Reliance on counting

There are key indicators of increased efficiency in mental calculation strategies.

• Counting based strategies start with treating sets as collections of ones.
  Students solve problems by counting every object.
  This includes joining sets (addition and multiplication) and separating a set (subtraction). Sharing is done one object at a time.
• Additive strategies involve treating sets as collections that can be partitioned and combined.
  The first evidence of this is counting on and back to solve addition problems, and skip counting to solve multiplication problems.
  When students learn that collections can be partitioned and recombined they use their knowledge of number facts to work out new results and solve more difficult problems.
• Multiplicative thinking involves using the properties of multiplication and division.
  Students distinguish which of the four operations to apply to situations.

Teaching for increased fluency and flexibility involves posing problems that take students beyond their preferred strategies. More advanced strategies can be developed by modelling solutions using materials, words and equations, and helping students to develop key knowledge.

You can read more in the article From Here to There: The Path to Computational Fluency with Multi-digit Multiplication on the AAMT website.