Download Fraction Actions

Promoting Mathematical Thinking
Fraction Actions:
Working with Fractions
as
Operators
John Mason
Calgary
Oct 2014
The Open University
Maths Dept
1
University of Oxford
Dept of Education
What Does it Mean?
3
5
3 divided by 5
Divide 3 by 5
The answer on dividing 3 by 5
The action of ‘three fifth-ing’
The result of ‘three fifth-ing’ of 1 on the numberline
The value of the ratio of 3 to 5
The equivalence class of all fractions with value three fifth’s (a number)
Place on the number line (number)
…
2
Different Perspectives
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3
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
Raise your hand when you can see
Flexibility in
choice of unit
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Something which is 2/5 of something
Something which is 3/5 of something
Something which is 2/3 of something
What others can you see?
Something which is 5/2 of something
Something which is 5/3 of something
Something which is 3/2 of something

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4

How did your
attention shift?
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
Doing & Undoing

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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’?

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5
What undoes ‘multiplying by ¾’?
 Two different expressions!
What operation undoes ‘dividing into 24’?
SWYS
Find things that are
Find something that is
Find something that is
6
1 , 1, 1
1 1 1
, , , of something
7 5 3 15 35 21
1
1
of of something
What is the same, and
3
7
what is different?
1
1
of of something
3
7
Presenting Fractions as Actions
7
Raise your hand when you can see …
Something that is 1/4 – 1/5
of something else
Did you look for
something that is 1/4 of something else
and for
something that is 1/5 of the same thing?
Commo
n
Measur
What did you have to do with
e
your attention?
What do you do with
your attention in order
to generalise?
8
Stepping Stones
R
9
1
1
1
=
R R + 1 R ( R + 1)
…
…
What needs to change so as
to ‘see’ that
R+1
r
1
1
=
R R + r R( R + r )
Elastic Multiplication
10
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?
time
distance
d
d
t1 =
t2 =
2v1
2v2
d
d
t =
+
2v1 2v2
11
t
t
d1 = v1 d2 = v2
2
2
2d
t=
v1 + v2
Frameworks
Doing – Talking – Recording
(DTR)
(MGA)
See – Experience – Master
(SEM)
Enactive – Iconic – Symbolic
(EIS)
12
Specialise …
in order to locate structural
Stuck?
What do I know?
relationships …
then re-Generalise for yourself
What do I want?
Mathematical Thinking


13
How describe the mathematical thinking you have
done so far today?
How could you incorporate that into students’
learning?
Possibilities for Action
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14
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
Human Psyche
Awareness (cognition)
Imagery
Will
Emotions
(affect)
Body (enaction)
Habits
Practices
15
Three Only’s
Language Patterns
& prior Skills
Imagery/Senseof/Awareness; Connections
Root Questions
predispositions
Different Contexts in which
likely to arise;
dispositions
Standard Confusions
& Obstacles
Techniques & Incantations
Emotion
16
Only Emotion is Harnessable
Only Awareness is Educable Only Behaviour is Trainable
Follow Up
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17
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)