Improving Mental Mathematics in Schools

1. Improving Mental Mathematics in Schools Dave Rowe Rob Perkes Tom Garner

2. Aims of Session: To familiarise ourselves with key mental calculation strategies that should be taught throughout the school; To develop an understanding of progression in mental calculation across the school; To learn how mental recall can be developed to facilitate mental calculation; To identify opportunities for and the importance of developing visualisation; To understand how to teach and not just test times tables.

3. Why is mental calculation so important? Activity 1: Discuss on your tables and be prepared to feedback.

4. What does working mentally in mathematics mean? What skills and attitudes do children need to be able to work mentally? What opportunities do children need in order to develop these skills?

5. Mental calculation is all about patterns and relationships between numbers. Children need to be able to learn how to solve problems by recognising which strategies and known facts to apply. How and where does mental calculation start?

6. FS  Yr1  Yr2  Yr3  Yr4  Yr5  Yr6 OFSTED – Understanding the scoreToo often pupils are expected to remember the methods, rules and facts without grasping the underpinning concepts, making connections with earlier learning and making sense of mathematics so they can use it independently.

7. Activity 2 – part 1: On your own have a go at writing down as many different mental calculation methods as you can. Activity 2 – part 2: Working with a partner have a go at filling in the grid. Which of the mental calculation methods you have thought of do you think go where?

8. Progression in mental calculation

9. Progression in mental calculation

10. Models and Images

11. ‘Teaching Children to Calculate Mentally’ publication. Activity 3: Have a look through the booklet. What is its main focus? What do you notice?

12. Teaching children to calculate mentally Addition and subtraction p4-7 Multiplication and division p8-11 Addition and subtraction strategies p26-50 Multiplication and division strategies p51-71

13. Ensuring mental and oral opportunities are planned across a range of mathematics: The 7 strands within the framework: Using and applying mathematics; Counting and understanding number; Knowing and using number facts; Calculating; Understanding shape; Measuring; Handling Data. Transum

14. The 6 Rs of Oral and Mental Work: Rehearse Recall Refresh Refine Read Reason

15. Rehearse To practise and consolidate existing skills, usually mental calculation skills, set in a context to involve children in problem-solving through the use and application of these skills, use of vocabulary and language of number, properties of shapes or describing and reasoning.

16. Recall To secure knowledge of facts, usually number facts, build up speed and accuracy, recall quickly names and properties of shape, units of measure or types of charts, graphs to represent data.

17. Refresh To draw on and revisit previous learning; to assess, review and strengthen children’s previously acquired knowledge and skills relevant to later learning; return to aspects of mathematics with which the children have had difficulty; draw out key points from learning.

18. Refine To sharpen methods and procedures; explain strategies and solutions; extend ideas and develop and deepen the children’s knowledge; reinforce their understanding of key concepts, build on earlier learning so that strategies and techniques become more efficient and precise.

19. Read To use mathematical vocabulary and interpret images, diagrams and symbols correctly; read number sentences and provide equivalents, describe and explain diagrams and features involving scales, tables or graphs; identify shapes from a list of their properties, read and interpret word problems and puzzles; create their own problems and lines of enquiry.

20. Reason To use and apply acquired knowledge, skills and understanding; make informed choices and decisions, predict and hypothesise; use deductive reasoning to eliminate or conclude, provide examples that satisfy a condition always, sometimes or never and say why.

21. The Six Rs of Oral and Mental Work Activity 4: Working with a partner, think of a mental oral starter for each of the six Rs. (The starter can be for any year group).

22. The Six Rs of Oral and Mental Work

23. Mathematical Language What language should be used in each year group? Is there a progression in the use of mathematical language?

24. Mathematical Language Mathematical Vocabulary booklets. Note: This was produced for the original NNS. Due to the fact that some objectives moved year groups in the 2006 Renewed Framework some vocabulary may need to be introduced in earlier year groups.

25. Developing Mathematical Language Fourbidden is a mathematical card game to promote the use of mathematical language devised by Phil Dodd and published by ATM. There are now two packs of Fourbidden cards, the latest designed with KS3 students in mind. There are 52 cards in each set, on each card a familiar mathematical term is printed on the left, with four related words shown on the right hand side of the card. There is a good explanation on different ways of using the pack. ATM - Fourbidden Cards.

26. Break Tea, coffee, fruit and biscuits available at the back of the hall.

27. How mental recall can be developed to facilitate mental calculation. There is a heavy reliance on known facts.

28. Conclusion: If children haven’t learnt the facts in the first place they can’t: • Recall them; • Use them to help them calculate; • Develop those facts further (i.e. for larger numbers).

29. Foundation Stones Activity 5: Look at the progression document again. Choose two year 6 objectives and identify earlier objectives from previous year groups that would need to be embedded for each objective to be understood.

30. How would you solve it?(calculation sorting) Don’t solve the calculation… …identify the most appropriate strategy.

31. How would you solve it?(calculation sorting) Activity 6: With a partner, work through the calculations on the yellow sheet in the middle of your table. DO NOT SOLVE THEM! For each one identify the most appropriate strategy that should be used to solve it.

32. How would you solve it?(calculation sorting) There is no right or wrong answer. The point is… About stopping and thinking; Making things easy for yourself; Using known facts to solve the problem (by doing as little maths as possible!)

33. How would you solve it?(calculation sorting) • Should be used to develop lateral thinking about strategies Should be introduced from year 3 • Should develop and build on . previously learned strategies.

34. Mental Maths Practise Tests

35. Mental Maths Practise Tests Use weekly: As a teaching opportunity to discuss strategies; To cover and practise a whole range of Maths; To practise rapid recall of facts/information; For speed.

36. The most important part isNOT the testing or the mark achieved, but the discussion that follows the test.

37. Children need regular timed practise to speed up their recall.

38. Other Strategies Times Tables (and division) Clubs Mathletics/Education City/RM Maths

39. Mental Oral Starters Pace is very important. Any recall of facts should be rapid.

40. Chris Moyles’ Quiz Night http://www.sheffieldmaths.co.uk/Chris%20Moyles%20Starters.html

41. What is Visualisation? Activity 7: Skim read the article ‘Thinking Through, and By, Visualising.

42. We rely on visualising when we solve problems. Sometimes we create an image of the situation that is being discussed in order to make sense of it; sometimes we need to visualise a model that can represent the situation mathematically before we can begin to develop it, and sometimes we visualise to see 'what will happen if ...?'. But are there other ways in which we visualise when solving mathematical problems and if so how can we encourage, value and develop visualising in our classrooms?

43. Children need to have had the opportunity to hold, turn, examine and work with objects before they can visualise them.

45. Progression Year 6 - - - - - - - - - - - - Year 1 Which column would number 12 be in? Find me 2 numbers that add up to 10. Give me a number that will appear in the middle column. What can you tell me about the numbers in each column? Find the sum of the first row. Is it a multiple of 3? Are there any other rows and columns with multiples of 3? Look at the first column. If extended, would 73 be included? Give me 2 numbers between 50 and 60 tat would appear in the final column. Will there be a multiple of 100 in the middle column?

46. Shape Visualisation • Make it into a smaller square. • Make it into a triangle. • What kind of triangle have you made? • How do you know? • How do you know it is not an equilateral triangle? Begin by visualising a square: With one fold make it into a rectangle. Describe the properties of the rectangle. • Fold it back into a square. Make it into a pentagon. • Can you visualise one fold to make a pentagon with one line of symmetry? • How do you know this is a line of symmetry?

47. Try using a post-it note to scaffold.

48. Shape Visualisation Imagine a pyramid. Walk around the pyramid. What can you see? Imagine you can fly and fly above it. What can you see now? Imagine you can lift it up with a magic spell. Spin it around, invert it and then put it back down again. What can you see now? Now talk to the person next to you and talk about the similarities and differences between your two pyramids.

49. Number visualisation Imagine the number five hundred and thirty two drawn in the air in front of you. Which digit is in the middle? Which is on the left? Which is on the right? Replace the middle digit with a four. What number can you see now? Swap over the middle digit with the one on the left. What number can you seen now? Remove the middle digit and push the two together so that they are next to each other. What number can you see now?

50. Conclusion: It is important for working out properties of shapes, positions of shapes or objects that have been reflected, rotated or translated and nets of 3D shapes. Visualisation is important in all areas of Maths. It is also important for measuring.