1 fraction actions: working with fractions as operators john mason calgary oct 2014 the open...
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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)