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Talk in MathsTalk in MathsLJMULJMU
2525thth January 2010 January 2010
Sue KillenSue Killen
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Williams ReportWilliams Report ““Talk in mathematics should not be seen simply as Talk in mathematics should not be seen simply as
a rehearsal in class of the vocabulary of a rehearsal in class of the vocabulary of mathematics, ... It should extend to high-quality mathematics, ... It should extend to high-quality discussion that develops children’s logic, discussion that develops children’s logic, reasoning and deduction skills, and underpins all reasoning and deduction skills, and underpins all mathematical learning activity. The ultimate goal mathematical learning activity. The ultimate goal is to develop mathematical understanding – is to develop mathematical understanding – comprehension ofcomprehension of mathematical ideas and mathematical ideas and applications.”applications.”
Williams Report 2008Williams Report 2008
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OpportunitiesOpportunities Child to childChild to child
Child to teacherChild to teacher
Teacher to childTeacher to child
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A Climate for LearningA Climate for Learning SafetySafety – ethos that encourages – ethos that encourages
‘having a go’ and mistakes are ok‘having a go’ and mistakes are ok RelationshipsRelationships – meet and greet – meet and greet LearningLearning - pair/share, VAKE - pair/share, VAKE TimeTime - wait time, down time - wait time, down time
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PuppetsPuppets&&
PropsProps
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Barrier gamesBarrier games&&
Clue cardsClue cards
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Child to childChild to child
Working collaboratively to Working collaboratively to solve problemssolve problems
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Data Handling Data Handling
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Kennel ClubKennel Club Four dozen dogs live in 6 different coloured Four dozen dogs live in 6 different coloured
kennels. kennels. The smallest kennel has 6 dogs and the orange The smallest kennel has 6 dogs and the orange
kennel is the largest with 10. kennel is the largest with 10. The yellow kennel and the green kennel are the The yellow kennel and the green kennel are the
only ones with the same number of dogs. only ones with the same number of dogs. The 13 youngest pups are in the red and blue The 13 youngest pups are in the red and blue
kennels housing the least number of dogs. kennels housing the least number of dogs. The purple kennel has 2 more dogs than the blue The purple kennel has 2 more dogs than the blue
kennel. How many dogs are in each kennel?kennel. How many dogs are in each kennel?
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10 8 8 6 7 9
I know that there are 48 dogs altogether and the orange kennel is
the largest with 10 dogs
13 pups live between red and blue, so one of these must have 6 and the other 7If blue has 6, that means purple would have 810 + 8 + 7 + 6 = 3148 – 31 = 17 You can’t divide 17 by 2 to get an equal number for green and yellowSo blue must have 7 and red must have 6This means purple has 9 (48 – 32 = 16 and 16 can be divided by 2 to give 8 in yellow and green
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Block ACounting,
partitioning and calculating
Block BSecuring number
Facts, understandingshape
Block CHandling
data and measures
Block DCalculating, measuring
and understanding shape
Block ESecuring number facts, calculationand relationships
Using and applying mathematicsCounting and understanding numberCalculating
Using and applying mathematicsKnowing and using number factsUnderstanding shape
Using and applying mathematics MeasuringHandling data
Using and applying mathematics
Using and applying mathematics
CalculatingMeasuring
Counting and understanding numberKnowing and using number factsCalculating
Understanding shape
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Links with APPLinks with APP
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Using and ApplyingUsing and Applying
Problem SolvingProblem Solving
CommunicatingCommunicating
ReasoningReasoning
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Child to teacherChild to teacher
Show what you knowShow what you know
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Key Stage 1Key Stage 1
17 + 817 + 8
Key Stage 2Key Stage 2
64 - 3764 - 37
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Teacher to childTeacher to child
ExplainingExplainingand and
QuestioningQuestioning
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ExplainingExplaining
Models and ImagesModels and Images
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Mathematical VocabularyMathematical Vocabulary Mathematical vocabulary versus everyday Mathematical vocabulary versus everyday
languagelanguage odd, difference, face, table, mean, operationodd, difference, face, table, mean, operation
Encourage children to explore the Encourage children to explore the meanings for themselvesmeanings for themselves
Incorporate mathematical vocabulary into Incorporate mathematical vocabulary into your displaysyour displays http://http://www.teachers.ash.org.au/jeather/maths/dictionary.htmlwww.teachers.ash.org.au/jeather/maths/dictionary.html
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Effective QuestioningEffective Questioningin Mathematicsin Mathematics
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Traditional teacher Traditional teacher questioning questioning
Teacher asks a questionTeacher asks a question
Children put their hands upChildren put their hands up
Teacher takes an answerTeacher takes an answer
Teacher accepts, rejects or develops the answerTeacher accepts, rejects or develops the answer
Teacher asks a further questionTeacher asks a further question
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Learning is enhanced when …Learning is enhanced when …
We ask fewer but better questionsWe ask fewer but better questions
Seek better answers, giving children Seek better answers, giving children sufficient ‘wait time’ to think and respondsufficient ‘wait time’ to think and respond
Sustain thinking during dialogue by using Sustain thinking during dialogue by using alternatives or extensions to questions alternatives or extensions to questions that challenge children’s thinkingthat challenge children’s thinking
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Strategies for encouraging Strategies for encouraging extended dialogueextended dialogue
Allow thinking time before taking responsesAllow thinking time before taking responses Use ‘think, pair, share’Use ‘think, pair, share’ Do not respond immediately after a child Do not respond immediately after a child
has replied – often they have more to sayhas replied – often they have more to say Challenge the response – ‘Tell me why…’, Challenge the response – ‘Tell me why…’,
‘But what about…?’‘But what about…?’ Ask the child to elaborate – ‘Can you say Ask the child to elaborate – ‘Can you say
more’more’
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Types of QuestionsTypes of Questions Recalling factsRecalling facts Applying factsApplying facts Hypothesising or predictingHypothesising or predicting Designing and comparing proceduresDesigning and comparing procedures Interpreting resultsInterpreting results Applying reasoningApplying reasoning
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Types of questioningTypes of questioningRecalling facts:Recalling facts:
What is 13 add 7?What is 13 add 7? What is double 30?What is double 30? How many sides does a pentagon How many sides does a pentagon
have?have?
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Applying facts:Applying facts:
Tell me two numbers which have a Tell me two numbers which have a difference of 12difference of 12
Tell me any multiple of 2Tell me any multiple of 2 Give me two numbers which total 20Give me two numbers which total 20 What are the factors of 42?What are the factors of 42?
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Hypothesising or predicting:Hypothesising or predicting:
Estimate the number of marbles in Estimate the number of marbles in this jarthis jar
What is the next number in this What is the next number in this sequence: 17, 15, 13, ? sequence: 17, 15, 13, ?
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Designing and comparing procedures:Designing and comparing procedures:
How might we count this pile of How might we count this pile of straws?straws?
How could you solve 46 + 43?How could you solve 46 + 43?
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Interpreting results:Interpreting results: What can we say about even What can we say about even
numbers?numbers? What can we say about multiples of What can we say about multiples of
5?5?
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Applying reasoning:Applying reasoning:
The seven coins in my purse total The seven coins in my purse total 23p. What could they be?23p. What could they be?
Complete this pattern to make it Complete this pattern to make it symmetricalsymmetrical
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Closed and open questionsClosed and open questions Count these cubesCount these cubes
A chew costs 3p, a lolly A chew costs 3p, a lolly costs 7p. What do they costs 7p. What do they cost altogether?cost altogether?
What are four threes?What are four threes?
How many centimetres How many centimetres are there in a metre?are there in a metre?
How could we count How could we count these cubes?these cubes?
A chew and a lolly cost A chew and a lolly cost 10p altogether. What 10p altogether. What could each sweet cost?could each sweet cost?
Tell me two numbers Tell me two numbers with a product of 12with a product of 12
Tell me three lengths Tell me three lengths that together make 1 that together make 1 metremetre
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Problem SolvingProblem Solving
How do we give children regular access How do we give children regular access to problems that will develop their ability to to problems that will develop their ability to use and apply their skills and knowledge use and apply their skills and knowledge
in the SATs tests, without it becoming in the SATs tests, without it becoming dull and arduous?dull and arduous?
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£ 1.48 85p
4 batteries 2 batteries
A shop sells batteries in packs of four and packs of two.
Task: Discuss…. What could the question be asking?
Consider how to extend your Y6 children into level 4/5 problem solving.
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A shop sells batteries A shop sells batteries in packs of four and packs of two.in packs of four and packs of two.
Simon and Nick want two batteries each. Simon and Nick want two batteries each. They buy a They buy a pack of four pack of four and share the cost equally.and share the cost equally. How much does each pay?How much does each pay?
Mary buys Mary buys 2 packs of two2 packs of two batteries. batteries. Hamid buys Hamid buys 1 pack of1 pack offourfour..How much How much moremore does Mary does Mary pay than Hamid?pay than Hamid?
£ 1 .48 85p
4 batteries 2 batteries
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Tem peraturein °C
30
25
20
15
10
5
0Jan Feb M ar Apr M ay Jun Jul Aug Sep O ct N ov D ec
M onth
××
××
××
×
××
×
××
Discuss: What could the graph be about?
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Tem peraturein °C
30
25
20
15
10
5
0Jan Feb M ar Apr M ay Jun Jul Aug Sep O ct Nov D ec
M onth
××
××
××
×
××
×
××
Abbie takes the temperature outside at midday on the first day of each month. The graph shows her results from January to December.
How many months on the graph show a temperature between 10°C and 20°C?
Find the difference in temperature shown on the graph between July and August.
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CalculationsCalculationsEffective:Effective: How many number sentences can you How many number sentences can you
give me using these numbers 7, 3 and give me using these numbers 7, 3 and 10?10?
Less effective:Less effective: 3 + 7 = ?3 + 7 = ? 7 – 3 = ?7 – 3 = ?
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CalculationsCalculationsEffective:Effective: Why do 17 + 9, 18 + 8 and 19 + 7 all Why do 17 + 9, 18 + 8 and 19 + 7 all
give the same answer?give the same answer?
Less effective:Less effective: What is 17 + 9, 18 + 8 and 19 + 7?What is 17 + 9, 18 + 8 and 19 + 7?
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Classifying questionsClassifying questions
Prompting questionsPrompting questions
Probing questionsProbing questions
Promoting questionsPromoting questions
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Prompting questions:Prompting questions: direct children to the knowledge and skills they direct children to the knowledge and skills they
have already learnedhave already learned help children to use knowledge to derive new facts help children to use knowledge to derive new facts
and skillsand skills help children to understand the problem and decide help children to understand the problem and decide
how they will begin to tackle ithow they will begin to tackle it draw children’s attention to alternative approaches, draw children’s attention to alternative approaches,
methods or ideas they have used previouslymethods or ideas they have used previously offer children simpler starting points and offer children simpler starting points and
illustrations of how they might tackle a problemillustrations of how they might tackle a problem
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Promoting questions:Promoting questions: set children challenges so they can apply their set children challenges so they can apply their
ideas and reasoning skills and deepen their ideas and reasoning skills and deepen their understandingunderstanding
encourage children to enquire by setting up and encourage children to enquire by setting up and testing hypotheses of their owntesting hypotheses of their own
stimulate discussion about efficiency and the stimulate discussion about efficiency and the merits of alternative strategiesmerits of alternative strategies
foster children’s ability to think, to review their foster children’s ability to think, to review their approaches and solutions, and identify other, approaches and solutions, and identify other, more efficient, strategies they might usemore efficient, strategies they might use
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Probing questions:Probing questions: establish the extent to which children’s establish the extent to which children’s
understanding is secureunderstanding is secure help children to identify and correct any errors they help children to identify and correct any errors they
have made and why they might have made themhave made and why they might have made them encourage children to review their ideas and encourage children to review their ideas and
methods and seek ways of improving their work methods and seek ways of improving their work support children in articulating their strategies and support children in articulating their strategies and
choices, and use correct mathematical vocabulary choices, and use correct mathematical vocabulary and notationand notation
sharpen children’s thinking skills and ability to sharpen children’s thinking skills and ability to hypothesise, test and justifyhypothesise, test and justify
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For children who are For children who are starting a piece of work:starting a piece of work:
How are you going to do this?How are you going to do this? What do you already know?What do you already know? What do you need to find out or do?What do you need to find out or do? Do you need to add or subtract?Do you need to add or subtract? Can you do it in your head or do you need Can you do it in your head or do you need
to use pencil and paper?to use pencil and paper? Do you need any equipment?Do you need any equipment? What do you think the answer will be? Can What do you think the answer will be? Can
you estimate?you estimate?
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For children who are stuck:For children who are stuck: Can you tell me what you have Can you tell me what you have
done so far?done so far? Is there something you already Is there something you already
know that might help?know that might help? Would a number Would a number
line/cubes/counters help?line/cubes/counters help? Could you draw a picture?Could you draw a picture?
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While children are working:While children are working: Can you tell me what you have Can you tell me what you have
done so far?done so far? Why did you decide to Why did you decide to
add/subtract?add/subtract? Could there be a quicker way of Could there be a quicker way of
doing this?doing this? What did you notice when…?What did you notice when…?
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Using ICTUsing ICT
Which Number Where?Which Number Where?Sum, Product, DifferenceSum, Product, Difference
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Maths Magic!Maths Magic!