1 / 103

Peter Sullivan

The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks. . Peter Sullivan. Abstract. While most students want to work on more challenging mathematics, there are still some who require substantial support.

rhondar
Télécharger la présentation

Peter Sullivan

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks.  Peter Sullivan Sullivan MAT Nov 2013

  2. Abstract • While most students want to work on more challenging mathematics, there are still some who require substantial support. • The workshop will explore examples of tasks with low "floors" but high "ceilings" that allow all students to engage with the tasks at some level, but which can be extended productively for those who are ready. • A particular lesson structure that supports the work of all students on such tasks will be presented and discussed. Sullivan MAT Nov 2013

  3. What are the challenges that you are experiencing in teaching mathematics? Sullivan MAT Nov 2013

  4. Some initial assumptions • Planning happens at 4 levels: the school, the year, the unit, the lesson • We are starting at the “planning the lesson” end • The goal is to improve the experience of students when learning mathematics • We will focus on a particular type of lesson structure (that is broadly applicable to many types of tasks) Sullivan MAT Nov 2013

  5. Even though such investigations can be made realistic and authentic… • The maximum gradient of a ramp exceeding 1520mm in length shall be 1:14. • Ramps shall be provided with landings at the top and bottom of the ramp and at 9m intervals for a ramp 1:14. • The length of landings shall be not less than 1200mm. • The gradient of ramps between landings will be consistent. • Ramps shall be provided with handrails on both sides which do not encroach on the 1000mm minimum clear width. • Angles of approach for ramps, walkways and landings is preferably zero degrees. Sullivan MAT Nov 2013

  6. Or even extended to • Design a ramp for some stairs at the school which do not yet have a ramp • And write a report for the School Council Sullivan MAT Nov 2013

  7. Nor are we focusing on games such as In turn, players roll a 10 sided die (numbered 0 to 9) and, after each roll, write the number rolled in one of the rectangles on a board that looks like ÷ The winner has the answer closest to 100 (for example). Sullivan MAT Nov 2013

  8. Even though such games can be extended to … How could you place 3, 4, 5 and 6 on a board like this, to make the answer closest to 100 ÷ Sullivan MAT Nov 2013

  9. And I am assuming that you already know how to structure lessons based on texts Sullivan MAT Nov 2013

  10. There are plenty of resources of great ways to teach mathematics • The Shell Centre Materials • http://www.mathshell.com/ • Formative Assessment Lessons and Tasks • http://map.mathshell.org/materials • nrich • http://nrich.maths.org/frontpage • transum • http://www.transum.org/ • hotmaths • http://www.hotmaths.com.au/ • tarsia - there is not actually a website with this name, but a number that offer software (example below)http://www.tes.co.uk/article.aspx?storyCode=6107407&s_cid=RESads_MathsTarsia Sullivan MAT Nov 2013

  11. The following are examples of tasks that exemplify the approach on which we will focus Sullivan MAT Nov 2013

  12. For year 8Drawing a single straight line, make two quadrilaterals with the same perimeter A 13 12 E B 9 11 C D 10 Sullivan MAT Nov 2013

  13. For Year 1Basketball scores How much did the Parrots win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

  14. What might make teaching a lesson based on one of those tasks difficult at your school? Sullivan MAT Nov 2013

  15. Proposition Set 1 • More of the same is not a feasible response • The pathway to improvement is teaching teams working collaboratively on planning, on teaching and on assessment • Each lesson sequence should ideally incorporate a variety of types of lessons, tasks and activities • All students need to make time (away from school) to develop their own fluency at the skills being taught (you might need to explain the rationale for this) Sullivan MAT Nov 2013

  16. Proposition Set 2 • Students benefit from working on tasks that they do not already know how to do • Students are more likely to connect ideas if they compare and contrast related ideas and build networks of concepts for themselves • These connections are the key to remembering and transferring knowledge • Asking students to solve and/or represent problems in more than one way helps them to build connections • There are risks if we build connections too slowly • The goal is that students come to know they can learn mathematics Sullivan MAT Nov 2013

  17. Should we start easy and wind it up or start at challenging or wind it back? • Students can benefit when they move from not knowing how to do something to knowing how to do it. • In other words, what they have learned is explicit to them. • This does not necessarily happen if they are working on the “known”. • When confronted with a task that they cannot do, students need to explore their existing mental structures and schemes, explore links, build connections and identify aspects that are unknown.   Sullivan MAT Nov 2013

  18. Where does the idea of “challenge” come from? • Guidelines for school and system improvement (see, e.g., City, Elmore, Fiarman, & Teitel, 2009) • The motivation literature (Middleton, 1995; 1999). Sullivan MAT Nov 2013

  19. This connects to “mindsets” • Dweck(2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset Sullivan MAT Nov 2013

  20. 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. Sullivan MAT Nov 2013

  21. 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. Sullivan MAT Nov 2013

  22. Teachers can change mindsets • the things they affirm (effort, persistence, co-operation, learning from others, flexible thinking) • the way they 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 posed Sullivan MAT Nov 2013

  23. In the video to follow • The first child says something like “when you are confused it means you are learning” • The second child says “the best part is being confused because you can think about what you can do” • The third child says you “learn from being confused” • The fourth child says “you can learn by yourself”

  24. Proposition Set 3 • Posing challenging tasks requires a different lesson structure • The lesson should foster the sense of a classroom community to which all students contribute with the intention that students learn from each other • The experience of engaging with the task happens before instruction • Few rather than many tasks • All students are given time to engage sufficiently to participate in the review Sullivan MAT Nov 2013

  25. This is relevant whether or not the students are grouped by their achievement • And is applicable with crowded (and even badly behaved) classrooms Sullivan MAT Nov 2013

  26. ability achievement Sullivan MAT Nov 2013

  27. The conventional mathematics lesson • Review homework • Explain the concept and model the techniques • Students practice the techniques • Solutions are corrected (by the teacher) • Homework is set Sullivan MAT Nov 2013

  28. Japanese Lesson Study and Lesson Structure Sullivan MAT Nov 2013

  29. How many squares? Sullivan MAT Nov 2013

  30. There are Japanese words for parts of lessons • Hatsumon • The initial problem • Kizuki • -what you want them to learn • Kikanjyuski • Individual or group work on the problem • Kikanshido – • thoughtful walking around the desks • Neriage • Carefully managed whole class discussion seeking the students’ insights • Matome • Teacher summary of the key ideas Sullivan MAT Nov 2013

  31. A five-component cyclic Chinese lesson structure • Reviewing • Bridging • Variation • Summarising, and • Reflection/Planning Sullivan MAT Nov 2013

  32. A revised lesson structureLappan et al. 2006 Launch Explore Summarise Sullivan MAT Nov 2013

  33. The summarise phaseSmith and Stein (2011) • anticipating potential responses • monitoring student responses interactively • selecting representative responses for later presentation • sequencing student responses • connecting the students’ strategies with the formal processes that were the intention of the task in the first place. Sullivan MAT Nov 2013

  34. A further 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) Sullivan MAT Nov 2013

  35. The notion of classroom culture • Rollard (2012) concluded from the meta analysis that classrooms in which teachers actively support the learning of the students promote high achievement and effort. Sullivan MAT Nov 2013

  36. Some elements of this active support : • the identification of tasks that are appropriately challenging for most students; • the provision of preliminary experiences that are pre-requisite for students to engage with the tasks but which do not detract from the challenge of the task; • the structuring of lessons including differentiating the experience through the use of enabling and extending prompts for those students who cannot proceed with the task or those who complete the task quickly; Sullivan MAT Nov 2013

  37. the potential of consolidating tasks, which are similar in structure and complexity to the original task, with which all students can engage even if they have not been successful on the original task; • the effective conduct of class reviews which draw on students’ solutions to promote discussions of similarities and differences; • holistic and descriptive forms of assessment that are to some extent self referential for the student and which minimise the competitive aspects; and • finding a balance between individual thinking time and collaborative group work on tasks. Sullivan MAT Nov 2013

  38. 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. Sullivan MAT Nov 2013

  39. The lessons consist of • One or more challenging task(s) • One or more consolidating task(s) (see Dooley, 2012) • preliminary experiences that are pre-requisite but which do not detract from the challenge of the tasks • supplementary tasks that offer the potential for differentiating the experience through the use of • enabling prompts (see Sullivan, et al., 2009) which can reduce the number of steps, simplify the complexity of the numbers, and vary the forms of representation for those students who cannot proceed with the task; • extending prompts for students who complete the original task quickly which often prompt abstraction and generalisation of the solutions. Sullivan MAT Nov 2013

  40. But if I try this, … • I will not have enough time for the rest of this topic • I do not have time to prepare lessons like this • My students will not persist enough to engage with the task • I am not sure I will be able to control the class • My students will not learn the mathematics by themselves. I need to tell them. • … Sullivan MAT Nov 2013

  41. Epmc perimeter Sullivan MAT Nov 2013

  42. A primary example Sullivan MAT Nov 2013

  43. There are many ways to find the difference between two numbers Sullivan MAT Nov 2013

  44. Basketball scores How much did the Parrots win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

  45. Basketball scores How much did the Wombats win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

  46. Enabling prompt(s) Sullivan MAT Nov 2013

  47. Basketball scores How much did the Eels win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

  48. Basketball scores How much did the Cats win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

  49. Extending prompt Sullivan MAT Nov 2013

  50. Darts scores How much did the Parrots win by? (Work out the answer in two different ways) Sullivan MAT Nov 2013

More Related