# Mathematical Thinking and Fluency

## Mathematical Thinking

Mathematical thinking is central to deep and sustainable learning of mathematics.

Taught ideas that are understood deeply are not just ‘received’ passively but worked on by the student. They need to be thought about, reasoned with and discussed.

Mathematical thinking involves:

looking for pattern in order to discern structure;

looking for relationships and connecting ideas;

reasoning logically, explaining, conjecturing and proving.

NCETM, The 5 Big Ideas of teaching for Mastery

"when children are engaged in mathematical activity (thinking), they are involved in manipulating one or more of these four key components of mathematical experience: concrete materials, symbols, language and pictures”

Derek Haylock and Anne Cockburn (2008), Understanding Mathematics for young children

Structure ⇔ mathematical relationship between elements

Emergent structure (involving analyzing/forming/ seeing local relationships)

Mathematical structure (involving analyzing/forming/ seeing general relationships)

ARCHITECTURE OF MATHEMATICAL STRUCTURE

HAMSA VENKAT, MIKE ASKEW, ANNE WATSON, JOHN MASON

The research suggests that mathematical thinking occurs when students are thinking about mathematical structure.

## Fluency

Fluency demands more of learners than memorisation of a single procedure or collection of facts. It encompasses a mixture of efficiency, accuracy and flexibility.

Quick and efficient recall of facts and procedures is important in order for learners’ to keep track of sub problems, think strategically and solve problems.

Fluency also demands the flexibility to move between different contexts and representations of mathematics, to recognise relationships and make connections and to make appropriate choices from a whole toolkit of methods, strategies and approaches.

NCETM, The 5 Big Ideas of teaching for Mastery