### Standard 2: Mathematics as Communication

In grades 9-12, the mathematics curriculum should include the continued development of language and symbolism to communicate mathematical ideas so that students can:
• Reflect on and clarify their own thinking about mathematical ideas and situations
• Benchmarks 11B (Common Themes: Models)
Grades 3-5, page 268
Seeing how a model works after changes are made to it may suggest how the real thing would work if the same thing were done to it. 1

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 6-8, page 269
Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly, or that are too vast to be changed deliberately, or that are potentially dangerous.

• Formulate mathematical definitions and express generalizations discovered through investigations
• Benchmarks 9E (The Mathematical World: Reasoning)
Grades 6-8, page 233
Sometimes people invent a general rule to explain how something works by summarizing observations. But people tend to over generalize, imagining general rules on the basis of only a few observations.

• Express mathematical ideas orally and in writing
• Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
Grades 6-8, page 37
Mathematicians often represent things with abstract ideas, such as numbers or perfectly straight lines, and then work with those ideas alone. The "things" from which they abstract can be ideas themselves (for example, a proposition about "all equal-sided triangles" or all odd numbers").

• Read written presentations of mathematics with understanding
• Benchmarks 12D (Habits of Mind: Communication Skills)
Grades 9-12, page 297
Write clear, step-by-step instructions for conducting investigations, operating something, or following a procedure.

Benchmarks 12D (Habits of Mind: Communication Skills)
Grades 9-12, page 297
Use and correctly interpret relational terms such as if ... then ..., and, or sufficient, necessary, some, every, not, correlates with, and causes.

Benchmarks 12D (Habits of Mind: Communication Skills)
Grades 9-12, page 297
Use tables, charts, and graphs in making arguments and claims in oral and written presentations.

• Ask clarifying questions related to mathematics they have read or heard about
• Benchmarks 12D (Habits of Mind: Communication Skills)
Grades 9-12, page 297
Participate in group discussions on scientific topics by restating or summarizing accurately what others have said, asking for clarification or elaboration, and expressing alternative positions.

• Appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas
• Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 9-12, page 29
As in other sciences, simplicity is one of the highest values in mathematics. Some mathematicians try to identify the smallest set of rules from which many other propositions can be logically derived.

Benchmarks 2B (The Nature of Mathematics: Mathematics, Science, and Technology)
Grades 9-12, page 33
Mathematics provides a precise language for science and technology to describe objects and events, to characterize relationships between variables, and to argue logically.