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1 Task Construction: lessons learned from 25 years of distance support for teachers John Mason Nottingham Feb 2012 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

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2 Outline Some Tasks OU Frameworks –MGA, DTR, Stuck, EIS, … –APC or ORA: Own experience, Reflection on parallels, Apply to classroom Systematics Frameworks What makes a task ‘rich’?

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3 Number Line Translations… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Translate the copy line to the right by 3 –Where does 7 end up? –Where does –2 end up? 012345678-2-3-4-5-6-7 Denote translation to the right by a, by T a What is T a followed by T b ? What about T b followed by T a ? I am thinking of a number … tell me how to work out where it ends up

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4 Number Line Scaling… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it 012345678-2-3-4-5-6-7 Denote scaling from 0 by a factor of s by S s I am thinking of a number … tell me how to work out where it ends up What is S a followed by S b ? Denote scaling from p by a factor of s by S p,s What is S p,s in terms of T and S s ?

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5 Number Line Scaling… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Scale the number line by a factor of 3 –(keeping 0 fixed) –Where does 2 end up? –Where does –3 end up? 012345678-2-3-4-5-6-7 Denote scaling from 0 by a factor of s by S s What is S a followed by S b ? Denote scaling from p by a factor of s by S p,s What is S p,s in terms of T and S s ? I am thinking of a number … tell me how to work out where it ends up

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6 Number Line Rotations… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Rotate the copy through 180° about the point 3 Rotate twice about 0 … … to see why R –1 R –1 = T 0 = S 1 and so (-1) x (-1) = 1 –Where does 7 end up? –Where does -2 end up? 012345678-2-3-4-5-6-7 I am thinking of a number … tell me how to work out where it ends up Denote rotating about the original point p by R p What is R p followed by R q ?

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7 Diamond Multiplication

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8 Differing Sums of Products Write down four numbers in a 2 by 2 grid Add together the products along the rows Add together the products down the columns Calculate the difference What other grids will give the answer 2? Choose positive numbers so that the difference is 7 That is the ‘doing’ What is an undoing? 45 37 28 + 15 = 43 20 + 21 = 41 43 – 41 = 2

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9 Differing Sums & Products Tracking Arithmetic 45 37 4x7 + 5x3 4x5 + 7x3 4x(7–5) + (5–7)x3 = (4-3) x (7–5) So in how many essentially different ways can 2 be the difference? What about 7? So in how many essentially different ways can n be the difference? = 4x(7–5) – (7–5)x3

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10 Patterns with 2 Embedded Practice (Gattegno & Hewitt)

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11 Structured Variation Grids TunjaFactoringQuadratic Double Factors Sundaram

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12 Put your hand up when you can see … Something that is 3/5 of something else Something that is 2/5 of something else Something that is 2/3 of something else Something that is 5/3 of something else ………… Something that is 1/4 – 1/5 of something else

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13 Remainders What is the remainder on dividing 5 by 3? What is the remainder on dividing -5 by 3? What question am I going to ask next? What is the remainder on dividing 5 by -3? What is the remainder on dividing -5 by -3?

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14 Task Purposes To introduce or extend contact with concepts To highlight awareness of human powers used mathematically To focus attention on mathematical themes To sharpen awareness of –study strategies –problem solving strategies (heuristics) –learning how to learn mathematics –evaluating own progress –exam technique Purpose for students Potential Utility (Ainley & Pratt)

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15 Learning from Tasks Tasks –> Activity –> Actions –> Experience But one thing we don’t seem to learn from experience … –is that we don’t often learn from experience alone! –> withdraw from action and reflect upon it –What was striking about the activity? –What was effective and what ineffective? –What like to have come-to-mind in the future? Personal propensities & dispositions? Habitual behaviour and desired behaviour? Fresh or freshened awarenesses & realisations?

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16 Task Design Content(Mathematics) Tasks Resources Interactions (as transformative actions) Post-parationPost-flection When does learning take place? Activity In sleep!!! Pre-paration Pre-flection Reflection

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17 Slogans A lesson without opportunity for learners … to generalise mathematically … is not a mathematics lesson! … is not a mathematics lesson! A lesson without opportunity for learners … to make and modify conjectures; to construct a narrative about what they have been doing; to use and develop their own powers; to encounter pervasive mathematical themes is not an effective mathematics lesson Trying to do for learners only what they cannot yet do for themselves

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18 Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing

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19 Teacher Student Content Expounding Student Content Teacher Examining Student Content Teacher Expressing Teacher Student Explaining Content Student Content Teacher Exploring Student Content Teacher Exercising

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20 Activity Goals, Aims, Desires, Intentions Tasks (as imagined, enacted, experienced, …) Resources: (physical, affective, cognitive, attentive) Initial State Affordances– Constraints–Requirements (Gibson)

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21 Potential Most it could be What builds on it (where it is going) Math’l & Ped’c essence Least it can be What it builds on (previous experiences) Affordances– Constraints–Requirements (Gibson) Directed–Prompted–Spontaneous Scaffolding & Fading (Brown et al) ZPD (Vygotsky)

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22 Thinking Mathematically CME –Do-Talk-Record (See–Say–Record) –See-Experience-Master –Manipulating–Getting-a-sense-of–Artculating –Enactive–Iconic–Symbolic –Directed–Prompted–Spontaneous –Stuck!: Use of Mathematical Powers –Mathematical Themes (and heuristics) –Inner & Outer Tasks

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23 Frameworks Doing – Talking – Recording Enactive– Iconic– Symbolic See– Experience– Master

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24 Example From NNP project, pattern sequences to be counted Stuck with providing first, second, third and only later recognising the dependency created Unlocking potential –Universality of the Frame Theorem (Gaussian Curvature and Betti Numbers) –Counting squares, counting sticks, … –Counting weights

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25 Example: Extending Mathematical Sequences Mathematically “What is the next term …?” only makes sense when... Mathmematical guarantee of uniqueness –Geometrical or other construction source –Some other constraint

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26 Provide two or more sequences in parallel Painted Wheel (Tom O’Brien) Someone has made a simple pattern of coloured squares, and then repeated it at least once more State in words what you think the original pattern was Predict the colour of the 100th square and the position of the 100th white square … … Make up your own: a really simple one a really hard one

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27 In how many different ways can you count them? Gnomon Border How many tiles are needed to surround the 137 th gnomon? The fifth is shown here What shapes will have the same Border Numbers?

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28 Extending Mathemtical Sequences Stress in Thinking Mathematically and later on ‘specifying the growth mechanism before trying to count things’ Uniquely Extendable Sequences Theorem Instance of general topological theorem (Betti numbers) Attempts in two Dimensions!

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29 Perforations How many holes for a sheet of r rows and c columns of stamps? If someone claimed there were 228 perforations in a sheet, how could you check?

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30 Gasket Sequences

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31 Attention Teahing and Learning is fundamentally about attention: –What is available or likely to come–to–mind when needed –What is available to be learned? variation –Use of powers –Use of themes –Use of resources (physical, mental, virtual) –Structure of attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

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32 Follow-Up Designing & Using Mathematical Tasks (Tarquin/QED) Thinking Mathematically (Pearson) Developing Thinking in Algebra, Geometry, Statistics (Sage) Fundamental Constructs in Mathematics Education (RoutledgeFalmer) Mathematics Teaching Practice: a guide for university and college lecturers (Horwood Publishing) Mathematics as a Constructive Activity (Erlbaum) Questions & Prompts for Mathematical Thinking (ATM) Thinkers (ATM) Learning & Doing Mathematics (Tarquin) j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3

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