Transcript Algebra in Trondheim

Generalisation in Mathematics:
who generalises what,
when, how and why?
John Mason
Trondheim
April 2009
1
Some Sums
1+2= 3
4+5+6= 7+8
= 13 + 14 + 15
9 + 10 + 11 + 12
16 + 17 + 18 + 19 + 20
= 21 + 22 + 23 + 24
Generalise
Say What You See
Justify
Watch What You Do
2
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?
3
+1
+2
+3
4
+6
One More
What numbers are one more than the
product of four consecutive integers?

Let a and b be any two numbers, one of them
even. Then ab/2 more than the product of any
number, a more than it, b more than it and a+b
more than it, is a perfect square, of the number
squared plus a+b times the number plus ab/2
squared.
4
CopperPlate
Calculations
5
Structured Variation Grids
6
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
7
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
8
What do you
have to do with
your attention?
Gnomon Border
How many tiles are needed to
surround the 137th gnomon?
The fifth is shown here
In how many different
ways can you count
them?
9
Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
10
If someone claimed
there were 228 perforations
in a sheet,
how could you check?
Honsberger’s Grid
5
1
2
6
3
7
4
8
9
4
11
1
7
1
8
1
9
20
1
0
1
1
1
2
1 2
3 1
1 22
4
1 23
5
1 24
6
25
3 43 57 73 91 11 133 147
1
1
Painted Cube
A
cube of wood is dropped into a
bucket of paint. When the paint dries
it is cut into little cubes (cubelets).
How many cubes are painted on how
many faces?
12
Attention
Holding
Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of
properties
13
The Place of Generality
A
lesson without the opportunity for
learners to generalise
mathematically, is not a mathematics
lesson
14
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?
15
New concepts
New actions
Mathematical themes
Use of mathematical powers
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
16
Arithmetic
– Uncovering and expressing the rules of arithmetic as the
rules of algebra
MGA & DTR
Doing – Talking – Recording
17
DofPV & RofPCh
Dimensions
of possible variation
– What can be varied and still something
remains invariant
Range
of permissible change
– Over what range can the change take
place and preserve the invariance
18
Some Mathematical Powers
Imagining
& Expressing
Specialising & Generalising
Conjecturing & Convincing
Stressing & Ignoring
Ordering & Characterising
19
Some Mathematical Themes
Doing
and Undoing
Invariance in the midst of Change
Freedom & Constraint
20
Consecutive Sums
Say What You See
21
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]
22