## Concepts

- Problems are solved with a heuristic, a repertoire of strategies, metacognition or reflection, and persistence.
- A series of generalized steps (heuristic) is helpful to know when solving problems. Following a heuristic is helpful to think about what you know or have done and what you need to find out or do when solving problems.
- Problems can be solved with different strategies (see strategies below).
- Monitor and reflect on the process of mathematical problem solving and regulate their actions.
- Have the habit and ability to monitor and regulate their thinking processes at each stage of the problem - solving process
- Use self talk, group discussion, to talk through a problem and problem solving process to reflect on all the decisions that are possible to better insure an accurate solution.
- The more problems I solve (persistence) the easier it is to solve problems and use mathematics.
- Some habits of mind are more conducive for solving problems than others.

## Strategies for solving problems

- Use of manipulatives to represent objects and actions in the problem.
- Work a simpler problem.
- Trial and error, guess and check.
- Work backwards
- Use smaller numbers
- Use systematic steps.
- Look for, recognize and describe patterns: quantity, AB/AB, ABBA/ABBA, size, area, volume, rotation, shading, shape, position, subtraction, addition, reflection, multiplication, analogy, and recursive
- Break a problem into two related problems and solve the original problem in two steps: one for each problem.
- Act out the problem. Physically or mentally.
- Use a pictures, graphical representation - model, drawing picture or diagram
- Problems can be solved with models and equations.
- Categorize information to find relationships and patterns that will assist reasoning and proof.
- Organize data to look for patterns sequence, chart, table, making a graph, Venn diagrams, and dichotomous key.
- Process of elimination or process of identification
- Write an open sentence
- Use algebraic reasoning
- Use logical reasoning: matrices, deductive, inductive, truth tables
- Brainstorming
- Use equivalent numbers 3/5, 6/10, 60/100, .6, 60%

## Teacher tools to facilitate mathematical problem solving

- Goals and Outcomes for the these Problem Solving concepts
- 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