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Learning Maths is a Step by Step Process.
Teaching new material to students that haven’t grasped fundamental subject knowledge is a recipe for disaster, yet that was what traditional teaching methods almost always encountered.
If a student doesn’t fully grasp the basics of addition, then trying to teach them times tables will always be a problem. Later, that student will learn about fractions, which will be even harder.
This issue happens again and again in classrooms up and down the country.
Real confidence and success in mathematics comes from mastering concepts before moving on to the next topic.
A true mastery approach is gives every student a deep and secure understanding of topics before advancing.
The Traditional Approach
Traditionally, each topic was taught and practised for a fixed length of time, irrespective of whether the students were comfortable with a topic or not.
As a result, a gap in understanding can appear. These gaps then undermine the students’ confidence in their mathematical ability as they progress through the school.
A well-used metaphor is building a house. If there are gaps in the foundations, then everything you build on top of those foundations is insecure.
A Maths Mastery Approach
Mastery requires all students to demonstrate a high level of understanding before the class moves on. The mastery approach works fundamentally on the principle that all learners, with differing levels of effort, will meet expectations.
Pupils who struggle should be provided with additional support such as tuition or homework (or access to Numeracy Intervention with Emile) until they succeed. Mastery learning can be particularly effective when pupils work in mixed-ability groups and take responsibility for supporting each peer’s progress.
Maths Mastery in the UK
In the UK, a maths mastery approach requires the subject matter to be broken into a series of units.
Each unit needs a clearly specified objective that is pursued until all students have achieved the objective.
In this way, students work through a unit and at the end all students must achieve the objective. This is typically demonstrated by all students achieving 80%+ in an end of unit test.
If some students do not achieve the objective, then the entire class revisit the subject until all students can demonstrate “mastery” of the subject.
A unit may well cover a few learning outcomes or a significant part of a strand from the national curriculum and usually takes 1-2 weeks for a teacher to cover and successfully test their class.
All this necessitates a teacher to have a plan for students who take longer to learn and a plan for those student who grasp concepts more quickly (extension activities).
Students who grasp concepts faster stay on the same content and are stretched with problems that challenge them to consolidate their understanding. These are known as extension activities (see below).
To develop fluency and understanding of concepts, students have to practise applying the learned concepts. This practise improves confidence and security with their ability in mathematics.
The 7 Essential Features of All Mastery Approaches
In my opinion, he is the leading voice on subject mastery on the UK, and so I believe his version of mastery is the best one to describe in more detail.
I’ve added in Summative Assessment to Mark’s own list as I think it’s implicitly expected.
The Mastery Learning Cycle by Mark McCourt, visuals by Oliver Caviglioli
1. Check the Foundations
The first stage is to carefully assess whether any students have any misconceptions or deficits in their understanding before the unit starts.
If there is an issue, then there must be some pre-teaching.
In practise, in primary maths lessons often it will be checking the previous year’s objectives are still remembered and understood.
2. Initial Instruction
The second stage is to start to explain the new subject matter. This should of course be high quality instruction using questions, demonstration and discussion.
3. Regular Assessment
Regular assessment is used to track progress of students in the understanding of a topic. In effect, it is a measure of how successful the second stage has been.
Students are expected to answer questions directed to them by the teacher and complete mini-quizzes for example (perhaps using Emile).
This Regular Assessment stage can be very useful in helping to embed understanding and if live feedback is available, to help students grasp concepts.
4. Corrective Instruction
The fourth stage is purely dependent on the outcomes of the Regular Assessments.
If a student has not grasped a concept, they need help. (Mark makes it clear that this is not reteaching i.e. taught the same way.)
A knowledgeable teacher will present the concept in another way that the student can grasp. Most commonly we see concepts in primary schools represented in a number of ways, such as bar models.
5. Second Assessment
This is the check that following the Corrective Instruction in stage 4 the student grasps the concept.
This is not the final assessment of this topic.
6. Summative Assessment
This is the end of unit assessment that determines if the class is ready to progress to the next unit.
ALL students need to demonstrate their understanding of the subject. This usually looks like 80% correct answers.
7. Extension Activities
Extension Activities are available for students who grasp a concept quickly to deepen their understanding. While those that have passed the assessment of the unit move on to this, those that fail will continue to receive corrective instruction until they pass.
Extension Activities do not move students on to new ideas. These are the students working at Greater Depth.
There are a lot of myths and misunderstandings about mastery propagating through the education world.
- there is a single meaning of ‘mastery’
- there is just one teaching approach which is ‘the’ mastery approach,
- there is a special curriculum which is ‘the’ mastery curriculum,
- there are textbooks which are ‘the’ mastery textbooks.
- using bar models does not necessarily mean a mastery approach
Greater Depth is NOT:
• working on content from the next year group
• practising the same concept with bigger numbers