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

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 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 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 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 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 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 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 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 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 Making Sense of the World

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

19 Differences Anticipating Generalising Rehearsing Checking Organising

20 Up & Down Sums x … + (2n–1) + … == n (2n–2) + 1 (n–1) 2 + n 2 = = Generalise! See generality through a particular

21 Kites

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

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

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

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 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 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 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 MGA

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 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  open.ac.uk