Download OAME Workshop - The Open University

The Open University
Maths Dept
University of Oxford
Dept of Education
Working Algebraically
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John Mason
OAME
Toronto
Feb 2009
1
Ways of Working

Everything said is a conjecture
– to be tested in experience
– to be modified as necessary
& said in order to ‘get it out’ so it can be looked at clearly and
closely
 When we disagree we offer a potential counter-example or
we invite someone to modify their conjecture
2
Expressing Generality
 “What
do we do with pencils, Johnnie?”
 If a number ends in 0 it is divisible by 10
 8yr old on train as it leaves a tunnel having
stopped at a station in the tunnel: “are all
stations tunnels?”
3
What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and
subtract one from the smaller
What then
would be
the difference?
What could
be varied?
4
What’s The Ratio?
÷
=
First, multiply each by 3
First,
multiply the larger by 2 and
divide the smaller by 3
What is the ratio?
What could
be varied?
5
Core
 Every
child comes to school having displayed
the powers necessary to think algebraically
 The question is:
– Am I getting children to use and develop their powers?
– Or am I usurping those opportunities?
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Another & Another
 Write
down a number that is one more than a
multiple of 5
 And another
 And another
 And one that is obscure as possible
 Multiply two of your numbers together
what do you notice?
 What can we change in this task?
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Variation
 Dimensions-of-possible-variation
 Range-of-permissible-change
 To
‘learn’ a concept is to discern & become
aware of
– Dimensions of possible variation
– Range of permissible change of each dimensions
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Counting & Actions
 If
I have 3 things & you have 5 things, how many
altogether?
– How else might we share them out?
 If
I have 3 more things than you do, and you have
5 more things than she has, how many more
things do I have than she has?
– Variations?
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Comparing
 If
you gave me 5 of your things then I would have
three times as a many as you then had, whereas if I
gave you 3 of mine then you would have 1 more than
2 times as many as I then had. How many do we
each have?
 If B gives A $15, A will have 5 times as much as B
has left. If A gives B $5, B will have the same as A.
[Bridges 1826 p82]
you take 5 from the father’s years and divide the
remainder by 8, the quotient is one third the son’s
age; if you add two to the son’s age, multiply the
whole by 3 and take 7 from the product, you will have
the father’s age. How old are they? [Hill 1745 p368]
 If
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Tracking Arithmetic
 If
you can check an answer, you can write down
the constraints (express the structure)
symbolically
 Check a conjectured answer BUT don’t ever
actually do any arithmetic operations that involve
that ‘answer’.
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Comparisons
 Which
–
–
–
–
–
–
is bigger?
83 x 27 or 84 x 26
8/0.4 or 8 x 0.4
867/.736 or 867 x .736
3/4 of 2/3 of something, or 2/3 of 3/4 of something
5/3 of something or the thing itself?
437 – (-232) or 437 + (-232)
 What
conjectured generalisations are being
challenged?
 What generalisations (properties) are being
instantiated?
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Central Issue
 It’s
not the ‘x’ that is the problem with algebra
 It’s learning to use symbols to stand for what is
as-yet-unknown or as-yet-unspecified
 You can’t learn arithmetic without thinking
algebraically!
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Tasks, Activity & Learning
 Tasks
initiate activity
 Activity provides experience
 Experience provides the basis for learning
 Learning involves
– Engaging in extended or new actions
– Attending differently; noticing freshly
– Internalising, integrating, so as to initiate actions for
oneself (ZPD)
 Withdrawing
of the action
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from action so as to become aware
Teaching
 Selecting
tasks
 Preparing Didactic Tactics and Pedagogic Strategies
 Prompting extended or fresh actions
 Being Aware of mathematical actions
 Directing Attention
Teaching takes
place in time;
Learning takes
place over time
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Extend My Sequence
 Make
a pattern of coloured unifix cubes
 Now repeat that pattern, twice.
 What did you have to do with your attention?
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Children’s Copied Patterns
model
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4.1 yrs
Marina Papic MERGA
Children’s Own Patterns
5.0 yrs
5.1 yrs
5.4 yrs
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Marina Papic MERGA
Extended Sequences
…
Someone has made a simple pattern of coloured squares,
and then repeated it a total of at least two times
 State in words what you think the original pattern was
 Predict the colour of the 100th square and the position of
the 100th white square

…
Make up your own:
a really simple one
a really hard one
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Attention
 Holding
Wholes (gazing)
 Discerning Details
 Recognising Relationships
 Perceiving Properties
 Reasoning on the basis of properties
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Sequencing
Describe a construction
rule for which the second
and fourth pictures are as
shown
#2
#4
+ 1 + + 2x
1 + 2x + 2x
2 + 1 + 2 + 2x2
1 + 2x4 + 2x4
(1 + 2x )(1 + 2x ) – 2( x2x )
4 + 1 + 4 + 2x4
1 + 4x4 + 4x4
(1 + 2x2)(1 + 2x2) - 2x(2x2x2)
2(1+2 ) - 1
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Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
1 + 3 + … + (2n – 1) + … + 3 + 1
=
22
(n – 1)2 + n2
= n (2n – 2) + 1
The Place of Generality
A
lesson without the opportunity for learners to
generalise mathematically, is not a mathematics
lesson
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Text Books
 Turn
to a teaching page
– What generality (generalities) are present?
– How might I get the learners to experience and express
them?
– For the given tasks, what inner tasks might learners
encounter?
New concepts
New actions
Mathematical themes
Use of mathematical powers
Rehearsal of developing skills and actions
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Progression
 Increasing
facility in expressing generality
– In words, diagrams, symbols
 Developing
disposition to see generality through
particulars
 Increasing propensity to recognise relationships
as instances of properties
 Increasing use of use of reasoning on the basis
of agreed properties to justify conjectures
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CopperPlate
Calculations
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THOANs
 Think
–
–
–
–
–
–
of a number
Add 3
Multiply by 3
Subtract 1 more than the number you first thought of
Divide by 2
Subtract the number you first thought of
Your answer is 4
Try 7:
7+3
3(7+3)= 3x7 + 9
(3x7 + 9) – (7 + 1) = 2x7 + 8
(2x7 + 8)/2 = 7 + 4
7+4–7=4
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Variations:
deduce start by hearing answer
steps on number line
journeys in the plane
Try :
+3
3( +3)= 3x + 9
3x + 9 – ( + 1) = 2x + 8
(2x + 8)/2 = + 4
+4– =4
Acknowledging Ignorance (Mary Boole)
 Admit
you don’t know;
denote what you don’t know by some symbol (I
recommend clouds);
 Express what you know until you have one or
more equations or inequalities
 Tracking Arithmetic
– If you can CHECK whether a conjectures answer is
correct
– You can express the constraints algebraically
– Check it but DO NOT actually carry out any arithmetic on
the conjectured ‘answer’; then replace it by a cloud or
letter
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Sometimes, Always, Never (SAN-tasks)
 Is
it sometimes (then when?), always, or never true
that:
–
–
–
–
–
Putting a 0 on the right hand end of a number multiplies it by 10
If I am thinking of a number, you can find a greater number
Between any two numbers there is another number
The sum of any two consecutive numbers is odd
The sum of any four consecutive numbers is divisible by 4
Justify your
conjecture!
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Raise Your Hand When You Can See
 Something
which is
1/4 of something
1/5 of something
1/4-1/5 of something
1/4 of 1/5 of something
1/5 of 1/4 of something
1/n – 1/(n+1) of
something
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What do you
have to do with
your attention?
Mystery 3 by 3 grid of numbers
Durham Maths Mysteries
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Magic Square Reasoning
2
2
6
7
2
1
5
9
8
Sum(
3
) – Sum(
What other
configurations
like this
give one sum
equal to another?
Try to describe
them in words
4
)
=0
Any colour-symmetric arrangement?
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More Magic Square Reasoning
Sum(
33
) – Sum(
) =0
MGA & DTR
Doing – Talking – Recording
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Some Mathematical Powers
Imagining
& Expressing
Specialising & Generalising
Conjecturing & Convincing
Stressing & Ignoring
Ordering & Characterising
35
Some Mathematical Themes
Doing
and Undoing
Invariance in the midst of Change
Freedom & Constraint
36
Roots of & Routes to Algebra
Expressing
Generality
– A lesson without the possibility of learners
generalising (mathematically) is not a mathematics
lesson
Multiple
Expressions
– Purpose and evidence for the ‘rules’ of algebraic
manipulation
Freedom
& Constraint
– Every mathematical problem is a construction task,
exploring the freedom available despite constraints
Generalised
Arithmetic
– Uncovering and expressing the rules of arithmetic as the
rules of algebra
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Attention
Holding
Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of properties
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Cutting Chocolate Bars
 How
many cuts
needed to release
all the squares?
How many folds needed
to make a grid?
 You can only cut
one current piece at
a time
Specialise:
use physical
objects
try simpler cases
In order to re-generalise for yourself
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Triangle Count
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For More Details
Thinkers (ATM, Derby)
Questions & Prompts for Mathematical Thinking
Secondary & Primary versions (ATM, Derby)
Mathematics as a Constructive Activity (Erlbaum)
Listening Counts (Trentham)
Structured Variation Grids
This and other presentations
http: //mcs.open.ac.uk/jhm3
[email protected]
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