Talk in MathsTalk in MathsLJMULJMU

2525thth January 2010 January 2010

Sue KillenSue Killen

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

OpportunitiesOpportunities Child to childChild to child

Child to teacherChild to teacher

Teacher to childTeacher to child

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

PuppetsPuppets&&

PropsProps

Barrier gamesBarrier games&&

Clue cardsClue cards

Child to childChild to child

Working collaboratively to Working collaboratively to solve problemssolve problems

Data Handling Data Handling

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?

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

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

Links with APPLinks with APP

Using and ApplyingUsing and Applying

Problem SolvingProblem Solving

CommunicatingCommunicating

ReasoningReasoning

Child to teacherChild to teacher

Show what you knowShow what you know

Key Stage 1Key Stage 1

17 + 817 + 8

Key Stage 2Key Stage 2

64 - 3764 - 37

Teacher to childTeacher to child

ExplainingExplainingand and

QuestioningQuestioning

ExplainingExplaining

Models and ImagesModels and Images

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

Effective QuestioningEffective Questioningin Mathematicsin Mathematics

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

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

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’

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

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?

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?

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, ?

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?

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?

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

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

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?

£ 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.

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

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?

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.

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 = ?

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?

Classifying questionsClassifying questions

Prompting questionsPrompting questions

Probing questionsProbing questions

Promoting questionsPromoting questions

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

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

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

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?

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?

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…?

Using ICTUsing ICT

Which Number Where?Which Number Where?Sum, Product, DifferenceSum, Product, Difference

Maths Magic!Maths Magic!