1 John Mason IMEC9 Sept 2007 Using Theoretical Constructs to Inform Teaching

2 Outline  Teaching Mathematics –Tasks, activities, experience, reflection  Teaching People To Teach Mathematics –Consistency –Awareness of the role of  Tasks, activities, experience, reflection

3 My Methods  Experiential What you get from this session is what you notice happens inside you you, and how you relate that to your own situation  Reflection –Linking to theories –Preparing to notice more carefully in future –Brief-but-vivid accounts The canal may not itself drink, but it performs the function of conveying water to the thirsty

4 One Sum  I have two numbers which sum to 1  Which will be larger: The square of the larger added to the smaller? The square of the smaller added to the larger? Don’t calculate!!! Conjecture! Only then Check!

5 One Sum Diagrams a a a2a2 (1-a) 2 a Anticipating, not waiting

6 Decimal  Write down a decimal number between 2 and 3  and which does NOT use the digit 5  which DOES use the digit 7  and which is as close to 5/2 as possible … …9 2.49…979…

7 Difference of Two  Write down two numbers which differ by 2  And another pair  And another pair which obscure the fact that the difference is & Fractions? Decimals? Negatives? … ?

8 How often do you set tasks for them where they need to do this? Do you encourage your learners to do this? What did you do first? Characterising  What numbers can be two more than the sum of four consecutive whole numbers? What numbers can be one more than the product of four consecutive numbers?

9 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

10 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

11 More Or Less Area & Perimeter Draw a rectangle moresameless more same less are a perimeter Same perim more area more perim same area more perim more area less perim more area less perim less area more perim less area same perim less area less perim same area When can it be done? When can it not be done?

12 Omar Khayam Myself when young did eagerly frequent Doctor and Saint, and heard great Argument About it and about: but evermore Came out by the same Door as in I went Pursuing knowledge in childhood we rise Until we become masterful and wise But if we look through the disguise We see the ties of worldly lies In childhood we strove to go to school, Our turn to teach, joyous as a rule The end of the story is sad and cruel From dust we came, and gone with winds cool.

13 MGA & DTR Doing Talking Recording

14 Powers  Specialising & Generalising  Conjecturing & Convincing  Imagining & Expressing  Ordering & Classifying  Distinguishing & Connecting  Assenting & Asserting

15 Themes  Doing & Undoing  Invariance Amidst Change  Freedom & Constraint  Extending & Restricting Meaning

16 Protases A sequence of experiences does not add up to an experience of that sequence One thing we do not often learn from experience, is that we do not often learn from experience alone Habit forming can be habit forming Absence of evidence is NOT evidence of absence

17 Implicit Contract  If learners ‘do’ the tasks they are set, then they will ‘learn’ what is required –Contrat didactique  The more clearly and specifically the teacher specifies the behaviour sought, the easier it is for learners to display that behaviour without encountering mathematics, without thinking mathematically –Didactic tension

18 Task & Activity  A task is what an author publishes, what a teacher intends, what learners undertake to attempt. –These are often very different  What happens is activity  Teaching happens in the interaction made possible by activity: performing familiar actions in new ways to make new actions  Learning happens through reflection and integrating new actions into functioning Teaching takes place in time Learning takes place over time

19 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic

20 Worlds, MGA, DTR  Enactive-Iconic-Symbolic –Three modes; three worlds  Manipulating–Getting-a-sense-of– Articulating  Doing–Talking–Recording

21 Further Reference  Mathempedia (  Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London (2004).  Designing and Using Mathematical Tasks. St. Albans: Tarquin. k