Welcome to the Early Years and KS1 Maths

workshop 21st February 2017

Aims of the workshop: • Understanding what underpins Maths teaching, the aims of the National Curriculum.

• The four operations – concrete, pictorial and abstract.

• Fluency, reasoning and problem solving.

• To appreciate the need for mastery within EYFS and KS1.

How do our children learn in Maths lessons? EYFS Carpet session each day linked to Maths Cross-curricular table top activities Lots of talking Thinking Self-discovery Problem solving Using manipulatives Asking questions Real-life learning Practical and engaging lessons – fascinators!

‘I hear and I forget. I see and I remember. I do and I understand.’ (A Chinese proverb)

Where mastery begins……

Counting:

• Number names to 20 in

correct order

• One to one principle (one

number for each item)

• Cardinal principle (last

number)

• Order irrelevance principle

(conservation)

Along with this: Knowing all about numbers to 10 Ability to subitise Numeral recognition

Key aspects of effective learning characteristics include children: • Being willing to have a go. • Being involved and concentrating. • Having their own ideas; • Choosing ways to do things. • Finding new ways. • Enjoying achieving what they set out to do.

These develop the children’s ability to reason and problem solve – these characteristics must be developed throughout a child’s education.

Early Years Foundation Stage handbook:

The story of a number…

4 + 0 = 4 3 + 1 = 4 2 + 2 = 4 1 + 3 = 4

4 – 3 = 1 4 – 2 = 2 4 – 1 = 3 4 – 0 = 4

Show me 4 in as many ways as you can. What do you know about 4? What is it greater than? What is it less than? Can you count out 4 from a larger group? Can you show the numeral? Number pairs for 4: part, part, whole model leading to number facts and generalisation.

4

3 1

4

3 1

• Counting is the starting point of number work. It forms an essential part of children’s developing understanding of numbers.

• However it is not the best foundation for

calculating. Children need to learn to manipulate numbers in particular ways.

• To calculate they need to move on from

counting. • Children who are still counting using their

fingers in Year 2 will struggle with mathematics in KS2.

Counting

Three key aspects of number sense

Counting: Knowing the

number names in order, forwards and backwards. Understanding how to count

objects, events or actions in ones

and also in twos, fives and tens.

Comparing: Having a feel for the relative

sizes of numbers

Putting numbers in order

Estimating

Counting 16 beads, 5 claps,

10 stairs Count down as 5 buns are eaten Count up money in 10p pieces

Know that 6 is smaller than 8 and

bigger than 2 Being able to see that a group of objects contains

about 10 Ordering 10, 4, 9, 2

from largest to smallest

Composition: Understanding how each number can be made up in different ways by

adding and subtracting Knowing how our

number system uses groups of tens and ones

Recognising that 6 = 2 + 4 or 7 – 1

Understand that 15 can be made from one ten and 5 ones, 15 ones

Numbers Some children enjoy and are good at counting. Others have less experience. All will do well if we provide a rich environment of situations and problems that the child finds engaging.

Concrete resources

Concrete – students should have the opportunity to use concrete objects and manipulatives to help them understand what they are doing.

Manipulatives/concrete resources are vital because they: • Help children to make sense of arithmetic • Help teachers see what children understand • Increase children’s engagement and enjoyment • Develop visual images and understanding • Help children to work together and share ideas • Are tools to help children solve problems,

investigate patterns and relationships, demonstrate results and reasoning.

Manipulatives and visual representations ‘open the door’ to conceptual understanding and should be used with all children. This will then lead to the mastery understanding that the new National Curriculum requires. Teaching rules alone does not give children the conceptual understanding that they need.

Embed manipulatives/Concrete into play activities inside…

Pictorial

Pictorial – students should then build on this concrete approach by using pictorial representations. These representations can then be used to reason and solve problems.

Children do not naturally think with mathematical concepts, they use their own ‘concept images’

Imagine a child in a Reception or KS1 class. What do you think his/her concept image of 5 will include?

Mathematical concepts

A concept image of 5

Challenge 1 • 12 + 15 =

• 12 + 11 = • 15 + 14 = • 17 + 12 = • 13 +13 = • 15 + 14 =

Challenge 2 • 16 + 15 =

• 17 + 18 = • 16 + 19 = • 19 + 13 = • 17 + 15 = • 15 + 17 =

Challenge 3 • 31 + 25 =

• 82 + 23 = • 66 + 25 = • 59 + 22 = • 177 + 146 = • 165 + 132 =

Abstract With the foundations firmly laid, students should be able to move to an abstract approach using numbers and key concepts with confidence.

Well known mental calculation strategies

• Partition and recombine • Doubles and near doubles • Use number pairs to 10 • Adding near multiples of ten and adjusting

• Using patterns of similar calculations

• Using known number facts • Counting on

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

Build it

Draw it

Write it

5 + 0 = 5 4 + 1 = 5 3 + 2 = 5 2 + 3 = 5

1 + 4 = 5

The four calculations

Addition Subtraction Multiplication Division

Understanding the equals sign

equivalent

balance

the same as

equal =

equivalent

balance

the same as

equal

Not the answer to a calculation!

Equivalence

10 = 6 +

Addition

Plus

Add

More

Commutitative

Using the numicon on your table can you show

me 10? A concrete resource for the children to use.

How many different ways can you make 20?

Pictorial

Whole

Part

Part

A pictorial resource.

The bar model

0 + 10 = 10 1 + 9 = 10 2 + 8 = 10

3 + 7 = 10 4 + 6 = 10 5 + 5 = 10

6 + 4 = 10 7 + 3 = 10 8 + 2 = 10

9 + 1 = 10 10 + 0 = 10

FILL IN THE MISSING NUMBERS!

Abstract

Here are some cubes. A concrete resource

for the children to use. A pictorial

representation on the white board.

At your tables show me how you would use them to work out 5 take away 1.

A concrete resource for the children to use.

How could you record this as a number sentence?

5 – 1 = 4 Well done!

After lots of practise, the children will be ready to move onto the abstract.

Say what you see! What if 2 swim away? What if 5 swim away?

What if 5 more come along? There are an odd number of fish, how

many could there be?

Exploring numbers 9 to 20

10 – 2 =

Using a number line on the white board is a pictorial method to help the children with the calculation.

Multiplication

Times

Groups

Array

Equal

How can you express the amount of apples?

1.

What is an equal group?

What is an Array?

Using the counters at your table can you show

me a array?

How can you express what you can see?

This a pictorial and abstract representation.

I can see 1, 2, 3, 4…12

I can see 3 rows of 4

If you go round the other side, it’s 4 rows of 3

I can see 4 and 4 and 4

I can see 3 and 3 and 3 and 3

It’s 4 trebled!

2

A pictorial representation of counting in 2’s.

4

6

8

Division

Share

Equal

Groups

Early fractions

Children have a basic understanding of fractions from reception age or younger. They can:

Make half a turn, Fill containers half full of sand and water, Share chocolates/counters/anything between two so have half each, Move a minute hand of a clock half way round the clock face Break a bar of chocolate into two equal pieces

We need to build on from these early experiences! Whenever we share we need to link to fractions. If we share 12 between 3, each has four or 1/3, two have 6 or 2/3

Get 10 cubes

How can you share them between you and your friend?

How can you share the sweets between you and your friend?

Now have a go at these

10 ÷ 2 = 15 ÷ 3 = 20 ÷ 5 =

What do we use to plan our schemes of learning?

https://vimeo.com/196410551

Teaching for Mastery (White rose)

• These overviews are designed to support a mastery approach to teaching and learning and have been designed to support the aims and objectives of the new National Curriculum.

• The overviews; • Have number at their heart. A large proportion

of time is spent reinforcing number to build competency

• Ensure teachers stay in the required key stage and Support the ideal of depth before breadth.

• Ensure students have the opportunity to stay together as they work through the schemes as a whole group.

• Provide plenty of time to build reasoning and problem solving elements into the curriculum.

Fluency

• By using the concrete, pictorial and abstract the children will become fluent when working with numbers and the methods.

5 + = 10 10 + = 20 17 + = 23

Match the words to the correct clock:

Reasoning

• The children will begin to explain their answers and give reasoning for the method they used.

Thinking Mathematically

Posing questions…

I wonder…

How many number sentences can you make?

0 1 2 3

How do you know you have all of them? + =

I have got all of the number sentences because…

How do you know?

Holly arrived at school at 8 o’clock. Megan arrived 9 minutes past 8. Holly says, “I arrived earlier.” Do you agree? Explain why.

Solving a problem involves…

• Identifying and understanding what the problem is

• Planning how to solve it • Working out the answer • Checking results

• Peter is eating his lunch at half past 12. Jane is eating her lunch half an hour later. Tick the clock which is showing when Jane eats her lunch.

Alan baked 16 cookies. He gave 14 of them away. How many are left?

What can you do at home?

• Fluency practise using suggested websites.

• Use maths in every day life. • Ask your child how they worked out the answer – what method did they use?

• Make sure they are thinking deeply.

• Make maths fun!