Greenwich - Thinking Mathematically and and Learning...

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Unformatted text preview: Thinking Mathematically and and Learning Mathematics Learning Mathematically John Mason Greenwich Oct 2008 1 Conjecturing Atmosphere  Everything said is said in order to Everything consider modifications that may be needed needed  Those who ‘know’ support those who are Those unsure by holding back or by asking revealing questions revealing 2 Up & Down Sums 1+3+5+3+ 1 = 22 + 32 = 3x4+1 1 + 3 + … + (2n–1) + … + 3 + 1 = 3 (n–1)2 + n2 = n (2n–2) + 1 One More  What numbers are one more than the product of four consecutive integers? product Let a and b be any two numbers, one of them even. Then ab/2 more than the product of any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared. 4 Remainders of the Day       Write down a number that leaves a reminder of 1 when Write divided by 3 divided and another and another Choose two simple numbers of this type and multiply Choose What is special about the ‘1’? What them together: what remainder does it leave when divided by 3? what Why? What is special What about the ‘3’? What is special about the ‘1’? 5 Primality  What is the second positive non-prime What after 1 in the system of numbers of the form 1+3n? form  100 = 10 x 10 = 4 x 25  What does this say about primes in the What multiplicative system of numbers of the form 1 +3n? form  What is special about the ‘3’? 6 Inter-Rootal Distances Sketch a quadratic for which the inter-rootal Sketch distance is 2. distance  and another  and another  How much freedom do you have?  What are the dimensions of possible variation What and the ranges of permissible change? and  If it is claimed that [1, 2, 3, 3, 4, 6] are the interrootal distances of a quartic, how would you rootal check? check?  7 Bag Constructions (1)  Here there are three bags. If Here 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. two For four bags, what is the least number of objects to meet the same constraint? constraint?  For four bags, what is the least number of colours to meet the same constraint? same  8 17 objects 3 colours Bag Constructions (2) Here there are 3 bags and two Here objects. objects.  There are [0,1,2;2] objects in There the bags with 2 altogether the  Given a sequence like [2,4,5,5;6] Given or [1,1,3,3;6] how can you tell if there is a corresponding set of bags? bags?  9 Statisticality  write down five numbers whose mean is write 5  and whose mode is 6  and whose median is 4 10 ZigZags Sketch the graph of y = |x – 1|| Sketch 1  Sketch the graph of y = | |x - 1| - 2| Sketch  Sketch the graph of Sketch y = | | |x – 1| – 2| – 3|  What sorts of zigzags can you make, and not What make? make?  Characterise all the zigzags you can make Characterise using sequences of absolute values like this. using  11 Towards the Blanc Mange function 12 Reading Graphs 13 Examples  Of what is |x| an example?  Of what is y = x2 and example? Of – y = b + (x – a)2 ? 14 Functional Imagining Imagine a parabola  Now imagine another one Now the other way up. the  Now put them in two planes Now at right angles to each other. other.  Make the maximum of the Make downward parabola be on the upward parabola the   15 Now sweep your downward Now parabola along the upward parabola so that you get a surface surface MGA 16 Getti Man Artic Getti Man Powers  Specialising & Generalising  Conjecturing & Convincing  Imagining & Expressing  Ordering & Classifying  Distinguishing & Connecting  Assenting & Asserting 17 Themes  Doing & Undoing  Invariance Amidst Change  Freedom & Constraint  Extending & Restricting Meaning 18 Teaching Trap   19 Learning Trap Doing for the learners what  they can already do for themselves themselves Teacher Lust: Teacher  – desire that the learner learn learn – desire that the learner desire appreciate and understand – Expectation that learner Expectation will go beyond the tasks as set set – allowing personal allowing excitement to drive behaviour behaviour Expecting the teacher to do for you what you can already do for yourself do Learner Lust: Learner – desire that the teacher teach teach – desire that learning will be desire easy easy – expectation that ‘dong the expectation tasks’ will produce learning learning – allowing personal reluctance/uncertainty to drive behaviour drive Human Psyche  Training Behaviour  Educating Awareness  Harnessing Emotion  Who does these? – Teacher? – Teacher with learners? – Learners! 20 Structure of the Psyche Awareness (cognition) Imagery Will Emotions Emotions (affect) (affect) Body (enaction) Habits Practices 21 Structure of a Topic Language Patterns & prior Skills Root Questions predispositions Standard Confusions & Obstacles Different Contexts in which likely to arise; dispositions Techniques & Incantations Aw ar ss en e Emoti on Only Emotion is Only Harnessable Harnessable Only Behaviour is Only Only Awareness is Only Trainable Trainable Educable Educable o o v vii ha ha Be ur Be ur 22 Imagery/Senseof/Awareness; Connections Didactic Tension The more clearly I indicate the behaviour sought from learners, the less likely they are to generate that behaviour for themselves (Guy Brousseau) 23 Didactic Transposition Expert awareness is transposed/transformed into instruction in behaviour (Yves Chevellard) 24 More Ideas For Students (1998) Learning & Doing Mathematics (Second revised edition), QED Books, York. (1982). Thinking Mathematically, Addison Wesley, London For Lecturers (2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester. (2008). Counter Examples in Calculus. College Press, London. 25 Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing Teacher Student Content Expounding Student Content Teacher Examining Teacher Content Student Student Teacher Content Explaining Exploring Content Teacher Student Expressing Content Student Teacher Exercising ...
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