1 Thinking Mathematically and Learning Mathematics Mathematically John Mason St Patrick’s College Dublin Feb 2010

2 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

3 Up & Down Sums x … + (2n–1) + … == n (2n–2) + 1 (n–1) 2 + n 2 = =

4 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?

5 Reading Graphs

6 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 ‘1’? What is special about the ‘3’?

7 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’?

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

9 MGA

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

11 Themes / Doing & Undoing / Invariance Amidst Change / Freedom & Constraint / Extending & Restricting Meaning

12 Teaching Trap Learning Trap  Doing for the learners what they can already do for themselves  Teacher Lust: – desire that the learner learn –desire that the learner appreciate and understand –Expectation that learner will go beyond the tasks as set –allowing personal excitement to drive behaviour  Expecting the teacher to do for you what you can already do for yourself  Learner Lust: – desire that the teacher tell me what to do –desire that learning will be easy –expectation that ‘dong the tasks’ will produce learning – allowing personal reluctance/uncertainty to drive behaviour

13 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)

14 Didactic Transposition Expert awareness is transposed/transformed into instruction in behaviour (Yves Chevellard)

15 More Ideas (2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester. (2008). Counter Examples in Calculus. College Press, London. (1998) Learning & Doing Mathematics (Second revised edition), QED Books, York. (1982). Thinking Mathematically, Addison Wesley, London For Lecturers For Students