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The Open University
Maths Dept
University of Oxford
Dept of Education
Working Algebraically
0-8
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?
6
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?
7
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
8
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?
9
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
10
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’.
11
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?
12
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!
13
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
14
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
15
Extend My Sequence
 Make
a pattern of coloured unifix cubes
 Now repeat that pattern, twice.
 What did you have to do with your attention?
16
Children’s Copied Patterns
model
17
4.1 yrs
Marina Papic MERGA
Children’s Own Patterns
5.0 yrs
5.1 yrs
5.4 yrs
18
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
19
Attention
 Holding
Wholes (gazing)
 Discerning Details
 Recognising Relationships
 Perceiving Properties
 Reasoning on the basis of properties
20
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
21
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
23
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
24
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
25
CopperPlate
Calculations
26
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
27
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
28
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!
29
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
30
What do you
have to do with
your attention?
Mystery 3 by 3 grid of numbers
Durham Maths Mysteries
31
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?
32
More Magic Square Reasoning
Sum(
33
) – Sum(
) =0
MGA & DTR
Doing – Talking – Recording
34
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
37
Attention
Holding
Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of properties
38
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
39
Triangle Count
40
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]
41