# Processes for the practice of mathematics

## Problem solving Tools to facilitate mathematical problem solving

- Problem solving in the
*mathematical knowledge base*: Concepts, misconceptions, outcomes, & standards - Development of problem solving
- Heuristic:
- 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
- Learners problem solving guide or checklist to use to as a guide with suggestions to solve problems
- Learners solving checklist - with before, during, and after suggestions
- Learners Solving Attitude Inventory with yes - no questions

- See also communication dimension -

## Reasoning and proof

### Teacher tools

- 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
- Imagining
- An Outline of Goals for a Critical Thinking Curriculum and Its Assessment

### Sample Problems for Reasoning and Proof

## Representation

### Teacher tools

### Area representations

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?

## Communication

Concepts and big ideas in the mathematical knowledge base

### Teacher tools

- 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

### Concepts

- Mathematical ideas build upon each other.
- Mathematical ideas are connected to other mathematical ideas.
- Mathematical ideas are connected to the world.

*Page Overview*

### Common sense & knowledge

*Counting in Winnebago*

- WI
^{n} - NO
^{n}BA - THABTHI
^{n} - DUBA
- SATO
^{n} - SHAPE
- PETHO
^{n}BA - PETHABTHI
^{n} - SHO
^{n}KA - GTHEBO
^{n}

### Interesting book

*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.

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