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Processes for the practice of mathematics

Problem solving


  • Problems are solved with a heuristic, a repertoire of strategies, metacognition or reflection, and persistence.
  • A heuristic is a series of generalized steps.
  • It is helpful to know and use a problem solving heuristic to solve problems, think about what you know or have done and what you need to find out or do.
  • Problems can be solved with different strategies.
  • Monitor and reflect on the process of mathematical problem solving and regulate their actions.
  • It's good to develop the habit and ability to monitor and regulate our thinking processes and actions when solving problems.
  • Metacogniton (self talk) and group discussion is helpful to talk through a problem to solve it and reflect on the accuracy of the process and 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.
  • Algorithm is a step by step list of instructions to solve a type of problems or task.

Strategies for solving problems

  • Use 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

Scoring guides and rubrics

Reasoning and proof

Teacher tools

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.

Teacher tools

Area representations

Lynne Outhred & Michael Mitchelmore. Young Children Intuitive Understanding of Rectangular Area Measurement, JRME. 2000. Vol 31, #2. 144-167.

area image

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.

Teacher tools

Six Ways to communicate mathematically: with sample of each

Graphing notes for line graphs.

Language of mathematical representation: nouns and verbs for 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.



Page Overview


Common sense & knowledge

Common sense Vs Knowledge




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Interesting book

book cover

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.