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Developing specific planning and pedagogies for improving mathematics and numeracy teaching

Developing specific planning and pedagogies for improving mathematics and numeracy teaching. Peter Sullivan. Program overview. Today’s program. Three content strands (nouns). Number and algebra Measurement and geometry Statistics and probability. Looking at patterns and algebra.

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Developing specific planning and pedagogies for improving mathematics and numeracy teaching

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  1. Developing specific planning and pedagogies for improving mathematics and numeracy teaching Peter Sullivan nmr day 1 program 2 2011

  2. Program overview nmr day 1 program 2 2011

  3. Today’s program nmr day 1 program 2 2011

  4. Three content strands (nouns) • Number and algebra • Measurement and geometry • Statistics and probability nmr day 1 program 2 2011

  5. Looking at patterns and algebra • Note that there is also a “linear and non-linear relationships” section in these years as well nmr day 1 program 2 2011

  6. 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 day 1 program 2 2011

  7. 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 day 1 program 2 2011

  8. 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 day 1 program 2 2011

  9. 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 day 1 program 2 2011

  10. 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 day 1 program 2 2011

  11. A meta analysis of many studies (Hattie, 2007) • Most important teacher influenced factors • Feedback • Instructional quality • Direct instruction • Remediation • Class environment • Challenging goals nmr day 1 program 2 2011

  12. 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 day 1 program 2 2011

  13. So far there is not much difference from what you are doing • It is the proficiencies that are different nmr day 1 program 2 2011

  14. In the past, we made the distinction between • Knowing how • (instrumental understanding) • Knowing why • (relational understanding) nmr day 1 program 2 2011

  15. The action words (proficiencies) • Understanding • knowing why, • Fluency • knowing how, • Problem solving • finding out how, • Reasoning • finding out why, what, where, … when, … when, … what, where, … nmr day 1 program 2 2011

  16. 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 day 1 program 2 2011

  17. 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 day 1 program 2 2011

  18. 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 day 1 program 2 2011

  19. Choosing a topic: Like terms nmr day 1 program 2 2011

  20. 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 day 1 program 2 2011

  21. 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 day 1 program 2 2011

  22. Task 1 • Choose some terms from the cloud and write an expression that is equivalent to 5 a + 9 a 4a 2a 7 3a 6 3 2 nmr day 1 program 2 2011

  23. Task 2 • Race to 10: • Start at 0, in turn add on either 1 or 2, first to 10 is the winner • Race to 5x + 5y • Start at 0, in turn add on either x, or y, or x + y,, first to 5x + 5y is the winner nmr day 1 program 2 2011

  24. A different issue:How might you support students experiencing difficulty? nmr day 1 program 2 2011

  25. How might you extend students who find an answer quickly? nmr day 1 program 2 2011

  26. Task 3 nmr day 1 program 2 2011

  27. 3a + 3 a + 3 3a + 6 + 3 - 3 + 2a + 3 -2a - 3 - 2a + 2a nmr day 1 program 2 2011

  28. + 3 + 2a - 3 3a + 3 - 2a a + 3 3a + 6 + 2a + 3 -2a - 3 nmr day 1 program 2 2011

  29. + 3 + 2a - 3 3a + 3 - 2a a + 3 3a + 6 + 2a + 3 -2a - 3 ???? ???? ???? nmr day 1 program 2 2011

  30. Task 4 • What might be the missing terms? 4x + 3 = __ + __ + __ nmr day 1 program 2 2011

  31. Task 5 • If x + 5 < 12 What might x be? nmr day 1 program 2 2011

  32. Focus on these five tasks collectively • What do you hope that students might learn? nmr day 1 program 2 2011

  33. What “Proficiencies” do these address? nmr day 1 program 2 2011

  34. Connecting the descriptions and proficiencies to six key strategies nmr day 1 program 2 2011

  35. 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 day 1 program 2 2011

  36. 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 day 1 program 2 2011

  37. 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 day 1 program 2 2011

  38. What would you say to the students were the goals of the Race to 5x + 5y game? • Would you write that on the board? • What would you say to the students about how you hope they would learn? nmr day 1 program 2 2011

  39. goals nmr day 1 program 2 2011

  40. 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 day 1 program 2 2011

  41. Part of this is using data nmr day 1 program 2 2011

  42. 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 day 1 program 2 2011

  43. Part of this is creating experience nmr day 1 program 2 2011

  44. How did that sequence connect with students’ experience? • Or • How could that sequence have connected with the students’ experience? nmr day 1 program 2 2011

  45. goals readiness nmr day 1 program 2 2011

  46. 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 day 1 program 2 2011

  47. How might those activities TOGETHER contribute to learning? nmr day 1 program 2 2011

  48. goals readiness engage nmr day 1 program 2 2011

  49. 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 day 1 program 2 2011

  50. Focusing on the “expressions and relationships” activity • How might we engage students who could experience difficulty with it? nmr day 1 program 2 2011

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