1 Where is the Reality of Algebra & Geometry ? John Mason Surrey HoDs Feb 2009 The Open University Maths Dept University of Oxford Dept of Education

2 Assumptions  Learning is extending the range of possible actions that come to mind in a situation  Discerning details  Recognising relationships  Perceiving properties  Reasoning on the basis of properties

3 Do you know any students who …  Don’t seem to participate in mathematics lessons  Say “I can’t …”  Ask “Why are we doing this?”  Appear to do the minimum to get through a lesson?

4 Tracks

5 Mechanism

6 Where Were You?  Where was your attention? –In or on the picture? –Imagining something moving? –In or on recognising relationships? –In or on perceiving instances of properties? –In or on reasoning about those relationships?

7 Imagine a Number-line …  Imagine a copy sitting on top;  Rotate the copy through 180° about the origin  Restore the copy.  Now rotate it through 180° about the point 3 on the original fixed number-line  Now rotate it through a further 180° about the point 1 on the original fixed number-line  What is special about the points 3 and 1  How could you develop or extend this task?

8 What ‘world’ were you occupying?  The world of the screen?  The world of your imagination?  A mathematical world?  The material world?

9 Kites

10 Where Were You?  Where was your attention? –In or on the paper? –In or on the folding? –In or on relationships? –Reasoning about those relationships?

11 Square Formation a b a+ba+b a+2b 2a+b2a+b a+3b 3a+b3a+b 3b-3a (3b-3a) = 3a+b 12a = 8b So a/b = 3/2 For an overall square 4a + 4b = 2a + 5b So 2a = b For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1)

12 Where Were You?  Where was your attention? –On material objects? –In or on some diagrammatic presentation? –What were you manipulating?

13 Triangle Count

14 Seven Circles How many different angles can you discern, using only the red points? How do you know you have them all? How many different quadrilaterals?

15 Four Consecutives  Write down four consecutive numbers and add them up  and another  Now be more extreme!  What is the same, and what is different about your answers?

16 Leibniz’s Triangle 1

17 Ages Ago  Two persons, A, and B were talking of their Ages: Says A to B, seven Years ago I was thrice as old as you at that Time; and seven Years hence I shall be just twice as old as you will be: I demand their present Ages? –[Mole 1788 problem V p129] Make a guess: A is 14 Now check: (14 – 7) = 3(9 – 7) so B is 9? And (14 + 7) ?=? 2(9 + 7) No … but (A – 7) = 3(B – 7) and (A+7) = 2(B + 7)

18 Sharing  A hungry hunter came upon two shepherds, one of whom had 3 loaves and the other 5, all of the same size. The loaves were divided equally among the three. The hunter paid 8 cents for his share. How should the shepherds divide the money?

19 Debbie’s Apples  Debbie bought apples at 25p each. She ate 2 and sold the rest for 32p each. Her profit was £1.88. How many apples did she buy? 25 32

20 Do you know any students who …  Don’t seem to participate in mathematics lessons  Say “I can’t …” –Convert can’t into didn’t  Ask “Why are we doing this?” –Customers want particular (price …) –Entrepreneurs & Managers need general (policy, …)  Appear to do the minimum to get through a lesson?

21 Are they being encouraged to …  use their natural powers to –Imagine & Express –Specialise & Generalise  make significant mathematical choices

22 Reality is …  Being intrigued, surprised, engaged  Using your own powers to –Imagine & Express –Specialise & Generalise –Conjecture & Convince –Extend & Restrict attention  It is NOT restricted to –Utility, or even purpose –Your own past experience –Some imagined use in the future  Reality is where your focal attention is One of the core contributions of schooling is … … experiencing ‘realities’ you might not otherwise encounter

23 Becoming Real  Holding Wholes (gazing)  Discerning details  Recognising relationships  Perceiving properties (as being instantiated)  Reasoning on the basis of properties

24 Worlds Material World World of Symbol s Inner World of Imagery enactive iconic symbolic Worlds during Modelling and Problem-Solving Aspects of Conceptual Development Modes of Attending Epistemological Stances

25 Follow-Up  These slides (and others) available on –  Contact me at