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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 and misconceptions; consider ways in which we might use these mistakes and misconceptions constructively to promote learning.
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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.
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.
Saira: Fractions and decimals • Confuses decimal and fraction notation.(0.25 = ) • Believes that numbers with more decimal places are smaller in value.(0.625 < 0.5). • Sees as involving the cutting of a cake into 8 parts but ignores the value of the numerator when comparing fractions.
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.
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.
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.
Why do learners make mistakes? • Lapses in concentration. • Hasty reasoning. • Memory overload. • Not noticing important features of a problem. or…through misconceptions based on: • alternative ways of reasoning; • local generalisations from early experience.
Generalisations made by learners • 0.567 > 0.85The more digits, the larger the value. • 3÷6 = 2Always divide the larger number by the smaller. • 0.4 > 0.62The fewer the number of digits after the decimal point, the larger the value. It's like fractions. • 5.62 x 0.65 > 5.62Multiplication always makes numbers bigger.
Generalisations made by learners • 1 litre costs £2.60; 4.2 litres cost £2.60 x 4.2;0.22 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.
Generalisations made by learners • If x + 4 < 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.
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.
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.”
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.
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?
