The principles and pragmatics of teaching maths to …...The principles and pragmatics of teaching...
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The principles and pragmatics of teaching maths to students with
learning difficulties and dyscalculia.
Steve ChinnOslo
Nov. 12th , 2018
www.mathsexplained.co.uk
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What do we know about dyscalculia, maths learning difficulties and teaching?
What affects learning?
What does maths demand from the learner?
Do beliefs and the culture of maths impact on learners?
example: Speed of working
Why don’t learners learn?
What is dyscalculia?
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Dyscalculia. (UK). 2001
Dyscalculia is a condition that affects the ability to acquire mathematical skills.
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Dyscalculic learners may have a difficulty
understanding simple number concepts, lack an
intuitive grasp of numbers, and have problems learning number facts and procedures.
Even if they produce a correct answer or use a
correct method, they may do so mechanically and without confidence.
Just dyscalculics?
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Learner skills Maths
An ability to work with
symbolsMaths vocabulary
STM
Long term maths
memory
WMSpeed of
processing
Flexible cognitive style
Sequencing/reversing
Generalising/patterns It’s complicated!
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anxiety
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Communication in the classroom.
An ability to work with
symbols (and their absence)Maths vocabulary
STM
Long term maths
memory
WMSpeed of
processing
Flexible cognitive style
Sequencing/reversing
Generalising/patterns
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anxiety
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Pre-natal number sense
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When does it begin?
Maths builds.
Concepts are developed and extended.
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Intervention 1
Going back. How far?
Linking to reinforce.
Inconsistencies.
Pre-empting/addressing misconceptions
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Intervention 2
What else are you teaching?
‘How did you do that?’
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‘How People Learn’ (2000) National Research
Council, USA
Key Finding 1
Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged they may fail to grasp the new concepts and information that they are taught, or they may learn them for the purposes of a test,
but revert to their preconceptions outside the classroom.
(Buswell and Judd, 1925)
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inhibiting
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‘Take the little number from the big number’
33
- 16
23
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Be careful what you teach.
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The beginnings.
One digit numbers.
0 1 2 3 4 5 6 7 8 9
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Subitising and counting
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counting
chunking
subitising
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Early teaching of number sense?
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Is it bigger or smaller?
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Maybe
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Numerical stroop and symbols
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8 2
4 9
7 5
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One digit numbers.Counting
0 1 2 3 4 5 6 7 8 9
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1 2 3 4 5 6 7 8 90
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Counting forwards. Adding 1 each time
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9 8 7 6 5 4 3 2 1 0
Counting back.
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Subtracting 1 each time
Reversing
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What else are you teaching?Maths develops.
Some key links between numbers
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For some students it is good to practise these links using counters
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A triangle 3 = 2 + 1
3 = 1 + 2
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3
The commutative property
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A square 4 = 2 + 2
4 = 2 x 2
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4
Linking + and x4 = 22
Symbols
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5 1+
6
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7
5 2+
Chunking.
Using the key numbers to
develop numbersense.
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Place value
2-digit numbers
and beyond
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Why is our system for writing numbers based on ten?
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How do we write numbers that are bigger than ten?
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Base ten (or Dienes) blocks
Pick your materials (and visuals) with care.
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Do your students understand what they represent?
Present the symbols alongside the materials
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Two key moments in the base 10 system
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nine ten 9 10
tennine 109
+1 10 - 19
We use two (or more) digits to show numbers that are bigger than 9
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A key link between 9 and 10
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ten 10
9 is 1 less than 10
nine 9
10 - 1 estimation
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ten ones = one ten. Visual images
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ten ones
1 tenVocabulary:
Renaming
Trading
Carrying
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The digits and where they are placed
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1 0
place value
and
zero
one ten
zero ones
one ten
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The digits and where they are placed
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place value
eleven
11
one ten one (one)
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Some (English) vocabulary issues: eleven and twelve
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one ten, one
one ten, two (ones)eleven (elleve)
11 twelve (tolv)
12
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The teen numbers 13, 14, 15, 16 …
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13
14
15
16
More vocabulary issues
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Backwards/Direction. Consistency.
Elephant Tnahpele
15
fifteen
(femten)
51
fifty-one
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The ‘teen’ numbers
tens ones
1
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3
1 4
1 5
1 6
thirteen (tretten)
fifteen (femten)
fourteen
sixteen
Place value columns
Scaffolding the concept
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Other 2-digit numbers
tens ones
2
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1
2 2
2 3
2 4
twenty one
twenty three
tjue to twenty two
tjue fire twenty four
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Hundreds 100 Tens 10 Ones 1
Subtracting across zero, an example: 103 - 28
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1 3
01
0
39- 2 8
910
7
1
5
1
1
90 + 13 = 103
5
7
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Subtracting across zero: 103 - 28
H T U1 0 3
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10 3
- 2 89 13
7 5
1 0 3
10 3
- 2 8
9 13
7 5
Mental arithmetic
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Basic addition and subtraction facts
and number sense
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‘How People Learn.’ NRC (2000).
Key Finding 2
• To develop competence in an area of enquiry, students must:
• (a) have a deep foundation of factual knowledge,
• (b) understand facts and ideas in the context of a conceptual framework, and
• (c) organise knowledge in ways that facilitate retrieval and application.
• (Use patterns and generalisations)
0 1 2 5 10 20 50 ……
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Why is maths a great subject?
• Reason 1
• Because of (b) and (c)
• Why can it be a bad subject?
• Because we don’t make enough use of (b) and (c)
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Number combinations for 10
5
5 + 5 = 10
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+ 5 = 10
0 1 2 5 10 …
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Number combinations for 10
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= 106
= 10+ 64
+ 4
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The number combinations for 10
10 + 0 =
9 + 1 =
8 + 2 =
7 + 3 =
6 + 4 =
5 + 5 =
10
0 + 10 =
1 + 9 =
2 + 8 =
3 + 7 =
4 + 6 =
5 + 5 =
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Number combinations for 10
Using only symbols
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
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Exam tip
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Number combinations for 10. Subtraction.
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= 610
10 - 6 = 4
- 4
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Doubles 7 + 7 = 14
14
10
4
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Doubles 9 + 9 = 18
18
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10 109 9
20 - 2
What else
are you
teaching?
9 + 7
9 + 1 + 6
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A break for some meta-cognition and
cognitive style.
www.mathsexplained.co.uk
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‘How People Learn’
Key Finding 3 (NRC, 2000)
• A ‘metacognitive’ approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.
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Cognitive (Thinking) Style
Riding and Rayner (1998) define
cognitive (thinking) style as:
• “A person’s preferred and habitual approach to organising and representing information.”
• Skemp. Instrumental and relational
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‘Thinking, Fast and Slow’ (2012) Kahneman, D., Penguin
• System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control.
• System 2 allocates attention to the effortful mental activities that demand it, including complex calculations. The operations of System 2 are often associated with the subjective experience of agency, choice and concentration.
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Time to check you out
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You are driving a bus from London to Pembroke:
At London 17 people get on
At Reading 6 people get off and 9 get on
At Swindon 2 people get off and 4 get on
At Cardiff 11 people get off and 16 get on
At Swansea 5 people get off and 3 get on
What is the name of the driver?
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Mental Arithmetic
223 + 98
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Mental Arithmetic
223 – 98
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Mental Arithmetic
2 x 4 x 3 x 5
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Mental Arithmetic
25 x 96
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Word Problem
•Red pens cost 17c. Blue pens cost 13c. If I buy two red pens and two blue pens, how much do I pay?
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There are 49 squares in this figure.
• How many are green?
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There are 25 squares in this figure.
• How many have an X ?
X
X XX
X X X
X X X X X
XXX
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The ‘Horse’
• How many light blue squares in the ‘horse’?
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‘How did you do that?’
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What else can you learn as an
assessor/teacher?
www.mathsexplained.co.uk
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The Inchworm
• Focuses on the parts and details… separates
• Looks for a relevant formula or procedure
• Constrained focus. Wants one method
• Works in serially ordered steps… forward
• Uses numbers exactly as given
• More comfortable with paper and pen… documents
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The benefits of algorithms (Usiskin, 1998)
• Power: An algorithm applies to a class of problems.
• Reliability and accuracy: Done correctly, an algorithm always provides the correct answer.
• Speed: An algorithm proceeds directly to the answer.
• A record: A paper and pencil algorithm provides a record of how the answer was determined.
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The Inchworm
• Unlikely to check or evaluate answers.
• If a check is done, it will be done using the same method.
• May not understand the method or procedure used. Works mechanically.
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Teacher-identified math weaknesses. Bryant,
Bryant and Hammill, JLD, 2000
• Ranked 4th….
• Fails to verify answers and settles for first answer
• Ranked 9th ….
• Reaches ‘unreasonable’ answers
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The Grasshopper
Better prepared for life?
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The Grasshopper
• Overviews, puts together, holistic
• Looks at the numbers and the facts to restrict down or estimate an answer. ‘This is not guesswork, it is controlled exploration.’
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The Grasshopper
• The methods used are flexible
• Often works back from a trial answer
• Adjusts, breaks down/builds up numbers, finding the easy numbers
• Performs calculation mentally. Rarely documents.
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6 year old pupil
• 32 - 19 = 27
• ‘Show how you worked out your answer’
27
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The Grasshopper
• Is likely to appraise and evaluate the answer
• Checks by a different method
• Has a good understanding of numbers, their inter-relationships and the operations and their inter-relationships.
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Cognitive Style – Non-Dyslexics
NL
England
Ireland
Inchworm
Grasshopper
Chinn et al (2001) Brit J of Special Ed, v28, #2copyright: steve chinn 2018
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Cognitive Style – Dyslexics
NL
England
Ireland
Inchworm
Grasshopper
Chinn et al (2001) Brit J of Special Ed, v28, #2copyright: steve chinn 2018
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Thinking Style and Dyscalculia
• Inchworm• Memories: Ltm, WM
• Speed of working
• Sequential skills
• Anxiety BUT they do document
• Grasshopper• Careless/inaccurate
• No documentation
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Back to teaching ……
Basic multiplication and division facts,
and operation sense
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expectations
unanxious
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An over-reliance
on rote learning
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Rote learning and self-voice echo.
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‘Nothing works for everyone.’
Modes of presentation.
Frequency of presentation.
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The four operations. + - x ÷
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56 ÷ 8 - 8 - 8 - 8- 8- 8 - 8- 8
6 x 6 6 + 6 + 6+ 6+ 6 + 6
vocabulary
chunk partial product
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To: 3x 4x 6x 7x 8x 9x 11x 12x
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From: 1x 2x 5x 10x
12 x 12
12 + 12 + 12+ 12 +12 + 12 + 12+ 12 +12 + 12 + 12+ 12
10 x 12 2 x 12
144
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Grouping the numbers: partial products
Using the ‘Key Facts’: 1x 2x 5x 10x
3 x 8 8 + 8 + 8
6 x 8 8 + 8 + 8 + 8 + 8 + 8
7 x 8 8 + 8 + 8 + 8 + 8 + 8 + 8
12 x 8 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8
9 x 8 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 +8 (-8)
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Multiplication as repeated addition and the area model
8 + 8 + 8 3 x 8
8 + 8 + 8 + 8 + 8 + 8 + 8 7 x 8copyright steve chinn 2018
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Multiplication facts: partial products
3 x 8 = 2x8 + 1x8
7 x 8 = 5 x 8 + 2x8
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Algebra for generalising
8 + 8 + 8 + 8 + 8 + 8 + 8 = 7 x 8
a + a + a + a + a + a + a = 7a
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The area model for y = ab
12 x 12
12 x 10 12 x 2
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The area model for y = ab
a x b
a x s a x t
b = s + t
a(s + t)
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Developing the procedures for
multiplication and division.
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Multiplication using key number partial products
67 x 34
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2034
67
34
67 x 20 =
2
134
2278
6701340
67 x 10 =67 x 2 =
67 x 2 =
= 20+ 10 + 2 + 2
10 2
134
2000
‘Everything can be made as simple as
possible, but not simpler than that.’
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Analyse the pre-requisites
What does the learner need?
So, let’s broaden our view of division
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Simple division
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72 ÷ 3
3 7 22
2 x 3 = 67 – 6 = 1
1110
How many 3s in 12?
12 ÷ 3
4
Vocabulary, language,
sequences, organisation
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72 ÷ 3 by chunking
10 x 3 = 30
72 – 30 = 42
42 – 30 = 12
2 x 3 = 6
12 – 6 = 6
6 – 6 = 0
2 x 3 = 6
Repeated
subtraction
10 x 3 = 3024
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Division and the number line
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How many 17s in 1207?
0 200 400 600 800 1000 1200 1400 1600 1800 2000
1700
1207 ÷ 17
1207
850
17 x 100 = 1700
17 x 50 = 850
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Division as repeated subtraction,using core/key fact partial products (chunks)
Using repeated subtraction of partial products:
1207
- 850 50 x 17
357
-340 20 x 17
17
-17 1 x 17
The total number of 17s is: 50 + 20 + 1 = 71
Initial jottings might be:
100 x 17 = 1700(too big)
50 x 17 (half of 100 x 17) = 850
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1207 ÷ 17
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Fractions.
Symbols, vocabulary and ‘inconsistencies’.
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half 1
2 quarter 1
4
third 1
3
Paper folding/cutting
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Fractions: getting smaller
1
2 1
1
3
1
21
3
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1
4
consistency
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The hidden ÷ symbol
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1
2
1 2
1 ÷ 21
2
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The way a fraction is written tells you what it means …. Symbols are sometimes invisible.
21
3
23
3
3
1
3
2
3
x
23
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half 1
2
quarter
two thirds 2
3
Paper folding/cutting. fraction x fraction
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half of a half
1 x 1 = 1
2 2 4
2 x 2
3 3
2 x 2 = 4
3 3 9
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Teach fractions, decimals and percentages together to link the
concepts and improve understanding
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Comparing and ordering whole numbers
0 1 2 3 4 5 6 7 8 9 10
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Comparing and ordering decimal numbers
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
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1
Comparing and ordering
fractions (by their value)
1
10
1
91
8
1
51
4
1
31
2
1
7
1
6
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Fractions and Decimals
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1/2 = 0.5
1/5 = 0.2
1/3 = 0.33
1/4 = 0.25
1/10 = 0.1
1
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Research and Education.‘Visible Learning’ Hattie, 2009
The highest effects accrued when teachers provided feedback data or recommendations to students. ‘How did you do that?’
The programmes with greatest effect were strategy based methods.
Least effective were using technology for independent practice, and the strategy of working within a peer group.
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The elephant in the classroom.
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Anxiety and the affective domain
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Why is maths a great subject?
Reason 2
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www.mathsexplained.co.uk
A collection of low cost on-line
video tutorials for maths.
For students,
parents,
adults,
teachers,
Learning Support Departments,
training.
copyright steve chinn 2018
A complete manual to
identifying and diagnosing
mathematical difficulties.
2nd edn. 2017