INSET: Carefully Crafted Lessons

Now we have 'done away' with groups we need to get even better at crafting lessons and task design to ensure stretch for high attainers and scaffolding for others. Teachers should be thinking precisely about every example we use and every question set for pupils. This ensures slow but steady progression, by varying just one component of the questions at a time. Pupil work time should be regularly punctuated by teacher-led discussion and examples, so frequently ‘reels the kite back in’ after letting children run with it in their own work. We used an example from Wroxham School in Potters Bar: https://vimeo.com/69709930

In the example lessons, students are given the input as a whole class 

Children then self-select the task they feel best suits their confidence with the content. 

If they feel the challenge they picked was too hard or too easy, they are free to go back and pick another one. 

Children were taught to challenge themselves but also not feel ashamed of picking the easier challenge if they want to. 

They can work through all 3 challenges during the lesson if they wish so can build confidence during a lesson.

'CPA' Examples... Concrete, Pictorial, Abstract Teaching and Learning in Maths

Some examples that we worked through on our Maths Inset of using concrete, pictorial and abstract methods when teaching Maths. We're hoping to ensure that 'CPA' methods will be used to support children's learning and secure concepts in the future. I think it's also going to be 'fun' - if that's a word we're allowed to use when talking about Maths!

Maths: Concrete, Pictorial and Abstract...?

Our recent Maths INSET was a focus on the use of 'manipulatives' which means practical resources in lessons. Ofsted’s 2012 report ‘Made to Measure’ suggests that although manipulatives are used in some primary schools to support teaching and learning they are not used as effectively or as widely as they might be. Research suggests the use of manipulatives give children a deep understanding of maths. We discussed the use of concrete, pictorial and abstract ways of learning which need to be considered with equal importance when learning new concepts. Concrete resources give time for pupils to investigate a concept first - and then make connections when formal methods are introduced.  The pictorial stage allows pupils to demonstrate and sustain their understanding of mathematical concepts and processes. Most importantly, the abstract stage should run alongside the concrete - pictorial stage to enable pupils to read mathematical statements and show their understanding using concrete resources or pictorial representations.

Learning Times Tables...

A very interesting article from the Primary & Early Years Magazine shared by the NCETM has looked at how children learn times tables. Children need to learn their tables facts to a required level of automaticity which can be challenging - committing a large number of facts to their long term memory which they can then recall to apply in different contexts and solve problems. One of the key messages here is that remembering stuff matters! How do we learn them? We might remember how we did it ourselves - perhaps by 'rote' or repetition? While this is really effective method, the article makes a key point that we need to develop understanding and 'reasoning' alongside the fluency. It's not just about remembering 7 x 8 = 56 but that I've got 7 people and 56 sweets for my party - how am I going to share them? Or what number between 50 and 60 has the factors of 7 and 8?
Here's another suggestion from the article:

ALL, SOME, NONE

Write three or four random numbers on the board. Pupils talk with a partner, using a whiteboard and pen if that helps, to construct three sentences about those numbers, using the words ‘all, some, none?’

E.g. 2, 7, 11

“All of the numbers are prime”

“Some of the numbers are odd”

“None of the numbers are factors of 15”

All these statements lead into some good dialogue about number.

Problem of the day for KS1 and 2!

The White Rose Maths Hub is providing teachers and parents with a free 'problem of the day' in March for KS1 and 2 pupils. An ideal addition to our daily routine as we approach SATs!
The link to the resource which will be complete by the end of the month is here: Problem of the day!

Outstanding Learning....

We're really impressed with this example of some outstanding learning in Year 4. One of our pupils is clearly developing her reasoning skills and increasing in confidence as a result. This is good to capture and share as we adopt some new approaches to our teaching Maths. We might have to work a little bit on the English too....?!

What makes effective teaching and Learning in Maths?

What do children need to know to add 196 and 10? And what happens if any of those gaps in understanding are present? Our INSET in February was based on the question above and we had several themes to consider.

• The principles and practices of teaching maths for ‘mastery’ or working at greater depth – thinking flexibly and creatively about number.
• A shared understanding of what this means and how we can spot it in mathematics lessons to help children move forward.
• How rich tasks can support and secure pupils’ understanding and our own assessment of their understanding.

One of the key considerations for teachers is building upon what already children know to further their mathematical understanding and how to 'plug' those gaps to ensure children make good progress. We believe it is through careful planning and coherent sequencing of Maths learning that children will best understand the concepts and master the procedures they need to progress. Providing rich tasks will allow them to problem solve and reason about their thinking and lead to 'mastery' and them 'knowing without hesitation'. 

Here's a good example. Have a go at this problem...

A 3 x 3 x 3 cube is painted red on the outside. If it is broken up into 1 x 1 x 1 cubes how many of these smaller cubes have 3 sides painted?

2 sides painted?

1 side painted?

No sides painted? 

Every Child Can Master Maths

Much is being made in the world of Maths teaching about the Shanghai method. The Chinese city which seems to top the league tables when its students take the 'Pisa' tests at 15 years old. The method is really an amalgamation of many ideas based on a deep understanding of mathematical concepts and how children learn them. One of the key principles for us is that everyone can be a maths 'master' given the opportunity to build on what they know in small steps. We've recently invested in tools and resources to support our thinking including the Maths: No problem books which we hope will impact on our teaching and learning. We'd recommend parents taking a look  - several staff are using them with their own children at home! We are really not on any sort of commission but if you would like more information head here.

Real Maths Stories....

Always on the look out for some inspiration, we were impressed with the approach to Maths held by colleagues at St Stephen's Junior in Bristol. The school has adopted a 'Maths Story' approach based on Singapore/Shanghai methods:
"Teaching of mathematics at St Stephen’s has links with Shanghai/Singapore maths. We have embedded the CPA (concrete, pictoral, abstract) approach to lessons. All concepts are introduced with concrete resources for children to feel and manipulate. As their conceptual understanding develops, they move towards the pictoral and abstract stages. These are recorded in their books as ‘the real story’ and ‘the maths story’. Children are not pushed to move through these stages until they have shown understanding gained by the teacher through skilful assessment." St Stephen's Junior
We've been actively considering our own approaches to teaching and learning Maths in line with some of the positive things we're hearing form the far East! For more details head to their website here.

Maths Exemplification Videos...

We are extremely grateful to the Head at Oakwood Primary School in Southampton for sharing their maths exemplification videos. These were put together by staff at Oakwood to support a DfE led project supporting teacher assessment in Maths. These are real children doing real Maths!

Mathematical thinking is about...

Some thinking from a recent Maths meeting.... As we are trying to improve our pupils' thinking, Teachers ask questions all the time. They serve a wide variety of purposes: to keep learners engaged during an explanation; to assess their understanding; to deepen their thinking or focus their attention on something. Unfortunately, there are many common pitfalls.

• asking questions with no apparent purpose;
• asking too many closed questions;
• asking several questions all at once;
• poor sequencing of questions;
• asking rhetorical questions;
• asking ‘Guess what is in my head’ questions;
• focusing on just a small number of learners;
• ignoring incorrect answers;
• not taking answers seriously.

In contrast, the research shows that effective questioning has the following characteristics:

• Questions are planned, well ramped in difficulty.
• Open questions predominate.
• A climate is created where learners feel safe.
• Probing follow-up questions are prepared.

The prompts above help us understand how to frame questions in a Mathematical context.

Working at the expected standard in Maths...

The DfE have begun to release the exemplification materials for the end of KS2. The first document we have eyes on is Maths. For the whole document head here. Pupils are being judged at the expected standard if they can demonstrate the following standards:

The pupil can demonstrate an understanding of place value, including large numbers and decimals (e.g. what is the value of the ‘7’ in 276,541?; find the difference between the largest and smallest whole numbers that can be made from using three digits; 8.09 = 8 + 9/?; 28.13 = 28 + ? + 0.03).

The pupil can calculate mentally, using efficient strategies such as manipulating
expressions using commutative and distributive properties to simplify the calculation
(e.g. 53 – 82 + 47 = 53 + 47 – 82 = 100 – 82 = 18; 20 × 7 × 5 = 20 × 5 × 7 = 100 × 7 = 700; 53 ÷ 7 + 3 ÷ 7 = (53 +3) ÷ 7 = 56 ÷ 7 = 8).

The pupil can use formal methods to solve multi-step problems (e.g. find the change from £20 for three items that cost £1.24, £7.92 and £2.55; a roll of material is 6m long: how much is left when 5 pieces of 1.15m are cut from the roll?; a bottle of drink is 1.5 litres, how many cups of 175ml can be filled from the bottle, and how much drink is left?).

The pupil can recognise the relationship between fractions, decimals and percentages and can express them as equivalent quantities (e.g. one piece of cake that has been cut into 5 equal slices can be expressed as 1/5 or 0.2 or 20% of the whole cake).

The pupil can calculate using fractions, decimals or percentages (e.g. knowing that 7
divided by 21 is the same as 7/21 and that this is equal to 1/3; 15% of 60; 11/2 + 3/4; 7/9 of 108; 0.8 x 70).

The pupil can substitute values into a simple formula to solve problems
(e.g. perimeter of a rectangle or area of a triangle).

The pupil can calculate with measures (e.g. calculate length of a bus journey given
start and end times; convert 0.05km into m and then into cm).

The pupil can use mathematical reasoning to find missing angles (e.g. the missing
angle in an isosceles triangle when one of the angles is given; the missing angle in a
more complex diagram using knowledge about angles at a point and vertically
opposite angles).

NCETM Assessment Materials

National Centre for Excellence in the Teaching of Mathematics is an excellent resource for teachers. While 'mastery' may mean many different things to many different people, members of NCETM are certainly doing some great thinking when it comes to improving Maths teaching. We've begun to dip into their assessment materials (available here) which might seem a little daunting at first but are a bank of really good activities and questions that are indicators of what children really understand about their learning in Maths. Remember to read the first page!

Teaching for Mastery...

We have begun some excellent CPD run by the Solent Maths Hub on teaching primary mathematics for mastery. The aims of the initial day were to develop our ideas on maths mastery and provide an introduction to the principles and practices of teaching for mastery. We are also developing a shared understanding of what mastery means and how we can spot it in mathematics lessons and how rich tasks can support teachers and children in their understanding.

A good quote from the day:
Mastery is not just being able to memorise key facts and procedures and answer test questions accurately and quickly. It involves knowing ‘why’ as well as knowing ‘that’ and knowing ‘how’. It means being able to use one’s knowledge appropriately, flexibly and creatively and to apply it in new and unfamiliar situations.

The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning so that:

• future mathematical learning is built on solid foundations which do not need to be re-taught; 
• there is no need for separate catch-up programmes due to some children falling behind; 
• children who, under other teaching approaches, can often fall a long way behind, are better able to keep up with their peers, so that gaps in attainment are narrowed whilst the attainment of all is raised.

Try and do the task above and consider the knowledge and skills that you would need to carry it out...

Ideas for Maths Reflection Questions...

1. Starter questions
These take the form of open-ended questions which focus the children's thinking in a general direction and give them a starting point. Examples:

• How could you sort these.......?
• How many ways can you find to ....... ?
• What happens when we ......... ?
• What can be made from....?
• How many different ....... can be found?

2. Questions to stimulate mathematical thinking
These questions assist children to focus on particular strategies and help them to see patterns and relationships. This aids the formation of a strong conceptual network. The questions can serve as a prompt when children become 'stuck'. (Teachers are often tempted to turn these questions into instructions, which is far less likely to stimulate thinking and removes responsibility for the investigation from the child).
Examples:

• What is the same?
• What is different?
• Can you group these ....... in some way?
• Can you see a pattern?
• How can this pattern help you find an answer?
• What do think comes next? Why?
• Is there a way to record what you've found that might help us see more patterns?
• What would happen if....?

3. Assessment questions
Questions such as these ask children to explain what they are doing or how they arrived at a solution. They allow the teacher to see how the children are thinking, what they understand and what level they are operating at. Obviously they are best asked after the children have had time to make progress with the problem, to record some findings and perhaps achieved at least one solution.
Examples:

• What have you discovered?
• How did you find that out?
• Why do you think that?
• What made you decide to do it that way?

4. Final discussion questions
These questions draw together the efforts of the class and prompt sharing and comparison of strategies and solutions. This is a vital phase in the mathematical thinking processes. It provides further opportunity for reflection and realisation of mathematical ideas and relationships. It encourages children to evaluate their work.
Examples:

• Who has the same answer/ pattern/ grouping as this?
• Who has a different solution?
• Are everybody's results the same?
• Why/why not?
• Have we found all the possibilities?
• How do we know?
• Have you thought of another way this could be done?
• Do you think we have found the best solution?

Ideas for Numeracy 'Working Walls'

Some notes from our meeting on using 'working walls in Maths:

· NUMBER LINE: Display a large number line either horizontally or vertically. This could be either numbered (all numbers displayed) or partly numbered (e.g. intervals of 10, intervals of 0.1). There should be a progression in the lines used throughout the school. The line should be at a level that can be reached by the children (so they are able to draw on the jumps). You could encourage the children to refer to the line regularly by displaying an activity next to it (e.g. Fibonacci the frog likes jumping along the line. Which number is he resting on? He started on number 20 and jumped to number 40. Each jump was exactly the same size. What size could his jumps have been? Fibonacci and his friend are each sitting on a number. What is the same about their two numbers?).

· NUMBER SQUARE: Display a large number square (e.g. 1 to 100 counting in ones, 0.01 to 1.0 counting in hundredths). Again it is useful if the children can reach the square. Encourage the children to become familiar with the square by displaying regularly changing questions next to it (e.g. Which numbers are covered up? Can you explain the pattern of green numbers? If you had to colour a pattern on the square, which numbers would you choose to colour? Numbers 7 and 12 are in the number sequence. What could it be?).

· NUMBER FACTS: Display facts that the children are working on. It is important that these are not on the wall too long as children can quickly become reliant on looking at them. Encourage children to interact with the display by posing different questions (e.g. Can you find pairs of numbers totalling 20? Can you sort the equations into ‘fact families’? If you know this fact, what else do you know? Can you draw a picture to model this number fact?). You could also have a ‘fact of the week’ for children to focus and comment on.

· VOCABULARY: Vocabulary should relate to the unit of work being covered. Again it is important that words are changed regularly and do not just become ‘wallpaper’. You could provide children with a ‘challenge’ at the end of the unit to focus them on the vocabulary (e.g. How many words do you know to describe ‘length’? How many different ways can you read ‘13+7=20’? Choose 3 words from the vocab board and give a definition of them to your partner.)

· CHALLENGE BOARD: Display a number or shape. The children write facts or questions about the number or shape on post-it notes and add these to the board over the week. The challenge board could also be a problem that the children have to try to solve.

· TARGETS: Class and group targets should be displayed, maybe with questions, prompts, resources and examples of work.

· PROMPT POSTERS: These should support children with the current unit of work. They could include ‘points to remember’, helpful diagrams, sentence starters, a problem solving flow chart or advice on what to do if you are stuck on a problem.

· CHILDREN’S WORK: This should include jottings as well as neat, finished work. Ensure there are displays of all maths strands over time as well as examples of maths across the curriculum. Displays could also include photos of children engaged in mathematical activities.

Am I a multiple of...?

How about trying this 'Mastery' activity in Maths....?

Did you know that if you double the last digit and subtract it from the number left by the remaining digits and if that equals 0 or is divisible by 7 then your starting number is a multiple of 7!  E.g. 672; 67 - 4 = 63 which is divisible by 7 so therefore 672 is a multiple of 7.  Challenge children to establish rules for finding multiples of numbers from 2 to 10.

A number is a multiple of:

2 if the last digit is even (0, 2, 4, 6, 8).

3 if the sum of the digits is divisible by 3.

4 if the last 2 digits are divisible by 4.

5 if the last digit is 0 or 5.

6 if the number is divisible by 2 (i.e. even) AND 3 (i.e. sum of digits is divisible by 3).

7 if you double the last digit and subtract it from the number made by the rest of the remaining digits and it equals 0 or is divisible by 7 e.g. 672; 67 - 4 = 63 which is divisible by 7.

8 if the last 3 digits are divisible by 8

9 if the sum of the digits is divisible by 9

10 if the number ends in a 0.