1 Making the Most of Mathematical Tasks John Mason Overton Jan 2011 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Aims  To develop strategies for promoting learning from experience  To develop questioning that promotes extension, variation, and generalisation  To consider a variety of tasks which can be used to stimulate reasoning

3 Teaching & Learning  Children are given mathematical tasks to do  Tasks stimulate activity  Activity provides experience –of the use of their powers –of mathematical themes –of mathematical topics, techniques, reasoning …  Experience may contribute to learning –especially when learners are prompted to withdraw from activity and reflect upon it

4 What did you have to do to accomplish this?  Make a copy of the following repeating pattern Reproduction

5 Children’s Copied Patterns 4.1 yrs Marina Papic MERGA model

6 Children’s Own Patterns Marina Papic MERGA yrs 5.0 yrs5.1 yrs

7 Patterned Wheels … An inked roller has made at least two full revolutions What colour is the 100 th square? Where is the 100 th red square?

8 Order! Order!  A, B, C, D, and E are in a queue –B is in front of C –A is behind E –There are two people between D and E –There is one person between D and C –There is one person between B and E BC EA BCEA BCEA BCEAD What did you do?

9 Say What You See  There are 16 canoes 5 asteroids 4 wedges 4 peaks and these account for the total area Also 6 arches; 6 troughs; What did you do?

10 Same & Different What distinguishes it from the others? Pick an entry.

11 Revealing Shapes Applet

12 And Another  Write down two numbers that differ by 3 –And another pair  Write down two numbers that differ by 3 that you think no-one else will write down  Write down two numbers that differ by 3 and that make that difference as obscure as possible

13 Smallest Unique  Write down a positive number that you think no-one else will write down  The ‘winner’ is the person who writes down the smallest such number!

14 What’s The Difference? What could be varied? –= First, add one to each First, add one to the first and subtract one from the second What then would be the difference?

15 What’s The Ratio? What could be varied? ÷= First, multiply each by 3 First, multiply the first by 2 and divide the second by 3 What is the ratio?

16 Marbles (Bob Davis)  I have a bag of marbles  I take out 7, then put in 3, then take out 4. What is the state of my bag now? –Variations?

17 Speed Reasoning  If I run 3 times as fast as you, how long will it take me compared to you to run a given distance?  If I run 2/3 as fast as you, how long will it take me compared to you?

18 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 4’?  What undoes ‘multiplying by ¾ ’? Two different expressions! Two different expressions!

19 Additive & Multiplicative Perspectives  What is the relation between the numbers of squares of the two colours?  Difference of 2, one is 2 more: additive  Ratio of 3 to 5; one is five thirds the other etc.: multiplicative

20 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 1/3 of 3/5 of something  Something which is 3/5 of 1/3 of something  Something which is 2/5 of 5/2 of something  Something which is 1 ÷ 2/5 of something

21 Why is (-1) x (-1) = 1?

22 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

23 More Magic Square Reasoning –= 0Sum( )Sum( )

24 Teaching  Selecting tasks  Preparing Didactic Tactics and Pedagogic Strategies  Prompting extended or fresh actions  Being Aware of mathematical actions  Directing Attention Teaching takes place in time; Learning takes place over time

25 The Place of Generality  A lesson without the opportunity for learners to generalise mathematically, is not a mathematics lesson

26 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

27 Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Organising & Characterising

28 Rich tasks, Rich Use of tasks  It may not be the task that is rich  But the way the task is used

29 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint

30 For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Thinking Mathematically (new edition) mcs.open.ac.uk/jhm3 Structured Variation Grids Revealing Shapes Other Publications This and other presentations