NCTM (National Council of Teachers of Mathematics) write the following concerning the activity of mathematics:
Guide to Mathematical Norms: From NCTM Standards for grades 6-8
Problem Solving:
- Share varied solutions
Reasoning and Proof:
- Examine patterns and structures and note regularities (possible generalizations)
- Make conjectures about possible generalizations
- Evaluate conjectures
- Use inductive and deductive reasoning to construct (formulate) and evaluate mathematical arguments
Communication:
- Express thinking orally and in writing
- Present/explain strategies used to solve problems
- Analyze, compare, contrast the meaningfulness, efficiency, and elegance of a variety of strategies (explanations should include mathematical arguments and rationales)
- Free to express one’s ideas honestly and openly, without fear of ridicule
Representation:
- Create, compare, and use various representations to organize and record thinking about mathematical ideas
- Representations consist of physical objects, drawings, charts, graphs, and symbols – these representations help communicate one’s thinking
Big Ideas in Geometry: From NCTM Standards for grades 6-8
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (e.g., investigating perimeters, areas, and side lengths)
- Specify locations and describe spatial relationships using coordinate geometry and other representational systems
- Apply transformations and use symmetry to analyze mathematical situations (“transformational geometry offers another lens through which to investigate and interpret geometric objects”
- Use visualization, spatial reasoning, and geometric modeling to solve problems
Space & Perspective 