# Processes for the practice of mathematics

## Problem solving Tools to facilitate mathematical problem solving

- Problem solving
**concepts**& list of**problem solving strategies** - Problems by categories (number value, algebra, geometry, measurement, probability, reasoning & problem solving)
- Goals and Outcomes for Problem Solving
- Year plan or curriculum starter for problem solving
- Heuristics - 4 step - 5 step - 8 step
- Vocabulary for Problem Solving
- Vocabulary for Calculators
- See also communication dimension - particularly Six ways to communicate problems
- Examples of Problems for different problem solving strategies
- Teacher's Role during problem solving sessions
- Steps to Help Students Solve Problems
- Features of a Good Problem
- Guiding Questions to Help Select Good Problems and Investigations
- Questions to use to assess students problem solving attitudes and skill
- Student Problem Solving Guide
- Problem Solving Checklist for students
- Student Attitudes Inventory for Mathematical Problem Solving

### Scoring guides and rubrics

- Problem solving scoring guide or rubric in three formats | See assessment for discussion on its creation |
- Problem solving check sheet with 8 categories and levels for doing with a prompt, without prompt, and unable to do without significant help
- Problem solving - with six categories: problem solving, reasoning, communication, representation, habits of mind or attitudes, and math understanding with four levels
- Problem solving - with three categories (understand problem, strategy slection, and accuracy) and three levels

## Reasoning and proof

### Teacher tools

- Reasoning and Proof Explanations: Mini article on reasoning and proof with examples to use with students and an extensive scoring guide
- Reasoning and proof big idea, concept, or generalization instructional planning map
- Development of reasoning from birth to maturity
- 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

### 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

### Teacher tools

#### Six Ways to communicate mathematically: with sample of each

- Pictures - Four ways to make 25 cents
- Manipulatives - use of - cubes Writing after Average student height
- Writing - Percentage - Sale problem
- Acting it out - The Door Bell Rang - from the book,
*The Door Bell Rang*by Pat Hutchins - Symbols - Pigs Go to Dinner - from the book,
*Pigs Will Be Pigs: Fun with Math and Money*. by Amy Axelrod, 1994 - Orally - talk it out.

Graphing notes for line graphs.

Language of mathematical representation: nouns and verbs for objects, use, and operations on objects

## 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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