1. NT 2012 Six principles of effective mathematics teaching Peter Sullivan Monash University

2. NT 2012 First, let us consider this task sequence The following sequence seeks to introduce students to – the nature of volume of prisms and cylinders (area of the end × length) – surface area of prisms (total area of the faces) – efficient methods of calculating volume and surface area of rectangular prisms – ways in which surface area and volume are different

3. NT 2012 The tasks are only intended to be illustrative

4. NT 2012 This is a rectangular prism made from cubes. What is the volume of this prism? What is the surface area?

5. NT 2012 A set of 36 cubes is arranged to form a rectangular prism. What might the rectangular prism look like? What is the surface area of your prisms?

6. NT 2012 A rectangular prism is made from cubes. It has a surface area of 22 square units. What might the rectangular prism look like?

7. NT 2012 Key teaching idea 1: Identify big ideas that underpin the concepts you are seeking to teach, and communicate to students that these are the goals of your teaching, including explaining how you hope they will learn

8. NT 2012 How might the AC help?

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11. Year 6 Year 7 Year 8

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14. Three content strands (nouns) Number and algebra Measurement and geometry Statistics and probability

15. NT 2012 Using the content descriptions Get clear in your mind what you want the students to learn Make your own decisions about how to help them learn that content – Overall what do these suggest are the overall goals, the big ideas, the important focus, etc

16. NT 2012 AC content descriptions: Using Units of measurement Year 6 – Connect volume and capacity and their units of measurement Year 7 – Calculate volumes of rectangular prisms Year 8 – Choose appropriate units of measurement for area and volume and convert from one unit to another – Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume Year 9 – Calculate the surface area and volume of cylinders and solve related problems – Solve problems involving surface area and volume of right prisms

17. NT 2012 So far there is not much difference from what you are doing It is the proficiencies that are different

18. NT 2012 In the Australian curriculum Understanding – (connecting, representing, identifying, describing, interpreting, sorting, …) Fluency – (calculating, recognising, choosing, recalling, manipulating, …) Problem solving – (applying, designing, planning, checking, imagining, …) Reasoning – (explaining, justifying, comparing and contrasting, inferring, deducing, proving, …)

19. NT 2012 The proficiencies – why do we change from “working mathematically”? These actions are part of the curriculum, not add ons Mathematics learning and assessment is more than fluency Problem solving and reasoning are in, on and for mathematics All four proficiencies are about learning

20. NT 2012 Choosing tasks will be a key decisions If we are seeking fluency, then clear explanations followed by practice will work If we are seeking understanding, then very clear and interactive communication between teacher and students and between students will be necessary If we want to foster problem solving and reasoning, then we need to use tasks with which students can engage, which require them to make decisions and explain their thinking

21. NT 2012 How is this represented in the AC?

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28. What would you say to the students were the goals the “36 cubes” task? Would you write that on the board? What would you say to the students about how you hope they would learn?

29. NT 2012 What proficiencies are associated with the 36 cubes task?

30. NT 2012 goals

31. NT 2012 Key teaching idea 2: Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning

32. NT 2012 This has two parts. Partly this is using data

33. NT 2012 What % of Year 5 Victorians can do this? A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres. What is the length of each short side? metres

34. NT 2012 56% of students could do this A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres. What is the length of each short side? metres

35. NT 2012 What % of Year 7 Victorians (no calculator) can do this?

36. NT 2012 12 % of Year 7 Victorians (no calculator) can do this

37. NT 2012 This tells us Around half of the year 5 students are ready for challenging tasks about perimeter Very few year 7 students have a sense of volume

38. NT 2012 Partly it is about connecting with story

39. NT 2012 1mm of rain on 1 sq m of roof is 1 L of water. Design a tank for this building that captures all of the rain that usually falls this month.

40. NT 2012 goals readiness

41. NT 2012 Key teaching idea 3 Engage students by utilising a variety of rich and challenging tasks, that allow students opportunities to make decisions, and which use a variety of forms of representation

42. NT 2012 How might those activities TOGETHER contribute to learning?

43. NT 2012 goals readiness engage

44. NT 2012 Key teaching idea 4: Interact with students while they engage in the experiences, and specifically planning to support students who need it, and challenge those who are ready

45. NT 2012 Focusing on the SA = 22cm 2 activity How might we engage students who could experience difficulty with it?

46. NT 2012 How might we extend students who finish quickly

47. NT 2012 goals readiness engage difference

48. NT 2012 Key teaching idea 5: Adopt pedagogies that foster communication, mutual responsibilities, and encourage students to work in small groups, and using reporting to the class by students as a learning opportunity

49. NT 2012 I watched a mathematics lesson when I was in Japan

50. NT 2012 First the teacher told a story about tatami mats that emphasised the notion of area as covering

51. NT 2012 Then the teacher posed the task

52. NT 2012 The students had a worksheet with TWO copies of the question on it that emphasised to the students it was the method, not the answer, that was the focus

53. NT 2012 How many squares?

54. NT 2012 … And that they were meant to go beyond counting the squares The students worked individually but talked with each other while working

55. NT 2012 The teacher selected students to share their work, giving them advance notice, an A3 sheet, and a pointer

56. NT 2012 Why do I tell you about this lesson? The lesson was connected to students’ experience – Relevance, engagement, utility It addressed at least one “big idea” of mathematics – Power of knowledge, building connections

57. NT 2012 The clear expectation is that students learn from each other – Culture, community, relationships The emphasis was on the process not on the answer – Quality of thinking, building capacity to learn

58. NT 2012 What are those big ideas?

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63. The Japanese have words for the parts of the lesson Hatsumon – The initial problem – Kizuki - what you want them to learn Kikanjyuski – Individual or group work on the problem – Kikan shido – thoughtful walking around the desks Neriage – Carefully managed whole class discussion seeking the students’ insights Matome – Teacher summary of the key ideas

64. NT 2012 goals lesson structure readiness engage difference

65. NT 2012 Key teaching idea 6 Fluency is important, and it can be developed in two ways – by short everyday practice of mental calculation or number manipulation – by practice, reinforcement and prompting transfer of learnt skills

66. NT 2012 7 43 +

67. 35 ?13 +

68. NT 2012 10 ?? +

69. NT 2012 ? + 6 5 4 + +

70. 12 43 X

71. NT 2012 39 ? 3 X

72. NT 2012 ? 345 X XX

73. 12 4 3 X 7 +

74. NT 2012 18 ? ? X 11 +

75. NT 2012 ? ? ? X 11 +

76. NT 2012 ? ? ? X +

77. ? ? ? X 0.7 +

78. NT 2012 ? ? ? X 7 +

79. -12 ? ? X +

80. NT 2012 ? x - 4 x + 3 X ? +

81. NT 2012 x 2 - 5x + 6 ? x - 2 X ? +

82. NT 2012 9x 2 - 49 ? ? X 6x6x +

83. NT 2012 1411 ? ? X 100 + ? ? X + -12 ? ? X +

84. NT 2012 goals lesson structure readiness practice engage difference

85. NT 2012 goals lesson structure readiness practice engage difference Collaborative teacher learning

86. NT 2012 AVAILABLE TO DOWNLOAD FREE FROM http://research.acer.edu.au/aer/13

87. NT 2012 peter.sullivan@monash.edu

88. NT 2012 Words are important … But why does the word Monosyllabic have 5 syllables

89. NT 2012 Words are important … but why is abbreviation such a long word

90. NT 2012 Words are important … but why is phonetic not spelled the way that it sounds

91. NT 2012 Words are important … but why is nostalgia not what it used to be

92. NT 2012 Words are important … but why is the person who invests your money called a BROKER

93. NT 2012 Words are important … but why is There only one word for thesaurus

94. NT 2012 And finally Why do they nail down the lid of a coffin?