Mental computation is dependent on a range of knowledge.
- Know that contains associations such as:
- basic number facts
- counting sequences
- the place value structure of whole numbers
- the meanings of words and symbols.
- Know how contains processes such as:
- rounding and compensating
- splitting numbers by place value and recombining them
- recording operations on an empty number line
- doubling and halving factors.
- Know why involves understanding the reasons that operations work and the connections between ideas. For example:
- collections can be regarded as units made of ones
- multiplication undoes division and vice versa
- sharing division problems can be solved by thinking "How many…?"
- Know why also involves recognising:
- the opportunity to use addition, subtraction, multiplication and division in different contexts
- the quantities that numbers refer to
- difference situations that can be solved by addition or subtraction
- that equal sharing or measuring involves division.
For further exploration of these ideas, you can read Numbers + magic = answer. Students explaining: Make the most of mental computation on the AAMT website.
It is important to understand that the structure of the place value system is based on grouping in tens. The additive and multiplicative nature of the system is useful in developing a sense of flexibility with numbers.
To perform simple addition and subtraction with sets organised in ones, students need to know forward and backward number sequences, starting at any counting number. Skip counting assists in the early development of multiplication.
Number facts: Addition and subtraction
Ready recall of number facts is important for easing the load of working memory when more complex calculations are required. Number facts using addition and subtraction can help students to see patterns and relationships, particularly place value structures and the properties of operations.
Number facts: Multiplication and division
Multiplication and division facts are best learned through a process of noticing patterns and relationships. Fluency of recall of number facts is important for easing the load of working memory when more complex calculations are required.