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PD2: Learning from mistakes and misconceptions

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PD2: Learning from mistakes and misconceptions Aims of the session This session is intended to help us to: reflect on the nature and causes of learners mistakes ... – PowerPoint PPT presentation

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Title: PD2: Learning from mistakes and misconceptions


1
PD2 Learning from mistakes and misconceptions
2
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.

3
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.

4
Saira Fractions and decimals
5
Saira Fractions and decimals
6
Saira Fractions and decimals
7
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.

8
Damien Multiplication and division
9
Damien Multiplication and division
10
Damien Multiplication and division
11
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.

12
Julia Perimeter and area
13
Julia Perimeter and area
14
Julia Perimeter and area
15
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.

16
Jasbinder Algebraic notation
17
Jasbinder Algebraic notation
18
Jasbinder Algebraic notation
19
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.

20
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.

21
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.

22
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.

23
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.

24
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.

25
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.

26
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.

27
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?
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