1 Thinking Mathematically and Learning Mathematics Mathematically John Mason Greenwich Oct 2008.
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Transcript of 1 Thinking Mathematically and Learning Mathematics Mathematically John Mason Greenwich Oct 2008.
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Thinking MathematicallyThinking Mathematicallyand and
Learning MathematicsLearning MathematicsMathematicallyMathematically
John MasonJohn Mason
GreenwichGreenwich
Oct 2008Oct 2008
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Conjecturing AtmosphereConjecturing Atmosphere
Everything said is said in Everything said is said in order to consider order to consider modifications that may be modifications that may be neededneeded
Those who ‘know’ support Those who ‘know’ support those who are unsure by those who are unsure by holding back or by asking holding back or by asking revealing questionsrevealing questions
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Up & Down SumsUp & Down Sums
1 + 3 + 5 + 3 + 1
3 x 4 + 122 + 32
1 + 3 + … + (2n–1) + … + 3 + 1
==
n (2n–2) + 1 (n–1)2 +
n2
==
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Remainders of the DayRemainders of the Day Write down a number that leaves a Write down a number that leaves a reminder of 1 when divided by 3reminder of 1 when divided by 3
and anotherand another and anotherand another Choose two simple numbers of this type Choose two simple numbers of this type and multiply them together: and multiply them together: what remainder does it leave when what remainder does it leave when divided by 3?divided by 3?
Why?Why? What is special What is special about the ‘3’? about the ‘3’?
What is special about the ‘1’?What is special about the ‘1’?
What is special about the ‘1’?
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PrimalityPrimality
What is the second positive What is the second positive non-prime after 1 in the system non-prime after 1 in the system of numbers of the form 1+3of numbers of the form 1+3nn??
100 = 10 x 10 = 4 x 25100 = 10 x 10 = 4 x 25 What does this say about primes What does this say about primes in the multiplicative system of in the multiplicative system of numbers of the form 1 +3numbers of the form 1 +3nn??
What is special about the ‘3’?What is special about the ‘3’?
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Inter-Rootal DistancesInter-Rootal Distances
Sketch a quadratic for which the Sketch a quadratic for which the inter-rootal distance is 2.inter-rootal distance is 2.
and anotherand another and anotherand another How much freedom do you have?How much freedom do you have? What are the dimensions of possible What are the dimensions of possible variation and the ranges of variation and the ranges of permissible change?permissible change?
If it is claimed that [1, 2, 3, 3, If it is claimed that [1, 2, 3, 3, 4, 6] are the inter-rootal 4, 6] are the inter-rootal distances of a quartic, how would distances of a quartic, how would you check?you check?
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Bag Constructions (1)Bag Constructions (1) Here there are three Here there are three bags. If you compare bags. If you compare any two of them, there any two of them, there is exactly one colour is exactly one colour for which the difference for which the difference in the numbers of that in the numbers of that colour in the two bags colour in the two bags is exactly 1.is exactly 1.
17 objects
3 colours
For four bags, what is For four bags, what is the least number of the least number of objects to meet the same objects to meet the same constraint?constraint? For four bags, what is For four bags, what is the least number of the least number of colours to meet the same colours to meet the same constraint?constraint?
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Bag Constructions (2)Bag Constructions (2) Here there are 3 bags and Here there are 3 bags and two objects.two objects.
There are [0,1,2;2] There are [0,1,2;2] objects in the bags with 2 objects in the bags with 2 altogetheraltogether
Given a sequence like Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there how can you tell if there is a corresponding set of is a corresponding set of bags?bags?
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StatisticalityStatisticality
write down five numbers whose write down five numbers whose mean is 5mean is 5
and whose mode is 6and whose mode is 6 and whose median is 4and whose median is 4
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ZigZagsZigZags
Sketch the graph of Sketch the graph of yy = | = |x x – 1|– 1| Sketch the graph of Sketch the graph of yy = | |x - 1| - = | |x - 1| - 2|2|
Sketch the graph of Sketch the graph of y = | | |y = | | |xx – 1| – 2| – 3| – 1| – 2| – 3|
What sorts of zigzags can you make, What sorts of zigzags can you make, and not make?and not make?
Characterise all the zigzags you can Characterise all the zigzags you can make using sequences of absolute make using sequences of absolute values like this.values like this.
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Towards the Blanc Mange Towards the Blanc Mange functionfunction
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Reading GraphsReading Graphs
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ExamplesExamples
Of what is |Of what is |xx| an example?| an example? Of what is Of what is yy = = xx22 and and example?example?– y = by = b + ( + (xx – – aa))22 ? ?
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Functional ImaginingFunctional Imagining
Imagine a parabolaImagine a parabola Now imagine another Now imagine another one the other way up.one the other way up.
Now put them in two Now put them in two planes at right planes at right angles to each other.angles to each other.
Make the maximum of Make the maximum of the downward parabola the downward parabola be on the upward be on the upward parabolaparabola
Now sweep your downward Now sweep your downward parabola along the upward parabola along the upward parabola so that you get a parabola so that you get a surfacesurface
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MGAMGA
aa
Getting-a-sense-ofManipulatingGetting-a-sense-ofArticulatingManipulating
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PowersPowers
Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Imagining & ExpressingImagining & Expressing Ordering & ClassifyingOrdering & Classifying Distinguishing & ConnectingDistinguishing & Connecting Assenting & AssertingAssenting & Asserting
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ThemesThemes
Doing & UndoingDoing & Undoing Invariance Amidst ChangeInvariance Amidst Change Freedom & ConstraintFreedom & Constraint Extending & Restricting Extending & Restricting MeaningMeaning
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Teaching Trap Teaching Trap Learning TrapLearning Trap
Doing for the learners Doing for the learners what they can already what they can already do for themselvesdo for themselves
Teacher Lust:Teacher Lust:– desire that the desire that the learner learnlearner learn
– desire that the desire that the learner appreciate learner appreciate and understand and understand
– Expectation that Expectation that learner will go learner will go beyond the tasks as beyond the tasks as setset
– allowing personal allowing personal excitement to drive excitement to drive behaviourbehaviour
Expecting the teacher to Expecting the teacher to do for you what you can do for you what you can already do for yourselfalready do for yourself
Learner Lust:Learner Lust:– desire that the teacher desire that the teacher teachteach
– desire that learning will desire that learning will be easybe easy
– expectation that ‘dong expectation that ‘dong the tasks’ will produce the tasks’ will produce learninglearning
– allowing personal allowing personal reluctance/uncertainty to reluctance/uncertainty to drive behaviourdrive behaviour
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Human PsycheHuman Psyche
Training BehaviourTraining Behaviour Educating AwarenessEducating Awareness Harnessing EmotionHarnessing Emotion Who does these?Who does these?
– Teacher?Teacher?– Teacher with learners?Teacher with learners?– Learners!Learners!
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Structure of the PsycheStructure of the PsycheImageryImageryAwareness (cognition)Awareness (cognition)
WillWill
Body (enaction)Body (enaction)
Emotions Emotions (affect)(affect)
HabitsHabitsPracticesPractices
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Structure of a TopicStructure of a TopicLanguage Patterns& prior Skills
Techniques & Incantations
Different Contexts in which likely to
arise;dispositions
Root Questionspredispositions
Only Behaviour is TrainableOnly Behaviour is Trainable
Only Emotion is HarnessableOnly Emotion is HarnessableOnly Awareness is Only Awareness is
EducableEducable
Behaviour
Behaviour
Behaviour
Behaviour
EmotionEmotionEmotionEmotion
Awareness
Awareness
Awareness
Awareness
Imagery/Sense-of/Awareness; Connections
Standard Confusions & Obstacles
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Didactic TensionDidactic Tension
The more clearly I indicate the behaviour sought from learners,
the less likely they are togenerate that behaviour for themselves
(Guy Brousseau)
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Didactic TranspositionDidactic Transposition
Expert awareness
is transposed/transformed into
instruction in behaviour(Yves Chevellard)
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More IdeasMore Ideas
(2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester.
(2008). Counter Examples in Calculus. College Press, London.
(1998) Learning & Doing Mathematics (Second revised edition), QED Books, York.
(1982). Thinking Mathematically, Addison Wesley, London
For Lecturers
For Students
http://mcs.open.ac.uk/[email protected]
Modes of interaction
Expounding
Explaining
Exploring
Examining
ExercisingExpressing
TeacherStudent
ContentExpounding
Student
ContentTeacher
Exploring
StudentContentTeacherExamining
StudentContent
TeacherExercising
Student
ContentTeacher
Expressing
Teacher
StudentExplaining
Content