Processes for the practice of mathematics
- Problem solving in the mathematical knowledge base: Concepts, misconceptions, outcomes, & standards
- Development of problem solving
- Strategies for solving problems with sample problems
- Planning & teaching tools pedagogical and curriculum
- Teacher's role in problem solving
- Suggestions to help learners solve problems
- Assessment sample questions to assess learners' mathematical problem solving
- Characteristics of a good problem ideas to consider when writing or selecting problems
- Planning questions to prepare to facilitate problem solving & investigating suggestions to consider when writing or planning to use problems
- Problem Solving Goals and Outcomes to review for the inclusion in appropriate curricular documents and plans
- Problem solving vocabulary to introduce to learners
- Dispositions to review & select to focus on
- Learner's materials
- See also communication dimension -
- Reasoning and Proof Explanations and development of reasoning, proof, & logic - information on reasoning and proof with examples of different types of proof, teaching suggestions, & an extensive scoring guide
- Reasoning and proof big idea, concept, or generalization instructional planning map
- Development of reasoning from birth to maturity
- Reasoning, conservation, and cognitive development tasks & activities
- Application of proof by elimination & verification with a card trick
- Bucket problems reasoning, proof, conjecture, combinations
- Decision making, critical thinging, and change processes
- Structure for Analyzing and Presenting Arguments
- Logical Reasoning and Reasoning Errors
- Position analysis
- Issue analysis
- Reasoning solutions for SUDOKU
- An Outline of Goals for a Critical Thinking Curriculum and Its Assessment
Sample Problems for Reasoning and Proof
Lynne Outhred and Michael Mitchelmore asked learners to illustrate the area of a rectangular shape. They found learners demonstrated four different strategies: incomplete covering, visual covering, concrete covering, and measurement. The drawings below show samples from each.
Representation of relative position of objects
Illustration of Internal Representation
Actual object, real world, or external representation
Representation of external objects with points on an other external object
What does this have to do with mathematical representations?
Concepts and big ideas in the mathematical knowledge base
- Different ways to communicate problems with sample problems
- Ways to communicate mathematical representations as objects, properties, actions, & operations
- Graphing with line graphs as communication
Connections and perspective
- Mathematical ideas build upon each other.
- Mathematical ideas are connected to other mathematical ideas.
- Mathematical ideas are connected to the world.
Common sense & knowledge
Counting in Winnebago
The Power of Logical Thinking: Easy Lessons in the Art of Reasoning... and Hard Facts About Its Absence in Our Lives, Marilyn Vos Savant (1996) ISBN 0-312-13985-3 Saint Martin's Press: New York.