Thinking Mathematically as Developing Students’ Powers

The Open University Maths Dept University of Oxford Dept of Education Thinking Mathematicallyas Developing Students’ Powers John Mason Oslo Jan 2009

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

Outline • Some tasks to work on together • Some remarks about what might have been noticed • Each task indicates: • a domain of similar tasks • a style or structure of tasks • More important than particular tasks: • ways of working with learners ON tasks

Imagining & Expressing Imagine a mathematical plane, and lying in it, a … Where can the centre get to? … circle Where can the centre get to? • … a fixed point P and a circle passing through P Where can the centre get to? • … two distinct fixed points P and Q and a circle passing through both points Where can the centre get to? • … three distinct points P, Q & R and a circle passing through all three points

One Sum I have two numbers which sum to 1 Which will be larger: • The square of the larger added to the smaller? • The square of the smaller added to the larger? Make a Conjecture! Only then Check!

2 + (1- ) + (1- )2 One Sum Diagrams 1 (1- )2 = 1 2 1- Anticipating,not waiting

Reading a Diagram x2 + (1-x)2 x3 + x(1–x) + (1-x)3 x2z + x(1-x) + (1-x)2(1-z) xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

Triangle Count

Variation • Dimensions-of-possible-variation • Range-of-permissible-change • Invariance in the midst of change

Structured Variation Grids

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

Reading Graphs • Imagine the graph of a cubic polynomial • Imagine also the graph of a quartic • Imaging also the graph of y = x • Now, imagine a point x on the x-axis; • proceed vertically up (or down) to the cubic; • proceed horizontally to the line y=x • proceed vertically up(or down) to the quartic • proceed horizontally untilyou are directly in verticalline with the x you started with

Cubical Property • Imagine a cubic • Imagine a chord, extended to a line;Find the midpoint of your chord • Imagine a second chord with the same midpoint; extend it to a line • What do you imagine will happen?

Chord-slopes • Imagine a quartic polynomial • Imagine an interval of fixed width on the x-axis • The interval determines a chord. The mid-point of the chord is marked • The slope of the chord is shown

Kites

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

Reflection • What did you notice happening for you mathematically? • What might you be able to use in an upcoming lesson? • Imagine yourself in the future, using or developing or exploring something you have experienced today!

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

Thinking Mathematically as Developing Students’ Powers