Maths

Confidence

No Maths Mastery lesson should be the same, as all should be planned according to the needs of the class with a focus on our core principles.

The first and most important message is that there is no such thing as a Mathematics Mastery lesson.  All lessons should be planned according to the specific context of the school and class, and therefore teachers should never feel compelled to ‘stick to’ specific lesson structures or ‘off the shelf’ activities.  While lessons always vary, there is an expectation that lessons reflect the core principles of Mathematics Mastery. As such, teachers should use the planning resources to support them to apply the approach in a way that suits their class

All students will benefit from receiving a curriculum underpinned by a focus on C-P-A representations.  Reinforcement of learning is achieved by going back and forth between these representations, and seeing them used in parallel.

• Concrete - the doing - At the concrete level, tangible objects are used to approach and solve problems. Almost anything students can touch and manipulate to help approach and solve a problem is used at the concrete level. This is a 'hands on' component using real objects and it is the foundation for conceptual understanding.
• Pictorial - the seeing - At the pictorial level, diagrams and other visual representations (e.g. bar models, number lines, diagrams, charts and graphs) are used to approach and solve problems. These can often be pictorial representations of the concrete manipulatives, in which case it is important for the teacher to explain this connection.
• Abstract - the symbolic - At the abstract level, symbolic representations are used to approach and solve problems. These representations can include numbers or letters. It is important for teachers to explain how symbols can provide a shorter often more and efficient way to represent mathematical relationships and operations.

Concrete manipulatives enable students to engage with the mathematics without the need for abstract representations. Even the most challenging algebraic concepts can be demonstrated using cubes and counters. Using these encourages students to really consider the underlying mathematics and develop their problem solving skills. Students should be asked to explain, justify and talk about what they are doing and how the manipulatives help them to do this.

Ambition

We believe that every student is entitled to a deep understanding of mathematics. We believe students should:

• Make connections between previous learning and their current thinking,
• Deepen understanding through the application of a concept,
• Research and seek information.

Learning is more effective when it is understood.  Whilst students can memorise facts, definitions and procedures without knowing why, such learning is neither efficient nor desirable. Whatever the age of the student, there is a temptation for the teacher to attempt to achieve a ‘quick win’ by teaching specific procedures, in the hope that understanding will come later. A ‘relational’ understanding of mathematics, however, is more powerful, long-lasting and useful than an ‘instrumental’ understanding.

We want students to think like mathematicians, not just DO the maths. We believe that students should:

• Explore, wonder, question and conjecture,
• Compare,classify, sort,
• Experiment, play with possibilities, vary aspects and see what happens,
• Make theories and predictions and act purposefully to see what happens.

It is important we support all students in developing their mathematical thinking, in order to improve their facility in learning key mathematical ideas and processes. Further and Higher education sectors alongside industry experts report that mathematical competence is an area of concern for our young people and focusing on empowering our students mathematically supports them in future career choices and independent living.  Our Core Maths provision to KS5 students is designed to facilitate further learning of maths, support with mathematical content in other disciplines and to prepare them for life after KS5.

Communication

Mathematicians often talk of the precise nature of their topic; they speak of the language of maths being rigorous, accurate and specific. However, students’ experiences in the classroom can often be quite different. A big problem in mathematics comes when students are expected to use their comprehension skills to decipher what mathematics is needed to solve a word problem. Problems in the calculations arise, not because students cannot do the maths, but because they cannot ascertain what maths to do.

The way students speak and write about mathematics has been shown to have an impact on their success in mathematics (Morgan, 1995; Gergen, 1995). We therefore use a carefully sequenced, structured approach to introducing and reinforcing mathematical vocabulary throughout maths lessons, so students have the opportunity to work with word problems from the beginning of their learning.

Every Mathematics Mastery lesson should provide opportunities for students to communicate and develop mathematical language. Students should revisit mathematical language from previous years and explore the concepts in greater depth. There should be opportunities for students to clarify vocabulary and explore activities that develop an understanding of the different concepts. Students should have the opportunity to practise using vocabulary and apply their knowledge and understanding in different contexts that require other areas of mathematics.

It is widely acknowledged that asking good questions can really develop students’ understanding and further learning in the classroom.

It is really important to model good questions and answers in lessons. Students should be encouraged to speak in full, correct sentences and should be prepared to explain and justify answers when prompted. When students struggle with this it can be tempting to think or say “I know what you’re trying to say…” and to move on, but this is not encouraged. Students should be pushed to explain themselves with clarity and precision as much as possible.

Questioning is a useful way of engaging, supporting and challenging students. In this context, good questions would prompt new thinking, probe understanding and promote discussion, pushing students outside of their comfort zone. Rather than accepting a given answer, teachers can always probe further by asking ‘how’ and ‘why’ questions.