An example of how the CPA Approach and Variation Theory can develop a Mastery Curriculum. Karen Sedgwick.

 Socio-economic gaps

 An attempt to close the gap  An acknowledgement that the National strategies have had limited success  Greater depth, greater consolidation, greater understanding  No extension onto content meant for older children (no more level 6 paper)  Challenge to provide access for all-not just for the more able Maths Curriculum: Closing the Gap.

Nick Gibb (Minister for State and School Reforms): the new National Curriculum is in line with that of countries which are high performing i.e. Shanghai and Singapore, and they use a mastery approach. Department for Education

The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning… What is Mastery?

 The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace... Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. (NC 2014:3) The National Curriculum:

 Changes in the way mathematics is taught  Changes in the mathematics taught  Changes in the way mathematics is assessed Mathematics reform calls for…

Support task: Level 1c ‘Knowing one more or less within 10’ How can we be sure that maths lessons meet the needs of all children? Main task: Level 1a ‘Solve problems adding and subtracting numbers up to 10’ Extension task: Level 2c ‘Solve practical number problems adding or subtracting numbers to 20. Why is this NOT a good model of differentiation for a Maths lesson?

 the ‘traditional’ way we differentiate i.e. putting the children into ability grouped tables and providing easier work for the less able and more challenging ‘extension’ work for the more able has ‘a very negative effect on mathematical attainment’  ‘one of the root causes for our low position in international comparisons’. Charlie Stripp (Director NCTEM)

 it damages the less able by fostering a negative mindset that they are no good at maths  in practice it results in the less able children being given a ‘reduced curriculum’.  it damages the more able because it encourages children to rush ahead or can ‘involve unfocused investigative work’  labelling the child as ‘able’ creates a fixed mindset so the child believes that they should find maths ‘easy’ and becomes unwilling to tackle demanding tasks for fear of failure. Stripp claims:

 ‘Do we assume our lower ability children cannot do it…?’ and, ‘Do we differentiate too far in maths?’ The National College of Teaching and Leadership (2014) Some are questioning current practice:

 New higher expectations  Mastery for all pupils  Regular use of manipulatives and representations  Differentiate through depth not extension  Widespread use of mathematical language  Connections made between different concepts  Access for all- not just the more able. Quality teaching that is 2014 National Curriculum compliant.

Teachers remain the single most important influence on children’s mathematics learning (Hattie, 2003)

Structuring Learning Active / Concrete Building visual images Abstract X X X X X X X X X X X +

Exploring models and representations used in teaching – Concrete Pictorial or diagramatic Symbolic representation or equation

 If parts of maths concepts are missing children can struggle to make sense of the knowledge. If pictorial references are used often enough children will have the depth to be able to think abstractly and fill in the gaps. For Mastery!!

How many Triangles do you see?

 For the untrained eye it is difficult to discern all triangles. For some they may not see the big triangle-  Others may not recognise the inner triangle as a triangle because the base is not situated at the bottom of the triangle. This might because a lack of variation in presentation when they learnt the concept of a triangle. Those who do recognise the inner triangle as well as the other ones show a simultaneous awareness of some of the critical features of the mathematical concept of a triangle. A triangle is a closed figure with three sides. A placement of a triangle in space is irrelevant and does not affect if the figure is a triangle or not.

 Based on Dienes' ideas (Dienes, 1960), systematic variation is that you vary the lesson through a series of examples that deal with the same problem/topic.  It is employed in several ways, including mathematical variability -where the learning of one particular mathematical concept is varied and perceptual variability -where the mathematical concept is the same, but the students are presented with different ways to perceive a problem.  Multiple embodiment-perceptual variability. Variation Theory- Zoltan Diennes (1960).

Examples of fraction additions to discuss theory of variability.  How is 2/5 + 2/5 different from 2/3 + 2/3?  Both are symbolic representation of fraction additions. But mathematically they are not the same. Dienes refer to this as mathematical variability.  In 2/3 + 2/3, students need to have the knowledge of mixed numbers and improper fractions.  Such knowledge is not needed in doing 2/5 + 2/5.  These two tasks possess mathematical variability and have implications when they are taught. Variation Theory- Zoltan Diennes (1960).

ZONE OF PROXIMAL DEVELOPMENT. VYgotsky (1986) TASK COMPLEXITY CONTINUUM ZONE OF PROXIMAL DEVELOPMENT Where learning occurs. Students should be here 80% of the time. What has been mastered and can be achieved without support. What can be mastered with support/scaffolding. What cannot yet be mastered with support/scaffold. Involves increasing or decreasing the structure to enable learners to access new tasks.

 Mastery is all about representing maths so that it makes sense to the children, so carefully plan the models, images and language that connect the maths.  Give all children the chance to access the learning with varying support when needed.  Add breadth and depth to the learning.  Scaffolded learning a feature Mastery

What would this look like? All Pupils. How is this different? 5+2=7 ‘Solve problems using adding and subtracting numbers up to 10’ + = 5

Lets look at an example.

 Discussing the equations  Representing them with diennes, bead strings or unifix  Making up stories to explain what happened.  Drawing pictorial representations on their mini whiteboards  Marking the additions on a number line. Developing Mastery. Increasing Scaffold.

 Matching  Finding the incorrect answer (an extra answer card)  Circling the representations to show the additions  Taking it turns to explain a question to their partner  Or representing in another abstract way. Developing Mastery.

 Filling in the missing answers  Representing the equations concretely (in a different way from their partner?  Discussing how many ones and tens there are in each part of the equation. How can we decrease scaffolding?

 Creating equations and representing them pictorially/concretely  Creating a pictorial representations so another table can complete the abstract representations  Checking their partner’s equations. How can we decrease the scaffolding?

Original task Increase Decrease Scaffolding

 Mathematical thinking- Pupils deepen their understanding by giving an example by sorting or by comparing or by looking for patterns and rules in the representations they are exploring the problem with.  Conceptual Understanding- Pupils deepen their understanding by representing concepts using objects and pictures, making connections between different representations and thinking about what different representations stress and ignore  Language and communication- Pupils deepen their understanding by explaining, creating problems, justifying and proving using mathematical language. This acts as a scaffold for their thinking, deepening their understanding further. Three Mathematical Principles for Mastery.