PMA Plenary

Activating thinking THEN consolidating learning

Peter Sullivan

PMA Plenary

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.

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I asked a mathematician …

• I would say both true, except "the capacity to explain solutions" is aspirational.

PMA Plenary

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?

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Note: these tasks are on concepts that are central to the curriculum

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LEARNING TASKWhat 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.

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What might this do?

• What is the mathematics?• What learning might be prompted by the

task?

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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

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What might this look like as a lesson?

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MISSING NUMBERS ON THE HUNDREDS CHART

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LEARNING TASKWhat 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.

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ENABLING PROMPT

What might be the missing numbers on this piece?

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EXTENDING PROMPT

Convince me that you have all of the possible combinations.

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CONSOLIDATING TASK

The numbers 62 and 84 are on the same jigsaw piece.

Draw what might that piece look like?

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TASK VARIATIONS TO ESTABLISH THE LEARNING

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SPOT THE MISTAKEThere are some mistakes in this hundreds chart. What are the mistakes? Explain how you found

them.1 2 3 4 5 6 7 8 9 10

11 12 12 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 44 35 36 37 38 39 40

41 42 43 44 35 46 47 49 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 68 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

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WHAT IS MISSING?This hundreds chart has not been completed.

Fill in the missing number1 2 3 4 5 6 7 8 9 10

11 12 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 36 37 40

41 42 43 44 46 47 49 50

51 54 59 60

61 64 66 67 68 69 70

71 74 75 76 77 78

81 82 83 84 85 86 87 98

91 92 93 94 95 96 97 98

101 102 103 104 115 106 107 108 109 110

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WHAT IS POSSIBLE?

Which of the following jigsaw pieces could be from a 100s chart, and which are not? Explain your reasoning.

33 34 35

36

31 32

41 42

51

51 52 53

42

78 79 80

69

77

109 110

99

109

119 60

40

50

30

70

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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

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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

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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).

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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.

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• 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).

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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)

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• 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)

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• Another example

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LEARNING TASKI 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.1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 49 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

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What might this do?

• What is the mathematics?• What learning might be prompted by the

task?

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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

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I AM THINKING OF TWO NUMBERS

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LEARNING TASKI 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.1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 49 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

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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?

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EXTENDING PROMPT

Show that you have all the possible answers (to the Learning task).

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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.

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TASK VARIATIONS TO ESTABLISH THE LEARNING

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EGGSSome 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?

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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?

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Pen and Pencil

Our goal

• We can represent solutions to problems in different ways, and see the connections between those representations.

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Show how you work this out

• A pen costs $2 more than a pencil. If the pen costs $8, how much is the pencil?

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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.

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

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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?

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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?

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And this

• At a party there are 230 people. There are 100 more adults than children. How many adults are there at the party?

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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?

Our goal

• We can represent solutions to problems in different ways, and see the connections between those representations.

PMA Plenary

PMA Plenary

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?