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John Mason MEI Keele June 2012

The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Transformations of the Number-Line an exploration of the use of the power of mental imagery and shifts of attention. John Mason MEI Keele June 2012. Challenge.

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John Mason MEI Keele June 2012

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  1. The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking Transformations of the Number-Linean exploration of the use of the power of mental imageryand shifts of attention John Mason MEIKeele June 2012

  2. Challenge • By the end of this session you will have engaged in the mathematucal and psychological actions necessary to solve and explain to someone else how to resolve: • Given a rotation through 180° of a number line about a specified point, followed by a scaling by a given factor from some other given point, to find a single point that can serve as the centre of rotation AND the centre of scaling, when this is possible.

  3. Assumptions That which enables action • Tasks –> (mathematical) Activity –> (mathematical) Actions –> (mathematical) Experience –> (mathematical) Awareness This requires initial engagement in activity • But …One thing we don’t seem to learn from experience … • is that we don’t often learn from experience alone • In order to learn from experience it is often necessary to withdraw from the activity-action and to reflect on, even reconstructthe action

  4. My Focus Today • The use of mental imagery • Shifts from Manipulating to Getting-a-sense-of to Articulating and Symbolising • All within a conjecturing atmosphere • What you get from today will be what you notice yourself doing … ‘how you use yourself’

  5. Imagine a Number-Line (T1) • Imagine a copy on acetate, sitting on top • Imagine translating the acetate number-line by 7 to the right: • Where does 3 end up? • Where does -2 end up? • Generalise Notation: T7 translates by 7 to the right T7(x) = x + 7 Notation: Tt translates by t Tt(x) = x + t

  6. Reflexive Stance • How did you work it out? • Lots of examples? • Recognising a Relationship in the particular? • Perceiving a Property in its generality?

  7. Two Birds Two birds, close-yoked companionsBoth clasp the same tree;One eats of the sweet fruit,The other looks on without eating. [Rg Veda]

  8. Imagine a Number-Line (T2) • Imagine a copy on acetate, sitting on top • Imagine translating the acetate number-line by 7 to the right; • Now translate the acetate number-line to the left by 4; • Where does 3 end up? • Where does -2 end up? • Generalise T-4 o T7 translates by 7 and then by -4 Does order matter? Ts Tt = Tt Ts = Ts+ t

  9. Reflection • How did you work it out? • Already familiar or expected? • Lots of examples? • Recognising a Relationship in the particular? • Perceiving a Property in its generality? • How fully do you understand and appreciate what you have done? • Could you reconstruct the sequence? • Could you explain it to someone else? • Could you do it without using a diagram?

  10. Imagine a Number-Line (R1) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine rotating the acetate number-line through 180° about the point 0: • Where does 3 end up? • Where does -2 end up? • Generalise Notation: R0 rotates about 0 R0 (x) = –x

  11. Reflexive Stance • How did you work it out? • Lots of examples? • Structurally? • Recognising a Relationship in the particular? • Perceiving a Property in its generality?

  12. Imagine a Number-Line (R2) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine rotating the acetate number-line through 180° about the point 5: • Where does 3 end up? • Where does -2 end up? • Generalise Notation: R5 rotates about 5 R5 (x) = Notation: Ra rotates about a Ra(x) = 5 – (x – 5) a – (x – a) Expressing relationships in general Tracking Arithmetic

  13. Reflexive Stance • How did you work it out? • Lots of examples? • Recognising a Relationship in the particular? • Perceiving a Property in its generality? Working from examples Many examples? One generic example? John Wallis 1616 - 1703 David Hilbert 1862-1943

  14. Imagine a Number-Line (R3) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine rotating the acetate number-line through 180° about the point 5; • Now rotate that about the point where 2 was originally • Where does 3 end up? • Where does -2 end up? • How are these results related? • Generalise! Given a succession of rotations about various points, when is there a single point that can act as the centre for all of them? Ra(x) = 2a – x Rb(Ra(x)) = Rb(2a – x) = 2b – (2a – x) = 2(b–a) + x = T2(b–a)(x)

  15. Reflective Stance • How fully do I understand? • Write down a pair of reflections in different points whose composite in one order is T6 • What is the composite in the other order? • Write down another such pair • And another • What action am I going to suggest you undertake now? • Express a generality! • What needs further work? • Could you reconstruct the sequence? • Could you explain it to someone else? • Could you do it without using a diagram?

  16. Imagine a Number-Line (R3a) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine rotating the acetate number-line through 180° about the point 5; • Now rotate that about the point where 2 now is. • Where does 3 end up? • Where does -2 end up? • Generalise Q5 rotates about where 5 currently is Qb(Qa(x)) = RR (b)(Ra(x)) a Qa(x) = Ra(x) = 2a – x = R2a–b(Ra(x)) = 2(2a–b) - (2a – x) = 2(a – b) + x Any resonances? = T2(a–b)(x)

  17. Meta Reflection • What mathematical actions have you carried out? • What cognitive actions have you carried out? • Holding wholes (gazing) • Discerning Details • Recognising Relationships in the particular • Perceiving Properties (generalities) • Reasoning on the basis of agreed properties (expressing generality; reasoning with symbols) • What affectual shifts have you noticed? • Surprise? • Doubt/Confusion? • Desire? • Shift from ‘easy!’ or ‘boring’ to intrigue?

  18. Imagine a Number-Line (S1) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine stretching the acetate number-line by a factor of 3/2 with 0 fixed: • Where does 3 end up? • Where does -2 end up? • Generalise Notation: S3/2(x : 0) scales from 0 by the factor 3/2 • S3/2(x : 0) = 3x/2 Suggestive: • Sσ(x : a) scales from a by the factor σ • Sσ(x : a) = σ(x–a) + a

  19. Imagine a Number-Line (S2) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 • Imagine a copy on acetate, sitting on top • Imagine scaling the acetate number-line from 2 by the factor of 3/2; • Now scale the acetate number-line from where -1 was originally, by a factor of 4/5; • Where does 3 end up? • Where does -2 end up? • Generalise • What about a succession of scalings about different points: when is there a single centre of scaling with the same overall effect?? • What about a succession of scalings each about the current position of a named point?

  20. Review • How did we start? • Imagining a number line • What actions did we carry out? • Translating the numberline: Ta(x) • Rotating the numberline through 180° • about painted points: Ra(x) • about current points: Qa(x) • Scaling the numberline • from painted points: Sσ(x : a) For exploration • from current points: Uσ(x : a) • How all the formula relate to each other

  21. Variation • A lesson without the opportunity for students to generalise … … mathematically, is not a mathematics lesson. • What was varied … • By me? • By you?

  22. Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) HabitsPractices Structure of the Psyche

  23. Meta-Reflection • Was there something that struck you,that perhaps you would like to work on or develop? • Imagine yourself as vividly as possiblein the place where you would do that work,working on it • Perhaps in a classroom acting in some fresh manner • Perhaps when preparing a lesson • Using your mental imagery to place yourself in the future, to pre-pare, is the single core technique that human beings have developed. • Researching Your Own Practice Using the Discipline of Noticing (Routledge Falmer 2002)

  24. To Follow Up http://mcs.open.ac.uk/jhm3 Presentations Applets Developing Thinking in Geometry j.h.mason@open.ac.uk

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