A useful presentation on misconceptions in mathematics for school teachers.

1. Misconceptions in mathematics and diagnostic teaching

2. Introduction Section 1

5. 1) Teaching is more effective when misconceptions are identified, challenged, and ameliorated. 2) Pupils face internal cognitive distress when some external idea, process, or rule conflicts with their existing mental schema. We accept the research evidence which suggests that the resolutions of these cognitive conflicts through discussion leads to effective learning.

6. The difference between a mistake and a misconception in mathematics Section 2

9. The pupil has shown that s/he i) understands the method to find the solution ii) knows how to open brackets but carelessly makes an error in the 2 nd bracket expansion: adding rather than multiplying. 1+5= 6 instead of 1 ×5=5 10-3=7 instead of 10 ×-3=-30

10. The pupil understands an algorithm but there is a computational error due to carelessness. Instead of using a calculator, as allowed in the examination, the pupil has, without due care, used mental methods. Source: QCA

11. A common pupil misconception: The pupil has transferred the algorithm for multiplying fractions to adding fractions.

12. The pupil has misapplied place value to interpret the conjunction of a number and a letter in algebra. If x = 5, 2 x has been interpreted to be 25. This is frequently observed. Source of pupil’s work: QCA

13. The pupil has a good understanding of addition of whole numbers but has misapplied it with respect to addition of decimal numbers. This is misapplication is observed repeatedly. Source : QCA

15. Misconception? Source: AQA GCSE Higher level text book, Heinemann

19. Some misconceptions in arithmetic Section 3

20. This a common misconception with pupils. Working with concrete objects in primary school the pupil is aware that if you have 2 apples you cannot possibly take 6 apples away. Thus when doing the subtraction in the tens column they presume that they must remove 2 from 6. This is sometimes called an inversion error .

21. Here the pupil has a correct interpretation and representation for 1/10: it is indeed 0.10. However the pupil here probably has misapplied the convention for fractions: 6 ½ means 6 + one half. So the pupil views 6 tenths to mean 6 + a tenth. Source: DfES

22. When multiplying fractions the numerators are multiplied as are the denominators. Here the pupil has misapplied the rule to the addition of fractions. Of course, the theory behind adding fractions is difficult. A teaching strategy is “ Can you add 2 Euros and 3 Pounds?” After getting agreement that this is only possible if the denominations are the same, the teacher can demonstrate that fractions can only be added if the denominators are the same.

23. There is (flawed) logic in this pupil’s answer. The fractions have different denominators. i) The pupil has the misconception that the larger the denominator the smaller the fraction . There is no acknowledgment of the role of the numerator. ii) The pupil groups them according to denominator then orders them in these 4 sets: note that 3/8 and 5/8 are correctly placed relative to each other. Source: DfES

24. Whilst the pupils answer looks like a mistake there may be an underlying misconception: that rounding is associative. The pupil may have rounded 15,473 to the nearest 10 first to obtain 15,470 Then may have rounded 15,470 to the nearest 100 to obtain 15, 500. Finally may have rounded 15,500 to the nearest 1000 to obtain 16, 000. Q. What is 15,473 to the nearest 1000? A. 16,000

27. Some misconceptions in algebra Section 4

28. The pupil here has misapplied the notion of letters as unknown variables. Here x, y, and n are indeed unknown variables – the pupil decides that, as such, she or he can make them each equal to a convenient number.

31. In KS3 and KS4, pupils are introduced to the algebraic analogue of the distributive law of arithmetic. For example, 2( a + b ) = 2 a + 2 b Then there follows the risk of over-generalising the rule to operations that are not distributive.

33. Some misconceptions in geometry Section 5

34. This misconception is frequently held by pupils as there is perceptual illusion between a larger turn and a larger space between the two lines making the angle. Despite the fact that the 45 0 angles are drawn on squared paper, the larger space of angle Y can lead a pupil to making this judgement. Q. Which angle is bigger? X Y A. Angle Y is bigger.

35. Pupils often believe that the rules of invariance that apply to algebra also apply to geometrical shapes. So there must be equality in all respects when A becomes B. Thus leading to misconception that the perimeters are the same.

36. Clearly x+ 3 and x are different: this lack of equality when something is taken away from an algebraic term can lead to pupils having this misconception about perimeters.

38. The concept of 2 dimensions can lead to conflicts with rules in the linear domains of number and linear equations. Multiplication by a factor k leads to all numbers and variables in the combination being scaled by the same factor k: E.g. k  (2 x+ 3) = 2 kx+ 3 k In the example k = 2 and the area is multiplied by this factor 2. This misconception is due to lack of awareness of the quadratic nature of area. Q 1. 3cm A 6cm B The two triangles are similar. The area of triangle A is 8 cm 2. Find the area of triangle B. A. 16 cm 2

39. Diagnostic teaching Section 6

44. Source : Swann, M : Gaining diagnostic teaching skills: helping students learn from mistakes and misconceptions , Shell Centre publications “ Traditionally, the teacher with the textbook explains and demonstrates, while the students imitate; if the student makes mistakes the teacher explains again. This procedure is not effective in preventing ... misconceptions or in removing [them]. Diagnostic teaching ..... depends on the student taking much more responsibility for their own understanding , being willing and able to articulate their own lines of thought and to discuss them in the classroom”.

45. M. Swann, Improving Learning in Mathematics, DFES

50. It is found that some pupils have the misconception that the larger the number of decimal places the smaller the number. Hence the pupil will group this set according to whether they have 4, 3, 2 or 1 digit and then try to order them set-wise. One way to contrast or challenge this misconception is to get agreement of the class via discussion that the order will remain the same if the numbers are all multiplied by 1000 (this being the multiplier that will make each decimal a whole number). This gives: 6250 2500 3753 1250 5000 Ordered: 1250 2500 3753 5000 6250 Decimal order: 0.125, 0.25, 0.3753, 0.5, 0.625 Source: DfES

51. After discussion with a pupil holding a cancelling misconception exhibited opposite you could find that it is based on a misapplication of simplifying ratio (the x coefficients ate in the ratio 1:3). One way to contrast or challenge this is to specialise the example with numbers – say with x = 2 – and show that the resulting equality is incorrect.

52. This pupil likely holds the misconception that a larger area implies a larger perimeter …probably based on the ‘naturalness’ of taking away. One way to contrast or challenge this misconception that a larger area implies a larger perimeter is to give the pupil examples where it is not true. E.g. 2  2 square and a 1  3 rectangle. Alternatively show that any staircase D made out of rectangle C has perimeter equal to C by a counting exercise. Source: DfES

53. Importance of dealing with misconceptions 1) Teaching is more effective when misconceptions are identified, challenged, and ameliorated. 2) Pupils face internal cognitive distress when some external idea, process, or rule conflicts with their existing mental schema. 3) Research evidence suggests that the resolutions of these cognitive conflicts through discussion leads to effective learning.

54. Dealing with misconceptions: Diagnosis: Get pupils to explain how they came to their answers or rules. Amelioration : If there is a misconception challenge it or contrast it with the faithful conception.