Designing and Using Tasks
Effectively
for Conceptual Development
Anne Watson
John Mason
Agder College
Kristiansand Norway
September 2006
1
Outline
Exercise
as object
Using structured variation
How many different ways …?
Construct a ….
2
Exercise as object
17 – 9 =
27 – 9 =
37 – 9 =
47 – 9 =
…
3
Learning from experience
Patterns
in layout
Patterns of digits
Familiarity
Generality
Going beyond mere answers
4
Conceptual development
 Tasks,
and the ways they are presented, mediate
formal mathematical ideas for learners
– Multiple examples
– Personal images
– Natural/scientific concepts
– Intuitive/formal understanding
– Further experience
5
Exercise as object
…
( x – 2 ) ( x + 1 ) = x2 - x - 2
( x – 3 ) ( x + 1 ) = x2 - 2x - 3
( x – 4 ) ( x + 1 ) = x2 - 3x - 4
…
6
Reflections
Going
beyond mere answers
Reflecting across the grain
Quasi-physical and visual repetition
How do learners know what to focus on?
7
Stars
8
Structured variation
Dimensions
of possible variation
Range of permissible change
9
Find the gradient between each pair
of points
(4,
3) and (8, 12)
(4, 3) and (7, 12)
(4, 3) and (6, 12)
(4, 3) and (5, 12)
10
(4, 3) and (4, 12)
(4, 3) and (3, 12)
(4, 3) and (2, 12)
(4, 3) and (1, 12)
Task elements
Quasi-physical,
visual, notational patterns
Dimensions of possible variation (DofPV)
Range of permissible change (RofPCh)
11
How many different ways …
… can a unit fraction be written as the
difference of unit fractions?
e.g.
12
1  1 1
4 3 12
 1 1
2 4
Unit fraction differences
1 = 1 – 1
2
1
2
1 = 1 – 1
3
2
6
1 = 1 – 1 = 1 – 1
4
3
12
2
4
1 = 1 – 1
5
4
20
13
1 = 1 – 1
7
6
42
1 = 1 – 1 = 1 – 1
8
7
56
6
24
= 1 – 1
4
8
Anticipating
Generalising
Rehearsing
Checking
1
1
1
1
1
1
1
1
1
=
–
=
–
=
–
=
–
6
5
30
2
3
3
6
4
12
Organising
Construct a …
QuickTime™
QuickTime™ and
and aa
TIFF
(Uncompressed)
decompressor
TIFF (Uncompressed) decompressor
are needed
needed to
to see
see this
this picture.
are
picture.
14
… pentagon
of area 20
by moving
only blue pegs
to other peg
positions
Sources of data;
Opportunities for learning
Who creates the data
What does the learner do?
Tasks:
27 – 9 =
(x – 2)(x + 1) = …
Tasks:
Unit fractions
Pentagon construction
Data: from teacher
Data: from learners
DofPV: constrained by task
DofPV: constrained by task
RofCh: suggested by teacher
RofCh: comes from learners
Scope and freedom of
what learner might think
about
How prompt learners to go
How prompt learners to go
beyond mere answers?
beyond mere construction?
15
Follow-up
16
Some Tools
for task design and use
 Structured
Variation
 How many different ways …?
 Construct a …
 Treating an exercise as a mathematical object
DofPV, who decides?
RofPC, who decides?
Openness of input
Openness of learning
17