1. Developing specific planning and pedagogies for improving mathematics and numeracy teaching Peter Sullivan nmr 2012 day 1

2. Program overview This program will build on the general strategies for improving mathematics and numeracy teaching suggested in AIZ numeracy programs, and will develop specific approaches to teaching selected mathematics concepts. We will use a particular aspect of the curriculum to plan teaching sequences that are engaging for students and which allow effective differentiation, and will plan individual lessons. nmr 2012 day 1

3. By being specific we will extend the initial exploration of the key principles for teaching numeracy and mathematics, and examine what these might be implemented in detail. We will also undertake an example of the Lesson Study approach to allow detailed study of what might be possible in mathematics teaching. This will involve planning, implementation, and review of a particular approach to teaching. This is a four day program. It is expected that participants will attend all of the four days. nmr 2012 day 1

4. The overall goal You will plan in detail a lesson sequence (in which you are trying something different) You (or a colleague) will teach the lesson sequence You will report back on what happened nmr 2012 day 1

5. Day 1: Teaching mathematics for curiosity and powerful learning This day will establish some principles of teaching that will inform the planning of units. In particular these include processes for differentiation, enquiry focused teaching especially the proficiencies from the Australian Curriculum, the importance of challenge and high expectations, lessons structures including grouping practices, and student motivation and engagement. Collaborative teacher learning processes will also be considered. nmr 2012 day 1

6. Day 2: Planning mathematics for curiosity and powerful learning This day will include goal setting, interpreting curriculum statements, accessing and adapting, resources, planning formats, planning for assessment for learning, including specifically designing assessments of students’ current knowledge for the units to be developed. nmr 2012 day 1

7. Day 3: Teacher planning This day will include opportunities for reviewing initial assessments of student learning, sharing of potential resources, and time for the planning and recording of the intended units of work. nmr 2012 day 1

8. Day 4: Presentation of units of work and reporting on their implementation This day is intended for all teachers involved in the planning of the units to present to the other schools, highlighting successes and challenges, and particular the learning from the process that can be generalised to future team based planning. It is expected that school leadership teams also attend these presentations. nmr 2012 day 1

9. As well as being used in the developing school, the units of work are intended to be shared with others and to be illustrative of what is possible. It is intended that the process for developing the units be reflected in future planning process in participating schools. We ask participating schools to arrange the mathematics curriculum for years 8 and 9 such that measurement topics are planned to be taught in late term 2 or term 3. nmr 2012 day 1

10. Today’s program Review of an approach to teaching building on the Australian Curriculum Review of the six principles for teaching mathematics Planning how we will work nmr 2012 day 1

11. A task to get us started nmr 2012 day 1 I am concerned that my car is not getting the best fuel economy. I filled up my car on 27 th April, noting the odometer as being 2345 km. When I filled the car next, I got this print out. What is the fuel economy of my car in L per 100 km?

12. What would be the point of asking a question like that? Engaging for boys Ability to analyse information Real life context, value for money Ratio calculations nmr 2012 day 1

13. What might make it difficult to ask such a question in your school? The decimals, proportions Access to calculators (and trusting their answers – rounding etc) Being exposed to more difficult numbers They give up Differences in readiness Reading, literacy level nmr 2012 day 1

14. 2/3 Draw this number line, and mark on it, as accurately as possible, 0 and 2 ½ Explain your reasoning. nmr 2012 day 1

15. What would be the point of asking a question like that? Frctions are numbers Thinking about different ways to solve problems Seeing fraction decimal relationships Value of numerator and denominator Proportions, number line Fractions and whole numbers Proving you are correct – being convincing nmr 2012 day 1

16. What might make it difficult to ask such a question in your school? Fractions Why do I need to do this Prior knowledge Relating to number line Have a go, record their work, explain their thinking, listen to others nmr 2012 day 1

17. Three content strands (nouns) Number and algebra Measurement and geometry Statistics and probability nmr 2012 day 1

18. Looking at patterns and algebra Note that there is also a “linear and non-linear relationships” section in these years as well nmr 2012 day 1

19. Year 6 “Patterns and algebra” Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence Explore the use of brackets and order of operations to write number sentences nmr 2012 day 1

20. Year 7 “Patterns and algebra” Introduce the concept of variables as a way of representing numbers using letters Write algebraic expressions and evaluate them by substituting a given value for each variable Extend and apply the laws and properties of arithmetic to algebraic terms and expressions nmr 2012 day 1

21. Year 8 “Patterns and algebra” Extend and apply the distributive law to the expansion of algebraic expressions Factorise algebraic expressions by identifying numerical factors Simplify algebraic expressions involving the four operations nmr 2012 day 1

22. Year 9 “Patterns and algebra” Extend and apply the index laws to variables, using positive integral indices and the zero index Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate nmr 2012 day 1

23. 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 nmr 2012 day 1

24. A meta analysis of many studies (Hattie, 2007) Most important teacher influenced factors – Feedback – Instructional quality – Direct instruction – Remediation – Class environment – Challenging goals nmr 2012 day 1

25. Feedback - better when they know … Where am I going? – “Your task is to …, in this way” How am I going? – “the first part is what I was hoping to see, but the second is not” Where to next? – “knowing this will help you with …” nmr 2012 day 1

26. So far there is not much difference from what you are doing It is the proficiencies that are different nmr 2012 day 1

27. In the past, we made the distinction between Knowing how – (instrumental understanding) Knowing why – (relational understanding) nmr 2012 day 1

28. The action words (proficiencies) Understanding – knowing why, Fluency – knowing how, Problem solving – finding out how, Reasoning – finding out why, nmr 2012 day 1 what, where, … when, … what, where, …

29. 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, …) nmr 2012 day 1

30. 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 nmr 2012 day 1

31. 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 nmr 2012 day 1

32. Another fractions tasks First do the task nmr 2012 day 1

33. If the blue rectangle represents 2/3, what fraction is represented by the red rectangle? nmr 2012 day 1

34. Examining a task in detail What does this task do? Where does it fit with the content descriptions? nmr 2012 day 1

35. A review of student working on the task What do you see in this video? nmr 2012 day 1

36. The shaded rectangle represents 3/4. What is the value of the whole square? StatisticValue Number of responses to this question82 Number of correct answers24 (29.3%) Please note that the results above do not include students that did not provide a response at all. For this question this was 5 students nmr 2012 day 1

37. Choosing a topic: Like terms nmr 2012 day 1

38. Is your planning sequence something like this? Identify the topic Examine curriculum content statements Use data to inform decisions on emphasis Select, then sequence, appropriate activities Identify the mathematical actions in which you want students to engage … nmr 2012 day 1

39. Using data to informing instruction From 2009 NAPLAN 2(2x – 3) + 2 + ? = 7x – 4 What term makes this equation true for all values of x ? 15% (Victorians) correct nmr 2012 day 1

40. Race to 10: – Start at 0, in turn add on either 1 or 2, – first to 10 is the winner INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

41. The videos nmr 2012 day 1

42. Task A Race to 5x + 5y – Starting at 0, you can add, in turn, x, or y, or x + y – The person who says 5x + 5y is the winner INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

43. What do you have to think about when playing that game? INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

44. Now play Race to 8x + 12 – Starting at 0, you can add, in turn, x, or 2x, 1, or 2 – The person who says 8x + 12 is the winner INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

45. What do you have to think about when playing that game? INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

46. Task B Choose some terms from the cloud and write some expression that equal 5 a + 9 3a 4a 2 6 a 3 2a 7 INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

47. Task C I want you to play Race to 32a 5 Starting at 1, you can multiply by a, or 2, or 2a The person who says 32a 5 is the winner But first … INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

48. These are different a + a + a + a + a + a = a × a × a × a × a × a = 6a a6a6 a to the power of 6 INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

49. How we say these x x2x2 x3x3 x4x4 x5x5 x6x6 INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

50. Write down the answer 3a + a = a 3 × a = INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

51. Now play Race to 32a 5 Starting at 1, you can multiply by a, or 2, or 2a The person who says 32a 5 is the winner INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

52. Now play Race to 32a 5 b 5 Starting at 1, you can multiply by a, b, ab, or 2 The person who says 32a 5 b 5 is the winner INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

53. What do you have to think about when playing that game? INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

54. Task D Choose some terms from the cloud and write some expressions that equal 6 a 3 3a 4a 2 6a a 3 2a 3a 2 nmr 2012 day 1

55. 3a + 3 Task E a + 3 3a + 6 + 3 - 3 + 2a + 3 + 2a - 2a -2a - 3 ???? nmr 2012 day 1

56. Make up your own INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

57. Task F What might be the missing terms? 4x + 3 = __ + __ + __ INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

58. The underline means that something is missing. What might be the bits missing in the following? __ + __ + __ + __ = 5x – 5y + 3 INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

59. The underline means that something is missing. What might be the bits missing in the following? 3( a + __ ) - __ = __ a + __ INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

60. The underline means that something is missing. What might be the bits missing in the following? __ ( a – 2c) = __ a + __ c INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS nmr 2012 day 1

61. Focus on these tasks collectively What can you say about the nature of those tasks? nmr 2012 day 1

62. Building from putting the answers to developing the questions Reversing the process Alignment of learning: same concept in different ways Tasks test depth of understanding We need to find ways of assessing student learning Learning the rules of maths through playing the game Develop a feeling of success, they can enter at their level Developing their own strategies Learning without a pen in their hand Does it engage the boys and girls differently Do students see this as the real learning? nmr 2012 day 1

63. Matching those tasks to the content descriptions nmr 2012 day 1

64. What can you say about the nature of those sequence? Get everyone’s attention Continuing the concept and adding additional complexity The jumps were progressive Variety Mode of delivery, structure of lessons varies Enjoyable (hopefully) – decisions are enjoyable as is success Can be differentiated readily The sequence allows all students to follow the same pathway They can go back if they struggle Games etc encourage checking appropriateness of the answer Accountable to their peers Opportunity to learn together (especially the explaining) nmr 2012 day 1

65. What might make teaching that sequence difficult in your school? Timetable Length of period Making connections, including the previous algebra experiences Getting all kids involved Grouping students Absenteeism Teacher buy in nmr 2012 day 1

66. What “Proficiencies” do these address? nmr 2012 day 1

67. Connecting the descriptions and proficiencies to six key strategies nmr 2012 day 1

68. What is the point of these six key strategies? We can all do these things better (although you will find many of them affirming of your current practice) Much advice is complex and hard to prioritise They can provide a focus to collaborative discussions on improving teaching They can be the focus of observations if you have the opportunity to be observed teaching nmr 2012 day 1

69. Improving teaching by thinking about pedagogy The following principles are a synthesis of: – Good, Grouws, and Ebmeier – Productive pedagogies – Principles of learning and teaching – Hattie – Clarke and Clarke – Anthony and Walshaw nmr 2012 day 1

70. 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 nmr 2012 day 1

71. What would you say to the students were the goals of the Race to 5x + 5y game? Would you write the goal(s) on the board? What would you say to the students about how you hope they would learn? nmr 2012 day 1

72. goals

73. What are the implications for our lesson sequence? Set clear goals for the sequence Set clear goals for the lessons Decide how to inform students Saying how they will learn Identifying big ideas nmr 2012 day 1

74. 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 nmr 2012 day 1

75. Part of this is using data nmr 2012 day 1

76. Using data to informing instruction From 2009 NAPLAN 2(2x – 3) + 2 + ? = 7x – 4 What term makes this equation true for all values of x ? 15% (Victorians) correct nmr 2012 day 1

77. Part of this is creating experience nmr 2012 day 1

78. How did that sequence connect with students’ experience? Or How could that sequence have connected with the students’ experience? nmr 2012 day 1

79. Partly about DIY experience nmr 2012 day 1

80. goals readiness

81. What are the implications for our lesson sequence? Where are they at, NAPLAN, VELS judgments, on demand, pre testing Formative assessment for learning Them finding out what they know and can build on and what they need to learn Identify common misconceptions Assessment/evaluation of their learning nmr 2012 day 1

82. Relating to experience Find out (or at least assume) what they are interested in Create a rationale for the learning, meaningful and relevant for them now Relate the topic to past and future topics Link to other studies Build experience nmr 2012 day 1

83. 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 nmr 2012 day 1

84. How might those activities TOGETHER contribute to learning? Progression, increasing complexity Going both ways, Start with concrete, moving to abstract Various activities within a lesson, appealing to different styles (the light bulbs can come at different times/rates) Kids are less bored, Rigorous nmr 2012 day 1

85. goals readiness engage

86. What are the implications for our lesson sequence? Different learning styles, including using a variety of types of experiences Decisions, choice, of pathway, destination, and form fo representation, including incorporating this into the assessment Don’t tell Explaining and justifying their thinking, strategies At least some of the tasks should be difficult nmr 2012 day 1

87. 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 nmr 2012 day 1

88. Focusing on the “expressions and relationships” activity How might we engage students who could experience difficulty with it? nmr 2012 day 1

89. How might we engage students who could experience difficulty with it? Mixed groups (and how to manage those groups) Like groups Try it in a row first Have only two expression cards, and a subset of the relationship cards Give students a role Have one with just numbers nmr 2012 day 1

90. What are enabling prompts? Enabling prompts can involve slightly varying an aspect of the task demand, such as – the form of representation, – the size of the numbers, or – the number of steps, so that a student experiencing difficulty, if successful, can proceed with the original task. This approach can be contrasted with the more common requirement that such students – listen to additional explanations; or – pursue goals substantially different from the rest of the class. nmr 2012 day 1

91. Factors contributing to difficulty It may not be clear which aspects may be contributing to a particular student’s difficulty, but by anticipating some of the factors, and preparing prompts that, for example, – reduce the required number of steps, – simplify the modes of representing results, – make the task more concrete, or – reduce the size of the numbers involved, the teacher can explore ways to give access to the task without the students being directed towards a particular solution strategy for the original task. nmr 2012 day 1

92. How might we extend students who finish quickly A harder one Create your own Assist strugglers (by asking questions) nmr 2012 day 1

93. goals readiness engage difference

94. What are the implications for our lesson sequence? Prepare to differentiate Commitment to interact with students Plan to interact Place a limit on textbook use nmr 2012 day 1

95. 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 nmr 2012 day 1

96. goals lesson structure readiness

97. I watched a mathematics lesson when I was in Japan nmr 2012 day 1

98. First the teacher told a story about tatami mats that emphasised the notion of area as covering nmr 2012 day 1

99. Then the teacher posed the task nmr 2012 day 1

100. 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 nmr 2012 day 1

101. How many squares? nmr 2012 day 1

102. … And that they were meant to go beyond counting the squares The students worked individually but talked with each other while working nmr 2012 day 1

103. The teacher selected students to share their work, giving them advance notice, an A3 sheet, and a pointer nmr 2012 day 1

104. Emphasises there ar many ways to come to a solution The teacher embraced the student solutions Focus on students explaining Open to scrutiny Student a got to see how student b did theirs Prepares them to later study in that much mathematics is about making choices of methods Practical organisation is well doen Visual cues nmr 2012 day 1

105. Solving one problem 5 different ways Orderly display of the solutions Sharing and celebrating the solutions Contextualising the task Conversation as a whole class rather than in small groups Student generated solutions Open-middled Easy entry, chance of making connections, including important mathematical properties Teacher allowed the students to learn and create and work through the challenge Focus was clear She knew he rstudents nmr 2012 day 1

106. What do you see as the connections to “curriculum”? The lesson was connected to students’ experience – Relevance, engagement, utility It addressed at least one “big idea” of mathematics – Power of knowledge, building connections nmr 2012 day 1

107. 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 nmr 2012 day 1

108. What are those big ideas? nmr 2012 day 1

113. The Japanese have better words 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 Nerige – Carefully managed whole class discussion seeking the students’ insights Matome – Teacher summary of the key ideas nmr 2012 day 1

114. goals lesson structure readiness engage difference

115. What are the implications for our lesson sequence? nmr 2012 day 1

116. 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 nmr 2012 day 1

117. After the equations task What task might you ask next? What might the next lesson look like? nmr 2012 day 1

118. goals lesson structure readiness practice engage difference

119. What are the implications for our lesson sequence? nmr 2012 day 1

120. goals lesson structure readiness practice engage difference Collaborative teacher learning

121. What are the implications for our lesson sequence? nmr 2012 day 1