PD2: Learning from mistakes and misconceptions

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PD2 Learning from mistakes and misconceptions
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Aims of the session

• This session is intended to help us to
• reflect on the nature and causes of learners
  mistakes and misconceptions
• consider ways in which we might use these
  mistakes and misconceptions constructively to
  promote learning.

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Assessing learners responses

• Look at the (genuine) examples of learners' work.
• Use the grid sheet to write a few lines
  summarising
• the nature of the errors that have been made by
  each learner
• the thinking that may have led to these errors.
• Discuss your ideas with the whole group.

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Saira Fractions and decimals
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Saira Fractions and decimals
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Saira Fractions and decimals
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Saira Fractions and decimals

• Confuses decimal and fraction notation.(0.25
  )
• Believes that numbers with more decimal places
  are smaller in value.(0.625 lt 0.5).
• Sees as involving the cutting of a cake into
  8 parts but ignores the value of the numerator
  when comparing fractions.

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Damien Multiplication and division
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Damien Multiplication and division
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Damien Multiplication and division
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Damien Multiplication and division

• Believes that one must always divide the larger
  number by the smaller (4 20 5).
• Appears to think that
• division 'makes numbers smaller
• division of a number by a small quantity reduces
  that number by a small quantity.

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Julia Perimeter and area
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Julia Perimeter and area
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Julia Perimeter and area
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Julia Perimeter and area

• Has difficulty explaining the concept of volume,
  which she describes as the 'whole shape.'
• Believes that perimeter is conserved when a shape
  is cut up and reassembled.
• Believes that there is a relationship between the
  area and perimeter of a shape.

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Jasbinder Algebraic notation
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Jasbinder Algebraic notation
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Jasbinder Algebraic notation
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Jasbinder Algebraic notation

• Does not recognise that letters represent
  variables. Particular values are always
  substituted.
• Shows reluctance to leave operations in answers.
• Does not recognise precedence of operations
  multiplication precedes addition squaring
  precedes multiplication.
• Interprets '' as 'makes ie a signal to evaluate
  what has gone before.

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Why do learners make mistakes?

• Lapses in concentration.
• Hasty reasoning.
• Memory overload.
• Not noticing important features of a problem.
• orthrough misconceptions based on
• alternative ways of reasoning
• local generalisations from early experience.

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Generalisations made by learners

• 0.567 gt 0.85 The more digits, the larger the
  value.
• 36 2 Always divide the larger number by the
  smaller.
• 0.4 gt 0.62The fewer the number of digits after
  the decimal point, the larger the value. It's
  like fractions.
• 5.62 x 0.65 gt 5.62Multiplication always makes
  numbers bigger.

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Generalisations made by learners

• 1 litre costs 2.60 4.2 litres cost 2.60 x
  4.20.22 litres cost 2.60 0.22.
• If you change the numbers, you change the
  operation.
• Area of rectangle ? area of triangleIf you
  dissect a shape and
• rearrange the pieces, you
• change the area.

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Generalisations made by learners

• If x 4 lt 10, then x 5.Letters represent
  particular numbers.
• 3 4 7 2 9 5 14.Equals' means
  'makes'.
• In three rolls of a die, it is harder to get 6,
  6, 6 than 2, 4, 6. Special outcomes are less
  likely than more representative outcomes.

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Some more limited generalisations

• What other generalisations are only true in
  limited contexts?
• In what contexts do the following generalisations
  work?
• If I subtract something from 12, the answer will
  be smaller than 12.
• The square root of a number is smaller than the
  number.
• All numbers can be written as proper or improper
  fractions.
• The order in which you multiply does not matter.
• You can differentiate any function.
• You can integrate any function.

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What do we do with mistakes and misconceptions?

• Avoid them whenever possible?
• "If I warn learners about the misconceptions
  as I teach, they are less likely to happen.
  Prevention is better than cure.
• Use them as learning opportunities?"I actively
  encourage learners to make mistakes and to learn
  from them.

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Some principles to consider

• Encourage learners to explore misconceptions
  through discussion.
• Focus discussion on known difficulties and
  challenging questions.
• Encourage a variety of viewpoints and
  interpretations to emerge.
• Ask questions that create a tension or cognitive
  conflict' that needs to be resolved.
• Provide meaningful feedback.
• Provide opportunities for developing new ideas
  and concepts, and for consolidation.

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Look at a session from the pack

• What major mathematical concepts are involved in
  the activity?
• What common mistakes and misconceptions will be
  revealed by the activity?
• How does the activity
• encourage a variety of viewpoints and
  interpretations to emerge?
• create tensions or 'conflicts' that need to be
  resolved?
• provide meaningful feedback?
• provide opportunities for developing new ideas?