1. 1 With and Across the Grain: making use of learners’ powers to detect and express generality London Mathematics Centre June 2006

2. 2 Outline  Orienting Questions  Tasks through which to experience and in which to notice  Reflections a lesson without the opportunity for learners to generalise is not a mathematics lesson! Conjecture

3. 3 Orienting Questions  What can be changed?  In what way can it be changed?  What is the inner task?  What pedagogical and didactic choices are being made? What you can get from this session What you can get from this session: what you notice inside yourself; ways of thinking and acting; useful distinctions.

4. 4 Seeing the general through the particular Up and Down Sums 1 + 2 + 3 + 4 + 3 + 2 + 1 = 1 + 2 + 3 + 2 + 1 = 1 + 2 + 1 = 1 = 12121212 22 22 22 22 32323232 42424242 … In how many different ways can you see WHY it works? What is the it? 1 + 2 + 3 + 4 + 3 + 2 + 1 = 4 x 4 = 4 2

5. 5 Seeing the general through the particular More Up and Down Sums 1 + 3 + 5 + 7 + 5 + 3 + 1 = 1 + 3 + 5 + 3 + 1 = 1 + 3 + 1 = 1 = 1x0 + 1 2x2 + 1 2x2 + 1 3x4 + 1 4x6 + 1 … In how many different ways can you see WHY it works? What is the it? 1 + 3 + 5 + 7 + 5 + 3 + 1 = 4x6 + 1

6. 6 Yet More Up and Down Sums 1 + 4 + 7 + 10 + 7 + 4 + 1 = 1 + 4 + 7 + 4 + 1 = 1 + 4 + 1 = 1 = 1x(-1) + 2 2x2 + 2 2x2 + 2 3x5 + 2 4x8 + 2 … In how many different ways can you see WHY it works? What is the it? 1 + 4 + 7 + 10 + 13 + 10 + 7 + 4 + 1 = 5x11 + 2 1 + 4 + 7 + 10 + 13 + 10 + 7 + 4 + 1 1 + 4 + 7 + 10 + 13 + 10 + 7 + 4 + 1 = 5x11 + 2

7. 7 Same and Different  What is the same and what is different about the three Up and Down Sum tasks?  What dimensions of possible variation are there? (what could be changed?)  What is the range of permissible change of each dimension (what values could be taken?) 1 + 4 + 7 + 4 + 1 = 3x5 + 2 1 + 3 + 5 + 3 + 1 = 3x4 + 1 1 + 2 + 3 + 2 + 1 = 32323232

8. 8 Reflections  What inner tasks might be associated with Up and Down Sums?  What pedagogic and didactic choices did you notice being made?

9. 9 Some Sums 1 + 2 = 4 + 5 + 6 = 9 + 10 + 11 + 12 = Generalise Justify Watch What You Do Say What You See 7 + 8 13 + 14 + 15 16+ 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24 3

10. 10 What’s The Difference? What could be varied? –= First, add one to each First, add one to the larger and subtract one from the smaller What then would be the difference?

11. 11 a b a+ba+b a+2b 2a+b2a+b a+3b 3a+b3a+b 3b-3a 2 3 5 8 7 9 1 3 3(3b-3a) = 3a+b 8b = 12a So a : b = 2 : 3 For an overall square 4a + 4b = 2a + 5b So 2a = b Oops! For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1) a : b = 3n – 1 : 3(n + 1) Square Reasoning

12. 12 Seeing the general through the particular; Expressing Generality Reflections  What might the inner tasks have been?  What pedagogic and didactic choices did you notice being made? In every lesson! Say What You See Can You See? With and Across the grain

13. 13 With & Across The Grain (part 2): Sequences and Grids Number Grid What sorts of orienting questions are you ready to use in this session?

14. 14 Experiencing Generalisation  Pleasure in use of powers; disposition: affective generalisation (Helen Drury)  Going across the grain: cognitive generalisation  Going with the grain: enactive generalisation

15. 15 Use of Powers on Inappropriate Data  If 10% of 23 is 2.3 –What is 20% of 2.3? –.23 ?!?  If to find 10% you divide by 10 –To find 20% you divide by –20 ?!? Perhaps some ‘wrong answers’ arise from inappropriate use of powers … so you can praise the use of powers but treat the response as a conjecture … which needs checking and modifying!

16. 16 Reflections  Arithmetic has an underlying structure/logic  What pedagogic choices are available?  What didactic choices are available?  What might you do if some learners ‘work it all out’ quickly?  What might you do if learners are reluctant to conjecture?

17. 17 Imagine …  A teaching page of a textbook, or a work card or other handout to learners used or encountered recently –What are its principal features? –What are learners supposed to get from ‘doing it’? Conjecture: it contains Examples of … A lesson without the opportunity for learners to generalise … is NOT a maths lesson! Problem type Technique Method Use Concept

18. 18 Descent to the Particular & the Simple  Research on problem solving, task setting, textbooks and use of ICT suggests that –tasks & problems get simplified ‘so learners know what to do’ –learners’ powers are often bypassed  Counteract this by trying to –do only for learners what they cannot yet do for themselves (even if it takes longer)

19. 19 Promoting Generalisation  Dimensions of Possible Variation Range of Permissible Change –What can be changed, and over what range?

20. 20 Toughy 12345678

21. 21 Summary  Expressing Generality –Enactive; Cognitive; Affective –with the grain; across the grain; disposition  Dimensions of Possible Variation Range of Permissible Change  Say What You See & Watch What You Do  What pedagogic and didactic choices could be made?  What is the inner task?

22. 22 Inner Task  What mathematical powers might be used? –Imagining & Expressing –Specialising & Generalising –Stressing & Ignoring –Conjecturing & Convincing  What mathematical themes might arise? –Invariance in the midst of change –Doing and Undoing  What personal dispositions might emerge to be worked on? –Tendency to dive in –Tendency to give up or seek help

23. 23 Consecutive Sums Say What You See Can You See …?

24. 24 For Further Exploration Mason, J. with Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage. Johnston-Wilder, S. & Mason, J. (Eds.) (2005). Developing Thinking in Geometry. London: Sage. Open University Courses on teaching Algebra, Geometry, Statistics and Mathematical Thinking Structured variation Grids available (free) on http://mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk