District Frameworks - Mathematics
Mathematics as Reasoning
"Reasoning is fundamental to the knowing and doing of mathematics." (NCTM, 1989, p. 81). The ability to reason enables students to solve problems, and it also contributes to their confidence that they can do mathematics by validating their own thinking.
Mathematics is characterized as much by particular types of reasoning as by particular types of content. Students need to learn to recognize and use deduction and induction, and develop the ability to apply these in numerical and spatial contexts. They should learn to evaluate arguments and establish their validity, to analyze what makes sense, and what is true, or plausible but not true.
Students should be encouraged to reflect on and to articulate their reasoning by questions such as, "How did you figure that out?" "Why is this a good way to solve the problem? "Are there other ways?" "Can you think of them?" "Can you be sure you have the right answer?" "Can there be more than one right answer?" "How would you explain your solution if you wanted to convince someone else that it was correct?" Students should be encouraged to make conjectures, test them, and determine whether they can be shown to be true. Asking students to share their thinking about how they worked on a problem often helps them identify their own mistakes or the flaws in their reasoning.
In grades K-4 the study of mathematics shall emphasize reasoning so that students will:
- Draw logical conclusions about mathematics;
- Use models, known facts, properties, and relationships to explain their thinking;
- Justify their answers and solution processes;
- Use patterns and relationships to analyze mathematical situations;
- Believe that mathematics makes sense.
In grades 5-8 the study of mathematics shall include opportunities to communicate so that students will:
- Recognize and apply deductive and inductive reasoning;
- Understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs;
- Validate their own thinking;
- Appreciate the pervasive use and power of reasoning as a part of mathematics.
In grades 9-12, the mathematics curriculum shall include numerous and varied experience that reinforce and extend-logical reasoning skills so that all students will:
- Make and test conjectures;
- Formulate counter examples;
- Follow logical arguments;
- Judge the validity of arguments;
- Construct simple valid arguments;
and, in addition, college-intending students will:
- Construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction.
ADULT BASIC EDUCATION
In the adult basic education classroom, curriculum design shall include approaches which emphasize mathematical reasoning so that the learner can:
- Draw logical conclusions from mathematical situations using concrete models and verbal skills to explain their thinking;
- Understand and apply deductive and inductive reasoning, proportional reasoning, with special attention to spatial and visual reasoning with proportions and graphs;
- Pose their own mathematical questions and evaluate their own arguments;
- Validate their own thinking and intuition, feel confident as math problem solvers, and see that mathematics makes sense.