1. 1 What is the Discipline of Mathematics Education? Essential Maths & Mathematical Essences John Mason Hobart 2007

2. 2 OutlineOutline  Justifying “a problem a day keeps the teacher in play”  What mathematics is essential?  What is mathematical essence?  Justifying “a problem a day keeps the teacher in play”  What mathematics is essential?  What is mathematical essence?

3. 3 Grid Movement 7 ? +3-3 x2 ÷2 ((7+3)x2)+3 is a path from 7 to ‘?’. What expression represents the reverse of this path? What values can ‘?’ have: if exactly one - and one ÷ are used? Max value? Min Value? What about other cells? Does any cell have 0? -7? Does any other cell have 7? Characterise ALL the possible values that can appear in a cell

4. 4 ReflectionsReflections  What variations are possible?  What have you gained by working on this task (with colleagues)?  What variations are possible?  What have you gained by working on this task (with colleagues)? +3 -3 7 ? x2 ÷2  What criteria would you use in choosing whether to use this (or any) task?  What might be gained by working on (a variant of) this task with learners? Tasks –> Activity –> Experience –> ‘Reflection’

5. 5 More Disciplined Enquiry  What is the point? ( Helen Chick ) –Outer task & Inner task  What is the line? ( Steve Thornton ) –Narrative for HoD, Head, parents, self  What is (the) plain? –What awarenesses? What ‘outcomes’?  What is the space? –Domain of related tasks –Dimensions of possible variation; ranges of permissible change  What is the point? ( Helen Chick ) –Outer task & Inner task  What is the line? ( Steve Thornton ) –Narrative for HoD, Head, parents, self  What is (the) plain? –What awarenesses? What ‘outcomes’?  What is the space? –Domain of related tasks –Dimensions of possible variation; ranges of permissible change

6. 6 DifferencesDifferences Rehearsing Checking Organising Anticipating Generalising

7. 7 Sketchy Graphs  Sketch the graphs of a pair of straight lines whose y-intercepts differ by 2  Sketch the graphs of a pair of straight lines whose x-intercepts differ by 2  Sketch the graphs of a pair of straight lines whose slopes differ by 2  Sketch the graphs of a pair of straight lines meeting all three conditions  Sketch the graphs of a pair of straight lines whose y-intercepts differ by 2  Sketch the graphs of a pair of straight lines whose x-intercepts differ by 2  Sketch the graphs of a pair of straight lines whose slopes differ by 2  Sketch the graphs of a pair of straight lines meeting all three conditions

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

9. 9 Chordal Midpoints  Where can the midpoint of a chord of your cubic get to? (what is the boundary of the region of mid-points?)  What about 1/3 points or 4/3 points?  Where can the midpoint of a chord of your cubic get to? (what is the boundary of the region of mid-points?)  What about 1/3 points or 4/3 points?

10. 10 Justifying ‘doing’ maths for oneself and with others  Sensitise myself to what learners may be experiencing  Refresh my awareness of the movements of my attention  Remind myself what it is like to be a learner  Experience the type of task I might use with learners  Sensitise myself to what learners may be experiencing  Refresh my awareness of the movements of my attention  Remind myself what it is like to be a learner  Experience the type of task I might use with learners

11. 11 AwarenessesAwarenesses Give a family a fish and you feed them for a day Show them how to fish, and you feed them until the stocks run out Give a family a fish and you feed them for a day Show them how to fish, and you feed them until the stocks run out Obtaining tasks and lesson plans gets you through some lessons … Becoming aware of affordances, constraints and attunements, in terms of mathematical themes, powers & heuristics enables you to promote learning

12. 12 More Or Less Altitude & Area Draw a scalene triangle moresameless more same less are a altitude Same alt more area more alt same area more alt more area less alt more area less alt less area more alt less area same alt less area less alt same area

13. 13 More Or Less Rectangles & Area moresameless more same fewer are a No. of rectangles same rects more area more rects same area more rects more area fewer rects more area fewer rects less area more rects less area same rects less area fewer rects same area Draw a rectilinear figure which requires at least 4 rectangles in any decomposition How many can have the same perimeter?

14. 14 More Or Less Percent & Value 50% of something is 20 moresameless more same less % Value 50% of 40 is 20 50% of 60 is 30 40% of 60 is 24 60% of 60 is 36 40% of 30 is 12 60% of 30 is 20 40% of 50 is 20 40% of 40 is 16 50% of 30 is 15

15. 15 More Or Less Whole & Part ? of 35 is 21 moresameless more same less Whole Part 3/5 of 35 is 21 3/4 of 28 is 21 6/7 of 35 is 30 3/5 of 40 is 24

16. 16 Magic Square Reasoning 519 2 4 6 83 7 –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

17. 17 More Magic Square Reasoning –= 0Sum( )Sum( )

18. 18 Graphical Awareness

19. 19 Multiplication as Scaling  If you stick a pin in Hobart in a map of Australia, and scale the map by a factor of 1/2 towards Hobart  And if a friend does the same in Darwin, scaling by 1/2 towards Darwin  What will be the difference in the two scaled maps?  If you stick a pin in Hobart in a map of Australia, and scale the map by a factor of 1/2 towards Hobart  And if a friend does the same in Darwin, scaling by 1/2 towards Darwin  What will be the difference in the two scaled maps? What if one of you scales by a factor of 2/3 towards Hobart and then by a further 1/2 towards Darwin, while the other scales by 1/2 towards Darwin and then by a further 2/3 towards Hobart?

20. 20 Raise Your Hand When You See … Something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/3 of something; 3/5 of 5/3 of something; 5/2 of 2/5 of something; 5/3 of 3/5 of something; 1 ÷ 2/5 of something; 1 ÷ 3/5 of something

21. 21 Essential Conceptual Awarenesses —Choosing the unit —Additive actions —Multiplicative actions —Scaling; multi-ply & many-fold, repetition, lots of; … —Coordinated actions —Angle actions —Combining —Translating —Measuring actions —Comparing lengths; areas; volum es; (unit) —Comparing angles —Discrete-Continuous —Randomness —Choosing the unit —Additive actions —Multiplicative actions —Scaling; multi-ply & many-fold, repetition, lots of; … —Coordinated actions —Angle actions —Combining —Translating —Measuring actions —Comparing lengths; areas; volum es; (unit) —Comparing angles —Discrete-Continuous —Randomness

22. 22 Essential Mathematical-nesses  Mathematical Awarenesses underlying topics  Movement of Attention  Mathematical Themes  Mathematical Powers  Mathematical Strategies  Mathematical Dispositions Ways of working on these constitute a (the) discipline of mathematics education

23. 23 Movement of Attention  Gazing (holding wholes)  Discerning Details  Recognising Relationships  Perceiving Properties  Reasoning on the Basis of Properties Compare SOLO & van Hiele

24. 24 Mathematical Themes  Doing & Undoing  Invariance in the midst of Change  Freedom & Constraint  Extending and Restricting Meaning  …

25. 25 Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Classifying & Characterising  …

26. 26 Mathematical Strategies/Heuristics  Acknowledging ignorance (Mary Boole)  Changing view point  Changing (re)presentation  Working Backwards  …

27. 27 Mathematical Dispositions  Propensity to ‘see’ the world math’ly  Propensity to pose problems  Propensity to seek structure  Perseverence  …

28. 28 Essential Pedgaogic Awarenesses  Tasks –initiate activity; –activity provides immediate experience; –learning depends on connecting experiences, often through labelling when standing back from the action  Tasks –initiate activity; –activity provides immediate experience; –learning depends on connecting experiences, often through labelling when standing back from the action  Mathematics develops from engaging in actions on objects; and those actions becoming objects, …  Actions need to become not just things done under instruction or guidance, but choices made by the learner

29. 29 ChoicesChoices  What pedagogic choices are available when constructing/selecting mathematical tasks for learners?  What pedagogic choices are available when presenting mathematical tasks to learners?  What criteria are used for making those choices?  What pedagogic choices are available when constructing/selecting mathematical tasks for learners?  What pedagogic choices are available when presenting mathematical tasks to learners?  What criteria are used for making those choices?

30. 30 What mathematics is essential?  Extensions of teaching-maths –Experience analogously something of what learners experience, but enrich own awareness of connections and utility  Extensions of own maths –Experience what it is like to encounter an unfamiliar topic  Extensions of teaching-maths –Experience analogously something of what learners experience, but enrich own awareness of connections and utility  Extensions of own maths –Experience what it is like to encounter an unfamiliar topic

31. 31 It is only after you come to know the surface of things that you venture to see what is underneath; but the surface of things is inexhaustible (Italo Calvino 1983)

32. 32 Human Psyche Awareness (cognition) Behaviour (enaction) Emotion (affect) Mental imagery Only awareness is educable Only behaviour is trainable Only emotion is harnessable

33. 33 What Can a Teacher Do?  Directing learner attention by being aware of structure of own attention (amplifying & editing; stressing & ignoring)  Invoking learners’ powers  Bringing learners in contact with mathematical heuristics & powers  Constructing experiences which, when accumulated and reflected upon, provide opportunity for learners to educate their awareness and train their behaviour through harnessing their emotions.  Directing learner attention by being aware of structure of own attention (amplifying & editing; stressing & ignoring)  Invoking learners’ powers  Bringing learners in contact with mathematical heuristics & powers  Constructing experiences which, when accumulated and reflected upon, provide opportunity for learners to educate their awareness and train their behaviour through harnessing their emotions.

34. 34 http://mcs.open.ac.uk/jhm3 I am grateful to the organisers for affording me the opportunity and impetus to contact, develop and articulate these ideas For this presentation and others and other resources see