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Mathematics, Teachers and Children: Finding meaning in mathematics learning and teaching

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Mathematics, Teachers and Children: Finding meaning in mathematics learning and teaching Mike Askew King s College London Independent writer and consultant – PowerPoint PPT presentation

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Title: Mathematics, Teachers and Children: Finding meaning in mathematics learning and teaching


1
Mathematics, Teachers and Children Finding
meaning in mathematics learning and teaching
  • Mike Askew
  • Kings College London
  • Independent writer and consultant

2
72 - 29 43
  • One day a man picked 72 bananas and gave away 29.
    How many were left?
  • A crocodile had 72 teeth and it was eating
    something when 29 teeth fell out. How many were
    left?
  • There were 72 sweets in a jar. 29 people guessed
    how many sweets there were and won. How many
    sweets did they each get?

3
72 - 29 43
  • This boy lived at 72 and his friend lived at 29
    and they went out to play and they ended up at 43
  • My brother is 29. My dad is 43 and my granddad is
    72
  • One day in school I was told to do this sum. 72 -
    29. I did it. It was right. I got a tick.

4
Key question
  • How can we harness childrens natural ability to
    learn to help them to make sense of mathematics
  • (as opposed to lessons)?

5
Success in maths
  • What do CHILDREN take as measures of success in
    maths?

6
Success in maths
  • What do CHILDREN take as measures of success in
    maths?
  • Speed
  • Minimal effort
  • Knowing what to do quickly
  • Right answers

7
Exclusion?
  • Time?
  • Pressure?
  • Pecking order?
  • Objectives driven lessons?
  • Rule following?
  • Abstract calculations?
  • Shame?

8
Inclusion - source of difficulty?
  • Curriculum
  • Maths arose out of problem solving
  • Children come to school as expert problem solvers
  • Problems motivate skill development and
    understanding
  • Children
  • Maths is a set of techniques to learn
  • Some children have difficulty accessing these
    techniques

9
MaTCh
  • Large urban school
  • Mixed intake racial heritage
  • Low socio-economic status
  • Failing school (as measured by national testing)

10
Culture
  • Lack of belief in childrens ability to think and
    be creative
  • Empty vessels model of teaching
  • Low expectations, especially about childrens
    language

11
Lesson narratives
  • Y2 Lesson began with looking at pairs of numbers
    that totalled 12 8 4, 6 6, 4 8
  • Children posed problems
  • 6 people go out for a Chinese meal. How many
    chopsticks do they use?
  • Here is a bag of 12 socks. How many pairs is that?

12
Molly
13
Dylan
14
Kirsten
15
Fractions
  • Pupils should be taught toFind fractions of
    numbers or quantities
  • As outcomes, Year 4 pupils should, for
    exampleBegin to relate fractions to division.
    For example
  • Recognise that when 1 whole cake is divided
    equally into 4, each person gets one quarter.

16
What fraction?
  • 2/5
  • 3/5
  • 2 1/2 1 1/2 2/3 1 2/3

17
1/2, 1/4, 1/8?
18
1/2, 1/4 or 1/8?
19
More fractions
  • Very low attaining Y4 at beginning of year
  • The hook - idea of nick-names
  • Colour rods - named by colour but also number
    nick-names that can change.

20
Learning paradox
  • Shift fromUnderstanding what we see
  • ToSeeing what we understand

21
Number nick-names
  • If I give the red rod the number nick-name 1,
    what can we call the white rod?
  • 2
  • 1
  • 1 1/2
  • 1/2

22
Can you find some others?
23
Can you find some others?
24
Ive found another
25
Ive found another
26
Pizzas
  • 12 friends went out for a pizza party.
  • It was near the end of the month, and when they
    put their money together, they had enough to pay
    for 8 pizzas.
  • They ordered 8 pizzas and shared them equally.
  • How much pizza did they each get?

27
Pizzas 1
28
Pizzas 2
29
Pizzas 3
30
Equivalence
31
Mikes party
  • Mike invited 4 people round for tea. So he
    bought 5 cakes.
  • Just as they were about to have tea, 3 more
    friends arrived.
  • Mike decided to share the cakes equally. How much
    cake did each friend get?

32
Mikes party 1
33
Mikes party 2
34
Teaching tripod
  • Tasks
  • Talk
  • Tools

35
Task and activity
  • TASK What the teacher or textbook asks for
  • ACTIVITYWhat the learner does

36
Think of a time
  • When the hands of an analogue clock are exactly
    at right angles to each other
  • And another
  • And another
  • ?

37
Activity too close to task
  • May encourage
  • Dependency on teacher
  • Performance orientation
  • Lack of meaning
  • SOLUTION? Treat problem solving as starting
    point, not end point

38
Artefacts and tools
  • Artefacts What the teacher presents or allows
  • Includes physical objects, symbols, diagrams
  • Tools Only when the learner interprets artefacts
    and can use them flexibly.
  • Working tools v thinking tools
  • Thinking tools affect the learner as well as the
    world

39
43 - 15
  • Jo wrote down 20 as his answer. How might he have
    got that?
  • Hint 1 - he used a 100 square
  • Hint 2 - Jo has a history of reversing digits.

40
Power of artefacts into thinking tools
  • Four-year-olds were trained to mark pairs of
    similar items with a yellow sticker.

41
Non-matching pair
  • They were trained to mark pairs of non-similar
    items with a red sticker.

42
Challenge
  • Once they were proficient they were challenged,
    without further training, to label pairs of pairs.

43
Red or yellow?
44
Red or yellow?
45
Red or yellow?
46
Red or yellow?
47
Success?
  • Yes, the subjects were able to use the artefacts
    to establish second level similarities and
    differences.
  • The artefacts had become thinking tools.
  • The subjects were chimpanzes

48
Key tools
  • 10 and 20 bead number strings
  • Empty number line
  • Arrays for multiplication
  • Ratio table for multiplication and division
  • TALK

49
10 bead string
50
10 bead string
51
Multiplication describing arrays
  •              

52
Torn grids (1)

53
Torn grids (2)

54
89 x 75
55
Reduced scale
56
15 x 14 - childs extension
57
9 x 22
58
34 x 9
59
Orientation
60
Ratio tables
  • I am putting apples into bags. There are six
    apples in each bag. I fill seven bags. How many
    apples is that?
  • Multiplication

61
Ratio tables
  • I am putting apples into bags. There are six
    apples in each bag. I have 42 apples. How many
    bags can I fill?
  • Division as repeated subtraction/grouping

62
Ratio tables
  • I am putting apples into bags. There are seven
    bags. I have 42 apples. If I put the same number
    of apples in each bag, how many is that?
  • Division as sharing

63
Marathon
  • Sam is running the London marathon
  • Every two miles she drinks 100 ml of water
  • How much does she drink over the 26 miles?

64
Talk
  • Paired cooperative work
  • Children expressing their thinking
  • Whole class explanations and discussions
  • Children articulating their thinking

65
Pupil needs
  • Attention
  • Not be patronised by easy work
  • Feel good by belittling each other

66
Developing cooperative tasks
  • Parallel pairs of calculation chains
  • Solver and recorder
  • Clue problems

67
Parallel calculation chains
  • Children unwilling to listen to other solutions
    if they had found an answer
  • Parallel calculation chains provided motive to
    explain
  • 137 6 147 5
  • 148 9 168 8
  • 356 5 447 6

68
Solver and recorder
  • One book and pen between two children
  • Take turns to explain to partner what to record
  • Whole class sharing of methods

69
Clue problems
  • Break set with fixed order of information given
    in written problems
  • Working in twos, threes or fours, each child
    given one clue.
  • Can share clues in any way, other than showing
    the others.

70
Ticket sales
  • Robbie Williams is performing in London.
  • Tickets sell quickly.
  • How many tickets are still on sale?
  • Clue 1
  • 5003 tickets are on sale.
  • Clue 2
  • 4997 tickets have been sold.

71
Promoting cognitive conflict?
  • Action of problem taking away5003 4997
  • Mathematically effectivefinding
    difference4997 5003

72
Ticket sales 1
73
Spontaneous reflection
  • The silliest thing that I did in that lesson was
    to agree with the first answer

74
Ticket sales 2
75
Ticket sales 3
76
Can that be correct?
  • Thats not taking away, thats finding the
    difference

77
MaTCh emphases
  • Cooperative problem solving based around simple
    contexts and accepting childrens initial
    solution methods
  • Introducing children to artefacts that can be
    tools for thinking bead strings, empty number
    lines, arrays and ratio tables
  • Children expressing themselves by explaining
    methods to peers - in small groups
  • Helping children articulate thinking and refining
    methods through presentation to class

78
Including childrens voices
  • Silence
  • External authority
  • Author(ity)

79
Maths is like...
  • cabbage
  • like it or loath it, it all depends on how
    it was served up to you as a child
  • tapioca
  • its unpleasant and you only ever get it in
    schools

80
Maths is like...
  • a woman
  • if you can understand it properly you
    might enjoy it more
  • a man
  • needs careful manipulation if you are to get
    any sensible result

81
Maths is like...
  • football on TV
  • lots of tiny figures moving around aimlessly
  • Viagra
  • the more you take, the harder it gets

82
Maths is like ... competition
  • mike_at_mikeaskew.net
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