Mike conference presentation

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

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

72 - 29 = 43: 

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?

72 - 29 = 43: 

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.

Key question: 

Key question How can we harness children’s natural ability to learn to help them to make sense of mathematics (as opposed to lessons)?

Success in maths: 

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

Success in maths: 

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

Exclusion? : 

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

Inclusion - source of difficulty?: 

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

MaTCh: 

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

Culture: 

Culture Lack of belief in children’s ability to think and be creative ‘Empty vessels’ model of teaching Low expectations, especially about children’s language

Lesson narratives: 

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?

Molly: 

Molly

Dylan: 

Dylan

Kirsten: 

Kirsten

Fractions: 

Fractions Pupils should be taught to: Find fractions of numbers or quantities As outcomes, Year 4 pupils should, for example: Begin to relate fractions to division. For example: Recognise that when 1 whole cake is divided equally into 4, each person gets one quarter.

What fraction?: 

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

1/2, 1/4, 1/8?: 

1/2, 1/4, 1/8?

1/2, 1/4 or 1/8?: 

1/2, 1/4 or 1/8?

More fractions: 

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.

Learning paradox: 

Learning paradox Shift from: Understanding what we see To: Seeing what we understand

Number nick-names: 

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

Can you find some others?: 

Can you find some others?

Can you find some others?: 

Can you find some others?

I’ve found another: 

I’ve found another

I’ve found another: 

I’ve found another

Pizzas: 

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?

Pizzas 1: 

Pizzas 1

Pizzas 2: 

Pizzas 2

Pizzas 3: 

Pizzas 3

Equivalence: 

Equivalence

Mike’s party: 

Mike’s 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?

Mike’s party 1: 

Mike’s party 1

Mike’s party 2: 

Mike’s party 2

Teaching tripod: 

Teaching tripod Tasks Talk Tools

Task and activity: 

Task and activity TASK: What the teacher or textbook asks for ACTIVITY: What the learner does

Think of a time:: 

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

Activity too close to task: 

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

Artefacts and tools: 

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

43 - 15: 

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.

Power of artefacts into thinking tools: 

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

Non-matching pair: 

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

Challenge: 

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

Red or yellow?: 

Red or yellow?

Red or yellow?: 

Red or yellow?

Red or yellow?: 

Red or yellow?

Red or yellow?: 

Red or yellow?

Success?: 

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

Key tools: 

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

10 bead string: 

10 bead string

10 bead string: 

10 bead string

Multiplication: describing arrays: 

Multiplication: describing arrays • • • • •  • • • • •  • • • • • 

Torn grids (1): 

Torn grids (1)

Torn grids (2): 

Torn grids (2)

89 x 75: 

89 x 75

Reduced scale: 

Reduced scale

15 x 14 - child’s extension: 

15 x 14 - child’s extension

9 x 22: 

9 x 22

34 x 9: 

34 x 9

Orientation: 

Orientation

Ratio tables: 

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

Ratio tables: 

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

Ratio tables: 

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

Marathon: 

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?

Talk: 

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

Pupil needs: 

Pupil needs Attention Not be patronised by ‘easy’ work Feel good by belittling each other

Developing cooperative tasks: 

Developing cooperative tasks Parallel pairs of calculation chains Solver and recorder ‘Clue’ problems

Parallel calculation chains: 

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

Solver and recorder: 

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

‘Clue’ problems: 

‘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.

Ticket sales: 

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.

Promoting cognitive conflict?: 

Promoting cognitive conflict? ‘Action’ of problem: taking away 5003 – 4997 Mathematically effective: finding difference 4997 +  = 5003

Ticket sales 1: 

Ticket sales 1

Spontaneous reflection: 

Spontaneous reflection ‘The silliest thing that I did in that lesson was to agree with the first answer’

Ticket sales 2: 

Ticket sales 2

Ticket sales 3: 

Ticket sales 3

Can that be correct?: 

Can that be correct? ‘That’s not taking away, that’s finding the difference’

MaTCh emphases: 

MaTCh emphases Cooperative problem solving based around simple contexts and accepting children’s 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

Including children’s ‘voices’: 

Including children’s ‘voices’ Silence External authority Author(ity)

Maths is like... : 

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

Maths is like...: 

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

Maths is like... : 

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

Maths is like ... competition: 

Maths is like ... competition mike@mikeaskew.net