Title: Mathematics, Teachers and Children: Finding meaning in mathematics learning and teaching
1Mathematics, Teachers and Children Finding
meaning in mathematics learning and teaching
- Mike Askew
- Kings College London
- Independent writer and consultant
272 - 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?
372 - 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.
4Key question
- How can we harness childrens natural ability to
learn to help them to make sense of mathematics - (as opposed to lessons)?
5Success in maths
- What do CHILDREN take as measures of success in
maths?
6Success in maths
- What do CHILDREN take as measures of success in
maths? - Speed
- Minimal effort
- Knowing what to do quickly
- Right answers
7Exclusion?
- Time?
- Pressure?
- Pecking order?
- Objectives driven lessons?
- Rule following?
- Abstract calculations?
- Shame?
8Inclusion - 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
9MaTCh
- Large urban school
- Mixed intake racial heritage
- Low socio-economic status
- Failing school (as measured by national testing)
10Culture
- Lack of belief in childrens ability to think and
be creative - Empty vessels model of teaching
- Low expectations, especially about childrens
language
11Lesson 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
15Fractions
- 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.
16What fraction?
- 2/5
- 3/5
- 2 1/2 1 1/2 2/3 1 2/3
171/2, 1/4, 1/8?
181/2, 1/4 or 1/8?
19More 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.
20Learning paradox
- Shift fromUnderstanding what we see
- ToSeeing what we understand
21Number 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
22Can you find some others?
23Can you find some others?
24Ive found another
25Ive found another
26Pizzas
- 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?
27Pizzas 1
28Pizzas 2
29Pizzas 3
30Equivalence
31Mikes 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?
32Mikes party 1
33Mikes party 2
34Teaching tripod
35Task and activity
- TASK What the teacher or textbook asks for
- ACTIVITYWhat the learner does
36Think of a time
- When the hands of an analogue clock are exactly
at right angles to each other - And another
- And another
- ?
37Activity too close to task
- May encourage
- Dependency on teacher
- Performance orientation
- Lack of meaning
- SOLUTION? Treat problem solving as starting
point, not end point
38Artefacts 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
3943 - 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.
40Power of artefacts into thinking tools
- Four-year-olds were trained to mark pairs of
similar items with a yellow sticker.
41Non-matching pair
- They were trained to mark pairs of non-similar
items with a red sticker.
42Challenge
- Once they were proficient they were challenged,
without further training, to label pairs of pairs.
43Red or yellow?
44Red or yellow?
45Red or yellow?
46Red or yellow?
47Success?
- 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
48Key tools
- 10 and 20 bead number strings
- Empty number line
- Arrays for multiplication
- Ratio table for multiplication and division
- TALK
4910 bead string
5010 bead string
51Multiplication describing arrays
52Torn grids (1)
53Torn grids (2)
5489 x 75
55Reduced scale
5615 x 14 - childs extension
579 x 22
5834 x 9
59Orientation
60Ratio tables
- I am putting apples into bags. There are six
apples in each bag. I fill seven bags. How many
apples is that? - Multiplication
61Ratio 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
62Ratio 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
63Marathon
- Sam is running the London marathon
- Every two miles she drinks 100 ml of water
- How much does she drink over the 26 miles?
64Talk
- Paired cooperative work
- Children expressing their thinking
- Whole class explanations and discussions
- Children articulating their thinking
65Pupil needs
- Attention
- Not be patronised by easy work
- Feel good by belittling each other
66Developing cooperative tasks
- Parallel pairs of calculation chains
- Solver and recorder
- Clue problems
67Parallel 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
68Solver and recorder
- One book and pen between two children
- Take turns to explain to partner what to record
- Whole class sharing of methods
69Clue 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.
70Ticket 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.
71Promoting cognitive conflict?
- Action of problem taking away5003 4997
- Mathematically effectivefinding
difference4997 5003
72Ticket sales 1
73Spontaneous reflection
- The silliest thing that I did in that lesson was
to agree with the first answer
74Ticket sales 2
75Ticket sales 3
76Can that be correct?
- Thats not taking away, thats finding the
difference
77MaTCh 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
78Including childrens voices
- Silence
- External authority
- Author(ity)
79Maths 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
80Maths 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
81Maths is like...
- football on TV
- lots of tiny figures moving around aimlessly
- Viagra
- the more you take, the harder it gets
82Maths is like ... competition