Thinking Mathematically and Learning Mathematics Mathematically John Mason

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 = 3 x 4+1 1 + 3 + … + (2 n– 1) + … + 3 + 1 = 3 (n– 1)2 + n 2 = n (2 n– 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 /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 nonprime after 1 in the system of numbers of the form 1+3 n? /100 = 10 x 10 = 4 x 25 /What does this say about primes in the multiplicative system of numbers of the form 1 +3 n? /What is special about the ‘ 3’? 7

Undoing Special Cases solves what else? what solves ? … 8

MGA 9

Powers /Specialising & Generalising /Conjecturing & Convincing /Imagining & Expressing /Ordering & Classifying /Distinguishing & Connecting /Assenting & Asserting 10

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 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/jhm 3 j. h. mason@open. ac. uk 15