TC 2

Depth NOT acceleration

TC 1The old national curriculum, measured in terms of levels, encouraged undue pace. Children were accelerated onto more complex concepts before really mastering earlier ones. Imagine the cubes to the left represent the building blocks of maths. As you move on swiftly from one topic to the next the tower gets taller and taller, until eventually it comes crashing down. By moving on before a child is ready, before they fully understand the concept, you are helping them build a tall tower, which at some point will become unstable.

TC 2

The new national curriculum encourages the study of fewer skills in greater depth in order to achieve mastery. By taking the same cubes and arranging them differently, the tower remains standing.

So by ensuring that the foundations are solid before moving on, children develop a more stable platform from which to build.

 

Curriculum: Sprial vs Linear

Many primary schools are still following a spiral curriculum where topics get revisited each term. The nature of this is that you have to move on quite quickly from one topic to the next, often spending only one week on something and not really having time to ensure that the children have properly and deeply undersood the concepts being taught. This suited the old national curriculum as it was content heavy and didn’t require any depth of understanding.

An alternative to this approach is following a linear curriculum in which topics are blocked together and covered only once in the year. Doing so allows more time for children to gain a deeper understanding of the key topics as there is less pressure to move on before they are ready. Evidence shows that children who have a deeper understanding of number and an enhanced ability to reason mathematically can progress through other topics more quickly.

TC 3

This is an example of how a linear curriculum might be structured. Note the emphasis that is placed on number in the first two terms. It is important to note that just because the curriculum is linear, it does not mean that the skills learnt in one block are not revisited again throughout the year as research clearly shows that interleaving ideas and revisting them aids long term retention. For example in the scheme above, during the geometry in the spring term you might look at perimeter and in doing so revisit addition and subtraction.

It is important at this stage in the year that schools consider whether the scheme they are following really provides their children with the most solid foundations.

There are many schemes availble on the market to suppot teaching for mastery (two of which we review in Textbooks: Compare and contrast). In addition to these, there is a FREE scheme that has been created by the White Rose Maths Hub, which provides an excellent starting point for any primary school wishing to adopt a new approach. We are currently working with the White Rose Maths Hub to further enhance these resources, details of what we are adding and expected timelines can be found here.

Depth is achieved through variation

There are three elements that are critical on the journey towards mastery in maths and that we need to develop in our children. Fluency, reasoning and problem solving. Without one, the next cannot follow and it is only by developing these three skills that children can move towards mastery.

TC 7

This flow of learning was first seen in Shanghai by the teachers who went on the exchange trip. Children must first gain fluency in whatever topic they are studying before they move on to reason mathematically and then begin to solve a variety of different problems which probe and challenge their depth of understanding.

Here’s a few examples to help illustrate how this development might look.

Example 1:

Fluency:

TC 8

Reasoning:

TC 9

Problem Solving:

TC 10

Example 2:

Fluency:

TC 4

Reasoning:

TC 6

Problem Solving:

TC 5

Can you see how each task builds on the previous one? The skills required are the same but the depth of understanding required is very different. When these activities are put in the context of whole-class, mixed-ability teaching where the aim is for all children to develop the skills of fluency, reasoning and problem solving at the same time, the slow progression through these three elements is even more critical. More will be written on this blog shortly regarding whole-class, mixed-ability teaching and how it can and does work.

Intelligent Practice

Variation is a phrase which has been banded about all too often since the first teachers returned from the Shanghai exchange visit. It think it has taken us a very long time to fully appreciate what is meany by variation…

Variation is NOT:

  • more of the same thing but a bit harder.
  • the same as variety!

When using intelligent practice, all tasks are selected and sequenced carefully with purpose, offering appropriate variation so that when viewed together they reveal something about the underlying mathematical structure, concept or process. Put simply, variation reveals concepts.

Variation is an approach to teaching. It is the art of sequencing similar but increasingly complex problems to “generate disturbance of some sort for the learner” Festinger (1957)

Consider the following example:

23 + 10 = [   ]

23 + 11 = [   ]

23 + 12 = [   ]

 

23 + 9 = [   ]

23 + 8 = [   ]

23 + 7 = [   ]

By changing just one small element at a time children are given the chance to develop their understanding of a concept rather than just learning a process to get the right answers. However, it is not as simple as just finding patterns. Care must be taken to ensure the structural concept is understood; not just a superficial procedure found, as it is this deep conceptual understanding that leads to mastery.

If you just rearrange the questions a little you can quickly see how big a difference the order the questions are in can make to the development of a child’s understanding.

23 + 9 = [   ]

23 + 12 = [   ]

10 + 23 = [   ]

23 + 7 = [   ]

23 + 11 = [   ]

8 + 23 = [   ]

It is this intelligent practice, the appropriate selection and sequencing of questions, that can make a fundamental difference to a child’s conceptual understanding.

Conceptual understanding leads to mastery

It is critical that children are taught maths in a way the develops a deep conceptual understanding as this is the only way of securing solid foundations. One way in which we can help children to do this is by using the concrete-pictorial-abstract (CPA) approach. This is based on research by psychologist Jerome Bruner, which suggests that there are three steps (or representations) necessary for pupils to develop understanding of a concept. Reinforcement is achieved by going back and forth between these representations.

CPAConcrete

In this stage a student is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating balls into groups of red ones and green ones or by sharing 10 biscuits among 5 children. This is a ‘hands on’ component using real objects and it is the foundation for conceptual understanding.

Pictorial

In this stage a student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of a division exercise this could be the action of circling objects.

Abstract

In this symbolic stage a student is now capable of representing problems by using mathematical notation, for example: 10 ÷ 2 = 5 Students only use abstract numbers and figures when they have enough context to understand what they mean This is the ‘final’ and most challenging of the three stages.

Example:

The following example demonstrates the principles of CPA. It starts with a pictorial representation of the problem using a clear diagram (the use of counters is important for any child still at the concrete stage), but it then builds the abstract alongside, so that the child can make that leap between the pictorial and the abstract as and when they are comfortable in doing so.

TC 11

Example:

In this second example the concrete is used as a starting point for the question. Two oranges are divided up into quarters, three of them are circled from each which leaves six quarters which are then put back together to make one and a half oranges. The pictorial stage sees the introduction of circles instead of oranges and then the final stage is the introduction of the abstract, the numerical calculation. By putting all three alongside each other students can see the progression and when they are ready, they will naturally make the transition from concrete to pictorial to abstract.

TC 12

In Summary…

  1. Teach fewer concepts in greater depth
  2. Intelligent practice gives children a rich appreciation of the concepts through variation
  3. If you help children unlock this depth of learning then, over time, whole class, mixed ability teaching will become even more successful.

 

Many of the ideas in this post have been adapted from ideas seen in Tom Collin’s session at the spring conference. Click here to download the powerpoint Session 1 – Teaching for Mastery

Mastery Specialists 2

Each of the 35 Maths Hubs have appointed 4 Primary teachers to become Primary Mastery Specialists.

Mastery Specialists 1The Mastery Specialists have all attended 3 residential training courses, run by the NCETM, during which they spent time unpicking what ‘Mastery’ means and looks like in practice. They have not only received excellent training on Teaching for Mastery but have now spent almost a whole year developing their own practice in the classroom using the ‘5 big ideas’ shown below. Central to these 5 elements is promoting the importance of making connections, not just within a series of questions, but across topics in Maths.

In the Surrey Plus Maths Hub, the Mastery Specialists are:

  • Jo Cullen – St Joseph’s Primary, Epsom
  • Tom Collins – St Joseph’s Primary, Guildford
  • Nicola Richards – Hook Junior School, Hook
  • Katie Breese – Kenyngton Manor, Sunbury

Mastery Specialists 3

Map: Blue stars shows the location of the 4 PMSTs, Red pins shows the location of the schools currently involved in TRGs

Each Mastery Specialist is currently running a set of TRG’s (Teacher Research Groups) with approx 12 teachers from 6 other schools. These meet once a half term and usually consist of:

1.Initial discussion of a lesson plan, specifically looking at pedagogy, questioning, variation etc.

2.Observe the learning in a lesson

3.Discussion of lesson afterwards, with emphasis on the learning observed.

These TRG’s have promoted discussion of key aspects of Teaching for Mastery, including: whole class teaching with no set differentiation/grouping; challenge through questioning and extension activities (depth not acceleration); access through pre-teach, practical resource and extra guidance.

The feedback from both participating teachers and the Mastery Specialists themselves has been extremely positive. Katie Breese, the Mastery Specialist based at Kenyngton Manor feels her teaching is still developing to include more aspects of Teaching for Mastery:

“I’ve been trying out mixed ability pairs to develop children’s discussion and reasoning – it’s made a difference already with their ability to explain their mathematical thinking.”

“The pace is much slower, and so now you can see the deeper understanding and connections being made, which before would have been skipped over.”

Mastery Specialists 2Fly the kite… and then reel them in.

This analogy has been used throughout the training received by the Primary Mastery Specialist Teachers. The idea is that you give students the opportunity to explore and develop their own ideas before pulling them back together to move forwards together as a class. For example, at the start of a lesson you might let the students explore a ‘real life’ problem that needs to be solved, give them time to develop their own ideas and then bring the class back together to teach them the skills they need to move forwards.

 

Click here to download the powerpoint from the spring conference Session 1 – Mastery Specialists

TC 3

This project is being undertaken by Deb Harper at South Farnham School and Tom Collins at St Joseph’s Catholic Primary School, Guildford.

Scope of project

To develop a primary mastery curriculum which will be accompanied by a variation document providing examples of how to extend students laterally.

  • Three strands:
    • Arrange the KS1 and KS2 primary curriculum objectives into a coherent and progressive mastery curriculum.
    • Identify which of these objectives can be taken out of numeracy and addressed in other topics
    • Create examples of variation to address these objectives and to allow teachers to extend students laterally (this does not need to be a full set of worksheets but key examples of different variations which can be shared and then further developed. More than one example may be needed for each variation to show how a variation can be progressed).

Background

By November, an outline of the order of topics had been drafted for each Year group. Teachers had begun to create resources that encouraged depth of learning and developed fluency, reasoning and problem solving skills. At this stage, the White Rose Maths Hub Schemes of Learning were discovered!

Although it was nice to see that another Hub was thinking along the same lines as us, it quickly became clear that the resources that their projet was considerably further ahead than ours, and there was no point in duplicating work!

  • White Rose Maths Hub
  • Based in Halifax, Yorkshire
  • Trinity Teaching School alliance
  • Published FREE scheme of learning for KS1 and KS2
    • Mastery Curriculum Model
    • Fluency, reasoning and problem solving questions
    • Assessments

Is it important to register with the White Rose Hub to ensure you are notified when new materials are produced. This package is not yet complete and further assessments and resources will be available throughout the summer term.

The White Rose Schemes of Learning give Overviews for each Year Group which detail the number of weeks that should be spent on each topic. The example below shows the overview for Year 5.

Curriculum 2

curriculum 1There is further information within the Schemes of Learning which details the specific learning objectives that should be taught in each topic section. The example here shows the objectives that should be covered in the 5 week fraction topic that takes place in the Spring term of Year 5.

Though this is extremely useful as a starting point, many Primary teachers we have spoken to are struggling to understand how to structure lesson plans so that these fractions objectives fill 25 lessons.

Collaborative Working

In order to address the areas highlighted by teachers, we will be working in collaboration with the White Rose Hub to produce some supplementary resources. Deb and Tom are working on producing a lesson by lesson breakdown for each topic to give teachers a better indication of how much time should be spent on each objective and how the learning could be scaffolded.

curriculum 4

The White Rose Schemes of Learning also give some examples of questions and tasks which develop student’s fluency, reasoning and problem solving skills of each of the national curriculum statements.

Curriculum 3

curriculum 6The White Rose documents have been extremely well received by Primary teachers, but some have commented that they would find powerpoint resources beneficial so they can be used within lesson slides. Deb and Tom are planning on linking the powerpoint resources developed by teachers at South Farnham and St Joseph’s to selected learning objectives, providing additional lesson resources. Here are some examples of some additional fluency, reasoning and problem solving questions for one fractions objective.curriculum 7curriculum 8

Textbook Mapping

The final aspect of this collaborative project is the development of a textbook mapping document. This will enable teachers to see how their current textbooks can be used to fit with the White Rose Schemes of Learning. The aim is to link exercises in 3 main textbooks (Busy Ants, Inspire and Maths No Problem) to specific learning objectives.

curriculum 5

End Goal!

The aim is to work with the White Rose Maths Hub to produce a FREE fully resourced package that includes the following:

  • Mastery Schemes of Learning for KS1 and KS2
  • Lesson by lesson learning objectives for all topics
  • Mapping to popular text books for all topics
  • Bank of questions for each learning objective
  • PowerPoint resources with fluency, reasoning and problem solving questions for selected topics
  • Termly assessments

We hope that the supplementary resources will be available for the Autumn term by June 2016.

Click here to download the powerpoint from the spring conference Session 1 – Curriculum Design