1. Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

2. Initial thoughts Thoughts about Mathematics Thoughts about teaching and learning Mathematics

3. Five ingredients to consider Starting with a rich challenge: low threshold, high ceiling activities Valuing mathematical thinking Purposeful activity and discussion Building a community of mathematicians Reviewing and reflecting

4. Starting with a rich challenge: Low Threshold, High Ceiling activity To introduce new ideas and develop understanding of new curriculum content

5. Making use of a Geoboard environmentGeoboard environment

6. Why might a teacher choose to use this activity in this way?

7. Some underlying principles Mathematics is a creative discipline, not a spectator sport Exploring → Noticing Patterns → Conjecturing → Generalising → Explaining → Justifying → Proving

8. Tilted Squares The video in the Teachers' Notes shows how the problem was introduced to a group of 14 year old students: http://nrich.maths.org/2293/note http://nrich.maths.org/2293/note

9. Some underlying principles Teacher’s role To choose tasks that allow students to explore new mathematics To give students the time and space for that exploration To bring students together to share ideas and understanding, and draw together key mathematical insights

10. Give the learners something to do, not something to learn; and if the doing is of such a nature as to demand thinking; learning naturally results. John Dewey

11. The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka!, but rather, “hmmm… that’s funny…” Isaac Asimov mathematics

12. There are many more NRICH tasks that make excellent starting points…

13. Number and Algebra Summing Consecutive Numbers Number Pyramids What’s Possible? What’s It Worth? Perimeter Expressions Seven Squares Attractive Tablecloths

14. Geometry and Measures Painted Cube Changing Areas, Changing Perimeters Cyclic Quadrilaterals Semi-regular Tessellations Tilted Squares Vector Journeys

15. Handling Data Statistical Shorts Odds and Evens Which Spinners?

16. …and for even more, see the highlighted problems on the Curriculum Mapping Document Curriculum Mapping Document

17. Time for reflection Thoughts about Mathematics Thoughts about teaching and learning Mathematics

18. Morning Break

19. Valuing Mathematical Thinking What behaviours do we value in mathematics and how can we encourage them in our classrooms?

20. As a teacher, do I value students for being… curious – looking for explanations – looking for generality – looking for proof persistent and self-reliant willing to speak up even when they are uncertain honest about their difficulties willing to treat ‘failure’ as a springboard to new learning

21. … and do I offer students sufficient opportunities to develop these “habits for success” when I set tasks to consolidate/deepen understanding to develop fluency to build connections

22. We could ask… Area = ? Perimeter = ? or we could ask … 6cm 4cm We could ask:

23. Perimeter = 20 cm Area = 24 cm² = 22 cm = 28 cm = 50 cm = 97 cm = 35 cm and we could ask …

24. Think of a rectangle Calculate its area and perimeter Swap with a friend – can they work out the length and breadth of your rectangle? or we could ask … …students to make up their own questions

25. Can you find rectangles where the value of the area is the same as the value of the perimeter?

26. Why might a teacher choose to use these activities in this way?

27. We could ask students to find… (x + 2) (x + 5) (x + 4) (x - 3) … or we could introduce them to…

28. Pair Products Choose four consecutive whole numbers, for example, 4, 5, 6 and 7. Multiply the first and last numbers together. Multiply the middle pair together… What might a mathematician do next?

29. We could ask students to… Identify coordinates and straight line graphs or we could introduce them to…

30. Route to Infinity Will the route pass through (18,17)? Which point will it visit next? How many points will it pass through before (9,4)? Route to Infinity

31. We could ask students to… List the numbers between 50 and 70 that are (a) multiples of 2 (b) multiples of 3 (c) multiples of 4 (d) multiples of 5 (e) multiples of 6 or we could ask students to play…

32. The Factors and Multiples Game A game for two players. You will need a 100 square grid. Take it in turns to cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out. The first person who is unable to cross out a number loses. Each number can only be crossed out once.

33. Why might a teacher choose to use these activities?

34. Some underlying principles Consolidation should address both content and process skills. Rich tasks can replace routine textbook tasks, they are not just an add-on for students who finish first.

35. There are many more NRICH tasks that offer opportunities for consolidation…

36. Number and Algebra What Numbers Can We Make? Factors and Multiples Game Factors and Multiples Puzzle Dicey Operations American Billions Keep It Simple Temperature Painted Cube Arithmagons Pair Products What’s Possible? Attractive Tablecloths How Old Am I?

37. Geometry and Measures Isosceles Triangles Can They Be Equal? Translating Lines Opposite Vertices Coordinate Patterns Route to Infinity Pick’s Theorem Cuboid Challenge Semi-regular Tessellations Warmsnug Double Glazing

38. Handling Data M, M and M Which List is Which? Odds and Evens Which Spinners?

39. …and for even more, see the highlighted problems on the Curriculum Mapping Document Curriculum Mapping Document

40. Time for reflection Thoughts about Mathematics Thoughts about teaching and learning Mathematics

41. Lunch

42. Promoting purposeful activity and discussion ‘Hands-on’ doesn’t mean ‘brains-off’

43. The Factors and Multiples Challenge You will need a 100 square grid. Cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out. Try to find the longest sequence of numbers that can be crossed out. Each number can only appear once in a sequence.

44. We could ask… 3, 5, 6, 3, 3 Mean = ? Mode = ? Median = ? or we could ask…

45. M, M and M There are several sets of five positive whole numbers with the following properties: Mean = 4 Median = 3 Mode = 3 Can you find all the different sets of five positive whole numbers that satisfy these conditions?

46. Possible extension How many sets of five positive whole numbers are there with the following property? Mean = Median = Mode = Range = a single digit number

47. What’s it Worth? Each symbol has a numerical value. The total for the symbols is written at the end of each row and column. Can you find the missing total that should go where the question mark has been put?

48. Translating Lines Each translation links a pair of parallel lines. Can you match them up?

49. Why might a teacher choose to use these activities?

50. Rules for Effective Group Work All students must contribute: no one member says too much or too little Every contribution treated with respect: listen thoughtfully Group must achieve consensus: work at resolving differences Every suggestion/assertion has to be justified: arguments must include reasons Neil Mercer

51. Developing Good Team-working Skills The article describes attributes of effective team work and links to "Team Building" problems that can be used to develop learners' team working skills. http://nrich.maths.org/6933

52. Time for reflection Thoughts about Mathematics Thoughts about teaching and learning Mathematics

53. Afternoon Break

54. Build a community of mathematicians by: Creating a safe environment for learners to take risks Promoting a creative climate and conjecturing atmosphere Providing opportunities to work collaboratively Valuing a variety of approaches Encouraging critical and logical reasoning

55. Multiplication square X12345678910 1 123456789 2 2 46 8 1214161820 3 3 69 12151821242730 4 48121620242832 3640 5 510152025303540 4550 6 6121824303642485460 7 7142128354249566370 8 81624 3240 4856647280 9 91827 3645 5463728190 10 2030405060708090100

56. The Challenge To create a climate in which the child feels free to be curious To create the ethos that ‘mistakes’ are the key learning points To develop each child’s inner resources, and develop a child’s capacity to learn how to learn To maintain or recapture the excitement in learning that was natural in the young child Carl Rogers, Freedom to Learn, 1983

57. There are many NRICH tasks that encourage students to work as a mathematical community…

58. Making Rectangles What’s it Worth? Steel Cables Odds and Evens M, M and M Odds, Evens and More Evens Tilted Squares Pair Products What’s Possible? Cyclic Quadrilaterals How Old Am I? Factors and Multiples Game

59. …and for even more, see the highlighted problems on the Curriculum Mapping Document Curriculum Mapping Document

60. Enriching mathematics website www.nrich.maths.org www.nrich.maths.org The NRICH Project aims to enrich the mathematical experiences of all learners by providing free resources designed to develop subject knowledge and problem-solving skills. We now also publish Teachers’ Notes and Curriculum Mapping Documents for teachers: http://nrich.maths.org/curriculum http://nrich.maths.org/curriculum

61. What next? Secondary CPD Follow-up on the NRICH site: http://nrich.maths.org/7768

62. Reviewing and reflecting There should be brief intervals of time for quiet reflection – used to organise what has been gained in periods of activity. John Dewey

63. “If I ran a school, I’d give all the average grades to the ones who gave me all the right answers, for being good parrots. I’d give the top grades to those who made lots of mistakes and told me about them and then told me what they had learned from them.” Buckminster Fuller, Inventor

64. Time for us to review…

65. Five strands of mathematical proficiency NRC (2001) Adding it up: Helping children learn mathematics

66. Conceptual understanding - comprehension of mathematical concepts, operations, and relations Procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic competence - ability to formulate, represent, and solve mathematical problems Adaptive reasoning - capacity for logical thought, reflection, explanation, and justification Productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

67. Alan Wigley’s challenging model (an alternative to the path-smoothing model) Leads to better learning – learning is an active process Engages the learner – learners have to make sense of what is offered Pupils see each other as a first resort for help and support Scope for pupil choice and opportunities for creative responses provide motivation

68. “…the ability to know what to do when they don’t know what to do” Guy Claxton

69. Guy Claxton’s Four Rs Resilience: being able to stick with difficulty and cope with feelings such as fear and frustration Resourcefulness: having a variety of learning strategies and knowing when to use them Reflection: being willing and able to become more strategic about learning. Getting to know our own strengths and weaknesses Relationships: being willing and able to learn alone and with others

70. What Teachers Can Do aim to be mathematical with and in front of learners aim to do for learners only what they cannot yet do for themselves focus on provoking learners to –use and develop their (mathematical) powers –make mathematically significant choices John Mason

71. Reflecting on today: the next steps Two weeks with the students or it’s lost…… Think big, start small Think far, start near to home A challenge shared is more fun What, how, when, with whom?

72. … a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Polya, G. (1945) How to Solve it

73. I don't expect, and I don't want, all children to find mathematics an engrossing study, or one that they want to devote themselves to either in school or in their lives. Only a few will find mathematics seductive enough to sustain a long term engagement. But I would hope that all children could experience at a few moments in their careers...the power and excitement of mathematics...so that at the end of their formal education they at least know what it is like and whether it is an activity that has a place in their future. David Wheeler

74. Recommended Reading Deep Progress in Mathematics: The Improving Attainment in Mathematics Project – Anne Watson et al, University of Oxford, 2003 Adapting and extending secondary mathematics activities: new tasks for old. Prestage, S. and Perks, P. London: David Fulton, 2001 Thinking Mathematically. Mason, J., Burton L. and Stacey K. London: Addison Wesley, 1982. Mindset: The New Psychology of Success. Dweck, C.S. Random House, 2006 Building Learning Power, by Guy Claxton; TLO, 2002

75. Final thoughts Thoughts about Mathematics Thoughts about teaching and learning Mathematics