In grades 9-12, the mathematics curriculum should include the continued development of language and symbolism to communicate mathematical ideas so that all students can-

• reflect upon and clarify their thinking about mathematical ideas and relationships;
• formulate mathematical definitions and express generalizations discovered through investigations;
• express mathematical ideas orally and in writing;
• read written presentations of mathematics with understanding;
• ask clarifying and extending questions related to mathematics they have read or heard about;
• appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas.

Focus

All students need extensive experience listening to, reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. Active student participation in learning through individual and small-group explorations provides multiple opportunities for discussion, questioning, listening, and summarizing. Using such techniques, teachers can direct instruction away from a focus on the recall of terminology and routine manipulation of symbols and procedures toward a deeper conceptual understanding of mathematics. It is not enough for students to write the answer to an exercise or even to "show all their steps." It is equally important that students be able to describe how they reached an answer or the difficulties they encountered while trying to solve a problem. Continually encouraging students to clarify, paraphrase, or elaborate is one means by which teachers can acknowledge the merit of students' ideas and the importance of their own language in explaining their thinking. Providing opportunities for discussions about issues, people, and the cultural implications of mathematics reinforces student understanding of the connection between mathematics and our society.

In grades 9-12, methods of mathematical communication become more formal and symbolic. Facility with mathematical language and notation enables students to more easily form multiple representations of ideas, express relationships within and among representation systems, and formulate generalizations. In fact, facility with the language of mathematics is an integral part of thinking mathematically, solving problems, and reflecting on one's own mathematical experiences.

Discussion

Although K-8 students will have many experiences with informal use of language and the construction of arguments, at the high school level these experiences are extended to the use of specialized symbolism associated with the various representation systems of mathematics. However, the introduction and use of technical symbolism should evolve as a natural extension and refinement of the students' own language. Students in grades 9-12 should build on their informal reasoning experiences in previous grades to write convincing arguments that validate their own generalizations. College-intending students should be able to extend those arguments to deductive proofs in which underlying inference schemes are made explicit.

The following example illustrates how the views of mathematics as problem solving, communication, and reasoning are inextricably connected.

Nine robots (fig. 2.1) are to perform various tasks at fixed positions along an assembly line. Each must obtain parts from a single supply bin to be located at some point along the line. Students are asked to investigate where the bin should be located so that the total distance traveled by all the robots is minimal.

Fig. 2.1. Assembly line with nine robots

The investigation should include an opportunity for class discussion of possible reasons for attempting to minimize this distance.

Students can address this problem by first experimenting with simpler cases, as in figure 2.2 They will determine that for n = 2, any point on line (or at a more sophisticated level, any P [ ]) will work. This conclusion can be expected to emerge only after extensive argument, since the "natural" point to consider is the midpoint. For n = 3, is the solution. For n = 4, any point on line will give the minimum distance. (Students may reason that for and, any point in [ ], will work. Similarly, for and , any point in [, ] will work. Thus, the solution is in any point in [ , ] [, ] = [, ].) For n = 5, similar reasoning yields as the optimal point, and so on. It follows that the solution of the original problem is to locate the bin at the position corresponding to .

Fig. 2.2. Simpler cases

All students should be encouraged to generalize their solutions to the case of n robots. The language and notation used by students will vary with their mathematical sophistication. Although all students should be expected to express their generalizations accurately (e.g., "at the middlemost robot's position or between the two middlemost robots"), college-intending students should additionally be able to use symbolic notation:

If n is even, then any P [ ] is optimal.
If n is odd, then is the optimal point.

Contextual situations and student experiences similar to these will serve to enhance students' appreciation of the value of mathematical activity and instill confidence in their ability to make sense of new problem situations.

In addition to the mathematical symbols related to concepts and operations developed in grades K-8, students in grades 9-12 need to use a variety of new symbols related to arrays, functions, and probability. This expanded symbol system extends and refines a student's ability to express quantitative ideas concisely. For example, the jeans-supply matrix in figure 2.3 provides an economical and well-ordered way of representing the size-by-brand information for a particular jeans department.

Fig. 2.3. Jeans-supply matrix

College-intending students would be expected to use more sophisticated notation associated with functions (including transformations), iterative algorithms, matrices, complex numbers, series, and limits in preparation for their continued study of mathematics.

In grades 9-12, student learning of mathematics becomes increasingly self-directed and dependent on textual materials. Appropriate symbolism and vocabulary should be used in all material presented to students, with the clear expectation of the appropriate use of such symbolism and notation by students. It cannot be assumed that even students who are skilled readers can read mathematical exposition effectively. All students will need specific instruction on how to read mathematical textbooks with understanding and how to use textbooks as valuable resources. Assignments that require students to read mathematics and respond both orally and in writing to questions based on their reading should be an integral part of the 9-12 mathematics program.

Techniques used to teach writing can be useful in teaching mathematical communication. The view of writing as a process emphasizes brainstorming, clarifying, and revising; this view can readily be applied to solving a mathematical problem. The simple exercise of writing an explanation of how a problem was solved not only helps clarify a student's thinking but also may provide other students fresh insights gained from viewing the problem from a new perspective.

Students could be encouraged to keep journals describing their mathematical experiences, including reflections on their problem-solving thought processes. Journal writing also can help students clarify feelings about mathematics or about a particular experience or activity in a mathematics classroom. These activities can foster students' positive attitudes about mathematics, particularly if the journal entries are accompanied by discussions about any negative feelings and ways to deal with unpleasant experiences.

Technology is yet another avenue for mathematical communication, both in transmitting and receiving information. Calculators and computers require students to use and understand accurate, concise language. To use a calculator, students must not only understand the underlying mathematics (e.g., the order of operations or the meaning of the fraction line) but also apply the specific syntax for the type of calculator being used. Using a computer language to implement a mathematical procedure requires translating the language of mathematics into the language of programming and then applying the syntax of the particular programming language. Interpreting the output of a computer program or a calculator display requires students to recognize equivalent forms of representation and to judge the reasonableness of results. Interpreting computer and calculator graphic displays additionally requires careful attention to the scales on the axes and an understanding of the effects of scaling on the characteristics of a graph.

Students whose primary language is not the language of instruction have unique needs. Specially designed activities and teaching strategies (developed and implemented with the assistance of language specialists) should be incorporated into the high school mathematics program so that all students have the opportunity to develop their mathematical potential regardless of a lack of proficiency in the language of instruction.