- Reflect on and clarify their own thinking about mathematical ideas and situations
- 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 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

*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.

*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.

*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").

*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.

*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.

*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.