Solving Mathematical Problems Mathematically

1. Solving Mathematical ProblemsMathematically John Mason ATM-MACornwall May2009

2. Assumptions • What you get from this session will be largely what you notice happening for you • If you do not participate, I guarantee you will get nothing! • I assume a conjecturing atmosphere • Everything said has to be tested in experience • If you know and are certain, then think and listen; • If you are not sure, then take opportunities to try to express your thinking • Learning is a maturation process, and so invisible • It can be promoted by pausing and withdrawing from the immediate action in order to get an overview

3. Kites

4. Visiting • People set off at the same time to walk between two towns, some in each direction. They all meet at noon. They have lunch together for an hour, then carry on as before; some reach their destination at 5pm, and the others at 7:15 pm. • At what time did they all set out? distance • Let the speeds be v and w, and the time before noon be h • The am distances are vh and wh • The pm distances are 4v and 25w/4 • So wh = 4v and vh = 25w/4 • Whence h2 = 25 and h = 5 • So they set out at 7:00 am wh vh time 4:00 12:00 h 6:15

5. Triangle Count

6. Max-Min

7. Max-Min • In a rectangular array of numbers, calculate • The maximum value in each row, and then the minimum of these • The minimum in each column and then the maximum of these • How do these relate to each other? • What about interchanging rows and columns? • What about the mean of the maxima of each row, and the maximum of the means of each column?

8. Imagining & Expressing Sliding Round and Round Get a sense ofthe freedom available Imagine a triangle • Imagine a second trianglewith one vertex on the edge of the first triangle ‘just touching’ • Now always keeping the second triangle just touching the first, slide it round the first. NO Rotating Allowed! • What is the length of the locus of the centroid of the sliding triangle?

9. Up & Down Sums 1 + 3 + 5 + 3 + 1 = 22 + 32 = 3 x 4 + 1 See generalitythrough aparticular Generalise! 1 + 3 + … + (2n–1) + … + 3 + 1 = = (n–1)2 + n2 n (2n–2) + 1

10. Interlude on Tasks • tasks are for initiating activity; • through engaging in activity people have the opportunity to experience things, particularly mathematical actions and their effects. • Teachers have intentions when they choose tasks: • it is not that the task itself will promote learning, but that the inner task will involve the directing or re-directing of attention • and perhaps the internalising or integrating of actions previously dependent on being triggered by some outside agency (ZPD). • It is valuable, even necessary to draw back from the action and to become aware of the actions and their effects (utility)

11. Differences AnticipatingGeneralising Rehearsing Checking Organising

12. Powers • Am I stimulating learners to use their own powers, or am I abusing their powers by trying to do things for them? • To imagine & to express • To specialise & to generalise • To conjecture & to convince • To stress & to ignore • To extend & to restrict

13. Reflections • Much of mathematics can be seen as studying actions on objects • Frequently it helps to ask yourself what actions leave some relationship invariant; often this is what is studied mathematically

14. More Resources • Questions & Prompts for Mathematical Thinking (ATM Derby: primary & secondary versions) • Thinkers (ATM Derby) • Mathematics as a Constructive Activity (Erlbaum) • Designing & Using Mathematical Tasks (Tarquin) • http: //mcs.open.ac.uk/jhm3 • j.h.mason @ open.ac.uk