mathematical substance. But even when this happens, the teacher and students can derive some benefit. It can be productive for the teacher to pick up on and probe incorrect or » incomplete responses. Only by examining misconceptions and errors can students deal with them appropriately.
Teachers can use class discussions as opportunities for ongoing assessment of their teaching and of students' learning. Making mental notes about missed teaching opportunities and about students' difficulties or confusions may help in making decisions about follow-up lessons.
Classroom communication can contribute to multiple goals. The lesson in the example generated at least four instructional directions. In the next few lessons, the teacher and her class explored the use of algebraic representations and solution methods for related problems as they sought a method that would work for a larger range of values for the areas and the ratios of the sides. The teacher later examined with the class how the area of a rectangle is affected when its length and width are each multiplied by the same factor. The teacher's informal assessment of the students' understanding during work on the task led her to take Lee and Randy aside later in the week to help them clarify their thinking. Finally, to acquaint her students with the scoring guidelines that would be used on the state proficiency test, she had them prepare written solutions for this problem and then score one another's solutions using the guidelines. In this way, they learned valuable lessons about the need for accuracy, precision, and completeness in their written communication.
Teachers can use oral and written communication in mathematics to give students opportunities to
To help students reflect on their learning, teachers can ask them to
write commentaries on what they learned in a lesson or a series of lessons
and on what remains unclear to them. To strive for clarity in explaining
their ideas, students can write a letter to a younger student explaining
a difficult concept (e.g., "Here's what it means for two figures to be
similar. Let me start with rectangles...."). In the example, journal writing
and individual homework offered all the students opportunities for individual
reflection and communication. Working in pairs also afforded opportunities
for communication. This approach is often very effective with students
in the middle grades because they can try out their ideas in the relative
privacy of a small group before opening themselves up to the entire class.
Even when students are working in small groups, the teacher has an important role to play in ensuring that the discourse contributes to the mathematics learning of the group members and helps to further the teacher's mathematical goals. For example, when students ask questions about the task requirements or about the correctness of their work, the teacher should respond in ways that keep their focus on thinking and reasoning rather than only on "getting the right answer." Teachers should resist students' attempts to have the teacher "do the thinking for » them." In the incident related in the example when the students were working in pairs, the teacher generally responded to questions with suggestions (e.g., "Try to think of some way to use a diagram") or her own questions (e.g., "What do you know about the relationship between the area of a rectangle and its length and width? How can you use what you know?"). Teachers also must be sure that all members are participating in the group and understanding its work.
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