MODULE 4 The role of challenging mathematical experiences in activating thinking of all students

2. MODULE 4 The role of challenging mathematical experiences in activating thinking of all students

3. If you are working through parts of this module by yourself or in a small group … • On some slides there is a link to additional audio information. • Listen by clicking on the speaker icon

4. Module 4 Overview Part 1(optional) The elements of the reSolve project Part 2 A rationale for offering challenging experiences to all students Part 3 Examining a task from the classroom resources and discussing the nature of the challenge Part 4 Encouraging students to take up challenges Part 5 Examining a task from the classroom resources to identify teacher actions that can encourage students to take up the challenge Part 6 Considering the implications for your own practice

5. Part 1: The reSolve Project Click here if you have already done a PD Module

6. Overview of the reSolve: Mathematics by Inquiry Project

7. Classroom Resources Exemplary classroom resources at every level from Foundation to Year 10 that embody a spirit of inquiry and enact the reSolve Protocol.

8. Classroom Resources

9. Special Topics Addressing identified needs or exploring new boundaries. Special Topics are significant resources that address the needs of 21st century learners. They: • are substantial units of work and accompanying resources that address major current gaps; • prioritise the Australian Curriculum proficiencies of reasoning and problem solving; • provide imaginative opportunities for creatively using new technologies and real world contexts; and • respond to the results of international assessments that show that solving real problems is a specific area of weakness for Australian students.

10. Special Topics

11. Professional Learning Modules Resources to support in-school leaders to address key issues and themes in the teaching and learning of mathematics through collegial professional learning programs.

12. Professional Learning Modules

13. reSolve Champions A community of more than 290 committed leaders across the country, who will use reSolve resources and approaches in professional learning programs they lead during and after the development phase of reSolve (finishes mid-2018).

14. The reSolve Protocol is the driver for all that the project is about, and does. In order to foster a spirit of inquiry in all mathematics teaching and learning: • reSolve mathematics ispurposeful • reSolve tasks are challenging yet accessible • reSolve classrooms have a knowledge-building culture

15. In particular, this module addresses these statements from the Protocol reSolve tasks are challenging yet accessible. reSolve contests a view that some students can “do” mathematics well and others cannot by: Activatingexisting knowledge, developing new knowledge and exploringrelationships between key ideas by workingon meaningful tasks. Engaging students in sustained inquiry, problem solving, decision making and communication.

16. The Protocol also includes the following statement reSolve learning environments promote a supportive knowledge-building culture.  reSolve contests a view that mathematics is best learned through copying and memorising by: Elicitingproductive dispositions, including productive struggleand the motivation and confidence to take risks.

17. This module addresses (in particular) the following AITSL Teaching Standards: Standard 3 – Plan for and implement effective teaching and learning • 3.1 Establish challenginglearning goals • Set learning goals that provide achievable challenges for students of varying abilities and characteristics. • 3.5 Use effective classroom communication • Demonstrate a range of verbal and non-verbal communication strategies to support student engagement.

18. At the end of the module we will discuss the following questions. Make notes on the handout as you work through the module. • If your plan is to allow students to struggle, what are some particular actions to which you might commit? • What might be the risks in challenging students and what might you do to minimise the risks?

19. Part 2: Considering the nature of “challenge” The purpose of this part of the module is • to examine some arguments and evidence from high level commentary; and • to reflect on the extent to which such approaches are part of current practice.

20. There is support world wide for posing challenges to students to support their learning of mathematics After some statements and quotes that support the notion of challenge, you will be asked to reflect on which ideas are already part of your practice and which might influence you to try something new.

21. Module 1 includes consideration of the spirit of inquiry • Students are not inquiring into a mathematical problem situation if they have been told how to solve it. • Students are also not inquiring if they know immediately how to solve the problem. • The implication is that most students in the class do not know how to solve the problem initially. • So problems and inquiries should be posed at a level that challengemost students in the class.

22. Module 2 emphasised the critical role of tasks in engaging students in learning mathematics “in the mathematics classroom, it is through tasks, more than in any other way, that opportunities to learn are made available to the students” (Anthony & Walshaw, 2009, p. 96).

23. And those tasks should be challenging Christiansen and Walther (1986) argued that non-routine tasks, because of the interplay between different aspects of learning, provide optimal conditions for cognitive development in which new knowledge is constructed relationally and items of earlier knowledge are recognised and evaluated.

24. PISA in Focus 37 use the term “cognitive activation”… • “Teachers’ use of cognitive-activation strategies, such as • giving students problems that require them to think for an extended time, • presenting problems for which there is no immediately obvious way of arriving at a solution, and • helping students to learnfrom their mistakes, • is associated with students’ drive.”

25. Of course “challenge” must be productive … “We use the word struggle to mean students expend effort to make sense of mathematics … We do not use struggle to mean needless frustration …” (Hiebert & Grouws, 2007, p. 387).

26. Productive struggle is not just for high achievers • It is stressed that productive struggle is for all students. • It is possible that removing the opportunity to struggle from students who are experiencing difficulty actually exacerbates their dependence on the teacher.

27. Does this argument make sense? reSolve PL Module 4 Contrasting Approaches video Click link to watch video

28. Which ideas from the previous slides are a) already central to your planning and practice? b) something you would consider trying? • Inquiry involves challenge • The type of task is important • Non-routine tasks activate learning • Cognitive activation builds positive motivation • Challenge needs to be productive • Challenge is beneficial for all students • Start with complexity moving to clarity

29. Part 3: Examining a task from the classroom resources and discussing the nature of the challenge The purpose of this part of the module is • using one of the tasks from the classroom resources, to discuss in what ways the task is challenging (for the level at which it is written); and • to consider teacher actions that might limit the challenge.

30. Two key issues in introducing challenging tasks to the students • The first is that a common language can be established not only for students to interpret the task appropriately but also so they can contribute to the subsequent discussion. • Second, teachers sometimes reduce the demand of tasks if they anticipate students experiencing difficulty and structure the activity so that the difficulty is reduced or even eliminated (Jackson, Garrison, Wilson, Gibbons, & Shahan, 2013)

31. Consider using the following phrases in discussing the following task Lower-level cognitive demand • Memorisation • Procedures without connections Higher-level cognitive demand • Procedures with connections • Doing mathematics (Smith & Stein, 2011)

32. Select one of the following tasks for a discussion of “challenge” Ten to One (for Year 4) Cutting Complex Polygons (for Year 8)

33. Ten to One • Flip between + and – symbols with the numbers so that the result is 27 • Flip between + and – symbols with the numbers so that the result is 15 • Flip between + and – symbols with the numbers so that the result is 12

34. The mathematical purpose of Ten to One This task is the second in a Year 4 sequence of lessons looking at the properties of odd and even numbers. The first task establishes that, in an addition, an odd number of odds will produce an odd total, while an even number of odds will produce an even total. The task asks students to consider how the question might be altered to produce an even total. They are also asked to consider if it is possible to have a collection of numbers that will produce an odd and even total.

35. In groups, discuss … • What is the nature of the challenge that most students might experience? • How might you introduce the task so that the opportunity for students to be challenged is not reduced? • How do you describe the level of demand of the task? Jump to Part 4

36. Cutting Convex Polygons How many ways can you cut the heptagon using one straight cut to make two new polygons with a total of 10 sides? What other side totals can you make?

37. The mathematical purpose of Cutting Convex Polygons The purpose is to identify and express algebraic relationships arising in a geometric context. In this case students develop two generalisations: (1) The total number of sides in two polygons that result from cutting a convex n-sided polygon, with one straight cut, into two polygons; and (2) The number of ways in which each total can be made.

38. In groups, discuss … • What is the nature of the challenge that most students might experience? • How might you introduce the task so that the opportunity for students to be challenged is not reduced? • How do you describe the level of demand of the task?

39. Part 4: Encouraging students to take up challenges The purpose of this part of the module is • to discuss actions you might take to encourage students to persist; and • to consider ways of responding if students give up.

40. There is widespread attention in schools to risk taking, resilience, persistence, … • What language is used by your school to encourage students to be willing to take risks in their learning, to be resilient, etc.? • What language is used by you to encourage students to be willing to take risks, to be resilient, and to persist in mathematics?

41. “Mindsets” offer us a helpful metaphor Dweck (2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve?language=en Click link to watch video

42. Students with a growth mindset: • remain focused on mastering skills and knowledge even when challenged; • do not see failure as an indictment on themselves; • tend to have a resilient response to failure; and • believe that effort leads to success.

43. Students with a fixed mindset: • 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; and • feel threats to self worth when effort does not lead to recognition.

44. You can have high achievers with a fixed mindset • You might have students who are usually successful with little effort who are uncomfortable with struggling on mathematics tasks. • Those students see quick and easy successes as an endorsement of their talent. • Such students experience more difficulty in upper primary and secondary years.

45. Teachers can change mindsets through… • 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”; and • the types of tasks posed and the ways lessons are structured.

46. How do you feel when you are watching this? https://www.youtube.com/embed/JHy6bBKu0j4?rel=0 Click link to watch video

47. WhatdoI do? What can I do? • What are some strategies you currently use or could use in the future to encourage students to persist, to take up challenges, to become more self reliant? • Are there things you say or do that might discourage risk taking?

48. Part 5: Examining a task from the classroom resources to identify teacher actions that can encourage students to take up the challenge You will work through an example of a challenge from the reSolve classroom resources and consider actions you can take to encourage students to take up the challenge.

49. Select one of these tasks taken from the Classroom Resources Attribute Trains (for Foundation) Pascal’s Angle Machine (for Year 9)

50. Attribute Trains Each of these blocks has two attributes: colour and shape.