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Mastering mathematics: developing generalising and proof

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:

 

  1. Getting started
  2. Working on the problem
  3. Digging deeper
  4. Concluding

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. 

 

Transferring
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Transferring

Age
5 to 11
This group of tasks offers opportunities for learners to transfer thinking from one example to another, new example (which might eventually lead to generalisation).
Generalising
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Generalising

Age
5 to 11
These tasks give learners chance to generalise, which involves identifying an underlying structure.
Proving
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Proving

Age
7 to 11
These tasks provide opportunities for learners to get better at proving, whether through proof by exhaustion, proof by logical argument, proof by counter example or generic proof.