**Deep Learning in KS2 Mathematics**

**Patterns of Mathematical Thinking**

**Grasp the problem**

**Mastery Structure**

Assess

**Top Talk**

**Creating a Mathematical Culture**

**Deep Representations**

Meta-Cognitive Thinking

Effective Feedback

Collaboration

Challenge

Meta-Cognitive Thinking

Effective Feedback

Collaboration

Challenge

Approach Independently

Read, think, draw, form strategy, attack

Focused Talk

Compare strategies, thoughts, outcomes

Self-Reflection

What have I learnt?

How does this compare to other questions?

**Have a growth mindset**

**Use your whole brain**

**Be active to understand**

**Adapt to the situation**

**See the connections**

Enjoy challenges, learn from mistakes, celebrate effort, control negative emotions

Explore creative ideas, work logically, check work

Take steps to help you see a problem more clearly

Use different strategies, make choices, try different ways

Think about how different mathematical ideas and strategies can be linked,

Address Misconceptions

Quick Tricks

Stretch Learning

target misconceptions

promote different approaches

physical pictorial abstract

learning journey

same idea, different ways

**Attack, attack, attack**

**Review the answer**

What was the key to solving the problem?

Did I use the easiest method? Can I see other ways? Did I switch strategies?

How was this question similar/different to other problems I have done?

Can I use the same method to solve other problems?

What have I learned in general about solving problems?

Understanding is an active process

If needed, try different methods and ideas

Do I need a new strategy? Do I trust my answer?

Make it my own: imagine, act, draw, label, simplify

Have I solved a problem like this before?

Can I use a similar approach?

Do the easy parts first

Is the answer realistic and in the right form?

Am I confident that each step is correct?

Is there a different route to check the answer?

Have a system with an order

Guess, check, learn

Work backwards

Make an 'easy steps' list

Answer an easy version of the problem

Narrow down possible answers

Extension Pathway

There are 500 paperclips in a box. The box weighs 400g when it is has all of the paperclips in, and 100g when it is empty. How much does each paperclip weigh?

(a) 1.67g

(b) 1.25g

(c) 0.6g

Target Common Misconceptions

Show Learning Journey

Same idea, different ways

A skydiver jumps from her helicopter when she is 3000m above ground level. She free-falls three-quarters of the way, then opens her parachute. How far above ground level is she when she opens her parachute?

Grasp the problem

*Narrow down possible answers

*Have a system with an order

*Narrow down possible answers

*Have a system with an order

I learn to...

hidden info

physical - pictorial - abstract

There are 4 fewer boys than girls in Mr Hill's class. There are 18 girls in the class. How many boys are there?

(level 2c)

In a class, 18 of the children are girls.

One-quarter are boys.

How many children are in the class?

(level 5)

multiple representations

reason and generalise

Multiplication structures

value strategies

(and know the limitations of counting)

The Mathematics

Apprenticeship

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.

Purpose of study:

Jo Boaler arguing against ability grouping at primary school:

Research has shown that when children get put in low sets 88% of them stay there until they leave school.

The fact that our children's future is decided for them at an early age derides the work of schools and contravenes basic knowledge about child development and learning.

Teaching to solve problems is an education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears. If the student

has no opportunity in school to familiarise himself with the varying emotions of the struggle for the solution his mathematical education failed at the most vital point.

'I know that 117-58=59. How can I use this to work out 115-56?

'If you divide a number by 3 three times, you will always get a decimal'

'All odd square numbers greater than one have 3 factors'

15 x 15 = 225

16 x 14 = 224 (-1)

17 x 13 = 221 (-4)

18 x 12 = 216 (-9)

19 x 11 = 209 (-16)

20 x 10 = 200 (-25)

*Make an 'easy steps' list

Raisins cost 70p for 100g. How much will a 320g bag of raisins cost?

100g = 70p

300g = £2.10

10g = 7p

20g = 14p

£2.10 + 14p = £2.24

What is the difference between -7 and 17?

-6 subtract -7

-16, -11, -6, -1, ___, ___

-67, ___, ___, ___, 13

Same idea, different ways

600, 593, 586...

What are the first two negative numbers in this sequence?

How many times would you have to halve 128 to get a negative number?

___+___+___+___ = 20

big

big

+1

Use real-world understanding

A toy bus is one-hundredth the size of a real bus. The toy bus is 4cm in height. How high is the real bus?

What is ¼ as a decimal?

£5 – £2.99

7 = ___ - 9

3/4 of a number is 60. What is the number?

Target Common Misconceptions

Grasp the problem

Raisins cost 60p for 100g. How much will a 250g bag of raisins cost?

*Make an 'easy steps' list

100g = 60p

200g = £1.20

50g = 30p

£1.20 + 30p = £1.50

*Narrow down possible answers

*Have a system with an order

Target Common Misconceptions

Use real-world understanding

A real London bus is ten times bigger than Harry's toy bus. The toy bus is 40cm in height. How high is the real bus?

(a) 10 degrees

(b) -50 degrees

(c) 50 degrees

Do an easy version of the problem

Do an easy version of the problem

Do an easy version of the problem

A bat and ball cost £1.10 in total.

The bat costs £1 more than the ball.

How much does the ball cost?

a + b = 150

a is twice as big as b

Calculate the values of a and b

First Class Maths

Example 1:

The 4-digit code works well for someone born on 1st December 1994 (5981), but less so for someone born on 31st January 1949 (456.25) both in that it is too small and produces a decimal. Code is clearly displayed with a good number of variables; just consider the use of 1/4.

Example 2:

I loved the use of the Greek alphabet code, particularly with a variable that is multiplied to the power 4 as it doesn't get too large. You only effectively had two variables, as B=A-1.

The code varies daily, which is great. It never gives more than a 4-digit product, but a 21 year-old on Friday would have the code 501 - just too small.

Example 3:

The four-digit code is amazing. Always a 4-digit code (great use of expiry date), with lowest possible 1408 and highest 9232, a great range. And with 3 variables, varying daily. Outstanding.

49p x 6

63 - 18

Reason and Generalise

Convince yourself

Convince a friend

Convince a sceptic

Ken Ken

Ken Ken

Nim-7

'Always leave your partner on a multiple of 3, then you have control.'

'If it was Nim-122, I would want to go first. If we we playing Nim-123, I would want to go second.'

Carl Freidrich Gauss

Use of Technology

Highly effective teachers of numeracy were characterised by a particular set of beliefs, which in turn led to a corresponding set of teaching approaches. The mathematical and pedagogical purposes behind particular classroom practices are as important as the practices themselves in determining effectiveness.

What makes great teaching? Sutton Trust Report 2014, citing research by Askew et al

What makes great teaching? Sutton Trust Report 2014, citing research by Askew et al

In particular, beliefs about the nature of mathematics and what it means to understand it, along with teachers' beliefs and theory about how children learn and about the teacher's role in promoting learning, are important distinguishing factors between those who were more or less effective.

Eight 8s

888 close to 1000

88 leaves 24

8x3=24

Need five 8s as units, therefore 5 numbers

The Bat and Ball Problem:

£0.05

£1.05

£1

Strike it out

Factors and multiples

youcubed.org

How many pencils are there in all of the primary schools in Middlesborough?

Test understanding

7x2+7

7x2+2

6+6+6+12

5+7+7+9

7x6 = 30+12

9x4 = 24+12

Address misconceptions

3 groups - how many in each group?

sharing

grouping

3 in each group - how many groups?

How many... in...

Minibuses can hold 23 people. How many minibuses are needed to transport 130 children?

Vary structure and depth

75 - 28 = __

__ - 36 = 25

45 - __ = 17 + __

23 - __ __ x 4

>

how many ways?

**Explore concepts**

**Intelligent practice**

**Active assessment**

Open tasks, potential for deepening

Visual representations

Mixed ability groupings

Purposeful questioning

Address Misconceptions

Response matched to need

Deepen learning

Upskill

What I know, identify the difficulty

Build on partner's understanding

Visualise: drawings or equipment

90p

£0.10

£1

16x9: how many ways?

10x9=90

6x9=54

90+54=144

16x10=160

160-16=144

16

16

16

16

16

16

16

16

+16

---

144

4x4x9=144

8x9x2=144

16x3x3=144

12x9=108

4x9=36

108+36=144

16x5=80

16x4=64

80+64=144

Dice estimation games

Target 300

Place value counters

What is the largest number that, when rounded to the nearest 100, is 2000?

Group question

500-200

500-201

500-199

499-198

Division strategies