Communication Standard for Grades 9–12

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.

What should communication look like in grades 9 through 12?

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 9–12 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.

Nancy: Watch this part of the floor (she outlines a 32 tile pattern like that shown and points to the points of intersection of the imagined string and the edges of the tiles). Here it crosses. Here it crosses. It hits the lines five times—and that comes from the 2 and the 3! (Nancy has counted the number of points of intersection of the string with the vertical and horizontal line segments and notices that the total is the same as the sum of 2 and 3.) » Ms. Koech: That's interesting! Have you seen that happen in other cases? Marta: Yes! It works for most small numbers. Ms. Koech: What do you mean—most? Marta: Well, it didn't work for this one. (Marta shows Ms. Koech a drawing of a 36 tiling pattern for which there are seven points of intersection. ) Nancy: Why is this one different? 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 tiles—each time it clicked, she touched another tile—they 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 attempts—first for specific cases such as the 32 and 34 configurations and then in general—the 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 mathematics publication. Fig. 7.34. A 96 tile configuration.

A formal 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.

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.