programs from prekindergarten through grade 12 should enable all students
Changes in the workplace increasingly demand teamwork, collaboration, and communication. Similarly, college-level mathematics courses are increasingly emphasizing the ability to convey ideas clearly, both orally and in writing. To be prepared for the future, high school students must be able to exchange mathematical ideas effectively with others.
However, there are more-immediate reasons
for emphasizing mathematical communication in high school mathematics.
Interacting with others offers opportunities for exchanging and reflecting
on ideas; hence, communication is a fundamental element of mathematics
learning. For that reason, it plays a central role in all the classroom
episodes in this chapter. Sharing ideas and building on the work of others
were essential ingredients in the ability of Ms. Rodriguez's class to
provide an analytic explanation of a surprising visual result (see the
discussion in the "Problem Solving" section). Communication was also important
in the episode about converting the unit of measurement for various statistics
in the "Reasoning and Proof" section where informal observations led to
discussions of specific cases that were ultimately abstracted and proved
as general results. Making connections among the various geometric examples
in the upcoming "Connections" section depends heavily on the exchange
of information. Students' written work is valuable for assessment, as
readers will see in the discussion in the "Representation" section, where
the students' incorrect distance-versus-time graph gives the teacher insights
into their misconceptions. In all these examples, the act of formulating
ideas to share information or arguments to convince others is an important
part of learning. When ideas are exchanged and subjected to thoughtful
critiques, they are often refined and improved (Borasi 1992; Moschkovich
1998). In the process, students sharpen their skills in critiquing and
following others' logic. As students develop clearer and more-coherent
communication (using verbal explanations and appropriate mathematical
notation and representations), they will become better mathematical thinkers.
In high school, there should be substantial growth in students' abilities to structure logical chains of thought, express themselves coherently » and clearly, listen to the ideas of others, and think about their audience when they write or speak. The relationships students wish to express symbolically and with graphs, as well as the notation and representations for expressing them, should become increasingly sophisticated. Consequently, communication in grades 912 can be distinguished from that in lower grades by higher standards for oral and written exposition and by greater mathematical sophistication.
High school students should be good critics and good self-critics. They should be able to generate explanations, formulate questions, and write arguments that teachers, coworkers, or mathematicians would consider logically correct and coherent. Whether they are making their points using spreadsheets, geometric diagrams, natural language, or algebraic symbols, they should use mathematical language and symbols correctly and appropriately. Students also should be good collaborators who work effectively with others.
Proofs should be a significant part of high school students' mathematical experience, as well as an accepted method of communication. The following episode, drawn from real experience, illustrates the part that mathematical communication plays in the development of increased understanding.
Marta and Nancy attended the biweekly mathematics competition in their school district's mathematics league and encountered the following problem:
They didn't solve
the problem, but they reported to their faculty sponsor, Ms. Koech,
that they had heard a fellow contestant say that he got the answer
by adding the two sides and subtracting their greatest common divisor.
Not content with a formula "dropped from the sky," Ms. Koech
encouraged the girls to see if this method always works and if so,
why. Over a period of days the girls developed some intuitions about
the problem and tried various ways of sharing them with their teacher.
Marta, who had a broken leg, demonstrated one emerging insight by
dragging her cast along the tiles on the classroom floor.
Ms. Koech suggested that the students pursue Nancy's question. Over a period of days they determined that the relationship breaks down when the string passes through a corner of a tile "inside" the pattern. Gradually and after pursuing some approaches that led nowhere, they realized that whenever the number of intersections was the same as the sum of the dimensions of the rectangle, the dimensions of the room are relatively prime. Thinking back to Marta's cast clicking over the tileseach time it clicked, she touched another tilethey realized that they could trace along the string and see how many times the string "exits" a tile. Except at the start of the path, the string exits a tile each time it crosses either a vertical or a horizontal line segment. Moreover, the string has to cross every vertical and horizontal line segment as it traverses the tiled room from one corner to the other. So the number of tiles the string crosses can be determined by the number of horizontal and vertical line segments in the tile configuration. After a number of attemptsfirst for specific cases such as the 32 and 34 configurations and then in generalthe students were able to show that if n and m (the dimensions of the rectangle) are relatively prime, the string passes over (m + n 1) tiles.
With this result established,
they returned to the general problem. A close look at the 96 tile configuration
(see fig. 7.34) shows that the area the string passes through can
be considered as three 32 configurations.
Thus the string passes over three times as many tiles in the 96 configuration as
it passes over in the 32 configuration.
This happens to be 3(2 + 3 1) = 6 + 9 gcd(6,
9) tiles, which conforms with the method they had heard from their
fellow contestant. From this analysis, the generalization to m
+ n gcd(m, n) was a natural
next step. The students wrote up their results for a regional student
explanation follows: Suppose n and m have a gcd of
g. The smaller, relatively prime grid is (n/g)(m/g).
The students discovered that the number of grid lines crossed there is
(n/g) + (m/g) 1, which indicates
that when a string passes through g of them, it passes through
g[(n/g) + (m/g) 1] = n
+ m g grid lines.
In this example, communication serves at least two purposes. First, it
is motivational: Marta and Nancy kept working at the problem in part because
it was a collaborative effort, and they were discussing their work. Second,
repeated attempts to explain their reasoning to each other and to their
teacher helped Marta and Nancy clarify their thinking and focus on essential
elements of the problem. Notice that » the
teacher offered some key observations and questions: "I'm not sure what
you mean." "Have you seen this happen in other cases?" "What do you mean,
most?" Nancy then asked the question that launched the girls' exploration:
"Why is this one different?" This focus was a major factor in their success
in eventually solving the problem.
High school teachers can help students use oral communication to learn
and to share mathematics by creating a climate in which all students feel
safe in venturing comments, conjectures, and explanations. Teachers must
help students clarify their statements, focus carefully on problem conditions
and mathematical explanations, and refine their ideas. Orchestrating classroom
conversations so that the appropriate level of discourse and mathematical
argumentation is maintained requires that teachers know mathematics well
and have a clear sense of their mathematical goals for students. Teachers
should help students become more precise in written mathematics and have
them learn to read increasingly technical text. In both written and oral
communication, teachers need to attend to their students and carefully
interpret what the students know from what they say or write.
Communication can be used in many ways as a vehicle for assessment and learning. Early in a mathematics course or when a new topic is being introduced, teachers can request information about the students' knowledge of the topic. At the beginning of a tenth-grade unit on circles, for example, a teacher might ask students to tell her everything they know about circles. The teacher could then compile the responses, distribute copies of them the next day, and ask the students to agree or disagree with each of the statements. Students should justify the stance they took. The teacher could look at students' incorrect observations and design a lesson to address those misconceptions. In this way, the students' knowledge becomes a starting point for instruction, and the teacher can establish the idea that the students are expected to have reasons for their mathematical opinions. Activities such as this can serve to establish a » classroom climate conducive to the respectful exchange of ideas. More generally, having students present work to the class at the chalkboard, overhead projector, or flip chartand having the class respond to what is presented rather than having only the teacher judge the correctness of what has been saidcan be a valuable way to foster classroom communication. Over time, such activities help students sharpen their ideas through attempts to communicate orally and in writing.
There are various ways that teachers can help students communicate effectively using written mathematics. Problems that require explanations can be assigned regularly, and the class can discuss and compare the adequacy of those explanations. The following exercises can help students sharpen their mathematical writing skills:
Writing is a valuable way of reflecting on and solidifying what one knows, and several kinds of exercises can serve this purpose. For example, teachers can ask students to write down what they have learned about a particular topic or to put together a study guide for a student who was absent and needs to know what is important about the topic. Students who have done a major project or worked on a substantial long-range problem can be asked to compare some of their early work with later work and explain how the later work reflects greater understanding. In these ways, teachers can help students develop skills in mathematical communication that will serve them well both inside and outside the classroom. Using these skills will in turn help students develop deeper understandings of the mathematical ideas about which they speak, hear, read, and write.
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