1. 1 Getting Students to Take Initiative when Learning & Doing Mathematics John Mason Oslo Jan 2009 The Open University Maths Dept University of Oxford Dept of Education

2. 2 Do You Know Any Students Who …  do the minimum to get through a lesson?  wait to be told what to do?  finish quickly and then mess around?  are content to assent to what is said and done, but rarely assert mathematically?

3. 3 When Do You Take Initiative?  When you are interested, engaged, involved  When you have a stake in getting something finished  When you are surprised or intrigued  When something is or becomes ‘real’ for you

4. 4 Fraction Construction  Write down two numbers that differ by 3/7  and another pair  And another pair that make the difference as obscure as possible

5. 5 Decimal Construction  Write down –A decimal number between 3 and 4 –that does not use the digit 5 –and that does use the digit 7 –and that is as close to 7/2 as possible

6. 6 Line Construction  Write down the equation of a straight line that passes through the point (1,0)  and another  Write them all down!

7. 7 More Line Constructions  Sketch the graph of two straight lines whose –x-intercepts differ by 2; and another … –y-intercepts differ by 2; and another … –slopes differ by 2; and another …  Sketch the graph of two straight lines meeting all three constraints

8. 8 Max-Min  In a rectangular array of numbers, calculate –The maximum value in each row, and then the minimum of these –The minimum in each column and then the maximum of these  How do these relate to each other?  What about interchanging rows and columns?

9. 9 Raise your hand when you can see …  Something which is 2/5 of something  Something which is 3/5 of something  Something which is 2/3 of something –What others can you see?   Something which is 2/5 of 5/3 of something; 3/5 of 5/3 of something;  Something which is 2/5 of 5/3 of something; –What part is it of your whole?  Something which is 1/3 of 3/5 of something; –What part is it of your whole?  Something which is 5/3 of 3/5 of something  Something which is 2/3 of 3/2 of something

10. 10 Getting Others To See … 1/4 – 1/5 = 1/20 1/4 – 1/20 = 1/5 1/5 – 1/20 = 1/4 1/a – 1/b = ?

11. 11 Doing & Undoing  What operation undoes ‘adding 3’?  What operation undoes ‘subtracting 4’?  What operation undoes ‘subtracting from 7’?  What are the analogues for multiplication?  What undoes multiplying by 3?  What undoes dividing by 2?  What undoes dividing by 3/2?  What undoes multiplying by 3/2? Now do it piecemeal!  What undoes ‘dividing into 12’?

12. 12 Remainder Construction  Write down a number that leaves a remainder of 1 on dividing by 3  and another  Write down two, multiply them together, and find the remainder on dividing by 3 What is special about the ‘1’? What is special about the ‘3’?

13. 13 Distributed Examples  Write down a number that leaves a remainder of 1 when divided by 7  Now write down one which is easy to see leaves a remainder of 1 on dividing by 7  Multiply by your number by the number of someone sitting beside you  Does the product have the same property?

14. 14 Remainders of the Day  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from that quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

15. 15 Remainders of the Day  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4  … … … …

16. 16 Making Sense of the World

17. 17 More Or Less Whole & Part ? of 35 is 21 moresameless more same less Whole Part 3/5 of 35 is 21 3/4 of 40 is 30 6/7 of 35 is 30 4/5 of 30 is 24

18. 18 Difference Divisions 4 – 2 = 4 ÷ 2 4 – 3 = 4 ÷ 3 1 2 1 2 5 – 4 = 5 ÷ 4 1 3 1 3 6 – 5 = 6 ÷ 5 1 4 1 4 7 – 6 = 7 ÷ 6 1 5 1 5 3 – 2 = 3 ÷ 2 1 1 1 1 0 – (-1) = 0 ÷ (-1) 1 -2 1 2 1 oops 1 – 0 = 1 ÷ oops 1 1 How does this fit in? Going with the grain Going across the grain

19. 19 Differences Anticipating Generalising Rehearsing Checking Organising

20. 20 Up & Down Sums 1 + 3 + 5 + 3 + 13 x 4 + 12 2 + 3 2 1 + 3 + … + (2n–1) + … + 3 + 1 == n (2n–2) + 1 (n–1) 2 + n 2 = = Generalise! See generality through a particular

21. 21 Kites

22. 22 Reacting & Responding  Do you know any students who jump at the first idea that comes to mind?  Do you know any students who react negatively when challenged by something unfamiliar?  Assenting ––> Asserting –conjecturing, trying, reasoning, …

23. 23 When Do You Take Initiative?  When you are interested, engaged, involved  When you have a stake in getting something finished  When you are surprised or intrigued  When something is or becomes ‘real’ for you

24. 24 When is Real-ity  Sense of purpose (engagement)  Sense of utility (present or future)  Use of own powers

25. 25 Strategies  Learners Making Significant Mathematical Choices –Learner Constructed Examples of Mathematical Objects –Learner Constructed Examples of Exercises –Learners deciding which exercises need doing –Distributed example construction

26. 26 ZPD  When students are ready to shift from –Reacting to cues and triggers –to initiating actions for themselves  Scaffolding & Fading –Directed, prompted, spontaneous use of strategies, powers, concepts, techniques

27. 27 Task Design  expert awareness is converted into instruction in behaviour – transposition didactique

28. 28 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 occasioned by activity Teaching takes place in time Learning takes place over time

29. 29 Tasks  Learners encounter variation  Learners build up example spaces  Learners rehearse other techniques while exploring  Learners encounter disturbances and surprises

30. 30 Purpose & Utility  whose purposes?  whose utility? –mathematics is useful  planning from objectives leads to dull lessons; planning from tasks may mean avoidance of mathematical ideas, thinking, etc.  Issue: how much do you tell learner in advance? –Inner and outer aspects of tasks Ainley & Pratt

31. 31 Teacher Aims and Goals  students to … –make use of their powers –experience mathematical themes –encounter mathematical concepts, topics –develop facility and fluency with techniques –use technical terms to express their conjectures and understandings

32. 32 Learner Aims & Goals  As learners, to –do as little as necessary to complete tasks adequately –attract as little (or as much) attention as possible –be stimulated, inspired, engaged

33. 33 Task Dimensions  How initiated – in silence; through phenomenon (shown or imagined)  How sustained –Group discussion; distributed tasks; individual  How concluded  How structured –Simple to complex; Particular to general –Complex simplified; General specialised

34. 34 MGA

35. 35 Reflection  What did you notice happening for you mathematically?  What might you be able to use in an upcoming lesson?  Imagine yourself in the future, using or developing or exploring something you have experienced this morning!

36. 36 More Resources  Questions & Prompts for Mathematical Thinking (ATM Derby: primary & secondary versions)  Thinkers (ATM Derby)  Mathematics as a Constructive Activity (Erlbaum)  Designing & Using Mathematical Tasks (Tarquin)  http: //mcs.open.ac.uk/jhm3  j.h.mason @ open.ac.uk