1

Progress Progress in in

Mathematical ThinkingMathematical Thinking

Portugal MScPortugal MSc

June 2010June 2010

The Open UniversityMaths Dept University of Oxford

Dept of EducationPromoting Mathematical Thinking

2

OutlineOutline

What is progress in mathematical What is progress in mathematical thinking?thinking?

Progress in …Progress in …– Performance (Performance (behaviourbehaviour))– Understanding; connection; being able to talk Understanding; connection; being able to talk

about (about (cognitioncognition))– Independence and initiative (Independence and initiative (affect + willaffect + will))– Ways of working (Ways of working (milieumilieu))

LanguageLanguage– for thinking and discussingfor thinking and discussing

Tasks that reveal progress and provide new language

3

Structure of human psyche: chariot Structure of human psyche: chariot metaphormetaphor

Behaviour

(enaction)

Emotion (affect)

Awareness

(cognition)

WillMental

Imagery

Habits

4

In BetweenIn Between How many circles could there be between the two shown?How many circles could there be between the two shown?

How many numbers could there be betweenHow many numbers could there be between

1.50 and 1.591.50 and 1.591.500 and 1.59871.500 and 1.5987

Range of permissi

ble change

Discrete&

Continuous

5

Difference of 2Difference of 2

write down 2 numbers with a difference of 2

Using coordinate notation, write down two points whose distance apart is two units

And anotherAnd another

And two numbers whose ‘distance apart’ is between 1.5 and 2

Primary Secondary

Progression is visible in the range of

choices exhibited;

in the richness of

the example space being

sampled

And another

And two points whose distance apart is between 1.5 and 2

And another

6

Progress through shiftsProgress through shifts

Every technical term indicates a shift in Every technical term indicates a shift in ‘ways of seeing’ ‘ways of seeing’

The name is a reminder of that shiftThe name is a reminder of that shift To use the term effectively, learners need to To use the term effectively, learners need to

experience the shiftexperience the shift

7

Seeing AsSeeing As

✎ Raise your hand when you can Raise your hand when you can see something that issee something that is1/3 of something; 1/3 of something;

again differentlyagain differently

A ratio of 1 : 2A ratio of 1 : 2

Range of permissi

ble change

Dimensions of

possible variation✎ What else can you see?What else can you see?

✎ What assumptions are you making?What assumptions are you making?

4/3 of 4/3 of somethingsomething

8

1

n1

n 1

1

n n 1

Seeing the general through an example

Can you see something that is:One fifth of somethingOne fourth of somethingOne fourth of something take away one fifth of the same thing

Now Generali

se!

1

4−

15

=120

9

What was your progress in that task?What was your progress in that task?

10

FractionsFractions

11

What was your progress in that task?What was your progress in that task?

12

Triangle CountTriangle Count

13

What was your progress in that task?What was your progress in that task?

14

Reading a Diagram: Seeing As …Reading a Diagram: Seeing As …

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

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)

15

TangentialTangential

At what point of y = xAt what point of y = x22 + 1 does the tangent + 1 does the tangent go through the origin?go through the origin?

What about y = 4xWhat about y = 4x22 + 1? + 1? What about y = 9xWhat about y = 9x22 + 1? + 1? When y = (λx)When y = (λx)22 + 1 , what is the locus of that + 1 , what is the locus of that

point (as λvaries) ?point (as λvaries) ? What about y = f(λx)?What about y = f(λx)?

16

Progress in mathematics means:Progress in mathematics means:– Getting better at …Getting better at …– Knowing more about ...Knowing more about ...– Being able to …Being able to …– Taking initiative to …Taking initiative to …– Contributing to an atmosphere (milieu) conducive Contributing to an atmosphere (milieu) conducive

to mathematical thinking …to mathematical thinking …

17

Progress in What?Progress in What? Use of abilityUse of ability

– To imagine & to expressTo imagine & to express– To specialise & to generaliseTo specialise & to generalise– To conjecture & to convinceTo conjecture & to convince– To stress & to ignoreTo stress & to ignore– To persist and to let goTo persist and to let go

Use of mathematical themes:Use of mathematical themes:– Doing & Undoing (inverses)Doing & Undoing (inverses)– Invariance and VariationInvariance and Variation– Freedom & ConstraintFreedom & Constraint– Extending & Restricting MeaningExtending & Restricting Meaning

18

My Website & Further ReadingMy Website & Further Reading

j.h.mason @ open.ac.ukj.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 go to mcs.open.ac.uk/jhm3 go to

PresentationsPresentations

Designing Mathematical Tasks (Tarquin)Designing Mathematical Tasks (Tarquin) Questions & Prompts (ATM)Questions & Prompts (ATM) Thinkers (ATM)Thinkers (ATM) Fundamental Constructs in Maths Edn (Sage)Fundamental Constructs in Maths Edn (Sage) Researching Your Own Practice (Routledge)Researching Your Own Practice (Routledge)