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Developing Mathematics Mastery standards with PG Online


PG Online are committed to providing outstanding teaching materials which are a blend of the best teaching pedagogies from around the world. In recognition of this, PG Online have developed a standards marque to indicate that materials bearing this design have been rigorously checked and developed to complement a Mastery approach. These teaching principles, including those of representation and procedural variation have been applied through our material to help enable students to develop a deeper understanding and more applied knowledge of mathematics. The gold is taken from the Challenge identifier within the unit content.


數學 are the traditional Chinese characters for Shù xué. This is literally translated as Number Learning, but widely recognised as meaning Mathematics. The Simplified Chinese 数学 is more widely spoken in Singapore. The repeating geometric patterns also represent three of the strands of mathematics including algebra, geometry and ratio.

Mastery Standards

What is Mastery?

Mastery is founded on the belief that every student can achieve in Mathematics, dismissing the notion that some ‘just don’t get it’. It has roots in Singaporean and Shanghai approaches whereby the focus of teaching is on achieving a long-term and secure understanding. The pace of learning is kept similar for all students. Only the depth of understanding varies.

Depth of learning

Mastery encourages the use of both pictorial and abstract representations at secondary level, including the use of bar modelling and number lines to reinforce learning.

Representation and variation theory

Techniques involving procedural variation and diagnostic questioning develop understanding by introducing new variation one step at a time. This encompasses those elements that change, as well as those that do not from one question to another. Questions also test misconceptions as well as the boundaries of understanding – further aiding the observational feedback to the teacher without the need for constant marking.


Whilst the teacher may commonly be stood at the front of a class in Singapore or Shanghai, the practice of Mastery is a far cry from the ‘chalk and talk’ image that this may conjure in the mind’s eye. Mastery involves posing constant questions to students. The philosophy of ‘Don’t tell when you can ask’ encourages students to think more about the answers to questions and to search themselves to reason with why something may be. This promotes reasoning in students and encourages their application of understanding to new concepts.

Teachers in both Singapore and Shanghai rely on vital resources to guide and inform their planning and delivery. This is commonly in the form of a Teachers’ guide to provide a structure for teachers to adapt and adopt in their own environments.

Free Mathematics resources Mathematics resources