ReThinking Stochastics

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Re-Thinking Stochastics
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• Professor Dave Pratt
• Institute of Education, University of London

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How would you test this out?
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To find the probability, I could

• Not use a classical approach
• I can not calculate the probability a priori as
  one could calculate the probability of throwing
  two sixes with a pair of dice without doing any
  experiments.
• Use a frequentist approach
• If I swim in the sea 100 times, how often do I
  get attacked by a shark?
• Use a subjectivist approach
• Before swimming, I thought that the probability
  of an attack is 0.05. But now I have swum safely,
  I think the probability has gone down to 0.048.

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A task for you

• Linda is 31 years old, single, outspoken and very
  bright. She majored in philosophy. As a student
  she was deeply concerned with issues of
  discrimination and social justice, and also
  participated in anti-nuclear demonstrations.
• Which of the following two statements about Linda
  is the more probable
• (i) Linda is a bank teller, or
• (ii) Linda is a bank teller who is active in the
  feminist movement.

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Kahneman and Tverskys Heuristics

• The availability heuristic
• e.g. Perhaps because of the publicity, we
  overestimate risks of flying, terrorism, being
  attacked in our homes (or indeed being attacked
  by a shark!)
• The representativeness heurisitic
• E.g. The gamblers fallacy
• Heuristics are liable to bias

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Konolds Outcome Approach

• Focussing on results rather than on strategy
• I won the card game. (It does not matter that I
  was lucky and usually would have lost playing it
  that way.)
• I safely crossed the road. (It does not matter
  that I ignored the nearby zebra crossing.)
• I survived the swim. (It does not matter the
  probability of being attacked by a shark was
  high.)

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Le Coutres Equiprobability Bias

• We tend to assume things are equally likely.
• Who knows? Its just chance
• I either get attacked by a shark or I do not. So
  the probability is 50/50.

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How might we respond to this?

• Classical statistics has depended upon strong
  understanding of probability? Can we simply avoid
  probability? (See Part One of this seminar.)
• Is it possible to help students gain a better
  appreciation of probability? (See Part Two of
  this seminar.)

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Part One

• Classical statistics has depended upon strong
  understanding of probability? Can we simply avoid
  probability?
• One response has been Exploratory Data Analysis.

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Exploratory Data Analysis (EDA)

• Perhaps the main idea behind EDA was to exploit
  the affordances of digital technology to enable
  students to draw inferences on the basis of
  organising and reorganising data without recourse
  to a strong appreciation of probabilistic
  concepts and ideas such as p-values.

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Software

• Two pieces of software stand out and we will look
  at each briefly.
• Fathom and its younger cousin, Tinkerplots.

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Tinkerplots

• Aimed at primary school children.
• Designed to be operated at an intuitive level.
• Designed to avoid over-simplification by making
  the complexity of graphing data manageable.

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Quoting Konold

• It is all too common in classrooms to find
  students succeeding at learning the small bits
  they are fed, but never coming to see the big
  picture nor experiencing the excitement of the
  enterprise Just as I have watched in frustration
  as students in traditional classrooms spend
  months learning to make simple graphs of single
  attributes and never get to a question they care
  about, I now have had the experience of watching
  students work through teacher-made worksheets to
  learn TinkerPlots operations one at a time...
  After class, I spoke with the teacher who had
  created the worksheets and gently offered the
  observation that students could discover and
  learn to use many of the commands he was drilling
  them on as a normal part of pursuing a question.
  He informed me that they didnt have time in
  their schedule to have students playing around.

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Fathom

• Aimed at post-11 years.
• It provides for dynamic exploratory data
  analysis.
• Far more powerful than Tinkerplots but
  conceptually more difficult.

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Fathom
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Part Two

• Is it possible to help students gain a better
  appreciation of probability?
• One response has been specailly designed software
  to build on what students DO know.

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What is the source of such thinking?

• Genetic Fallibility?
• Are our brains hardwired wrongly?
• Inexperience?
• Do we not gain the appropriate experiences
  through our everyday or school-based activity?

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Different perspectives

• Psychologists have tended to catalogue the many
  misconceptions people have about chance.
• More recently, educationalists are studying what
  students CAN do under more favourable conditions,
  and in some cases exploiting technology to this
  end.

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Why is this important?

• What experiences should we give students to
  enable them to construct more sophisticated
  intuitions for chance?
• People might otherwise
• Over-protect their children
• Be scared to travel
• Fail to interpret properly information about
  weather or investments
• Be ineffective in games and sports
• Be poor jurors!

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Piagets Genetic Epistemology

• Random mixtures are special
• They can not be explained in terms of the
  deterministic
• We each invent probability as a means of handling
  randomness
• This is a very late development

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Fischbeins Intuitions

• Even pre-school children have intuitions related
  to probability
• Schools focus too much on the deterministic

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Software

• Two related pieces of software.
• ChanceMaker and its older cousin Basketball.

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Pratt on Childrens Local Resources

• Children around age 11 years regard random
  phenomena as
• Unpredictable
• Irregular
• Uncontrollable
• Fair
• In fact, much as experts do!

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Pratt on Childrens Global Resources

• But children around age 11 years do not
  spontaneously relate to the aggregated patterns
  that appear in long term randomness
• However, it is possible to provide certain types
  of computer-based experience that enables them to
  create situated understandings of long term
  randomness, the so-called Law of Large Number

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ChanceMaker
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New knowledge

• The Large Number resource, N
• the larger the number of trials, the more even
  the pie chart.
• The Distribution resource, D
• the more frequent an outcome in the workings
  box, the larger its sector in the pie chart.
• Co-ordination of N and D
• the more frequent an outcome in the workings
  box, the larger its sector in the pie chart,
  provided the number of trials is large.

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A trace of Anne and Rebeccas webbing
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Motivating ideas for Basketball

• Students must use current knowledge in order to
  make sense of randomness.
• Somehow they must mobilise knowledge of the
  deterministic.
• Distribution has two faces data-centric and
  modelling.
• We would like students to coordinate these two
  faces of distribution.

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Basketball
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Exploring other successful throwing positions

• The students observed variation in the graphs
  created by the manual changing of the parameters
  during the running of the simulation.
• They acted as agents of variation

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Exploring the arrows

• The students were introduced to the error buttons
  and the two arrows appeared either side of the
  handle on the corresponding slider.
• They were trying to explore the effect of moving
  these arrows.

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Arrows as agents

• The students explored the effect of the position
  of the arrows (close together, wide apart,
  symmetrically or asymmetrically around the
  handle).
• They recognised that the throws were being
  chosen randomly from values between the two
  arrows.
• They saw the arrows as agents of the variation.

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Modelling with the arrows

• The students were challenged to simulate more or
  less skilled players.
• When the arrows are closer together, the chance
  of a successful throw is increased.

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Modelling with the arrows

• They had an appreciation of the effect of
  creating an asymmetry in the arrows (skewness).
• They were able to explain the bimodal green
  distribution.
• Can you?

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Conclusions

• We must find ways of helping our students to
  think about the key ideas in intuitive ways,
  building on current knowledge.
• Carefully designed environments can facilitate
  learning about stochastics by
• Allowing the creation of many cases
• Reducing calculation overload
• Allowing students to test out their personal
  conjectures
• Experimenting with new ideas to see their
  explanatory power.