Promoting Mathematical Thinking Making the Most of Mathematical

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

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 2

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 3

Reproduction /Make a copy of the following repeating pattern What did you have to do to accomplish this? Make up your own repeating pattern 4

Children’s Copied Patterns model 5 4. 1 yrs Marina Papic MERGA 30 2007

Children’s Own Patterns 5. 0 yrs 5. 1 yrs 5. 4 yrs 6 Marina Papic MERGA 30 2007

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? 7

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 What did you do? C A 8 B E C B A C A B E A E C B D E

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

Same & Different 10 10 12 15 6 Pick an entry. What distinguishes it from the others?

Revealing Shapes Applet 11

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 12

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! 13

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

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

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? 16

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? 17

Doing & Undoing /What operation undoes ‘adding 3’? /What operation undoes ‘subtracting 4’? /What operation undoes ‘subtracting from 7’? /What are the analogues for multiplication? IWhat undoes ‘multiplying by 3’? IWhat undoes ‘dividing by 4’? undoes ‘multiplying by ¾’? üTwo different expressions! IWhat 18

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 19

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 20

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

Magic Square Reasoning 2 2 7 2 1 5 9 8 Sum( 22 6 3 What other configurations like this give one sum equal to another? Try to describe them in words 4 )– Sum( ) =0

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

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 24

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

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

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

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

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

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) Structured Variation Grids Revealing Shapes Other Publications This and other presentations mcs. open. ac. uk/jhm 3 30 j. h. mason@open. ac. uk