Transcript Wits Variation

Variation
as a
Pedagogical Tool
in Mathematics
John Mason & Anne Watson
Wits
May 2009
1
Pedagogic Domains
 Concepts
 Topics
– Arithmetic  Algebra
 Techniques
 Tasks
2
(Exercises)
Topic: arithmetic  algebra
 Expressing
Generality for oneself
 Multiple Expressions for the same thing
leads to algebraic manipulation
– Both of these arise from becoming aware of variation
– Specifically, of dimensions-of-possible-variation
3
What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and
subtract one from the smaller
4
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
5
What is the ratio?
What could
be varied?
Counting & Actions
 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?
 If
Anne gives me one of her marbles, she will
then have twice as many as I then have, but if I
give her one of mine, she will then be 1 short of
three times as many as I then have.
Do your expressions
express what you
mean them to
express?
6
Construction before Resolution
Working down
start with 12 and 8
and up,
– 12
8
12
8
keeping sum
invariant,
– 11
9
13
7
looking for a
– 10
10
14
4
multiplicative
relationship
–
15
5
 So if Anne gives John 2, they will then have the
same number; if John gives Anne 3, she will then
have 3 times as many as John then has
 Construct one of your own
Translate into
I
– And another
– And another
7
‘sharing’
actions
Principle
 Before
showing learners how to answer a typical
problem or question, get them to make up
questions like it so they can see how such
questions arise.
–
–
–
–
8
Equations in one variable
Equations in two variables
Word problems of a given type
…
Four Consecutives
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
 Write
Alternative:
I have 4 consecutive numbers in
mind.
They add up to 42. What are they?
9
+1
4
+2
+1
+3
+2
+6
4
D of P V?
R of P Ch?
+2
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,

10
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
11
Tunja Sequences
12
-1 x -1 – 1 =
-2 x 0
0x0–1=
-1 x 1
1x1–1=
0x2
2x2–1=
1x3
3x3–1=
2x4
4x4–1=
3x5
With
the
Grain
Across the
Grain
Lee Minor’s Mutual Factors
x2 + 5x + 6 = (x + 3)(x +
2)
2
x
= (x
+ 6)(x
– +
x2 +
+ 5x
13x–+630
= (x
+ 10)(x
1)
3)
2
x
x2 +
+ 13x
25x –
+ 30
84 =
= (x
(x +
+ 15)(x
21)(x –
+ 4)
2)
x2 + 25x – 84 = (x + 28)(x – 3)
x2 + 41x + 180 = (x + 36)(x + 5)
x2 + 41x – 180 = (x + 45)(x – 4)
13
14
43
44
45
46
47
48
49
42
21
22
23
24
25
26
41
20
7
8
99
10
27
40
19
6
1
2
11
28
39
18
5
4
3
12
29
38
17
16
15
14
13
30
37
36
35
34
33
32
31
50
64
36
37
38
39
40
41
42
43
44
35
14
15
16
17
18
19
20
45
34
13
2
3
4
21
46
33
12
11
10
1
5
22
47
32
31
30
9
8
7
6
23
48
29
28
27
26
25
24
49
50
15
81
Triangle Count
16
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
=
17
(n–1)2 + n2
= n (2n–2) + 1
Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
18
If someone claimed
there were 228 perforations
in a sheet,
how could you check?
Differences
19
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
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’.
THOANs
Think of a number
Add 3
Multiply by 2
Subtract your first number
Subtract 6
You have your starting
20 number
7
7+3
2x7 + 6
2x7 + 6 – 7
2x7 – 7
7
+3
2x
2x
2x
+6
+6–
–
Ped
Doms
Concepts
 Name
–
–
–
–
–
some concepts that students struggle with
Eg perimeter & area;
slope-gradient;
annuity (?)
Multiplicative reasoning
Algebraic reasoning
 Construct
an example
– Now what can vary and still that remains an example?
Dimensions-of-possible-variation; Range-ofpermissible-change
21
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
variations can you produce?
 What conjectured generalisations are being
challenged?
 What generalisations (properties) are being
instantiated?
22
Powers
 Specialising
& Generalising
 Conjecturing
 Imagining
 Ordering
& Convincing
& Expressing
& Classifying
 Distinguishing
 Assenting
23
& Connecting
& Asserting
Teaching Trap
Doing for the learners what they can already do
for themselves
 Teacher Lust:

– desire that the learner learn
– allowing personal excitement to drive behaviour
24
Mathematical Themes
 Doing
& Undoing
 Invariance
 Freedom
& Constraint
 Extending
25
Amidst Change
& Restricting Meaning
Protases
Only awareness is educable
Only behaviour is trainable
Only emotion is harnessable
26
Didactic Tension
The more clearly I indicate
the behaviour sought from learners,
the less likely they are to
generate that behaviour for themselves
27
Pedagogic Domains
 Concepts
– What do examples look like?
What in an example can be varied? (DofPV; RofPCh)
 Topics
Learners constructing examples (Solving as Undoing of
building)
Learners experiencing variation (DofPV, RofPCh)
Learners constructing variations (Doing & Undoing)
 Techniques
(Exercises)
– See above!
– Structured exercises exposing DofPV & RofPCh
 Tasks
– Varying DofPV; exposing RofPCh
28
Variation
 Object(s)
of Learning
– Key understandings; Awarenesses
– Intended; Perceived-afforded; Enacted
– Encountering structured variation
Varying to enrich Example Spaces
 Actions
performed
– Tasks  activity  experience
 Reconstruction
& Reflection on Action
(efficiency, effectiveness)
 Use of powers &
Exposure to mathematical themes
– Affective: disposition
 Psyche
– awareness, emotion, behaviour
29
 DofPV
& RofPCh