Getting Students to Take Initiative when Learning

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Getting Students to Take InitiativewhenLearning
Doing Mathematics
John Mason Oslo Jan 2009
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Do You Know Any Students Who

• do the minimum to get through a lesson?
• wait to be told what to do?
• finish quickly and then mess around?
• are content to assent to what is said and done,
  but rarely assert mathematically?

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When Do You Take Initiative?

• When you are interested, engaged, involved
• When you have a stake in getting something
  finished
• When you are surprised or intrigued
• When something is or becomes real for you

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Fraction Construction

• Write down two numbers that differ by 3/7
• and another pair
• and another pair
• And another pair that make the difference as
  obscure as possible

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Decimal Construction

• Write down
• A decimal number between 3 and 4
• that does not use the digit 5
• and that does use the digit 7
• and that is as close to 7/2 as possible

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Line Construction

• Write down the equation of a straight line that
  passes through the point (1,0)
• and another
• and another
• Write them all down!

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More Line Constructions

• Sketch the graph of two straight lines whose
• x-intercepts differ by 2 and another
• y-intercepts differ by 2 and another
• slopes differ by 2 and another
• Sketch the graph of two straight lines meeting
  all three constraints

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Max-Min

• In a rectangular array of numbers, calculate
• The maximum value in each row, and then the
  minimum of these
• The minimum in each column and then the maximum
  of these
• How do these relate to each other?
• What about interchanging rows and columns?

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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 2/5 of 5/3 of something 3/5
  of 5/3 of something
• Something which is 2/5 of 5/3 of something
• What part is it of your whole?
• Something which is 1/3 of 3/5 of something
• What part is it of your whole?
• Something which is 5/3 of 3/5 of something
• Something which is 2/3 of 3/2 of something

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Getting Others To See
1/4 1/5 1/20
1/4 1/20 1/5
1/5 1/20 1/4
1/a 1/b ?
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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 dividing by 3/2?
• What undoes multiplying by 3/2? Now do it
  piecemeal!
• What undoes dividing into 12?

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Remainder Construction

• Write down a number that leaves a remainder of 1
  on dividing by 3
• and another
• and another
• Write down two, multiply them together, and find
  the remainder on dividing by 3

What is special about the 3?
What is special about the 1?
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Distributed Examples

• Write down a number that leaves a remainder of 1
  when divided by 7
• Now write down one which is easy to see leaves a
  remainder of 1 on dividing by 7
• Multiply by your number by the number of someone
  sitting beside you
• Does the product have the same property?

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Remainders of the Day

• Write down a number which when you subtract 1 is
  divisible by 2
• and when you subtract 1 from that quotient, the
  result is divisible by 3
• and when you subtract 1 from that quotient the
  result is divisible by 4
• Why must any such number be divisible by 3?

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Remainders of the Day

• Write down a number which is 1 more than a
  multiple of 2
• and which is 2 more than a multiple of 3
• and which is 3 more than a multiple of 4

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Making Sense of the World
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More Or Less Whole Part
? of 35 is 21
Part
Whole
3/4 of 40 is 30
3/5 of 35 is 21
6/7 of 35 is 30
4/5 of 30 is 24
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DifferenceDivisions
4 2 4 2
How does this fit in?
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Differences
AnticipatingGeneralising
Rehearsing
Checking
Organising
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Up Down Sums
1 3 5 3 1
3 x 4 1
22 32


See generalitythrough aparticular
Generalise!
1 3 (2n1) 3 1

n (2n2) 1
(n1)2 n2

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Kites
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Reacting Responding

• Do you know any students who jump at the first
  idea that comes to mind?
• Do you know any students who react negatively
  when challenged by something unfamiliar?
• Assenting gt Asserting
• conjecturing, trying, reasoning,

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When Do You Take Initiative?

• When you are interested, engaged, involved
• When you have a stake in getting something
  finished
• When you are surprised or intrigued
• When something is or becomes real for you

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When is Real-ity

• Sense of purpose (engagement)
• Sense of utility (present or future)
• Use of own powers

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Strategies

• Learners Making Significant Mathematical Choices
• Learner Constructed Examples of Mathematical
  Objects
• Learner Constructed Examples of Exercises
• Learners deciding which exercises need doing
• Distributed example construction

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ZPD

• When students are ready to shift from
• Reacting to cues and triggers
• to initiating actions for themselves
• Scaffolding Fading
• Directed, prompted, spontaneoususe of
  strategies, powers, concepts, techniques

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Task Design

• expert awareness is converted into instruction
  in behaviour
• transposition didactique

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Task Activity

• A task is what an author publishes, what a
  teacher intends, what learners undertake to
  attempt.
• These are often very different
• What happens is activity
• Teaching happens in the interaction occasioned
  by activity

Teaching takes place in timeLearning takes place
over time
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Tasks

• Learners encounter variation
• Learners build up example spaces
• Learners rehearse other techniques while
  exploring
• Learners encounter disturbances and surprises

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Purpose Utility
Ainley Pratt

• whose purposes?
• whose utility?
• mathematics is useful
• planning from objectives leads to dull
  lessonsplanning from tasks may mean avoidance
  of mathematical ideas, thinking, etc.
• Issue how much do you tell learner in advance?
• Inner and outer aspects of tasks

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Teacher Aims and Goals

• students to
• make use of their powers
• experience mathematical themes
• encounter mathematical concepts, topics
• develop facility and fluency with techniques
• use technical terms to express their conjectures
  and understandings

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Learner Aims Goals

• As learners, to
• do as little as necessary to complete tasks
  adequately
• attract as little (or as much) attention as
  possible
• be stimulated, inspired, engaged

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Task Dimensions

• How initiated
• in silence through phenomenon (shown or
  imagined)
• How sustained
• Group discussion distributed tasks individual
• How concluded
• How structured
• Simple to complex Particular to general
• Complex simplified General specialised

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MGA
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Reflection

• What did you notice happening for you
  mathematically?
• What might you be able to use in an upcoming
  lesson?
• Imagine yourself in the future, using or
  developing or exploring something you have
  experienced this morning!

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More Resources

• Questions Prompts for Mathematical Thinking
  (ATM Derby primary secondary versions)
• Thinkers (ATM Derby)
• Mathematics as a Constructive Activity (Erlbaum)
• Designing Using Mathematical Tasks (Tarquin)
• http //mcs.open.ac.uk/jhm3
• j.h.mason _at_ open.ac.uk