Central to mastering mathematics is understanding its underlying structures. This involves being fluent at generalising and proof. We also see this as part of the problem-solving process, which can usually be thought of as having four stages:
- Getting started
- Working on the problem
- Digging deeper
The third stage, 'Digging deeper', takes place once the problem has been thoroughly explored and some solutions may have been found. Learners can be challenged to dig deeper by finding generalisations or a proof. In England, this is encouraged by the current National Curriculum (2014), which says:
The expectation is that the majority of children will move through the programmes of study at broadly the same pace. … Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before acceleration through new content.
The article and tasks below will support you in helping learners get better at generalising and, ultimately, at proving.