# 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