Task Construction: lessons learned from 25 years of distance support for teachers

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Task Construction lessons learned from 25 years
of distance support for teachers
Promoting Mathematical Thinking
John Mason Nottingham Feb 2012
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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?

Denote translation to the right by a, by Ta What
is Ta followed by Tb? What about Tb followed by
Ta?
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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

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)

I am thinking of a number tell me how to work
out where it ends up

• Where does 2 end up?
• Where does 3 end 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

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?

Denote rotating about the original point p by
Rp What is Rp followed by Rq?
Rotate twice about 0
to see why R1R1 T0 S1and 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

28 15 43

• Add together the products down the columns

20 21 41
43 41 2

• Calculate the difference

• That is the doingWhat is an undoing?

• What other grids will give the answer 2?
• Choose positive numbers so that the difference is
  7

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Differing Sums Products

• Tracking Arithmetic

4x7 5x3
4x5 7x3
4x(75) (57)x3
4x(75) (75)x3
(4-3) x (75)

• So in how many essentially different ways can 2
  be the difference?
• What about 7?

• So in how many essentially different ways can n
  be the difference?

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Patterns with 2
Embedded Practice (Gattegno Hewitt)
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Structured Variation Grids
Tunja
Factoring
Quadratic Double Factors
Sundaram
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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 1/5of 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

• Tasks gt Activity gt Actions gt Experience
• But one thing we dont seem to learn from
  experience
• is that we dont often learn from experience
  alone!
• gt 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?

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Task Design
Post-paration Post-flection
Pre-parationPre-flection
Content (Mathematics)
Reflection
Interactions (as transformative actions)
Tasks
Resources
Activity
When does learning take place?
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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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(No Transcript)
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Activity
Goals, Aims,Desires, Intentions
Tasks(as imagined, enacted,experienced, )
Resources(physical, affective, cognitive,
attentive)
Initial State
Affordances ConstraintsRequirements (Gibson)
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Potential
What builds on it(where it is going)
Most it could be
Mathl Pedcessence
Least it can be
What it builds on (previous experiences)
Affordances ConstraintsRequirements (Gibson)
DirectedPromptedSpontaneous Scaffolding
Fading (Brown et al) ZPD (Vygotsky)
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Thinking Mathematically

• CME
• Do-Talk-Record (SeeSayRecord)
• See-Experience-Master
• ManipulatingGetting-a-sense-ofArtculating
• EnactiveIconicSymbolic
• DirectedPromptedSpontaneous
• Stuck! Use of Mathematical Powers
• Mathematical Themes (and heuristics)
• Inner Outer Tasks

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Frameworks
Enactive Iconic Symbolic
Doing Talking Recording
See Experience Master
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Example

• From NNP project, pattern sequences to be counted
• Stuck with providing first, second, third and
  only later recognising the dependency created
• Unlocking potential
• Universality of the Frame Theorem (Gaussian
  Curvature and Betti Numbers)
• Counting squares, counting sticks,
• Counting weights

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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 OBrien)

• 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

Make up your own a really simple one a
really hard one
Provide two or more sequences in parallel
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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

• Stress 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!

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Perforations
If someone claimedthere were 228 perforationsin
a sheet, how could you check?
How many holes for a sheet of r rows and c
columns of stamps?
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Gasket Sequences
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Attention

• Teahing and Learning is fundamentally about
  attention
• What is available or likely to cometomind 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
  DetailsRecognising RelationshipsPerceiving
  PropertiesReasoning 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 _at_ open.ac.uk mcs.open.ac.uk/jhm3