1. Activating thinking THEN consolidating learning Peter Sullivan PMA Plenary

2. Abstract Thinking like a mathematician involves making connections between ideas, approaching problems creatively, adapting known methods in new ways, and transferring learning to new contexts. Working like a mathematician involves persistence, willingness to take risks, and the capacity to explain solutions. PMA Plenary

3. I asked a mathematician … I would say both true, except "the capacity to explain solutions" is aspirational. PMA Plenary

4. Abstract Thinking like a mathematician involves making connections between ideas, approaching problems creatively, adapting known methods in new ways, and transferring learning to new contexts. Working like a mathematician involves persistence, willingness to take risks, and the capacity to explain solutions. None of this can happen in schools if students are always being shown what to do. Students can benefit if they work on problems that they have not been shown how to solve, and explain to others their own strategies. This presentation will give some examples of such problems that activate the learning of important mathematical ideas and stimulate creative ways of working. It will also consider the subsequent challenge: how can learning through problem solving be consolidated? PMA Plenary

5. Note: these tasks are on concepts that are central to the curriculum PMA Plenary

6. LEARNING TASK What might be the numbers on the L Shaped piece? I know that one of the numbers is 65. Give as many possibilities as you can. PMA Plenary

7. What might this do? What is the mathematics? What learning might be prompted by the task? PMA Plenary

8. Assuming that this task is posed with NO instruction, vote … 1 if this task is much too simple 2 if this task is too simple 3 if this task is just right 4 if this task is too hard 5 if this task is much too hard PMA Plenary

9. What might this look like as a lesson? PMA Plenary

10. MISSING NUMBERS ON THE HUNDREDS CHART PMA Plenary

11. LEARNING TASK What might be the numbers on the L Shaped piece? I know that one of the numbers is 65. Give as many possibilities as you can. PMA Plenary

12. ENABLING PROMPT What might be the missing numbers on this piece? PMA Plenary

13. EXTENDING PROMPT Convince me that you have all of the possible combinations. PMA Plenary

14. CONSOLIDATING TASK The numbers 62 and 84 are on the same jigsaw piece. Draw what might that piece look like? PMA Plenary

15. TASK VARIATIONS TO ESTABLISH THE LEARNING PMA Plenary

16. SPOT THE MISTAKE There are some mistakes in this hundreds chart. What are the mistakes? Explain how you found them. PMA Plenary 12345678910 1112 14151617181920 21222324252627282930 31323344353637383940 4142434435464749 50 51525354555657585960 61626364656668 6970 71727374757677787980 81828384858687888990 919293949596979899100 101102103104105106107108109110

17. WHAT IS MISSING? This hundreds chart has not been completed. Fill in the missing number PMA Plenary 12345678910 1112181920 21222324252627282930 31323334363740 4142434446474950 51545960 61646667686970 717475767778 8182838485868798 9192939495969798 101102103104115106107108109110

18. WHAT IS POSSIBLE? Which of the following jigsaw pieces could be from a 100s chart, and which are not? Explain your reasoning. PMA Plenary 33 34 35 36 31 32 41 42 51 52 53 42 78 79 80 69 77 109110 99 109 119 60 40 50 30 70

19. The rationale The proposition is that students will learn mathematics best if they engage in lessons that enable them to build connections between mathematical ideas for themselves (prior to instruction from the teacher) at the start of a sequence of learning rather than at the end. Above all else, we want students to know they can learn mathematics But such learning requires risk taking and persistence PMA Plenary

20. At the same time, we are addressing the classroom implementation of … problem solving approaches reasoning and critical thinking mathematical communication inquiry approaches in mathematics metacognitive strategies student resilience and persistence the connection between effort and achievement (growth mindsets) productive values, attitudes and beliefs dealing with difference PMA Plenary

21. Tasks are important Anthony and Walshaw (2009) in a research synthesis, concluded that “in the mathematics classroom, it is through tasks, more than in any other way, that opportunities to learn are made available to the students” (p.96). PMA Plenary

22. 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. PMA Plenary

23. Kilpatrick, Swafford, and Findell (2001) suggested that teachers who seek to engage students in developing adaptive reasoning and strategic competence (or problem solving) should provide them with tasks that are designed to foster those actions. Such tasks clearly need to be challenging and the solutions are ideally developed by the learners. This notion of appropriate challenge also aligns with the Zone of Proximal Development (ZPD) (Vygotsky, 1978). PMA Plenary

24. Some support from the literature National Council of Teachers of Mathematics (NCTM) (2014) noted: – Student learning is greatest in classrooms where the tasks consistently encourage high-level student thinking and reasoning and least in classrooms where the tasks are routinely procedural in nature. (p. 17) PMA Plenary

25. This approach was described in PISA in Focus (Organisation for Economic Co-operation and Development (OECD) (2014) as: – 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 learn from their mistakes, is associated with students’ drive. (p. 1) PMA Plenary

26. Another example PMA Plenary

27. LEARNING TASK I am thinking of two numbers on the hundreds chart. One number is 15 more than the other. The numbers are two rows apart. One of the numbers has a 3 in it. What might be my two numbers? Give as many answers as you can. PMA Plenary 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 4142434445464749 50 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100 101102103104105106107108109110

28. What might this do? What is the mathematics? What learning might be prompted by the task? PMA Plenary

29. Assuming that this task is posed with NO instruction, vote … 1 if this task is much too simple 2 if this task is too simple 3 if this task is just right 4 if this task is too hard 5 if this task is much too hard PMA Plenary

30. I AM THINKING OF TWO NUMBERS PMA Plenary

31. LEARNING TASK I am thinking of two numbers on the hundreds chart. One number is 15 more than the other. The numbers are two rows apart. One of the numbers has a 3 in it. What might be my two numbers? Give as many answers as you can. PMA Plenary 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 4142434445464749 50 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100 101102103104105106107108109110

32. ENABLING PROMPT (S) I am thinking of two numbers on the hundreds chart. One number is 5 more than the other. One of my numbers has a 3 in it. What might be my two numbers? PMA Plenary

33. EXTENDING PROMPT Show that you have all the possible answers (to the Learning task). PMA Plenary

34. CONSOLIDATING TASK I am thinking of two numbers on the hundreds chart. They are two rows apart. The sum of the numbers is 52. What might be the numbers? Give as many answers as you can. PMA Plenary

35. TASK VARIATIONS TO ESTABLISH THE LEARNING PMA Plenary

36. EGGS Some egg cartons hold 10 eggs. Amy has some full cartons and some loose eggs. Becky has 6 full cartons and some loose eggs. Becky has two more full cartons than Amy does. Amy has 15 fewer eggs that Becky. How many eggs might Amy and Becky have? PMA Plenary

37. PENCILS Boxes of pencils hold 10 pencils. I have 4 full boxes and some extra pencils. My friend had 16 more pencils than me. How many boxes and how many extra pencils might my friend have? PMA Plenary

38. Pen and Pencil PMA Plenary

39. Our goal We can represent solutions to problems in different ways, and see the connections between those representations. PMA Plenary

40. Show how you work this out A pen costs $2 more than a pencil. If the pen costs $8, how much is the pencil? PMA Plenary

41. The Learning task A pen and a pencil together cost $7. The pen costs $6 more than the pencil. How much does the pencil cost? Represent your solution using two DIFFERENT methods. PMA Plenary

42. If you are stuck A drink and a snack costs $10. The drink costs $2 more than the snack. How much does the drink cost? Ask the students to show their solution in two different ways PMA Plenary

43. If you are finished A book and a ruler and an eraser costs $20. The book and the ruler costs $16, the ruler and the eraser cost at least $12. What can you say about the cost of the book, the ruler and the eraser? PMA Plenary

44. Now try this A hat and a pair of sunglasses cost $110. The sunglasses cost $100 more than the hat. How much does the hat cost? PMA Plenary

45. And this At a party there are 230 people. There are 100 more adults than children. How many adults are there at the party? PMA Plenary

46. And this I had a dream that Australia and NZ reach the final. The total of the runs scored was 400. One team scored 150 runs more than the other. What might each team have scored? PMA Plenary

47. Our goal We can represent solutions to problems in different ways, and see the connections between those representations. PMA Plenary

48. Abstract Thinking like a mathematician involves making connections between ideas, approaching problems creatively, adapting known methods in new ways, and transferring learning to new contexts. Working like a mathematician involves persistence, willingness to take risks, and the capacity to explain solutions. None of this can happen in schools if students are always being shown what to do. Students can benefit if they work on problems that they have not been shown how to solve, and explain to others their own strategies. This presentation will give some examples of such problems that activate the learning of important mathematical ideas and stimulate creative ways of working. It will also consider the subsequent challenge: how can learning through problem solving be consolidated? PMA Plenary