While the emphasis in the Numeracy Project is on using mental strategies to solve mathematical problems, it is important that we realise that written recording is an important part of numeracy. There are two main reasons that we use written recording:
- to reduce the mental load when solving a problem
- to communicate ideas to others.
Reducing mental load
If students are unable to solve a problem mentally they need to choose another way to solve the problem; often written recording can provide this alternative. Written recording as a means of reducing mental load can range from scribbled notes to ‘hold’ part of a problem while the student is carrying out a mental calculation, through to formalised standard algorithms. While using pen and paper to aid in solving problems is useful at all stages, students should not be introduced to standard algorithms before they have a firm grasp of part-whole strategies. Introducing standard algorithms before students have firmly established partitioning strategies can slow down the development of mental problem solving skills and number sense.
Communicating ideas to others
Being able to communicate mathematical ideas is important. Written recording provides the basis for both storing working and solutions to refer back to at a later time, and also a means of communicating more complicated ideas to others. Diagrams and equations often provide a more efficient means of communicating ideas than textual explanations.
Encouraging students to record their thinking
- Provide good examples of a variety of forms of written recording
Ensure that your teaching models examples of a variety of forms of written recording to your students. Explicitly teaching key forms of recording such as number lines, ratio tables, and equations will ensure that all students have a wide range of methods of recording available to them.
- Expect students to use their own methods of recording as well as those taught to them
Recording their ideas on paper with words, numbers, symbols, diagrams and pictures is part of students' exploration of number. Both informal and formal methods of recording contribute to students' developing understanding of number. Encourage students to share their recording with the class and to discuss why they chose to use different methods.
Types of recording
This document (PDF, 125KB) illustrates some of the methods students at different stages may use for recording or communicating their ideas. The types of methods described include written explanations, equations, use of calculators and both formal and informal diagrams.
- Written explanations are an extension of oral explanations and may be no more than the spoken explanation written out in full.
- Equations use symbols to represent quantities, unknowns and operations and are a more efficient way of recording mathematical operations.
- Calculators can perform much the same role as standard algorithms in that they allow the student to find an answer without requiring a real understanding of the operation being performed. When a calculator or standard algorithm has been used to obtain an answer encourage students to use a mental strategy to explain the reasonableness of the answer.
- Formal diagrams, such as number lines and ratio tables can be taught to students, and use conventions which are widely understood.
- Informal diagrams include students’ own illustrations of problems to help them clarify what calculations need to be carried out.
Click here to download a file (PDF, 125KB) with examples of the types of recording that students at different stages are likely to use.
Note on standard algorithms:
While standard written algorithms are a valuable tool for solving problems too difficult to be worked out mentally, it is important that students first develop their part-whole thinking to enable them to recognise whether the answers they obtain are reasonable. Introducing written working forms too early may also discourage students from developing mental strategies by providing them with a single method that always works for a given operation. While it is true that a properly applied standard algorithm will always provide a solution there are many instances in which it is not the most appropriate way to approach a problem, for example 99 x 5, or 798 + 99.