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Thinking Mathematically and Learning Mathematics Mathematically John Mason St Patrick’s College Dublin Feb 2010 1 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 revealing questions 2 Up & Down Sums 1+3+5+3+ 1 22 + 3 2 = = 3x4+1 1 + 3 + … + (2n–1) + … + 3 + 1 = 3 (n–1)2 + n2 = n (2n–2) + 1 Doing & Undoing Whenever you find you can ‘do’ something, ask yourself how to ‘undo’ it. – If doing is ‘subtract from 100’, what is the undoing? – If undoing is ‘divide 120 by’, what is the undoing? – If doing is find the roots of a polynomial, what is the undoing? 4 Reading Graphs 5 Remainders of the Day Write down a number that leaves a reminder of 1 when divided by 3 and another and another Choose two simple numbers of this type and multiply them together: what remainder does it leave when divided by 3? Why? What is special about the ‘3’? 6 What is special about the ‘1’? Primality What is the second positive non-prime 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’? 7 Undoing Special Cases e d e x x solves f ' f dx 1 what solves f ' ? f what solves f ' f 2 ? … 8 what else? MGA 9 Powers Specialising & Generalising Conjecturing Imagining Ordering & Convincing & Expressing & Classifying Distinguishing Assenting 10 & Connecting & Asserting Themes Doing & Undoing Invariance Freedom & Constraint Extending 11 Amidst Change & Restricting Meaning Teaching Trap Learning Trap Expecting the teacher to Doing for the learners do for you what you can what they can already do for themselves already do for yourself Teacher Lust: Learner Lust: – desire that the learner – desire that the teacher learn tell me what to do – desire that the learner – desire that learning will appreciate and be easy understand – expectation that ‘dong – Expectation that learner the tasks’ will produce will go beyond the tasks learning as set – allowing personal – allowing personal excitement to drive reluctance/uncertainty behaviour to drive behaviour 12 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) 13 Didactic Transposition Expert awareness is transposed/transformed into instruction in behaviour (Yves Chevellard) 14 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. http://mcs.open.ac.uk/jhm3 [email protected] 15