With and Across the Grain: making use of learners’ powers to detect and express generality

1. With and Across the Grain:making use of learners’ powersto detect and express generality London Mathematics CentreJune 2006

2. a lesson without the opportunityfor learners to generaliseis not a mathematics lesson! Conjecture Outline • Orienting Questions • Tasks through which to experience and in which to notice • Reflections

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 notice inside yourself; ways of thinking and acting; useful distinctions.

4. Seeing the general through the particular Seeing the general through the particular Seeing the general through the particular Up and Down Sums In how many different ways can you see WHY it works? What is the it? 1 = 12 1 + 2 + 1 = 22 1 + 2 + 3 + 2 + 1 = 32 1 + 2 + 3 + 4 + 3 + 2 + 1 = 42 … 1 + 2 + 3 + 4 + 3 + 2 + 1 = 4 x 4 = 42

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

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

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 + 2 + 3 + 2 + 1 = 32 1 + 3 + 5 + 3 + 1 = 3x4 + 1 1 + 4 + 7 + 4 + 1 = 3x5 + 2

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

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

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

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

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

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

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

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. 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. Problem type Technique Method Use Concept 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’? A lesson without the opportunity for learners to generalise … is NOT a maths lesson! Conjecture: it contains Examples of …

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. Promoting Generalisation • Dimensions of Possible VariationRange of Permissible Change • What can be changed, and over what range?

20. Toughy 1 2 3 4 5 6 7 8

21. Summary • Expressing Generality • Enactive; Cognitive; Affective • with the grain; across the grain; disposition • Dimensions of Possible VariationRange 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. 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. Consecutive Sums Say What You See Can You See …?

24. For Further Exploration Open University Courses on teachingAlgebra, Geometry, Statistics and Mathematical Thinking 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. Structured variation Grids available (free) onhttp://mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk