Download Th`g Math`ly & Powers

Thinking Mathematically
as
Developing Students’ Powers
John Mason
West Berkshire & Reading
Oct 2008
1
Conjecturing Atmosphere
Everything
said is said in order to
consider modifications that may be
needed
Those who ‘know’ support those
who are unsure by holding back or
by asking informative questions
2
Conjectures
The
richness of mathematical tasks
does NOT lie in the task itself
NOR does it lie in the format of
interactions
It DOES lie in the teacher’s ‘being’,
manifested in
– teacher-learners relationships
– Teacher’s mathematical awareness
3
More Conjectures
The
richness of learners’ mathematical
experience depends on
– Opportunities to use and develop their
own powers
– Opportunities to make significant
mathematical choices
– Being in the presence of mathematical
awareness
4
Pattern Continuation
…
…
5
Children’s Copied Patterns
model
6
4.1 yrs
Marina Papic MERGA 30 2007
Children’s Own Patterns
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are needed to see this picture.
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5.0 yrs
5.1 yrs
5.4 yrs
7
Marina Papic MERGA 30 2007
Extending Patterns
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6 yr olds
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R: Can you tell me what is happening each time we make the
square bigger?
Ch: Yeh, here it has one, then it has 2 and 2 lines and it’s
bigger. Then this one has three and three lines and then
four and four lines.
R: What do you mean four and four lines.
Ch: See there’s four in each line.
R: So what would the next one in my pattern be?
Ch: Umm … five and five lines.
8
Marina Papic MERGA 30 2007
Remainders of the Day
 Write
down a number that leaves a reminder of 1
when divided by 3
 and another
 and another
 Multiply two of these numbers together:
what remainder does it leave when divided by 3?
 Why?
 What is special about the ‘3’?
9
Primality
What
does ‘prime’ mean in the system of
numbers leaving a remainder of 1 when
divided by 3?
What is the second positive non-prime
after 1 in the system of numbers of the
form 1+3n?
100 = 10 x 10 = 4 x 25
What does this say about primes in the
multiplicative system of numbers of the
form 1 +3n?
What is special about the ‘3’?
What is special about the ‘1’?
10
Triangle Count
11
Bag Constructions (1)
Here
there are three bags.
If you compare any two of
them, there is exactly one
colour for which the
difference in the numbers
of that colour in the two
bags is exactly 1.
For four bags, what is the
least number of objects to meet
the same constraint?
 For four bags, what is the
least number of colours to
meet the same constraint?

12
17 objects
3 colours
Bag Constructions (2)
Here
there are 3 bags and
two objects.
There are [0,1,2;2] objects in
the bags with 2 altogether
Given a sequence like
[2,4,5,5;6] or [1,1,3,3;6] how
can you tell if there is a
corresponding set of bags?
13
What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and
subtract one from the smaller
14
What then
would be
the difference?
What could
be varied?
What’s The Ratio?
÷
=
First, multiply each by
3
First,
multiply the larger by 2 and
divide the smaller by 3
15
Which will be
What is the ratio? ?
larger?
What could
be varied?
Magic Square Reasoning
2
6
7
2
1
5
9
8
Sum(
16
3
) – Sum(
4
What other
configurations
like this
give one sum
equal to another?
Try to describe
them in words
) =0
More Magic Square Reasoning
Sum(
17
) – Sum(
) =0
Doing & Undoing
What
operation undoes ‘adding 3’?
What operation undoes ‘subtracting 4’?
What operation undoes
‘subtracting from 7’?
What are the analogues for multiplication?
What undoes multiplying by 3?
What undoes dividing by 2?
What undoes multiplying by 3/2?
What undoes dividing by 3/2?
18
Composite Doing & Undoing
I
am thinking of a number
I add 8 and the answer is 13.
I add 8 then multiply by 2;
the answer is 26.
I add 8, multiply by 2, subtract 5;
the answer is 21.
I add 8, multiply by 2, subtract 5, divide by 3.
The answer is 7
HOW do you turn +8, x2, -5, ÷3 answer 7
into a solution?

19
Additive & Multiplicative Perspectives
What
is the relation between the
numbers of squares of the two
colours?
Difference of 2, one is 2 more:
additive
Ratio of 3 to 5; one is five thirds the
other etc.:
multiplicative
20
Raise your hand when you can
see
 Something
which is 2/5 of something
 Something which is 3/5 of something
 Something which is 2/3 of something
– What others can you see?
 Something which is 1/3 of 3/5 of something
 Something which is 3/5 of 1/3 of something
 Something which is 2/5 of 5/2 of something
 Something which is 1 ÷ 2/5 of something
21
What fractions can you ‘see’?
What
relationships between
fractions can you see?
22
Grid Sums
To move to the right
one cell you add 3.
To move up one cell
you add 2.
??
7
In how many different ways can you work out a value for
the square with a ‘?’ only using addition?
Using exactly two subtractions?
23
Grid Movement
((7+3)x2)+3
is a path from 7 to ‘?’.
What expression
represents the reverse of
this path?
What values can ‘?’ have:
x2
÷2
?
7
- if only + and x are used
- if exactly one - and one ÷
are used, with as many
+ & x as necessary
What about other cells?
Does any cell have 0? -7?
Does any other cell have 7?
-3
24
+3
Characterise ALL the possible
values that can appear in a cell
Reflections
x2
÷2
?
7
What
variations are possible?
What have you gained by
-3 +3
working
on this task (with colleagues)?
What criteria would you use in
choosing whether to use this (or any) task?
What might be gained by working on (a
variant of) this task with learners?
Tasks –> Activity –> Experience –> ‘Reflection’
25
Attention
Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of agreed
properties
26
CopperPlate
Multiplication
27
796
7964455
64789
64789
30
2420
361635
54242840
4236423245
28634836
497254
5681
63
5160119905
Inter-Rootal Distances
Sketch
a quadratic for which the interrootal distance is 2.
and another
and another
How much freedom do you have?
What are the dimensions of possible
variation and the ranges of permissible
change?
If it is claimed that [1, 2, 3, 3, 4, 6] are the
inter-rootal distances of a quartic, how
would you check?
28
Reflections
 Initiating Activity
 Sustaining
Activity
 Transcending Activity
29