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Deep Learning in KS2 Mathematics

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Gareth Metcalfe

on 25 September 2015

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Transcript of Deep Learning in KS2 Mathematics

Deep Learning in KS2 Mathematics
Patterns of Mathematical Thinking
Grasp the problem
Mastery Structure
Top Talk
Creating a Mathematical Culture
Deep Representations
Meta-Cognitive Thinking
Effective Feedback

Approach Independently
Read, think, draw, form strategy, attack
Focused Talk
Compare strategies, thoughts, outcomes
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
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
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
'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
Need five 8s as units, therefore 5 numbers
The Bat and Ball Problem:
Strike it out
Factors and multiples
How many pencils are there in all of the primary schools in Middlesborough?
Test understanding



7x6 = 30+12

9x4 = 24+12
Address misconceptions
3 groups - how many in each group?
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
What I know, identify the difficulty
Build on partner's understanding
Visualise: drawings or equipment
16x9: how many ways?
Dice estimation games
Target 300
Place value counters
What is the largest number that, when rounded to the nearest 100, is 2000?
Group question
Division strategies
Full transcript