1. Solving Mathematical Problems Mathematically
John Mason
ATM-MA Cornwall
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 First: chasing sides using A4 paper
Second: angle chasing from bottom right

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? h
4:00
6:15 wh vh
12:00
distance
time
• 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
Note that rotating the graph through 90° anticlockwise reveals a ‘crossed ladder’ diagram:
Given heights of a and b for ladders crossed in an alleyway, the crossing point is at h where 1/a + 1/b = 1/h, the harmonic sum.
Here we are told a-h and b-h, say A and B, so 1/(h+A) + 1/(h+B) = 1/h leading to h =√ab, the geometric mean!

5. Triangle Count
Specialising; Generalising; Imagining; Expressing (to self, to others)
When will crossings align?

6. Max-Min 2 5 6 8 3 4 1 7 9 For each row, take the maximum in that row4 1 7 9
For each row, take the maximum in that row
For each column take the minimum in that column
Now take the max of the min’s and the min of the max’s: which is greater?
Replace either the min-action or tha max-action by the arithmetic mean.
Which greater, the max of the mean’s or the mean of the max’s?
Which is greater, the mean of the min’s or the min of means’s?

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?
Distributed examples: each table has several examples to contemplate looking for a generality
Do you need to try an example? … helps to find what relationship might be, but then you need to find a way to reason in general

8. Imagining & Expressing
Sliding Round and Round
Get a sense of the freedom available
Imagine a triangle
Imagine a second triangle with 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!
Freedom & Constraint
What is the length of the locus of the centroid of the sliding triangle?

9. See generality through a particular
Up & Down Sums = =
3 x 4 + 1
See generality through a particular
Generalise!
Specialising in order to generalise
Seeing things differently (Watch What You Do; Say What You See)
Seeing this as 2^2 + 3^2 (inner and outer square on) leads to Draw the diagram for and ‘see’ 3^2 + 4^2 = 5^2
… + (2n–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. Anticipating Generalising
Differences
Anticipating Generalising
If n = pqr then 1/n = 1/pr(q-r) - 1/pq(q-r) so there are as many ways as n can be factored as pqr with q>r
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
open.ac.uk