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Processes for the practice of mathematics

Problem solving Tools to facilitate mathematical problem solving

Reasoning and proof

Teacher tools

Sample Problems for Reasoning and Proof


Concepts and big ideas

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.

area image

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

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?



Concepts and big ideas in the mathematical knowledge base

Teacher tools

Connections and perspective

Concepts and big ideas


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



Content dimensions:

Common sense & knowledge

Common sense Vs Knowledge




Counting in Winnebago

  1. WIn
  2. NOnBA
  4. DUBA
  5. SATOn
  6. SHAPE
  9. SHOnKA
  10. GTHEBOn



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.