1 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 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Outline  Translating the numberline; –compositions  Rotating the numberline through 180°; –Compositions (absolute & relative)  Combining Rotations and Translations  Scaling the numberline from 0; from a  Combining Rotations, Translations and Scaling  Translations in 2D  Reflctions in 2D  Rotations in 2D  Something Different! (Rolling Polygons)

3 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.

4 Assumptions  Tasks –> (mathematical) Activity –> (mathematical) Actions –> (mathematical) Actions –> (mathematical) Experience –> (mathematical) Experience –> (mathematical) Awareness –> (mathematical) Awareness That which enables action 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 reconstruct the action and to reflect on, even reconstruct the action

5 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’

6 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: T 7 translates by 7 to the right T 7 (x) = Notation: T t translates by t T t (x) = x + 7 x + t

7 Reflexive Stance  How did you work it out? –Lots of examples? –Recognising a Relationship in the particular? –Perceiving a Property in its generality?

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

9 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 T 7 translates by 7 and then by -4 Does order matter? T s T t = T t T s = T s + t

10 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?

11 Imagine a Number-Line (R1)  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: R 0 rotates about 0 R 0 (x) = –x

12 Reflexive Stance  How did you work it out? –Lots of examples? –Structurally? –Recognising a Relationship in the particular? –Perceiving a Property in its generality?

13 Imagine a Number-Line (R2)  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: R 5 rotates about 5 R 5 (x) = Notation: R a rotates about a R a (x) = 5 – (x – 5) a – (x – a) Expressing relationships in general Tracking Arithmetic

14 Reflexive Stance  How did you work it out? –Lots of examples? –Recognising a Relationship in the particular? –Perceiving a Property in its generality? Many examples? John Wallis David Hilbert One generic example? Working from examples

15 Imagine a Number-Line (R3)  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? R a (x) = 2a – x R b (R a (x)) = = 2b – (2a – x) R b (2a – x) = 2(b–a) + x = T 2(b–a) (x)

16 Reflective Stance  How fully do I understand? –Write down a pair of reflections in different points whose composite in one order is T 6 –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?

17 Imagine a Number-Line (R3a)  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 Q 5 rotates about where 5 currently is Q b (Q a (x)) = Q a (x) = R a (x) = 2a – x = R 2a–b (R a (x)) = 2(2a–b) - (2a – x) = 2(a – b) + x = T 2(a–b) (x) R R (b) (R a (x)) a Any resonances?

18 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?

19 Imagine a Number-Line (S1)  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: S 3/2 (x : 0) scales from 0 by the factor 3/2 S 3/2 (x : 0) = x/2 Suggestive: S σ (x : a) scales from a by the factor σ S σ (x : a) = σ(x–a) + a

20 Imagine a Number-Line (S2)  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?

21 Review  How did we start? –Imagining a number line  What actions did we carry out?  Translating the numberline: T a (x)  Rotating the numberline through 180° … about painted points: R a (x) … about current points: Q a (x)  Scaling the numberline … from painted points: S σ (x : a) For exploration …from current points: U σ (x : a)  How all the formula relate to each other

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

23 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

24 Meta-Reflection  Was there something that struck you, that perhaps you would like to work on or develop?  Imagine yourself as vividly as possible in 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)

25 To Follow Up Developing Thinking in Geometry