Thinking Mathematically and Learning Mathematics Mathematically John Mason

Thinking Mathematically and Learning Mathematics Mathematically John Mason Greenwich Oct 2008 1

Conjecturing Atmosphere /Everything said is said in order to consider modifications that may be needed /Those who ‘know’ support those who are unsure by holding back or by asking revealing questions 2

Up & Down Sums 1+3+5+3+ 1 = 22 + 3 2 = 3 x 4+1 1 + 3 + … + (2 n– 1) + … + 3 + 1 = 3 (n– 1)2 + n 2 = n (2 n– 2) + 1

Remainders of the Day /Write down a number that leaves a reminder of 1 when divided by 3 /and another /Choose two simple numbers of this type and What is special about the ‘ 1’? multiply them together: what remainder does it leave when divided by 3? /Why? /What is special about the ‘ 3’? What is special about the ‘ 1’? 5

Primality /What is the second positive nonprime after 1 in the system of numbers of the form 1+3 n? /100 = 10 x 10 = 4 x 25 /What does this say about primes in the multiplicative system of numbers of the form 1 +3 n? /What is special about the ‘ 3’? 6

Inter-Rootal Distances /Sketch a quadratic for which the interrootal distance is 2. /and another /How much freedom do you have? /What are the dimensions of possible variation and the ranges of permissible change? /If it is claimed that [1, 2, 3, 3, 4, 6] are the inter-rootal distances of a quartic, how would you check? 7

Bag Constructions (1) /Here there are three bags. If you compare any two of them, there is exactly one colour for which the difference in the numbers of that colour in the two bags is exactly 1. For four bags, what is the least number of objects to meet the same constraint? / For four bags, what is the least number of colours to meet the same constraint? / 8 17 objects 3 colours

Bag Constructions (2) /Here there are 3 bags and two objects. /There are [0, 1, 2; 2] objects in the bags with 2 altogether /Given a sequence like [2, 4, 5, 5; 6] or [1, 1, 3, 3; 6] how can you tell if there is a corresponding set of bags? 9

Statisticality /write down five numbers whose mean is 5 /and whose mode is 6 /and whose median is 4 10

Zig. Zags /Sketch the graph of y = |x – 1| /Sketch the graph of y = | |x - 1| - 2| /Sketch the graph of y = | | |x – 1| – 2| – 3| /What sorts of zigzags can you make, and not make? /Characterise all the zigzags you can make using sequences of absolute values like this. 11

Towards the Blanc Mange function 12

Reading Graphs 13

Examples /Of what is |x| an example? /Of what is y = x 2 and example? – y = b + (x – a )2 ? 14

Functional Imagining /Imagine a parabola /Now imagine another one the other way up. /Now put them in two planes at right angles to each other. /Make the maximum of the downward parabola be on the upward parabola /Now sweep your downward 15 parabola along the upward parabola so that you get a surface

MGA 16

Powers /Specialising & Generalising /Conjecturing & Convincing /Imagining & Expressing /Ordering & Classifying /Distinguishing & Connecting /Assenting & Asserting 17

Themes /Doing & Undoing /Invariance /Freedom & Constraint /Extending 18 Amidst Change & Restricting Meaning

Teaching Trap Learning Trap / Expecting the teacher to Doing for the learners do for you what you can what they can already do for themselves already do for yourself / Teacher Lust: / Learner Lust: – desire that the learner – desire that the teacher learn teach – desire that the learner – desire that learning will appreciate and be easy understand – expectation that ‘dong – Expectation that learner the tasks’ will produce will go beyond the tasks learning as set – allowing personal excitement to drive reluctance/uncertainty behaviour to drive behaviour 19 /

Human Psyche /Training Behaviour /Educating Awareness /Harnessing Emotion /Who does these? – Teacher with learners? – Learners! 20

Structure of the Psyche Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices 21

Structure of a Topic Language Patterns & prior Skills Imagery/Senseof/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise; dispositions Techniques & Incantations Emotion ur vio ha Be Aw ar en es s Standard Confusions & Obstacles Only Emotion is Harnessable Only Awareness is Educable 22 Only Behaviour is Trainable

Didactic Tension The more clearly I indicate the behaviour sought from learners, the less likely they are to generate that behaviour for themselves (Guy Brousseau) 23

Didactic Transposition Expert awareness is transposed/transformed into instruction in behaviour (Yves Chevellard) 24

More Ideas For Students (1998) Learning & Doing Mathematics (Second revised edition), QED Books, York. (1982). Thinking Mathematically, Addison Wesley, London For Lecturers (2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester. (2008). Counter Examples in Calculus. College Press, London. http: //mcs. open. ac. uk/jhm 3 j. h. mason@open. ac. uk 25

Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing

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