One aspect of Mathematical Knowledge for Teaching is a teacher’s use of a theoretical foundation to guide instructional decisions, rather than relying on imitation of another teacher or trial and error. Therefore, it is important for teachers to know factors that are significant in the teaching and learning of mathematics (Rowland et al., 2009; Ball, Lubienski & Mewborn, 2001).

**Descriptors/Aspects of TUP (Theoretical Underpinning of Pedagogy)**

**"Strong" examples:**

Draws on knowledge of well-established results in mathematics education research regarding the way pupils learn the topic in hand to underpin the planning and/or delivery of the lesson (whether this is implicit through lesson observation or revealed explicitly later through post-observation interview).

Draws on knowledge of well-established results in mathematics education research regarding pupils’ misconceptions to underpin the planning and/or delivery of the lesson (whether this is implicit through lesson observation or revealed explicitly through post-observation interview).

Draws on knowledge of well-established results in mathematics education research regarding the ‘big ideas’ of mathematics to underpin the planning and/or delivery of the lesson (whether this is implicit through lesson observation or revealed explicitly through post-observation interview).

Lesson underpinned by beliefs relating to the teaching and learning of mathematics and to the nature of mathematics itself which are in accordance with beliefs that research evidence suggests are associated with highly effective mathematics teaching.

Highlighting importance of fundamental ideas that cut across the subject

Revealing teacher’s beliefs about teaching mathematics

Conscious use of appropriate mathematical language to express ideas

**"Weak" examples:**

The planning and/or delivery of the lesson does not take into account and hence may run contrary to well-established results in mathematics education research regarding the way pupils learn the topic in hand.

The planning and/or delivery of the lesson does not take into account and hence may run contrary to well-established results in mathematics education research regarding pupils’ misconceptions.

The planning and/or delivery of the lesson does not take into account and hence may run contrary to well-established results in mathematics education research regarding the ‘big ideas’ of mathematics.

Lesson underpinned by beliefs relating to the teaching and learning of mathematics and to the nature of mathematics itself which are in accordance with beliefs that research evidence suggests are associated with less or moderately effective mathematics teaching.

Introduction of subtraction is not consistent with what research tells us about the nature of subtraction

Lack of consistency between teacher’s conception of comparing, take away and differences and that of the children