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Fostering and Sustaining Mathematical Thinking ATM-MA meeting Leicester Nov 2008 1 John Mason Open University & University of Oxford Conjecturing Atmosphere Everything said is said in order to consider modifications that may be needed Those who ‘know’ support those who are unsure by holding back or by asking informative questions 2 Conjectures The richness of mathematical tasks does NOT lie in the tasks themselves NOR does it lie in the format of interactions It DOES lie in – Opportunities afforded matching student attunements – Possibilities for use of student powers including variations and extensions – Access to encountering mathematical themes It DEPENDS on teachers’ ‘being’, manifested in – Teacher-Learners relationships – Teacher’s mathematical awareness – Working milieu 3 More Conjectures The richness of learners’ mathematical experience depends on – Opportunities to use and develop their own powers – Opportunities to make significant mathematical choices – Being in the presence of mathematical awareness 4 Pattern Continuation … … 5 Children’s Copied Patterns model 6 4.1 yrs Marina Papic MERGA 30 2007 Children’s Own Patterns QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. 5.0 yrs 5.1 yrs 5.4 yrs 7 Marina Papic MERGA 30 2007 Four Consecutives What numbers arise as sum one more than the product of 4 consecutive numbers? n, n + a, n+b, n + (a+b) So natural to specialise! Try some examples … in order to see what is going on … to re-generalise 8 Triangle Count 9 Up & Down Sums 1+3+5+3+ 1 22 + 3 2 = = 3x4+1 1 + 3 + … + (2n–1) + … + 3 + 1 = 10 (n–1)2 + n2 = n (2n–2) + 1 What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 11 What then would be the difference? What could be varied? What’s The Ratio? ÷ = First, multiply each by 3 First, multiply the larger by 2 and divide the smaller by 3 12 Which will be What is the ratio? ? larger? What could be varied? Specific Centre of Gravity When a can of fizzy drink is full, the centre of gravity is in the middle of the can; When the can is empty, the centre of gravity is in the middle of the can (it is a mathematical can!) How does the centre of gravity of the can move as liquid is taken out, and when is it at its lowest point? 13 Averaged Speed At some construction on a motorway, the sign said speed limit 50; average speed calculated I notice that for a specified number of minutes I was going 60. For how long do I have to go at 30? At 35? I notice that for a specified distance I was going 60. How far do I have to go at 30? At 35? 14 Averaged Speed Graphs Know: distance 50mph D =V T D+d =w T+t d =v t d D Also know: w, T, V, v time TV + tv = w(T + t) T t Want: t t(w – v) = V – w Also know: w, D, V, v D+d = w w d( 1 – 1 ) w v 15 d ) v D + V Want: d D + d = w( D + V d v = D( 1 – )1 w V Remainders of the Day Write down a number that leaves a reminder of 1 when divided by 3 and another and another Multiply two of these numbers together: what remainder does it leave when divided by 3? Why? What is special about the ‘3’? 16 Primality What does ‘prime’ mean in the system of numbers leaving a remainder of 1 when divided by 3? What are the first three positive nonprimes after 1 in the system of numbers of the form 1+3n? 100 = 10 x 10 = 4 x 25 What does this say about primes in the multiplicative system of numbers of the form 1 +3n? What is special about the ‘3’? What is special about the ‘1’? 17 Fractional What can be said about fractions where numerator and denominator both leave a remainder of 1 when divided by 3? 18 Bag Constructions (1) Here there are three bags. If you compare any two of them, there is exactly one colour for which the difference in the numbers of that colour in the two bags is exactly 1. For four bags, what is the least number of objects to meet the same constraint? For four bags, what is the least number of colours to meet the same constraint? 19 17 objects 3 colours Bag Constructions (2) Here there are 3 bags and two objects. There are [0,1,2;2] objects in the bags with 2 altogether Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there is a corresponding set of bags? 20 Magic Square Reasoning 2 6 7 2 1 5 9 8 Sum( 21 3 ) – Sum( 4 What other configurations like this give one sum equal to another? Try to describe them in words ) =0 More Magic Square Reasoning Sum( 22 ) – Sum( ) =0 Composite Doing & Undoing I am thinking of a number I add 8 and the answer is 13. I add 8 then multiply by 2; the answer is 26. I add 8, multiply by 2, subtract 5; the answer is 21. I add 8, multiply by 2, subtract 5, divide by 3. The answer is 7 HOW do you turn +8, x2, -5, ÷3 answer 7 into a solution? 23 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 multiplying by 3/2? Two different expressions! What undoes dividing by 3/2? Two different expressions! 24 Fractional Increase and Decrease 1 1 (1 + ) (1 – ) 3 2 2 2 (1 + ) (1 – ) 5 7 a (1 + ) (1 – b 25 =1 By how much do I have to decrease in order to undo an increase by one-half? =1 By how much do I have to increase in order to undo a decrease by two-sevenths? ) =1 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 26 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 27 What fractions can you ‘see’? What relationships between fractions can you see? 28 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties 29 Reflections Initiating Activity Sustaining Activity Transcending Activity 30 Conjectures The richness of mathematical tasks does NOT lie in the tasks themselves NOR does it lie in the format of interactions It DOES lie in – Opportunities afforded matching student attunements – Possibilities for use of student powers including variations and extensions – Access to encountering mathematical themes It DEPENDS on teachers’ ‘being’, manifested in – Teacher-Learners relationships – Teacher’s mathematical awareness – Working milieu 31 More Conjectures The richness of learners’ mathematical experience depends on – Opportunities to use and develop their own powers – Opportunities to make significant mathematical choices – Being in the presence of mathematical awareness 32 Powers Specialising & Generalising Conjecturing Imagining Ordering & Convincing & Expressing & Classifying Distinguishing Assenting 33 & Connecting & Asserting Themes Doing & Undoing Invariance Freedom & Constraint Extending 34 Amidst Change & Restricting Meaning For More Stimulation Starting Points (ATM) Thinking Mathematically (Pearson) MA NRICH website These slides available on http://mcs.open.ac.uk/jhm3 [workshops] [email protected] 35