Transcript Tasks Nottingham JM

Promoting Mathematical Thinking
Task Construction:
lessons learned from 25 years of
distance support for teachers
John Mason
Nottingham
Feb 2012
The Open University
Maths Dept
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University of Oxford
Dept of Education
Outline
 Some
Tasks
 OU Frameworks
– MGA, DTR, Stuck, EIS, …
– APC or ORA:
Own experience,
Reflection on parallels,
Apply to classroom
 Systematics Frameworks
 What makes a task ‘rich’?
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Number Line Translations…
 Imagine
a number line with the integers marked on it
 Imagine a copy of the number line sitting on top of it
 Translate the copy line to the right by 3
I am thinking of a number …
tell me how to work out where
it ends up
– Where does 7 end up?
– Where does –2 end up?
-7 -6 -5 -4 -3 -2 -1 0
1
2
3 4
Denote translation to the right by a, by Ta
What is Ta followed by Tb?
What about Tb followed by Ta?
3
5
6
7 8
Number Line Scaling…
 Imagine
a number line with the integers marked on it
 Imagine a copy of the number line sitting on top of it
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Denote scaling from 0 by a factor of s by Ss
I am thinking of a number …
tell me how to work out where it ends up
What is Sa followed by Sb?
Denote scaling from p by a factor of s by Sp,s
What is Sp,s in terms of T and Ss?
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Number Line Scaling…
 Imagine
a number line with the integers marked on it
 Imagine a copy of the number line sitting on top of it
 Scale the number line by a factor of 3
– (keeping 0 fixed)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
– Where does 2 end up?
– Where does –3 end up?
I am thinking of a number …
tell me how to work out where
it ends up
Denote scaling from 0 by a factor of s by Ss
What is Sa followed by Sb?
Denote scaling from p by a factor of s by Sp,s
What is Sp,s in terms of T and Ss?
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Number Line Rotations…
 Imagine
a number line with the integers marked on it
 Imagine a copy of the number line sitting on top of it
 Rotate the copy through 180° about the point 3
-7 -6 -5 -4 -3 -2
-1 0
– Where does 7 end up?
– Where does -2 end up?
1
2
3 4
5
6
7 8
I am thinking of a number …
tell me how to work out where
it ends up
Denote rotating about the original point p by Rp
What is Rp followed by Rq?
Rotate twice about 0 …
… to see why R–1R–1 = T0 = S1
and so (-1) x (-1) = 1
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Diamond Multiplication
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Differing Sums of Products
Write
down four numbers in a
2 by 2 grid
Add together the products
along the rows
4 7
5 3
28 + 15 = 43
Add
together the products down
20 + 21 = 41
the columns
43 – 41 = 2
Calculate the difference
is the ‘doing’
What is an undoing?
 That
What
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other grids will give the answer 2?
Choose positive numbers so that the difference
is 7
Differing Sums & Products
 Tracking
Arithmetic
4x7 + 5x3
4 7
5 3
4x5 + 7x3
4x(7–5) + (5–7)x3
= 4x(7–5) – (7–5)x3
= (4-3) x (7–5)
 So
in how many essentially different ways can 2 be
the difference?
 What about 7?
So
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in how many essentially different ways can n be
the difference?
Patterns with 2
Embedded Practice
(Gattegno & Hewitt)
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Structured Variation Grids
Tunja
Factoring
Sundaram
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Quadratic Double
Factors
Put your hand up when you can see …
 Something
that is 3/5 of something else
 Something that is 2/5 of something else
 Something that is 2/3 of something else
 Something that is 5/3 of something else
…
Something that is 1/4 – 1/5
of something else
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Remainders
 What
is the remainder on dividing 5 by 3?
 What is the remainder on dividing -5 by 3?
What question
am I going to
ask next?
 What
is the remainder on dividing 5 by -3?
 What is the remainder on dividing -5 by -3?
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Task Purposes
 To
introduce or extend contact with concepts
 To highlight awareness of human powers used
mathematically
 To focus attention on mathematical themes
 To sharpen awareness of
– study strategies
– problem solving strategies (heuristics)
– learning how to learn mathematics
– evaluating own progress
– exam technique
Purpose for students
Potential Utility
(Ainley & Pratt)
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Learning from Tasks
–> Activity –> Actions –> Experience
 But one thing we don’t seem to learn from experience …
– is that we don’t often learn from experience alone!
 –> withdraw from action and reflect upon it
– What was striking about the activity?
– What was effective and what ineffective?
– What like to have come-to-mind in the future?
 Personal propensities & dispositions?
 Habitual behaviour and desired behaviour?
 Fresh or freshened awarenesses & realisations?
 Tasks
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Task Design
Pre-paration
Pre-flection
Post-paration
Post-flection
Content
(Mathematics)
Reflection
Interactions
(as transformative actions)
Tasks
Resources
Activity
When does learning take place?
In sleep!!!
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Slogans
A lesson without opportunity for learners …
to generalise mathematically
… is not a mathematics lesson!
A lesson without opportunity for learners …
to make and modify conjectures;
to construct a narrative about what they have been doing;
to use and develop their own powers;
to encounter pervasive mathematical themes
is not an effective mathematics lesson
Trying to do for learners
only what they cannot yet do for themselves
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Modes of interaction
Expounding
Explaining
Exploring
Examining
Exercising
Expressing
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Teacher
Student
Content
Expounding
Teacher
Content
Student
Explaining
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Student
Teacher
Content
Exploring
Student
Content
Teacher
Examining
Content
Student
Teacher
Exercising
Content
Teacher
Student
Expressing
Activity
Goals, Aims,
Desires,
Intentions
Tasks
(as imagined, enacted,
experienced, …)
Resources:
(physical, affective,
cognitive, attentive)
Initial State
Affordances– Constraints–Requirements
(Gibson)
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Potential
Most it
could be
What builds on it
(where it is going)
Least it
can be
What it builds on
(previous experiences)
Math’l & Ped’c
essence
Affordances– Constraints–Requirements
(Gibson)
Directed–Prompted–Spontaneous
Scaffolding & Fading (Brown et al)
ZPD (Vygotsky)
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Thinking Mathematically
 CME
–
–
–
–
–
–
–
–
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Do-Talk-Record (See–Say–Record)
See-Experience-Master
Manipulating–Getting-a-sense-of–Artculating
Enactive–Iconic–Symbolic
Directed–Prompted–Spontaneous
Stuck!: Use of Mathematical Powers
Mathematical Themes (and heuristics)
Inner & Outer Tasks
Frameworks
Enactive– Iconic– Symbolic
Doing – Talking – Recording
See– Experience– Master
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Example: Extending Mathematical
Sequences Mathematically
 “What
is the next term …?” only makes sense when ...
 Mathmematical guarantee of uniqueness
– Geometrical or other construction source
– Some other constraint
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Painted Wheel (Tom O’Brien)
…
Someone has made a simple pattern of coloured squares, and
then repeated it at least once more
 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

…
Provide two or more
sequences in parallel
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Make up your own:
a really simple one
a really hard one
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?
What shapes will have the
same Border Numbers?
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Extending Mathemtical Sequences
in Thinking Mathematically and later on ‘specifying
the growth mechanism before trying to count things’
 Uniquely Extendable Sequences Theorem
 Instance of general topological theorem (Betti numbers)
 Attempts in two Dimensions!
 Stress
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Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
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If someone claimed
there were 228 perforations
in a sheet,
how could you check?
Gasket Sequences
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Attention
 Teahing
and Learning is fundamentally about attention:
– What is available or likely to come–to–mind when needed
– What is available to be learned?
variation
–
–
–
–
Use of powers
Use of themes
Use of resources (physical, mental, virtual)
Structure of attention
Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of agreed properties
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Follow-Up









Designing & Using Mathematical Tasks (Tarquin/QED)
Thinking Mathematically (Pearson)
Developing Thinking in Algebra, Geometry, Statistics (Sage)
Fundamental Constructs in Mathematics Education
(RoutledgeFalmer)
Mathematics Teaching Practice: a guide for university and
college lecturers (Horwood Publishing)
Mathematics as a Constructive Activity (Erlbaum)
Questions & Prompts for Mathematical Thinking (ATM)
Thinkers (ATM)
Learning & Doing Mathematics (Tarquin)
j.h.mason @ open.ac.uk
mcs.open.ac.uk/jhm3
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