john mason mei keele june 2012
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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. - PowerPoint PPT PresentationTRANSCRIPT
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Transformations of the Number-Line
an exploration of the use of the power of mental imagery
and shifts of attention
John MasonMEI
KeeleJune 2012The Open University
Maths Dept University of OxfordDept of Education
Promoting Mathematical Thinking
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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.
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Assumptions Tasks –> (mathematical) Activity –> (mathematical) Actions –> (mathematical) Experience –> (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
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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’
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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 rightT7(x) = Notation:
Tt translates by tTt(x) =
x + 7
x + t
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Reflexive Stance How did you work it out?
– Lots of examples?– Recognising a Relationship in the particular?– Perceiving a Property in its generality?
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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]
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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
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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?
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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:R0 rotates about 0R0 (x) = –x
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
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Reflexive Stance How did you work it out?
– Lots of examples?– Structurally?– Recognising a Relationship in the particular?– Perceiving a Property in its generality?
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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:R5 rotates about 5 R5 (x) =
Notation:Ra rotates about a Ra (x) = 5 – (x – 5) a – (x – a)
Expressing relationships in generalTracking Arithmetic
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
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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 1616 - 1703
David Hilbert 1862-1943
One generic example?
Working from examples
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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?
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
Ra(x) = 2a – x
Rb(Ra(x)) = = 2b – (2a – x)
Rb(2a – x)
= 2(b–a) + x= T2(b–a)(x)
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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?
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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
Q5 rotates about where
5 currently isQb(Qa(x)) = Qa(x) = Ra(x) = 2a –
x= R2a–b(Ra(x)) = 2(2a–b) - (2a – x) = 2(a – b) + x = T2(a–b)(x)
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
RR (b)(Ra(x)) a
Any resonances?
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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?
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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:S3/2(x : 0) scales from 0 by the factor 3/2S3/2(x : 0) =
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
3x/2 Suggestive:Sσ(x : a) scales from a by the factor σSσ(x : a) = σ(x–a) + a
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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?
-1 1 2 3 4 5 6 7 8 9 10
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-2-3-4-5-6-7-8-9-10
-11
0-12
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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
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Variation A lesson without the opportunity for students to
generalise …… mathematically, is not a mathematics lesson.
What was varied …– By me?– By you?
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Structure of the Psyche
ImageryAwareness (cognition)
Will
Body (enaction)
Emotions (affect)
HabitsPractices
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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)
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To Follow Up
http://mcs.open.ac.uk/jhm3Presentations
AppletsDeveloping Thinking in Geometry