Report

Promoting Mathematical Thinking From Teaching Procedures To Thinking Mathematically: Making Use of Students’ Natural Powers John Mason Gothenberg Nov 30 2012 The Open University Maths Dept 1 University of Oxford Dept of Education Conjectures 2 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! Tasks 3 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 Responsible teaching Articulate about justifying choices of – – – – 4 tasks ways of initiating mathematical thinking ways of sustaining mathematical thinking ways of concluding mathematical thinking Learning (Mathematics) 5 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 My Way of Working Phenomenological-Experiential – Try to generate an experience, – draw attention to it – label it in some way 6 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 7 Generalising From Thomas Lingefjård 8 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. More or Less grids Perimeter Area More Same Less More Same Less 9 With as little change as possible from the original! 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? A B C D E 1 2 3 4 5 9 8 7 6 10 … If that object is elimated, you start again from the ‘next’. Which object is the last one left? 10 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? 11 Substitution Relationships W WBB WBB 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 12 Gasket Sequences 13 Sundaram’s Sieve 16 27 38 49 60 71 82 13 22 31 40 49 58 67 10 17 24 31 38 45 52 7 12 17 22 27 32 37 4 7 10 13 16 19 22 Claim: N will appear in the table iff 2N + 1 is composite What number will appear in the Rth row and the Cth column? 14 Circle Round a Square 15 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? Numberline Movements 16 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. ThOANs 17 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! Ride & Tie 18 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 17th 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? Ride & Tie Imagine, then draw a diagram! Seeking Relationships / 19 Does the diagram make sense (meet the constraints)? 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 time distance d d t1 = t2 = 2v1 2v2 d d t = + 2v1 2v2 20 t t d1 = v1 d2 = v2 2 2 2d t= v1 + v2 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? 21 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! 22 What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 23 What then would be the difference? What could be varied? Ride & Tie Imagine, then draw a diagram! Seeking Relationships / 24 Does the diagram make sense (meet the constraints)? Understanding Division 25 234234 is divisible by 13 and 7 and 11; What is the remainder on dividing 23423426 by 13? By 7? By 11? Make up your own! More or Less grids Perimeter Area More Same Less More Same Less 26 With as little change as possible from the original! Put your hand up when you can see … 27 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? Put your hand up when you can see … Something that is 1/4 – 1/5 of something else Did you look for something that is 1/4 of something else and for something that is 1/5 of the same thing? What did you have to do with your attention? Can you generalise? 28 1 1 1 - = c -1 c c ( c -1) 1 1 c-r - = r c rc 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 time distance d d t1 = t2 = 2v1 2v2 d d t = + 2v1 2v2 29 t t d1 = v1 d2 = v2 2 2 2d t= v1 + v2 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? 30 One Sum Diagrams 1 (1- )2 1 2 1- Anticipating, not waiting 31 Reading a Diagram x3 + x(1–x) + (1-x)3 x2z + x(1-x) + (1-x)2(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) 32 x2 + (1-x)2 xz + (1-x)(1-z) yz + (1-x)(1-z) Outer & Inner Tasks Outer Task – – – – Inner Task – – – – – 33 What author imagines What teacher intends What students construe What students actually do 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? Imagining Basis of Geometric Thinking Basis of Anticipating Basis of ‘Realising’ Basis of Accessing & Enriching Example Spaces Basis of Planning Geometric Images ATM 34 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? 35 Mathematical Themes 36 Doing & Undoing Invariance in the midst of change Freedom & Constraint Restricting & Expanding Meaning 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? – – – – 37 Powers Themes Heuristics The nature of their own attention Attention / / / / / 38 Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of properties Motivation Motivation is not a thing – Sense of gap or disturbance – Appropriate challenge + Trust in teacher 39 Phenomena to explain using mathematics Mathematical phenomena to explain & appreciate 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 – … 40 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 41 Follow Up mcs.open.ac.uk/jhm3 j.h.mason @ open.ac.uk Thinking Mathematically (new edition) Designing and Using Mathematical Tasks (Tarquin) Questions and Prompts … (from ATM) Thinkers (from ATM) 42