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1 John Mason Mathsfest Cork Oct 2012 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking Reasoning Mathematically Reasoning Mathematically

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2 Specific Aims for Ordinary Level an understanding of mathematical concepts and of their relationships an understanding of mathematical concepts and of their relationships confidence and competence in basic skills confidence and competence in basic skills the ability to solve problems the ability to solve problems an introduction to the idea of logical argument an introduction to the idea of logical argument appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems

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3 Conjectures Everything said here today is a conjecture … to be tested in your experience Everything said here today is a conjecture … to be tested in your experience The best way to sensitise yourself to learners The best way to sensitise yourself to learners –is to experience parallel phenomena yourself So, what you get from this session is what you notice happening inside you! So, what you get from this session is what you notice happening inside you!

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4 Tasks Tasks promote activity; activity involves actions; actions generate experience; Tasks promote activity; activity involves actions; actions generate experience; –but one thing we dont learn from experience is that we dont often learn from experience alone Something more is required Something more is required

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5 Secret Places One of the places around the table is a secret place. One of the places around the table is a secret place. If you click near a place, the colour will tell you whether you are hot or cold: If you click near a place, the colour will tell you whether you are hot or cold: –Hot means that the secret place is within one place either way –Cold means that it is at least two places away Homage to Tom OBrien (1938 – 2010) What is your best strategy to locate the secret place?

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6 Counting Out In a selection game you start at the left and count forwards and backwards until you get to a specified number (say 37 or 177). Which object will you end on? In a selection game you start at the left and count forwards and backwards until you get to a specified number (say 37 or 177). Which object will you end on? ABCDE … If that object is eliminated, you start again from the next. Which object is the last one left? 10

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7 Alternating Square Sums Imagine a triangle Imagine a triangle Imagine a point inside the triangle Imagine a point inside the triangle Drop perpendiculars to the three sides of the triangle Drop perpendiculars to the three sides of the triangle Each side of the triangle comprises two segments Each side of the triangle comprises two segments On each segment of each edge, construct a square On each segment of each edge, construct a square Conjecture: the sum of the areas of the yellow squares is the sum of the areas of the cyan squares. Conjecture: the sum of the areas of the yellow squares is the sum of the areas of the cyan squares. For what hexagons is this the case? For what hexagons is this the case? Alternately colour the squares yellow and cyan around the triangle Alternately colour the squares yellow and cyan around the triangle

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8 Selective Sums Add up any 4 entries, one taken from each row and each column. Add up any 4 entries, one taken from each row and each column. The answer is (always) 6 The answer is (always) 6 Why? Why? Example of (use of) permutations Example of seeking invariant relationships Example of focusing on actions preserving an invariance Opportunity to generalise

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9 Selective Sums Add up any 4 entries, one taken from each row and each column. Add up any 4 entries, one taken from each row and each column. Is the answer always the same? Is the answer always the same? Why? Why?

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10 Chequered Selective Sums Choose one cell in each row and column. Choose one cell in each row and column. Add the entries in the dark shaded cells and subtract the entries in the light shaded cells. Add the entries in the dark shaded cells and subtract the entries in the light shaded cells. What properties makes the answer invariant? What properties makes the answer invariant? What property is sufficient to make the answer invariant? What property is sufficient to make the answer invariant?

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11 Circles in Circles How are the red and yellow areas related? red orange yellow

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12 Carpet Theorems In a room there are two carpets whose combined area is the area of the room. In a room there are two carpets whose combined area is the area of the room. –The area of overlap is the area of floor uncovered In a room there are two carpets. They are moved so as to change the amount of overlap. In a room there are two carpets. They are moved so as to change the amount of overlap. –The change in the area of overlap is the change in area of uncovered floor

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13 Rectangular Room with 2 Carpets How are the red and blue areas related?

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14 Perimeter Projections The red point traverses the quadrilateral The red point traverses the quadrilateral The vertical movement of the red point is tracked. The vertical movement of the red point is tracked. What shape is the graph? What shape is the graph? Given a graphical track of the vertical movement and the horizontal movement, Given a graphical track of the vertical movement and the horizontal movement, What is the shape of the polygon? What is the shape of the polygon?

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15 Square Deduction Could these all be squares?

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16 Square Deduction: tracking arithmetic x4 2x x4 3x3+4 Track the 3 and the 4: Replace the 3 by a and the 4 by b(3x3+4)/3 3x4-3x3

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17 Square Deduction: acknowledging ignorance a b a+ba+ba+ba+b a+2b 2a+b2a+b2a+b2a+b a+3b 3a+b3a+b3a+b3a+b 3a+b = 3(3b-3a) 12a = 8b So 3a = 2b For an overall square 4a + 4b = 2a + 5b So 2a = b For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1) But not also 2a = b (3a+b)/3 3b-3a

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18 Reflection What aspects of reasoning… What aspects of reasoning… –Stood out for you? –Involved some struggle What actions … What actions … –Did you undertake? –Were ineffective (why?) –Were effective (why?)

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19 Tasks Tasks promote activity; activity involves actions; actions generate experience; Tasks promote activity; activity involves actions; actions generate experience; –but one thing we dont learn from experience is that we dont often learn from experience alone It is not the task that is rich … It is not the task that is rich … – but whether it is used richly What matters more than the particular answer is … What matters more than the particular answer is … – how do you know? – what can you vary and still the same approach works?

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20 Reminder Do–Talk–Record Do–Talk–Record –Generating need to communicate Provoking Engagement Provoking Engagement –Surprise –Challenge (trust) Promoting Mathematical thinking Promoting Mathematical thinking –How do you know? … –Why must …? Active Students Active Students –Constructing –Extending

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21 Follow Up mcs.open.ac.uk/jhm3 open.ac.uk mcs.open.ac.uk/jhm3 Thinking Mathematically (new edition) Designing and Using Mathematical Tasks Questions and Prompts … (Primary version from ATM) Thinkers Thinking Mathematically (new edition) Designing and Using Mathematical Tasks Questions and Prompts … (Primary version from ATM) Thinkers Institute of Mathematical Pedagogy August mcs.open.ac.uk/jhm3

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