Mathematics Mastery’s mission is to enable all learners to enjoy and succeed in mathematics. We want learners to think about maths beyond what is tested in national examinations and to be equipped with an understanding of mathematics that will be relevant and useful in their future studies and/or in the world of work.
Our programme has been designed on principles to provide learners with a deep conceptual understanding of mathematical principles, the ability to confidently communicate in precise mathematical language, while becoming mathematical thinkers. The programme can be delivered with confidence in the knowledge that if a student understands the core principles, they will be able to remember more and do more maths, in whatever context they encounter it.
Our principles and how they’re delivered:
• Mathematics tasks are about constructing meaning and making sense of relationships. Learners deepen their understanding by representing concepts using objects, pictures, symbols and words.
• Different representations stress and ignore different aspects of a concept and so moving between representations and making explicit links between them allows learners to construct a comprehensive conceptual framework that can be used as the foundation for future learning.
• We use the content of the national curriculum as the starting point for our curriculum but this is expanded upon by making explicit the foundational knowledge that learners need to understand in order to access this.
• Tasks are sequenced to help learners build a narrative through different topics. These topics are then sequenced in a logical progression that allows learners to establish connections and draw comparisons.
• Multiple representations are carefully selected so that they are extendable within and between different areas of mathematics. Using these rich models encourages learners to develop different perspectives on a concept.
• Mathematical language strengthens conceptual understanding by enabling pupils to explain and reason. This must be carefully introduced and reinforced through frequent discussion to ensure it is meaningfully understood.
• The more learners use mathematical words the more they feel themselves to be mathematicians. Talk is an essential element of every lesson and time is dedicated to developing confidence with specific vocabulary as well as verbal reasoning.
• The content of our curriculum carefully progresses in order to induct learners into the mathematical community. A large part of this community is confident use of the language, signs and symbols of mathematics. Verbal and non-verbal communication is part of every sequence of learning in the curriculum.
• This often starts with more informal language initially, building up to formal and precise mathematical language.
• Talk tasks are part of every lesson in the curriculum to help with this development.
• By the time they reach school, all pupils have demonstrated a significant range of innate ways of thinking that can be harnessed in the classroom to develop mathematical thinking.
• We must support pupils to develop mathematical ‘habits of mind’ – to be systematic, generalise and seek out patterns.
• The creation of a conjecturing environment and considered use of questions and prompts are important elements of encouraging learners to think like mathematicians.
• Our curriculum is designed to give learners the opportunities to think mathematically. Throughout the curriculum you will see tasks that require learners to specialise and generalise, to work systematically, to generate their own examples, to classify and to make conjectures.
• This is aided by our prompts for thinking which help make these important parts of mathematics more explicit.
At Mathematics Mastery, problem solving is at the heart of our curriculum as the essence of everything we do as mathematicians. Problem solving should not be an add-on at the end of a maths lesson or a weekly investigation lesson.
Pupils must be given every opportunity to explore, recognise patterns, hypothesise and be empowered to let problem solving take them on new and unfamiliar journeys. Even the most straightforward tasks can be an opportunity for pupils to investigate, seek solutions, make new discoveries and reason about their findings.
You could consider using some of the suggestions below to help integrate problem-solving in your lessons:
Mindset is defined as a set of beliefs that determine somebody’s behaviour and outlook in life, and can be split into two types – a fixed mindset and a growth mindset. At Saint Patrick's, we encourage growth mindset through phrases like I can not do it yet or I do not know it yet. The addition of the word yet encourages the children to be more resilient and to try alternative methods to work out the problem. We encourage our children to use a range of methods and pictorial representations to problem solve and to show their full maths understanding.
Fixed mindset - this is someone who believes ability and intelligence are things that you are born with. They believe natural talent alone creates success and one doesn't need to put much effort into achieving things they are naturally good at. People with a fixed mindset tend to give up easily with tasks, as they get upset by mistakes, and are afraid of challenges and failure.
Growth mindset - this is someone who believes intelligence and ability can be developed over time through effort, dedication and hard work. They tend to persevere with tasks and enjoy challenges due to the belief that effort needs to be expended to learn. People with a growth mindset believe they can be successful if they apply effort and hard work, and are more likely to continue working hard despite setbacks.
So how does this relate to maths? In the UK, our attitude towards maths is very much in a fixed mindset. We often hear people say they are ‘rubbish at maths’, but if children hear this, it could encourage them to believe that maths isn’t important.
At Mathematics Mastery, we promote a growth mindset belief – that all children can achieve regardless of their background. To encourage children to develop a growth mindset around maths, the way in which we speak to pupils is very important.
How does our mission inform the structure of our curriculum?
We have made decisions on how the curriculum is structured. These decisions are driven by the overarching principles described above and to enable all learners to enjoy and succeed in mathematics.
• Mathematics Mastery provides a single curriculum map that all learners are expected to follow. This means that all learners have the same access to the curriculum content and there is no ceiling imposed on what learners can achieve.
• While there is only one curriculum, we recognise that not all learners come to each lesson at the same starting point. For this reason, we provide additional resources designed to help learners access the main curriculum and also provide planning resources designed to help teachers adapt lessons to provide scaffolding and depth according to the needs of their learners.
• In the MM curriculum extended time is spent within a single area of mathematics. This allows teachers to spend more time developing learners’ conceptual understanding. It also provides opportunities to go into greater depth within a concept area and make connections with other areas of mathematics.
• The MM curriculum is organised to be cumulative. This means that mathematical concepts that are taught earlier in the curriculum are revisited in the context of a new area of mathematics.
• This helps learners to make connections between different mathematical concepts. Retrieving, using and applying concepts regularly, transferring to new contexts helps develop fluency as well as conceptual understanding.
• A six-part lesson gives a structure in which to implement the pedagogic principles of the curriculum. The different parts of the lesson allow teachers to bring the different dimensions of depth to the foreground. Having a consistent structure for each lesson ensures that learners are exposed to the pedagogies associated with each dimension.
What should assessment look like in implementation?
• Evidence points to high quality formative assessment leading to the greatest learning gains.
• We provide opportunities and guide teachers in asking questions that will reveal learners’ understanding of a concept.
• Most importantly we provide opportunities for meaningful dialogue to take place in lessons. It is by giving learners opportunities to talk, and by listening carefully to what they say, that we gather some of the richest data on their understanding, in order to influence teachers’ next moves.
• We recommend that schools are cautious when using the results of summative assessments. This is because the domain from which test items are sampled is usually much larger in a summative assessment compared with a formative assessment.
• This makes it impossible to make a reliable inference about a student’s learning within a subcategory of this larger domain. We also urge caution when linking the results of summative assessments to any future performance in public examinations.
• Due to their low impact on future teaching, we recommend that the number of summative assessments in a year should be relatively low and to thoroughly interrogate the quality of the assessment that is used.
What does it mean to know more, remember more and be able to do more mathematics?
• In order for learners to make sense of a new idea or relationship learners need to incorporate it into their current understanding and see how it connects with ideas and relationships they have encountered previously.
• The greater their understanding of what has been taught previously, the more sense-making they will be able to do in the future with increasingly complex mathematics. Therefore, we believe that the key to knowing more mathematics lies in understanding.
• We also believe that learners who make sense of the mathematics they are learning have more memorable and enjoyable experiences that are more likely to be remembered in the long term. They will also be able to do more as they understand how to push the boundaries of what they know and apply it to solve problems.
Skills Progression :
Due to school closures this term we have created support for primary teachers to deliver four terms of mathematical content in three terms during the academic year 2020-21. This has been achieved through creating ‘abridged’ curriculum maps. We have produced carefully constructed curriculum maps for each year group from Year 2 to Year 6 which identify the learning that will have been missed during the summer term 2020 and where it links directly to the units of work for which the knowledge is next required. This has been exemplified in one overall abridged Yearly plan
What does a mastery lesson look like?
The Do Now
The purpose of the Do Now task is to consolidate previous learning. This could be recapping on what was learnt the day before or a topic from a previous unit that is necessary for the current lesson. This is a key part of our curriculum for depth – that learning should be cumulative and concepts should be built on throughout the year.
Do Now tasks should be independent work that the children do at their tables for about 5 minutes. The material is a recap of previous learning, so every child should be able to complete the work with ease, without too much instruction from the teacher.
Alternatively, teachers may like to teach Fluency First in this section of the lesson. This is a teacher led task that may consolidate previous learning or introduce new concepts. This is an opportunity to rehearse, reinforce and consolidate mental calculation skills.
This Do Now section is a great way to make sure all children are concentrating for when they come together for the lesson’s New Learning section that follows. In between tasks, all pupils will take part in transition activities to ensure the class stay focused on the lesson.
The New Learning section introduces the main learning for the lesson, beginning by sharing the lesson’s key vocabulary with the pupils. This segment will require clear explanations and modelling of tasks to be completed throughout the lesson, especially the Talk task.
Each unit focuses on a well-known theme or children’s story so the pupils can engage with characters and objects to learn mathematical concepts throughout the unit. The theme is introduced to the pupils through a Big Picture, depicting a scene that allows many opportunities for mathematical discussion. The Big Picture is usually introduced in the New Learning segment of a lesson early on in the unit, but is then referred to in other segments.
The New Learning section usually lasts around 10 to 15 minutes, and may involve partner discussions and answering questions, so teachers can check pupils’ understanding before moving onto the next parts of the lesson.
As language is such an important feature of Mathematics Mastery, Talk Tasks are imperative. This segment allows talking about maths and comprehension to be developed, and provides opportunities to use mathematical language.
The main focus here is on the children working together in pairs or small groups and talking in full sentences about maths. Developing pupils’ language is an important feature of the Mathematics Mastery approach, and taking turns and listening are important to children’s development.
Pupils will need training in how Talk Tasks will look. Ensure that you are always using the correct language structures when modelling questions and responses, and insist that pupils respond fully at all times when they are replicating your language use. This may take time at first but don’t give up! Assessment of understanding should take place during this segment by listening to pupils’ explanations and discussions.
An alternative approach to this section is Let's Explore - this is an opportunity for pupils to apply the skills they have learnt previously, by discussing and reasoning mathematically.
The Develop Learning segment is designed to mirror the New Learning earlier in the lesson, but aims to move the pupils’ learning on further and deepen their understanding.
Learning could be developed by introducing different resources, adding a problem solving element, or encouraging further good language use following the Talk Task. This is a great opportunity to assess progress and understanding, and deal with any common misconceptions before pupils start independent work.
This segment gives pupils the opportunity to practise their Develop Learning by working independently and demonstrating what they have understood and learnt. Although this is an independent task, this does not mean that the children must work alone, in silence, as they should be encouraged to discuss mathematical concepts together using the key vocabulary of the lesson.
Here, tasks will often need to be adapted so they challenge everyone in the class. Consider how you can deepen mathematical thinking for those pupils who have accessed the learning with ease. What modifying, generalising and comparisons can be made using the task sheets provided, so that exercises are differentiated for all?
The final part of the lesson is used to reflect on learning, gather evidence for assessments and plan for future learning. It should sum up what the children have learnt during the lesson, consolidating all learning, address any common misconceptions, and pose a question for the next lesson.
The Plenary should be based on the needs of the pupils after the previous segments have been taught, through continual assessment of progress and understanding throughout the lesson. In some lesson guides a Plenary will be suggested, but often these cannot be planned for until the lesson has begun.