Six principles of effective teaching of mathematics


Published on

Connect with Maths Webinar presented by Professor Peter Sullivan: Six Principles of Effective Mathematics Teaching
There are many recommendations on how to teach mathematics but fewer about the teaching of mathematics’ classes with Indigenous students. This webinar will examine how six principles for effective mathematics teaching were adapted to advice for teachers of schools with high numbers of Indigenous students.

Published in: Education, Technology

Six principles of effective teaching of mathematics

  1. 1. Finding a focus for ongoing teacher learning in mathematics: Six key principles Peter Sullivan 6 principles
  2. 2. Overview • Some stories • Six principles 6 principles
  3. 3. A story from a Grade 3 lesson I taught at Bathurst Island 6 principles
  4. 4. Football scores Saints 105 Bombers 98 6 principles How much are the Saints winning by? (Work out the answer in two different ways)
  5. 5. Football scores Saints 25 Bombers 18 6 principles How much are the Saints winning by? (Work out the answer in two different ways)
  6. 6. From a grade 3 lesson in the Kimberleys 6 principles
  7. 7. How many ways to make $10 6 principles
  8. 8. 6 principles
  9. 9. A suggestion as part of a geometry unit written for MiTK 6 principles
  10. 10. • Have pile of cubes. Ask the students to: – Build something using 15 cubes. Describe what you have built. – Build something which is 3 cubes high and 3 cubes wide made with 15 cubes. Describe what you have built. – Something like “make a tower of 3 cubes, put a yellow cube to the left, and blue cube to the north, …” • Repeat, adapting to the level of success of the students. 6 principles
  11. 11. A lesson I plan to teach to a year 8 class next Tuesday at CMS (Alice Springs) 6 principles
  12. 12. ONE HECTARE PARK • A conservation park is to be enclosed by a fence that has exactly 6 internal right angles. What might the park look like? • The total area of the park is 1 hectare. What might be the perimeter of the park? (give two different answers) 6 principles
  13. 13. What are the common characteristics of those tasks? 6 principles
  14. 14. Six principles as the basis of sustainable improvement 6 principles
  15. 15. What is the point of these six key principles ? • 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 • The principles can provide a focus to collaborative discussions on improving teaching • The principles can be the focus of observations if you have the opportunity to be observed teaching 6 principles
  16. 16. 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 6 principles
  17. 17. Key principle 1: • Identify important 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 6 principles
  18. 18. 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 …” 6 principles
  19. 19. In terms of learning intentions, we know • It is much more difficult to describe the purpose of lessons than we think • The learning intention should – not restrict – nor lower the ceiling – but provide focus to the students – and the teacher 6 principles
  20. 20. In terms of the subtraction lesson • There are many ways to find the difference between two numbers 6 principles
  21. 21. In this lesson, I need you to • show how you get your answers • keep trying even if it is difficult (it is meant to be) • explain your thinking • listen to other students 6 principles
  22. 22. 6 principles goals
  23. 23. Key principle 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 6 principles
  24. 24. Part 1: Using data 6 principles
  25. 25. • It is more important to know what the students know than what they do not • Learning mathematics is not a hierarchy of sequential steps on a ladder, but a network of interconnected ideas • Students can work on tasks that are beyond what they know – Students at GP 2 can work on GP 4 tasks 6 principles
  26. 26. Part 2: Connecting with “story” 6 principles
  27. 27. • A chameleon has a tongue that is half as long as its body ... • … how long would your tongue be if you were a chameleon? 6 principles
  28. 28. Part 3: Creating experience 6 principles
  29. 29. 6 principles
  30. 30. 6 principles goals readiness
  31. 31. Key Principle 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 6 principles
  32. 32. 6 principles For students to learn, two sets of factors must align • The first set of factors include that the: –students have the requisite prior knowledge; –curriculum is relevant to them; –classroom tasks match their expectations; –classroom tasks help them make connections –pedagogies use their knowledge and experience; –assessment regimes measure their learning.
  33. 33. 6 principles The second set of factors relates to • whether the students –are motivated to learn –see participation in schooling as creating opportunities –are willing to persist –connect effort and success
  34. 34. Why challenge? • Learning will be more robust if students connect ideas together for themselves, and determine their own strategies for solving problems, rather than following instructions they have been given. • Both connecting ideas together and formulating their own strategies is more complex than other approaches and is therefore more challenging. • It is potentially productive if students are willing to take up such challenges. 6 principles
  35. 35. 6 principles This connects to “mindsets” • Dweck (2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset
  36. 36. 6 principles Students with growth mindset: • Believe they can get smarter by trying hard • Such students – tend to have a resilient response to failure; – remain focused on mastering skills and knowledge even when challenged; – do not see failure as an indictment on themselves; and – believe that effort leads to success.
  37. 37. 6 principles Students with fixed mindset: • Believe they are as smart as they will even get • Such students – seek success but mainly on tasks with which they are familiar; – avoid or give up quickly on challenging tasks; – derive their perception of ability from their capacity to attract recognition.
  38. 38. Teachers can change mindsets • This connects to – the things we affirm (effort, persistence, co- operation, learning from others, flexible thinking) – the way we affirm • You did not give up even though you were stuck • You tried something different • You tried to find more than one answer – the types of tasks we pose 6 principles
  39. 39. Getting started “zone of confusion” “four before me” •representing what the task is asking in a different way such as drawing a cartoon or a diagram, rewriting the question … •choosing a different approach to the task, which includes rereading the question, making a guess at the answer, working backwards … •asking a peer for a hint on how to get started •looking at the recent pages in the workbook or textbook for examples. 6 principles
  40. 40. Key Principle 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 6 principles
  41. 41. Related to those 4 tasks above .. • To what extent – Are they challenging? – Are they engaging? – Do they allow student decision making – Do they encourage different representations? 6 principles
  42. 42. 6 principles goals readiness engage
  43. 43. Key Principle 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 6 principles
  44. 44. Enabling prompt 6 principles
  45. 45. Basketball scores Eels 18 Carp 13 6 principles How much did the Eels win by? (Work out the answer in two different ways)
  46. 46. Basketball scores Cats 8 Dogs 3 6 principles How much did the Cats win by? (Work out the answer in two different ways)
  47. 47. Extending prompt 6 principles
  48. 48. Darts scores Parrots 1005 Galahs 988 6 principles How much did the Parrots win by? (Work out the answer in two different ways)
  49. 49. 6 principles goals readiness engagedifference
  50. 50. Key Principle 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 6 principles
  51. 51. A revised lesson structure • In this view, the sequence – Launch (without telling) – Explore (for themselves) – Summarise (drawing on the learning of the students) • … is cyclical and might happen more than once in a lesson (or learning sequence) shepp secondary Launch ExploreSummarise
  52. 52. Football scores Saints 18 Bombers 13 6 principles How much are the Saints winning by? (Work out the answer in two different ways)
  53. 53. 6 principles goals lesson structure readiness engagedifference
  54. 54. 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 6 principles
  55. 55. One aspect is transfer 6 principles
  56. 56. Another aspect is fluency 6 principles