Talk in Maths LJMU 25 th January 2010 Sue Killen

Williams Report “Talk in mathematics should not be seen simply as a rehearsal in class of the vocabulary of mathematics,... It should extend to high-quality discussion that develops children’s logic, reasoning and deduction skills, and underpins all mathematical learning activity. The ultimate goal is to develop mathematical understanding – comprehension of mathematical ideas and applications.” “Talk in mathematics should not be seen simply as a rehearsal in class of the vocabulary of mathematics,... It should extend to high-quality discussion that develops children’s logic, reasoning and deduction skills, and underpins all mathematical learning activity. The ultimate goal is to develop mathematical understanding – comprehension of mathematical ideas and applications.” Williams Report 2008

Opportunities Child to child Child to child Child to teacher Child to teacher Teacher to child Teacher to child

A Climate for Learning Safety – ethos that encourages ‘having a go’ and mistakes are ok Safety – ethos that encourages ‘having a go’ and mistakes are ok Relationships – meet and greet Relationships – meet and greet Learning - pair/share, VAKE Learning - pair/share, VAKE Time - wait time, down time Time - wait time, down time

Puppets & Props

Barrier games & Clue cards

Child to child Working collaboratively to solve problems

Data Handling

Kennel Club Four dozen dogs live in 6 different coloured kennels. Four dozen dogs live in 6 different coloured kennels. The smallest kennel has 6 dogs and the orange kennel is the largest with 10. The smallest kennel has 6 dogs and the orange kennel is the largest with 10. The yellow kennel and the green kennel are the only ones with the same number of dogs. The yellow kennel and the green kennel are the only ones with the same number of dogs. The 13 youngest pups are in the red and blue kennels housing the least number of dogs. The purple kennel has 2 more dogs than the blue kennel. How many dogs are in each kennel?

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 7 If blue has 6, that means purple would have = – 31 = 17 You can’t divide 17 by 2 to get an equal number for green and yellow So blue must have 7 and red must have 6 This means purple has 9 (48 – 32 = 16 and 16 can be divided by 2 to give 8 in yellow and green

Block A Counting, partitioning and calculating Block B Securing number Facts, understanding shape Block C Handling data and measures Block D Calculating, measuring and understanding shape Block E Securing number facts, calculation and relationships Using and applying mathematics Counting and understanding number Calculating Using and applying mathematics Knowing and using number facts Understanding shape Using and applying mathematics Measuring Handling data Using and applying mathematics Calculating Measuring Counting and understanding number Knowing and using number facts Calculating Understanding shape

Links with APP

Using and Applying Using and Applying Problem Solving Problem Solving Communicating Communicating Reasoning Reasoning

Child to teacher Show what you know

Key Stage Key Stage

Teacher to child ExplainingandQuestioning

Explaining Models and Images Models and Images Models and Images

Mathematical Vocabulary Mathematical vocabulary versus everyday language Mathematical vocabulary versus everyday language odd, difference, face, table, mean, operation odd, difference, face, table, mean, operation Encourage children to explore the meanings for themselves Encourage children to explore the meanings for themselves Incorporate mathematical vocabulary into your displays Incorporate mathematical vocabulary into your displays

Effective Questioning in Mathematics

Traditional teacher questioning Teacher asks a question Teacher asks a question Children put their hands up Children put their hands up Teacher takes an answer Teacher takes an answer Teacher accepts, rejects or develops the answer Teacher accepts, rejects or develops the answer Teacher asks a further question Teacher asks a further question

Learning is enhanced when … We ask fewer but better questions We ask fewer but better questions Seek better answers, giving children sufficient ‘wait time’ to think and respond Seek better answers, giving children sufficient ‘wait time’ to think and respond Sustain thinking during dialogue by using alternatives or extensions to questions that challenge children’s thinking Sustain thinking during dialogue by using alternatives or extensions to questions that challenge children’s thinking

Strategies for encouraging extended dialogue Allow thinking time before taking responses Allow thinking time before taking responses Use ‘think, pair, share’ Use ‘think, pair, share’ Do not respond immediately after a child has replied – often they have more to say Do not respond immediately after a child has replied – often they have more to say Challenge the response – ‘Tell me why…’, ‘But what about…?’ Challenge the response – ‘Tell me why…’, ‘But what about…?’ Ask the child to elaborate – ‘Can you say more’ Ask the child to elaborate – ‘Can you say more’

Types of Questions Recalling facts Recalling facts Applying facts Applying facts Hypothesising or predicting Hypothesising or predicting Designing and comparing procedures Designing and comparing procedures Interpreting results Interpreting results Applying reasoning Applying reasoning

Types of questioning 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 have? How many sides does a pentagon have?

Applying facts: Tell me two numbers which have a difference of 12 Tell me two numbers which have a difference of 12 Tell me any multiple of 2 Tell me any multiple of 2 Give me two numbers which total 20 Give me two numbers which total 20 What are the factors of 42? What are the factors of 42?

Hypothesising or predicting: Estimate the number of marbles in this jar Estimate the number of marbles in this jar What is the next number in this sequence: 17, 15, 13, ? What is the next number in this sequence: 17, 15, 13, ?

Designing and comparing procedures: How might we count this pile of straws? How might we count this pile of straws? How could you solve ? How could you solve ?

Interpreting results: What can we say about even numbers? What can we say about even numbers? What can we say about multiples of 5? What can we say about multiples of 5?

Applying reasoning: The seven coins in my purse total 23p. What could they be? The seven coins in my purse total 23p. What could they be? Complete this pattern to make it symmetrical Complete this pattern to make it symmetrical

Closed and open questions Count these cubes Count these cubes A chew costs 3p, a lolly costs 7p. What do they cost altogether? A chew costs 3p, a lolly costs 7p. What do they cost altogether? What are four threes? What are four threes? How many centimetres are there in a metre? How many centimetres are there in a metre? How could we count these cubes? A chew and a lolly cost 10p altogether. What could each sweet cost? Tell me two numbers with a product of 12 Tell me three lengths that together make 1 metre

Problem Solving How do we give children regular access to problems that will develop their ability to use and apply their skills and knowledge in the SATs tests, without it becoming dull and arduous?

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 in packs of four and packs of two. Simon and Nick want two batteries each. They buy a pack of four and share the cost equally. How much does each pay? How much does each pay? Mary buys 2 packs of two batteries. Hamid buys 1 pack of four. How much more does Mary pay than Hamid?

Discuss: What could the graph be about?

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.

Calculations Effective: How many number sentences can you give me using these numbers 7, 3 and 10? How many number sentences can you give me using these numbers 7, 3 and 10? Less effective: = ? = ? 7 – 3 = ? 7 – 3 = ?

Calculations Effective: Why do , and all give the same answer? Why do , and all give the same answer? Less effective: What is , and ? What is , and ?

Classifying questions Prompting questions Probing questions Promoting questions

Prompting questions: direct children to the knowledge and skills they have already learned direct children to the knowledge and skills they have already learned help children to use knowledge to derive new facts and skills help children to use knowledge to derive new facts and skills help children to understand the problem and decide how they will begin to tackle it help children to understand the problem and decide how they will begin to tackle it draw children’s attention to alternative approaches, methods or ideas they have used previously draw children’s attention to alternative approaches, methods or ideas they have used previously offer children simpler starting points and illustrations of how they might tackle a problem offer children simpler starting points and illustrations of how they might tackle a problem

Promoting questions: set children challenges so they can apply their ideas and reasoning skills and deepen their understanding set children challenges so they can apply their ideas and reasoning skills and deepen their understanding encourage children to enquire by setting up and testing hypotheses of their own encourage children to enquire by setting up and testing hypotheses of their own stimulate discussion about efficiency and the merits of alternative strategies stimulate discussion about efficiency and the merits of alternative strategies foster children’s ability to think, to review their approaches and solutions, and identify other, more efficient, strategies they might use foster children’s ability to think, to review their approaches and solutions, and identify other, more efficient, strategies they might use

Probing questions: establish the extent to which children’s understanding is secure establish the extent to which children’s understanding is secure help children to identify and correct any errors they have made and why they might have made them help children to identify and correct any errors they have made and why they might have made them encourage children to review their ideas and methods and seek ways of improving their work encourage children to review their ideas and methods and seek ways of improving their work support children in articulating their strategies and choices, and use correct mathematical vocabulary and notation support children in articulating their strategies and choices, and use correct mathematical vocabulary and notation sharpen children’s thinking skills and ability to hypothesise, test and justify sharpen children’s thinking skills and ability to hypothesise, test and justify

For children who are 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 to use pencil and paper? Can you do it in your head or do you need to use pencil and paper? Do you need any equipment? Do you need any equipment? What do you think the answer will be? Can you estimate? What do you think the answer will be? Can you estimate?

For children who are stuck: Can you tell me what you have done so far? Can you tell me what you have done so far? Is there something you already know that might help? Is there something you already know that might help? Would a number line/cubes/counters help? Would a number line/cubes/counters help? Could you draw a picture? Could you draw a picture?

While children are working: Can you tell me what you have done so far? Can you tell me what you have done so far? Why did you decide to add/subtract? Why did you decide to add/subtract? Could there be a quicker way of doing this? Could there be a quicker way of doing this? What did you notice when…? What did you notice when…?

Using ICT Which Number Where? Which Number Where? Sum, Product, Difference Sum, Product, Difference

Maths Magic!