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

2 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

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

4 One More  What numbers are one more than the product of four consecutive integers? Let a and b be any two numbers, one of them even. Then ab/2 more than the product of any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared.

5 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 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’?

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

7 Inter-Rootal Distances  Sketch a quadratic for which the inter- rootal 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?

8 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 objects 3 colours  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?

9 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?

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

11 ZigZags  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.

12 Towards the Blanc Mange function

13 Reading Graphs

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

15 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 parabola along the upward parabola so that you get a surface

16 MGA

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

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

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

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

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

22 Structure of a Topic Language Patterns & prior Skills Techniques & Incantations Different Contexts in which likely to arise; dispositions Root Questions predispositions Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness Imagery/Sense- of/Awareness; Connections Standard Confusions & Obstacles

23 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)

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

25 More 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

Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing

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