Transcript Making Use of Students’ Natural Powers to Think Mathematically John Mason

The Open University
Maths Dept
University of Oxford
Dept of Education
Making Use of
Students’ Natural Powers
to Think Mathematically
John Mason
Grahamstown
May 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
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Consecutive Sums
Say What You See
3
CopperPlate
Calculations
4
0
Difference
Divisions
1
2
4–2=4÷2
1
2
–3=4
5
1
3
–4=5
÷4
6
1
4
–5=6
1
÷
4
7
1
5
–6=7
÷6
4
5
1
÷
2
3
1
– (-1) =
-2
1
–0=1
-1
1
oops
1
–2=3
1
1
÷-2 (-1)
0
1
÷
-1
oops
1
÷
1
2
3
How does this fit in?
1
3
1
5
5
Going with the grain
Going across the grain
Leibniz’s Triangle
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1
2
1
3
1
4
1
5
1
6
6
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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Remainders of the Day (1)
Write
down a number which when
you subtract 1 is divisible by 5
and another
and another
Write down one which you think noone else here will write down.
7
Remainders of the Day (2)
Write down a number which when
you subtract 1 is divisible by 2
 and when you subtract 1 from the
quotient, the result is divisible by 3
and when you subtract 1 from that
quotient the result is divisible by 4
Why must any such number be
divisible by 3?

8
Remainders of the Day (3)
Write down a number which is 1 more
than a multiple of 2
 and which is 2 more than a multiple
of 3
 and which is 3 more than a multiple
of 4
…

9
Remainders of the Day (4)
Write down a number which is 1
more than a multiple of 2
 and 1 more than a multiple of 3
 and 1 more than a multiple of 4
…

10
Assumptions
What
you get from this session will be largely
what you notice happening for you
If you do not participate, I guarantee you will
get nothing!
I assume a conjecturing atmosphere
– Everything said has to be tested in experience
– If you know and are certain, then think and listen;
– If you are not sure, then take opportunities to try to
express your thinking
Learning
is a maturation process, and so
invisible
– It can be promoted by pausing and withdrawing from
the immediate action in order to get an overview
11
Triangle Count
12
Max-Min
13
2
5
6
8
3
2
4
1
7
7
6
1
2
9
4
6
8
9
5
8
9
8
2
5
9
7
2
1
9
8
3
7
1
9
6
9
Max-Min
In
a rectangular array of numbers,
calculate
– The maximum value in each row, and then the
minimum of these
– The minimum in each column and then the
maximum of these
How
do these relate to each other?
What about interchanging rows and
columns?
What about the mean of the maxima of
each row, and the maximum of the means
of each column?
14
Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
See
generality
through a
particular
Generalise!
1 + 3 + … + (2n–1) + … + 3 + 1
=
15
(n–1)2 + n2
= n (2n–2) + 1
Differences
16
1  1 1
1 1 1
7 6 42
2 1 2
1 11
1 1 1  1 1  11
3 2 6
8 7 56 6 24 4 8
Anticipating
1  1 1  1  1
Generalising
4 3 12 2 4
Rehearsing
1 1 1
5 4 20
Checking
1  1  1  1 1 1 1  1  1
Organising
6 5 30 2 3 3 6 4 12
Powers
Am
I stimulating learners to use their own
powers, or am I abusing their powers by
trying to do things for them?
–
–
–
–
–
17
To imagine & to express
To specialise & to generalise
To conjecture & to convince
To stress & to ignore
To extend & to restrict
Reflections
Much
of mathematics can be seen
as studying actions on objects
Frequently it helps to ask yourself
what actions leave some relationship
invariant; often this is what is
studied mathematically
18
More Resources
Questions
& Prompts for Mathematical
Thinking
(ATM Derby: primary & secondary versions)
Thinkers (ATM Derby)
Mathematics as a Constructive Activity
(Erlbaum)
Designing & Using Mathematical Tasks
(Tarquin)
http: //mcs.open.ac.uk/jhm3
j.h.mason @ open.ac.uk
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81
Gasket Sequences
22
Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
23
If someone claimed
there were 228 perforations
in a sheet,
how could you check?