1 Phenomenal Mathematics Phenomenal Mathematics John Mason AAMT-MERGA Alice Springs July The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Ground Rules  EVERYTHING said in this room is a conjecture: –It must be tested in your own experience  When we are unsure about what is said or done: –We ask (or squeak!) !  When we disagree we say –“I invite you to modify your conjecture”

3 Phenomena, Engagement & Motivation  What is it that engages attention? –Mathematical Story Telling –Placing in a meaningful context –Learners’ own questions –Surprise (Nitsa Movshovits-Hadar 1988) –Desire to explain or make sense of ??

4 Outline  Encounter some phenomena  Try to notice how we ‘use’ ourselves to make sense of them mathematically  Apply this to working with learners  Consider the ‘phenomenal conjecture’

5 Phenomenal Conjecture  Every mathematical topic in school and early university –Can be introduced through a phenomenon which provokes a desire to explain –Can be reviewed by using it to explain a phenomenon  While I shall be working on phenomena as triggers to activity –You may be caught up in the activity and may not catch the moment of stimulation!

6 Differing Sums of Products  Write down four numbers in a 2 by 2 grid  Add together the products along the rows  Add together the products down the columns  Calculate the difference  Now choose different numbers so that the difference is again = = – 41 = 2 Action! Undoing the Action!

7 Differing Sums of Products  Tracking Arithmetic x7 + 5x3 4x5 + 7x3 4x(7–5) + (5–7)x3 = (4-3)x (7–5) = 4x(7–5) – (7-5)x3 So how can an answer of 2 be guaranteed? In how many essentially different ways can 2 be the difference? What is special about 2?

8 Making a Phenomenon Phenomenal  A lesson without an opportunity for learners to generalise (mathematucally) –is not a mathematics lesson  What turns an incident into a phenomenon is the act of: –seeing through the particular to a generality –recognising relationships in a situation and perceiving these as instantiations of properties

9 Trained to watch Trains How long is the train? How fast is it going?

10 Fairground Rides

11 Fairground Rides What difference would it make to the experience of the ride if the short arm was attached to the tower, and the long arm attached to the end of the short arm?

12 More Fairground Rides

13 Rolling Perpendiculars  Drop perpendiculars from point P to diameters  Move P round the circle  What will you happen?

14 Cross Ratio

15 Ratio Sums AP PB AQ QC + Where in the diagram might you find might you find ?

16 Combining Functions Making Math’l Sense Math’l Reasoning Using Coordinates to ‘coordinate’ graphs Getting a sense of composite functions

17 Box Ribbon Where does the ribbon have to be in order to be tight? Which uses more ribbon?

18 Ribbons Not only is the diagonal ribbon always tight, but it uses less ribbon!

19 Structured Variation Grids Number Grid Fraction Grid

20 Sundaram’s Grid Conjecture: a number appears in this infinite grid if and only if one more than double it is not prime

21 Recognising Transformations

22 De-Constructing Scalings & Rotations  Given a triangle and a rotated copy, how can you find the centre of rotation?  Given a triangle and a scaled copy, how can you find the centre of scaling?  Given a triangle and a rotated and scaled copy, is there a single point for the centre of rotation and scaling?

23 Secret Places  One of the places around the table is special but secret.  If you click on a place you will be told whether or not the secret place is within one place. How few choices can be made to guarantee finding the secret place? 2D 14 10

24 Differences Anticipating Generalising Rehearsing Checking Organising

25 Magic Square Reasoning – = 0 Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

26 More Magic Square Reasoning – = 0 Sum( ) Sum( )

27 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

28 Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Organising & Characterising

29 Reflection  Tasks –> Activity –> Experience –But one thing we don’t seem to learn from experience … –Is that we don’t often learn from experience alone  What provoked activity?  What was effective?  What powers did I use?  What themes did I encounter?  What could I vary and still use the same ideas?

30 Phenomenal Conjecture  Every mathematical topic in school and early university –Can be introduced through a phenomenon which provokes a desire to explain –Can be reviewed by using it to explain a phenomenon  Were you able to catch the moment of being provoked, surprised or otherwise stimulated?

31 Follow-Up Open.AC.UK mcs.open.ac.uk/jhm3 Applets & Animations SVGrids This and other presentations Thinking Mathematically (new edition 2010) Researching Your Own Practice (Discipline of Noticing) Questions & Prompts; Thinkers;