1. 1 Before We Start! If only learners would or could … Teacher s On your card please complete the relevant sentence AND please predict what the others will say If only teachers would or could … Learner s Indicate your perspective with T for teacher, S for student, S+T for both

2. 2 If Only’s  If only learners would/could... –review their notes between lectures –remember what they had been taught –carry out techniques correctly –appreciate the big ideas –know why they are here –like mathematics –…–…–…–…

3. 3 Mathematical Pedagogy John Mason PIMS Celebration Lecture SFU 2007

4. 4 Definitions  Mathematical Pedagogy –Strategies for teaching maths; useful constructs  Mathematical Didactics –Tactics for teaching specific topics or concepts  Pedagogical Mathematics –Mathematical explorations useful for, and arising from, pedagogical considerations

5. 5 Mathematics & Mathematics Education Definitions rarely precise; Lots of different words for similar things; Use of the same words for different things; Multiple ways of perceiving & interpreting; No theorems; There are several differences between mathematics and mathematics education: Any generality can be contradicted (including this one?)

6. 6 Holisitic ––> Strategic studying Learning-Teaching Tension  Procedural & Conceptual...  Instrumental & Relational... (Skemp)  Surface & Deep...  Syntactic & Semantic...  Metonymic & Metaphoric... approach... understanding... competence... appreciation

7. 7 Inter-Rootal Note the task structure: imagine; another & another  Imagine a quadratic function with an inter-rootal distance of 2  and another  what can be varied and still preserve the inter-rootal distance?  Imagine all possible ones

8. 8 Constrained Construction  Sketch a cubic which goes through the origin and which also has a local maximum and which has only one real root and which also has a positive inflection slope Note the task structure: use of a constraint to challenge usual/familiar examples

9. 9 Chords  What is the locus of midpoints of chords of your cubic? Multiple variation: Control one thing; Control another

10. 10 Reflection  In what ways have you been acting procedurally, instrumentally, syntactically, metonymically?  In what ways have you been acting conceptually, relationally, semantically, metaphorically?  How would you describe your approach on a scale of surface ---- deep?

11. 11 Functions (1)  Write down a continuous function which is differentiable everywhere on R except at one point –and another Dimensions of possible variation Range of permissible change and another, but not-differentiable for a different reason How many different reasons can you invoke?

12. 12 Exemplification Paradox In order to appreciate a generality, it helps to have examples; In order to appreciate a generality, it helps to have examples; In order to appreciate something as an example, it is necessary to know the generality being exemplified; In order to appreciate something as an example, it is necessary to know the generality being exemplified; so, I need to know what is exemplary about something in order to see it as an example of something! so, I need to know what is exemplary about something in order to see it as an example of something! Example-Space What can change and what must stay the same, to preserve examplehood?

13. 13 Conjecture  When learners construct their own examples of mathematical objects they: –Extend and enrich their own example spaces –Become more engaged with and more confident about their studies –Make use of their own mathematical powers –Experience mathematics as a constructive and creative enterprise

14. 14 Pedagogical Issues  what examples to show learners?  what needs to be done with learners for them to appreciate what is exemplary?  Where does their attention need to be?  How can one probe learners’ sense of examplehood, effectively? Dimensions of possible variation Range of permissible change Invariance in the midst of change

15. 15 Functions (2)  Write down a differentiable function on [-1, 1] which is strictly increasing on the interval but has derivative 0 at 0. [Edinburgh Chpt 3 tutorial sheet] –and another –how tricky can you be? –how often could such a function have 0 derivative?

16. 16 Functions (3)  Find two functions f and g differentiable on R for which f g’ everywhere e -x 2e -x -e -x -2e -x

17. 17 Probing by Reversing What can you assert about f if x –> 2 f(x) – 5 x – 2 lim = 3 ? Note the task structure: reversing a standard task probes meaning rather than technique

18. 18 Probing by Exploring behaviour of polynomials for |x| large  The ‘tangent-power’ of a point P relative to a twice differentiable function f is the number of tangents to f through P.  Describe or characterise the regions of the plane with the same tangent- power relative to a function f.

19. 19 Probing Tasks  Given a non-singular linear transformation T from R n to R n, which matrices can represent T?  Construct a limit problem which requires three uses of L’Hôpital’s theorem to work it out.  Given a (finite) group G, which subsets of form a group under element-wise multiplication of subsets?  Given a (finite) group G, which subsets of P (G) form a group under element-wise multiplication of subsets?

20. 20 Conjecture  When learners construct their own examples of mathematical objects they: –Extend and enrich their own example spaces –Become more engaged with and more confident about their studies –Make use of their own mathematical powers –Experience mathematics as a constructive and creative enterprise –Are therefore more likely to function ‘mindfully’

21. 21 Observations  Learners often depend on visual images which may be misleading.  Learners are developing their example spaces and building concept images  To ‘enrich my example space’ means: –Not just ‘more examples’ (cf practice) –but also Explicit Construction Tools –and organisation-categorisation of the space (abstraction; structural awareness) Tasks and Activity are not sufficient for learning!

22. 22 Tasks & Teaching  Tasks are only a vehicle for engaging in mathematical thinking  Learners need to be guided, directed, prompted, and stimulated to make sense of their activity: to reflect –To manifest a reflection geometrically as a rotation, you need to move into a higher dimension! –The same applies to mathematical thinking!

23. 23 Procedural-Instrumental Conceptual-Relational  Human psyche is an interweaving of behaviour (enaction) emotion (affect) awareness (cognition)  Behaviour is what is observable  Teaching: –Expert awareness is transposed into instruction in behaviour –The more clearly the teacher indicates the behaviour expected, the easier it is for learners to display it without generating it from and for themselves transposition didactique didactic tension

24. 24 Pedagogic Structure of a Topic/Concept  Awareness: concept images, example spaces; obstacles  Behaviour: language patterns; techniques, procedures & incantations  Emotions:root problems; dispositions; other uses Three interwoven strands corresponding to three strands of the psyche: behaviour (enaction) affect (emotion) cognition (awareness)

25. 25 Probing Awareness  What is meant by ‘conceptual understanding’ or appreciation?  How might a learner display this?  How might we probe for this?  Justifying or explaining 'as if to a peer'

26. 26 Reconceptualising Assessment  Opportunity to display evidence of becoming a ‘pro’: –Production of examples and counter-examples, conjectures and justifications; –Proficiency in carrying out specific techniques; –Profundity & Profusion of concept images, example spaces; appreciation of mathematical themes, heuristics and interconnectivity; –Propensity to pose problems, make and test conjectures, and to see opportunities for mathematical thinking outside of mathematics; –Probing of own awareness of concepts, definitions, theorems, etc.; –Processing unfamiliar mathematics; –Promoting mathematical thinking among others; –Progress in all of these.

27. 27 Potential follow-up  Ask your students … – What examples can you remember from the lectures/work on topic X? – What do you do with the examples given in lectures? In the text?  Please send any findings to j.h.mason @ open.ac.uk j.h.mason @ open.ac.uk