Reasoning and Proof
Students should be able to reason about situations in the real world and throughout mathematics. Spatial reasoning gives insight into geometric results, especially in two- and three-dimensional geometry. Probabilistic reasoning is helpful in analyzing the likelihood that an event will occur. Statistical reasoning allows students to assess risks and make generalizations about a population using representative samples drawn from that population. Algebra is conducive to symbolic reasoning. Students who can use many types of reasoning and forms of argument will have resources for more effective reasoning in everyday situations. Students should see mathematics as reasoned and reasonable. They should develop a repertoire of increasingly sophisticated methods of reasoning and proof. In high school their standards for accepting explanations should become more stringent. Students should develop an appreciation of mathematical justification in the study of all mathematical content. They should develop the skill of seeking and finding explanations for the patterns they observe. (Adapted from NCTM, 2000.)
Implications for Curriculum, Instruction, and Assessment Reasoning and proof should be addressed throughout curriculum, instruction, and assessment. These skills should be taught as an integral part of classroom instruction in all areas of mathematics. As the context for reasoning and proof, teachers should choose problems rich in mathematical content and accessible and challenging to all students. Students build confidence in their abilities to develop and defend their own arguments as they solve problems in a classroom environment that supports questioning, discussion, and listening. In such a supportive, inquiry-based classroom environment students will use their mathematical knowledge to make conjectures about problems. Students will analyze various approaches to investigate their conjectures. They will develop a carefully reasoned mathematical argument to support their conclusion. This justification of their conjecture will be communicated through interactions with classmates and teacher and validated against conventional arguments.
Essential Skills in Reasoning and Proof: