Knowledge takes many forms.
- Knowing that involves associations such as 6 + 4 = 10.
- Knowing how is about carrying out a process, such as counting in fives.
- Knowing why is about connecting knowledge to justify why an action is valid or invalid.
- Knowing to is about being able to apply an idea when it is appropriate, such as recognising that sharing problems can be solved by division.
Understanding usually refers to justifying actions (knowing why) and being able to apply concepts to unfamiliar situations (knowing to).
Many issues related to learning mathematics are relevant to mental calculation. These issues include load on working memory, number sense, and pattern and structure.
Fluent and flexible mental computation depends on foundational understanding of counting, place value and the properties of the four operations.