Task Construction: lessons learned from 25 years of distance support for teachers

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

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’?

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 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 • I am thinking of a number … tell me how to work out where it ends up • Where does 7 end up? • Where does –2 end up? Denote translation to the right by a, by Ta What is Ta followed by Tb? What about Tb followed by Ta?

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 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Denote scaling from 0 by a factor of s by Ss • I am thinking of a number … tell me how to work out where it ends up • What is Sa followed by Sb? • Denote scaling from p by a factor of s by Sp,s • What is Sp,s in terms of T and Ss?

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) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 • I am thinking of a number … tell me how to work out where it ends up • Where does 2 end up? • Where does –3 end up? • Denote scaling from 0 by a factor of s by Ss • What is Sa followed by Sb? • Denote scaling from p by a factor of s by Sp,s • What is Sp,s in terms of T and Ss?

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 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 • I am thinking of a number … tell me how to work out where it ends up • Where does 7 end up? • Where does -2 end up? • Denote rotating about the original point p by Rp • What is Rp followed by Rq? Rotate twice about 0 … … to see why R–1R–1 = T0 = S1and so (-1) x (-1) = 1

7. Diamond Multiplication

8. Differing Sums of Products 4 7 • Write down four numbers in a 2 by 2 grid 3 5 • Add together the products along the rows 28 + 15 = 43 • Add together the products down the columns 20 + 21 = 41 43 – 41 = 2 • Calculate the difference • That is the ‘doing’What is an undoing? • What other grids will give the answer 2? • Choose positive numbers so that the difference is 7

9. Differing Sums & Products 4 7 • Tracking Arithmetic 3 4x7 + 5x3 5 4x5 + 7x3 4x(7–5) + (5–7)x3 = 4x(7–5) – (7–5)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?

10. Patterns with 2 Embedded Practice (Gattegno & Hewitt)

11. Structured Variation Grids Tunja Factoring Quadratic Double Factors Sundaram

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 – 1/5of something else

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?

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)

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?

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

17. Slogans A lesson without opportunity for learners … to generalise mathematically … 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

18. Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing

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

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

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

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

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

24. 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

25. 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 Provide two or more sequences in parallel

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

27. 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!

28. Perforations If someone claimedthere were 228 perforationsin a sheet, how could you check? How many holes for a sheet ofr rows and c columns of stamps?

29. Gasket Sequences

30. 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 DetailsRecognising RelationshipsPerceiving PropertiesReasoning on the basis of agreed properties

31. 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