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Fraction Actions:Working with Fractions

as Operators

John MasonCalgary

Oct 2014The Open University

Maths Dept University of OxfordDept of Education

Promoting Mathematical Thinking

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What Does it Mean?

3 divided by 5Divide 3 by 5The answer on dividing 3 by 5The action of ‘three fifth-ing’The result of ‘three fifth-ing’ of 1 on the numberlineThe value of the ratio of 3 to 5The equivalence class of all fractions with value three fifth’s (a number)Place on the number line (number) …

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Different Perspectives

What is the relation between the numbers of squares of the two colours?

Difference of 2, one is 2 more: additive thinking

Ratio of 3 to 5; one is five thirds the other etc.: multiplicative thinking

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Raise your hand when you can see

Something which is 2/5 of something

Something which is 3/5 of something

Something which is 2/3 of something

How did your attention shift?

Flexibility in choice of

unit

Something which is 1/3 of 3/5 of something

Something which is 3/5 of 1/3 of something

Something which is 2/5 of 5/2 of something

Something which is 1 ÷ 2/5 of something

Something which is 5/2 of something

Something which is 5/3 of something

Something which is 3/2 of something

What others can you see?

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Doing & Undoing

What operation undoes ‘adding 3’? What operation undoes ‘subtracting 4’? What operation undoes ‘subtracting from

7’? What are the analogues for

multiplication? What undoes ‘multiplying by 3’? What undoes ‘dividing by 4’? What undoes ‘multiplying by ¾’?

Two different expressions! What operation undoes ‘dividing into 24’?

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SWYS

Find things that are , , , , , of something

Find something that is of of something

Find something that is of of something

What is the same, and what is different?

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Presenting Fractions as Actions

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Raise your hand when you can see …

Something that is 1/4 – 1/5of something else

What did you have to do with your attention?

What do you do with your

attention in order to

generalise?

Did you look for something that is 1/4 of something else

and forsomething that is 1/5 of the same thing?

Common Measure

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Stepping Stones

…

…R

R+1

What needs to change so as to ‘see’ that

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Elastic Multiplication

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Two Journeys Which journey over the same distance at two

different speeds takes longer:– One in which both halves of the distance are done at

the specified speeds?– One in which both halves of the time taken are done

at the specified speeds?

distance time

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FrameworksDoing – Talking – Recording

(DTR)

Enactive – Iconic – Symbolic

(EIS)

See – Experience – Master(SEM)

(MGA)

Specialise … in order to locate structural

relationships …then re-Generalise for

yourself

What do I know?What do I want?

Stuck?

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Mathematical Thinking

How describe the mathematical thinking you have done so far today?

How could you incorporate that into students’ learning?

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Possibilities for Action

Trying small things and making small progress; telling colleagues

Pedagogic strategies used today Provoking mathematical thinking as happened

today Question & Prompts for Mathematical Thinking

(ATM) Group work and Individual work

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Human Psyche

ImageryAwareness (cognition)

Will

Body (enaction)

Emotions (affect)

HabitsPractices

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Three Only’s

Language Patterns& prior Skills

Imagery/Sense-of/Awareness; Connections

Different Contexts in which likely to arise;

dispositions

Techniques & Incantations

Root Questionspredispositions

Standard Confusions

& Obstacles

Only Behaviour is TrainableOnly Emotion is Harnessable

Only Awareness is Educable

Behaviour

Emotion

Awar

enes

s

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Follow Up

j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Presentations Questions & Prompts (ATM) Key ideas in Mathematics (OUP) Learning & Doing Mathematics (Tarquin) Thinking Mathematically (Pearson) Developing Thinking in Algebra (Sage)