Processes for the practice of mathematics
- 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
- 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
- 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
- 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
- Representations help organize, record, and communicate mathematical ideas.
- Representations help solve problems.
- Mathematical ideas are represented externally and internally.
- Mathematical ideas are represented with object and actions with those objects.
- The representation, null and zero are special representations of the lack of objects or ideas.
- All mathematical representations are connected to physical entities.
- When representing values graphically the use of a scale and units helps to visualize that representation and to do so more accurately and proportionally. For example the difference between six feet and ten feet is four. The difference between 72 inches and 120 inches is 48. How each are represented makes a difference as to communicating the equality or not.
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?
- Mathematical ideas can be communicated.
- Mathematical ideas can be communicated with written narrative, spoken words, pictures, manipulatives, symbols, and movements.
- Combining different ways of communication can make for more efficient or better communication.
- Charts and graphs can be used to communcate relationships.
- There are different wasy to communicate mathematically.
- Six Ways to communicate mathematically: samples of written and spoken words, pictures, manipulatives, symbols, and movements help communication. Includes, Door bell problem - from the book, The Door Bell Rang by Pat Hutchins which can be acted out to assist communication and Using symbols for a problem in the book, Pigs Will Be Pigs: Fun with Math and Money. by Amy Axelrod, 1994 problem
- Graphing notes for line graphs.
- Language of mathematical representation: nouns and verbsfor objects, use, and operations on objects
Connections and perspective
- Mathematical ideas build upon each other.
- Mathematical ideas are connected to other mathematical ideas.
- Mathematical ideas are connected to the world.
- Problem solving
- Stragegies for problem solving
- Teacher tools to facilitate problem solving
- Reasoning and proof
- Connections and perspective
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.