1 From Teaching Procedures To Thinking Mathematically: Making Use of Students’ Natural Powers The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking John Mason Gothenberg Nov

2 Conjectures  Everything said here today is a conjecture … to be tested in your experience  The best way to sensitise yourself to learners … … is to experience parallel phenomena yourself  So, what you get from this session is what you notice happening inside you!

3 Tasks  Tasks promote Activity;  Activity involves Aactions;  Actions generate Experience; –but one thing we don’t learn from experience is that we don’t often learn from experience alone  It is not the task that is rich … – but whether it is used richly

4 Responsible teaching  Articulate about justifying choices of –tasks –ways of initiating mathematical thinking –ways of sustaining mathematical thinking –ways of concluding mathematical thinking

5 Learning (Mathematics)  What is Avaibale to be learned (what is varying and in what ways)  What Actions are Initiated  What Dispositions are Evoked  What Powers are called upon

6 My Way of Working  Phenomenological-Experiential –Try to generate an experience, –draw attention to it –label it in some way

7 One More Than  What numbers can be presented as one more than the product of four consecutive numbers?  One natural response is to use algebra (if that is confidence-inspiring) –But that runs into obstacles  One natural response is to try some specific examples… –In order to locate a relationship that might be an instance of a property! Specialising Generalising

8 From Thomas Lingefjård  Given the numbers 1, 3, 4, and 6 - try to construct all numbesr from 20 to 30 by simple arithmetic (addition, subtraction, multiplication and division). No other way of combining or using numbers as power of is allowed. For instance: 1*6*3 + 4 = 22. In every calculation, all four digits must be present.  Try to find a number which consists of 769 digits, the sum of all the digits is 3693, every pair of consecutive digits is either a multiple of 17 or of 23 and all multiples of 17 or 23 in two digits is in the number.

9 More or Less grids MoreSame Less More Same Less Perimeter Area With as little change as possible from the original!

10 Counting Out  In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37). Which object will you end on? ABCDE … If that object is elimated, you start again from the ‘next’. Which object is the last one left? 10

11 Substitution Pattern Generating W –> WB B –> W How many squares will there be? How many white squares will there be? How many blue squares will there be?

12 Substitution Relationships WWBBWBB BW BW WBB BW BW BW WBB BW WBB WBB BW BW BW WBB BW WBB BW WBB WBB BW BW BW WBB WBB BW BW

13 Gasket Sequences

14 Sundaram’s Sieve What number will appear in the R th row and the C th column? Claim: N will appear in the table iff 2N + 1 is composite

15 Circle Round a Square  Imagine a Square  Now imagine a circle in the same plane as the square, so that the two are touching at a single point  Now imagine the circle rolling around the outside of the square, always staying in touch  Pay attention to the centre of the circle as it rolls  What is the path the centre takes, and how long is it?

16 Numberline Movements  Imagine you are standing on a number line somewhere facing the positive direction. (Make a note of where you are!)  Go forward three steps;  Now go backwards 5 steps  Now turn through 180°  Go backwards 3 steps  Go forwards 1 step  You should be back where you started but facing to the left.

17 ThOANs  Think of a number between 0 and 10  Add six  Multiply by the first number you thought of  Add 4  Subtract twice the number you first thought of  Take the square root (positive!)  subtract the number you first thought of  You (and everybody else) are left with 2!

18 Ride & Tie  Imagine that you and a friend have a single horse (bicycle) and that you both want to get to a town some distance away.  In common with folks in the 17 th century, one of you sets off on the horse while the other walks. At some point the first dismounts, ties the horse and walks on. When you get to the horse you mount and ride on past your friend. Then you too tie the horse and walk on…  Supposing you both ride faster than you walk but at different speeds, how do you decide when and where to tie the horse so that you both arrive at your destination at the same time?

19 Ride & Tie  Imagine, then draw a diagram!   Does the diagram make sense (meet the constraints)? Seeking Relationships

20 Two Journeys  Which journey over the same distance at two different speeds takes longer: –One in which both halves of the distance are done at the specified speeds –One in which both halves of the time taken are done at the specified speeds distance time

21 Named Ratios  Now take a named ratio (eg density) and recast this task in that language  Which mass made up of two densities has the larger volume: –One in which both halves of the mass have the fixed densities –One in which both halves of the volume have the same densities?

22 Counter Scaling  Someone has placed 5 counters side-by-side in a line  Someone else has made a similar line with 5 counters but with one counter-width space between counters.  By what factor has the length of the original line been scaled?  How many counters would be needed so that the scale factor was 15/8? “Fence-post Reasoning” Generalise!

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

24 Ride & Tie  Imagine, then draw a diagram!   Does the diagram make sense (meet the constraints)? Seeking Relationships

25 Understanding Division  is divisible by 13 and 7 and 11;  What is the remainder on dividing by 13?  By 7? By 11?  Make up your own!

26 More or Less grids MoreSame Less More Same Less Perimeter Area With as little change as possible from the original!

27 Put your hand up when you can see …  Something that is 3/5 of something else  Something that is 2/5 of something else  Something that is 2/3 of something else  Something that is 5/3 of something else  What other fraction-actions can you see? How did your attention shift?

28 Put your hand up when you can see … Something that is 1/4 – 1/5 of something else What did you have to do with your attention? Can you generalise? Did you look for something that is 1/4 of something else and for something that is 1/5 of the same thing?

29 Two Journeys  Which journey over the same distance at two different speeds takes longer: –One in which both halves of the distance are done at the specified speeds –One in which both halves of the time taken are done at the specified speeds distance time

30 Named Ratios  Now take a named ratio (eg density) and recast this task in that language  Which mass made up of two densities has the larger volume: –One in which both halves of the mass have the fixed densities –One in which both halves of the volume have the same densities?

31 One Sum Diagrams 1 1 (1- ) 2 Anticipating, not waiting 1- 2

32 Reading a Diagram x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

33 Outer & Inner Tasks  Outer Task –What author imagines –What teacher intends –What students construe –What students actually do  Inner Task –What powers might be used? –What themes might be encountered? –What connections might be made? –What reasoning might be called upon? –What personal dispositions might be challenged?

34 Imagining  Basis of Geometric Thinking  Basis of Anticipating  Basis of ‘Realising’  Basis of Accessing & Enriching Example Spaces  Basis of Planning Geometric Images ATM

35 Powers  Every child that gets to school has already displayed the power to – imagine & express – specialise & generalise – conjecture & convince – organise and categorise  The question is … – are they being prompted to use and develop those powers? –or are those powers being usurped by text, worksheets and ethos?

36 Mathematical Themes  Doing & Undoing  Invariance in the midst of change  Freedom & Constraint  Restricting & Expanding Meaning

37 Reflection  Tasks promote activity; activity involves actions; actions generate experience; –but one thing we don’t learn from experience is that we don’t often learn from experience alone  It is not the task that is rich –but the way the task is used  Teachers can guide and direct learner attention  What are teachers attending to? –Powers –Themes –Heuristics –The nature of their own attention

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

39 Motivation  Motivation is not a thing –Sense of gap or disturbance –Appropriate challenge + Trust in teacher  Phenomena to explain using mathematics  Mathematical phenomena to explain & appreciate

40 The Problem about Problem Solving  It is not simply a Friday afternoon entertainment  It is not a ‘thing’ you (or the students) do  It is an orientation to learning and doing mathematics  Change of Vocabulary: –Teaching using exploration as one mode of interaction among many –‘teaching Investigatively’ –Using Stdeunts’ Powers to teach Mathematics –…–…

41 Pedagogic Strategies & Didactic Tactics  In how many different ways can you …  Do as many exercises as you need to do in order to be able to do any uestion of this type –Construct an easy, hard, peculiar, general question of this type  What is the same and what different about …  If this is the answer, what questions of this type would give the same answer?  What sorts of answers can you get to questions of this type?  Presentation –Particular  General –General –> Particular –> Re-Generalise –Partly General –> Particular –> Re-Generalise

42 Follow Up mcs.open.ac.uk/jhm3 open.ac.uk mcs.open.ac.uk/jhm3 open.ac.uk Thinking Mathematically (new edition) Designing and Using Mathematical Tasks (Tarquin) Questions and Prompts … (from ATM) Thinkers (from ATM) Thinking Mathematically (new edition) Designing and Using Mathematical Tasks (Tarquin) Questions and Prompts … (from ATM) Thinkers (from ATM)