Transcript Where is the Reality of Algebra & Geometry ?

The Open University
Maths Dept
University of Oxford
Dept of Education
Where is the Reality
of
Algebra & Geometry
?
John Mason
Surrey HoDs
Feb 2009
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Assumptions
Learning
is extending the range of
possible actions that come to mind
in a situation
Discerning details
Recognising relationships
Perceiving properties
Reasoning on the basis of properties
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Do you know any students who …
seem to participate in
mathematics lessons
Say “I can’t …”
Ask “Why are we doing this?”
Appear to do the minimum to get
through a lesson?
Don’t
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Tracks
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Mechanism
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Where Were You?
Where
was your attention?
– In or on the picture?
– Imagining something moving?
– In or on recognising relationships?
– In or on perceiving instances of
properties?
– In or on reasoning about those
relationships?
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Imagine a Number-line …
 Imagine
a copy sitting on top;
 Rotate the copy through 180° about the origin
 Restore the copy.
 Now rotate it through 180° about the point 3 on
the original fixed number-line
-5
-4
 Now
-3
-2
-1
0
1
2
3
4
rotate it through a further 180° about the
point 1 on the original fixed number-line
 What is special about the points 3 and 1
 How could you develop or extend this task?
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What ‘world’ were you occupying?
The
world of the screen?
The world of your imagination?
A mathematical world?
The material world?
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Kites
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Where Were You?
Where
was your attention?
– In or on the paper?
– In or on the folding?
– In or on relationships?
– Reasoning about those relationships?
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Square Formation
3b-3a
a+3b
3a+b
a b
2a+b
a+b
a+2b
3(3b-3a) = 3a+b
12a = 8b
So a/b = 3/2
For an overall square
4a + 4b = 2a + 5b
So 2a = b
3
9
7
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For n squares upper left
n(3b - 3a) = 3a + b
So 3a(n + 1) = b(3n - 1)
1
1
2 3
5
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Where Were You?
Where
was your attention?
– On material objects?
– In or on some diagrammatic
presentation?
– What were you manipulating?
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Triangle Count
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Seven Circles
How many
different angles
can you discern,
using only the red
points?
How do you
know you have
them all?
How many
different
quadrilaterals?
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Four Consecutives
Write
down four consecutive
numbers and add them up
and another
and another
Now be more extreme!
What is the same, and what is
different about your answers?
+1
+2
+3
4
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+6
Leibniz’s Triangle
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1
2
1
3
1
4
1
5
1
6
16
1
6
1
12
1
20
1
30
1
2
1
3
1
12
1
30
1
60
1
4
1
20
1
60
1
5
1
30
1
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Ages Ago
 Two
persons, A, and B were talking of their Ages:
Says A to B, seven Years ago I was thrice as old
as you at that Time; and seven Years hence I
shall be just twice as old as you will be: I demand
their present Ages?
– [Mole 1788 problem V p129]
Make a guess: A is 14
Now check: (14 – 7) = 3(9 – 7) so B is 9?
And (14 + 7) ?=? 2(9 + 7) No … but
(A – 7) = 3(B – 7) and (A+7) = 2(B + 7)
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Sharing
A
hungry hunter came upon two
shepherds, one of whom had 3
loaves and the other 5, all of the
same size. The loaves were divided
equally among the three. The hunter
paid 8 cents for his share. How
should the shepherds divide the
money?
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Debbie’s Apples
Debbie
bought apples at 25p each. She
ate 2 and sold the rest for 32p each. Her
profit was £1.88. How many apples did
she buy?
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Do you know any students who …
seem to participate in mathematics
lessons
Say “I can’t …”
Don’t
– Convert can’t into didn’t
Ask
“Why are we doing this?”
– Customers want particular (price …)
– Entrepreneurs & Managers need general
(policy, …)
Appear
to do the minimum to get through
a lesson?
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Are they being encouraged to …
use
their natural powers to
– Imagine & Express
– Specialise & Generalise
make
significant mathematical
choices
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Reality is …
Being
intrigued, surprised, engaged
Using your own powers to One of the core
–
–
–
–
It
Imagine & Express
contributions
of schooling
Specialise & Generalise
… experiencing
is …
Conjecture & Convince
‘realities’
Extend & Restrict attention
you might not
is NOT restricted to
– Utility, or even purpose
– Your own past experience
– Some imagined use in the future
Reality
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otherwise
encounter
is where your focal attention is
Becoming Real
Holding
Wholes (gazing)
Discerning details
Recognising relationships
Perceiving properties (as being
instantiated)
Reasoning on the basis of
properties
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Worlds
Inner
World of
Imagery
Material
World
enactive
iconic
World of
Symbol
s
symbolic
Worlds during Modelling and Problem-Solving
Aspects of Conceptual Development
Modes of Attending
Epistemological Stances
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Follow-Up
These
slides (and others) available
on
– http://mcs.open.ac.uk/jhm3
Contact
me at
– [email protected]
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