1. 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. Representations of a half Q 8 from Sats paper

5. Repesentations of a half Misconception: –that two halves of something need to look exactly the same Causes: –Lack of variety in representations to which pupils have been exposed –Limiting of fractions to ‘pizza’ representations –Lack of experience of fractions of quantities

6. Parallel and perpendicular Q 17 from Sats paper

7. Parallel and perpendicular Believes that parallel means two lines at 90 degrees to one another Causes –Introduction of parallel and perpendicular at the same time –Lack of variety of examples and representations with parallel usually being shown parallel to the edge of the page

8. Co-ordinates Q 19 from Sats paper

9. Co-ordinates Misconception –The label of the axis refers to the value of the label –Muddling x and y co-ordinates in ordered pairs Causes –Lack of clarity in examples used –Confusing numbers and labels

10. Probability Q 23 from Sats paper

11. Probability and properties of numbers Misconception –Confusion about certain, possible and impossible when related to events –Lack of understanding of the general rules of number such as the sum of two odd numbers is and even number Causes –Lack of experiences of assessing probabilities of ordinary events –Lack of experiences of generalising about numbers

12. 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.

13. Generalisations made by learners 0.567 > 0.85 The more digits, the larger the value. 3÷6 = 2 Always divide the larger number by the smaller. 0.4 > 0.62 The fewer the number of digits after the decimal point, the larger the value. It's like fractions. 5.62 x 0.65 > 5.62 Multiplication always makes numbers bigger.

14. Generalisations made by learners Area of rectangle ≠ area of triangle If you dissect a shape and rearrange the pieces, you change the area.

15. 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. –All numbers can be written as proper or improper fractions. –The order in which you multiply does not matter.

16. 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.

17. Look at suggestions from Maths4Life booklets 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?

18. Follow up work Identify a misconception that you have seen your children show or pick a common one Try using some ideas from the resources to address the misconception Be prepared to share your experiences with the rest of the group –Note what children say and how they react –Bring some examples of their work before and after your intervention