Using rich tasks to support fluency and understanding in arithmetic Dr Jenni Back Host Schools Project Lead

Key thresholds in mathematical development in arithmetic KS1 entry: conservation and counting KS2 entry: addition/subtraction, number bonds to 20, place value KS3 entry: multiplication/division, multiplication tables KS4 entry: proportional reasoning

The issues in years 3 and 4 Securing conservation and counting Additive reasoning, fluent recall and application of number bonds to 20, understanding place value Developing ideas of multiplication and division Intuitive understanding of proportional reasoning

The Big Ideas in arithmetic in Y3 & 4 Doing and undoing The number system and place value: exchange Equivalence and comparison Building and using a repertoire of known facts

Strike it out

Progression from EYFS to KS2 and beyond Recognising number symbols to 10 – restrict the number line and use concrete apparatus KS1 Developing skills in addition and subtraction of numbers to 20 Lower KS2 Recording results systematically Conjecturing about possible choices Relating addition and subtraction facts Upper KS2 Building fluency with number bonds to 20 Predicting results, developing strategies Beyond KS2 Developing mathematical thinking: exemplify, visualise, generalise, do/undo, conjecture, prove

Rich tasks: low threshold, high ceiling Accessible to all at the start of the lesson Allows the most confident/ competent to go as far as they can and offers the opportunity to ask their own questions Plenty of supporting activity of those who will benefit from it Allows different types of thinking and different approaches Article on NRICH website at http://nrich.maths.org/7701 Rich tasks offer a meaningful context that provokes the students to learn in order to solve it

Rich tasks and high attainers Rich tasks offer high attainers opportunities to think deeply about mathematics High attainers are offered opportunities to tackle problems which can really challenge their conceptions and help them to develop higher order thinking skills such as working systematically or devising a winning strategy without moving beyond KS2 concepts Collaborating with others and explaining their thinking helps them to cement their understanding through taking a teaching role Posing their own questions develops their deep and broad understanding of mathematics

Understanding place value: a task The task Show your partner 3, 5, 12, 7 : take it in turns (everyone has 9 counters) Now show them 16, 23, 18, 20 What can you do when you haven’t got enough counters? Ask the children to choose their own numbers to make and explain their strategies

Progression from EYFS to KS2 and beyond Accurate counting to ten KS1 Partitioning numbers and using addition and subtraction facts Developing an understanding of place value Lower KS2 Secure understanding of place value Using a range of know addition and subtraction facts Solving problems involving number Upper KS2 Building fluency with number bonds to 20 Predicting results, developing strategies Beyond KS2 Developing mathematical thinking: exemplify, visualise, generalise, do/undo, conjecture, prove

Understanding place value: notes for teachers Give every pair of children in the class a set of 9 double sided counters which are a different colour on the reverse. Assigning places for tens, hundreds etc is analogous to assigning different values to the counters. Key questions: Have you run out of counters? What can you do now? How about letting one blue counter be worth 2? Explain to your partner how those counters show that number.

Noah

Noah Can be found at http://nrich.maths.org/136 and the poster here 17/09/2018 Noah Can be found at http://nrich.maths.org/136 and the poster here Can be introduced at Key Stage 1 How might it be extended to stretch the highest attainers in KS2?

Progression from EYFS to KS2 and beyond 17/09/2018 Progression from EYFS to KS2 and beyond EYFS Using number names and counting concrete objects KS1 Counting accurately to 12 Applying notions of counting to a real context Making models and using them to solve mathematical problems Lower KS2 Recording results and reasoning about numbers Multiplication as successive addition and considering grouping Upper KS2 Extending task to higher targets, predicting results, posing questions Developing mathematical thinking in relation to numbers: exemplify, visualise, generalise, do/undo, conjecture, reason

Another task: factors and multiples Give me a factor of 132, and another, and another Give me a factor of 108, and another, and another Give me a multiple of 4 bigger than 100, and another, and another Give me a multiple of 3 bigger than 100, and another, and another

Variations: adapting the task for different children Support Extension Prior learning assumed

Progression in this task EYFS Counting up and down KS1 Reciting sequences of odd or even numbers, counting in 10s and 5s Lower KS2 Derive and recall multiplication facts for the 2, 3, 4, 5, 6, and 10 times table Explore the properties of the numbers in these tables Upper KS2 Express general rules about the properties of multiples and factors of a give number verbally Beyond KS2 Generalise these rules and express them using algebra

The key place of talk Emphasis on prompts and questions that promote mathematical reasoning IRF or IRQRF? Children as problem posers

Getting stuck and ‘unstuck’ It is good for children to get stuck The role of the teacher is to model being stuck and help children to develop strategies for getting ‘unstuck’ What might they be?

Getting ‘unstuck’ Talk about the problem with your friends Use practical resources to model the problem Try a simpler case Try smaller numbers Look at what you know and what you are trying to find out Write down or talk about everything you know so far Consider the kind of solution you are looking for – one, many, a proof Convince yourself, a friend, a sceptic

Thank you for listening Dr Jenni Back jenni.back@ncetm.org.uk Telephone 07850100074