A pupil actively constructs their own mathematical knowledge by interacting new ideas with existing ideas, which can lead to misconceptions. Diagnostic teaching is important as it involves identifying misconceptions, challenging them through discussion to resolve conflicts, and replacing misconceptions with correct understanding. The teacher must understand the source of the misconception to effectively challenge it, and research shows this diagnostic approach promotes better learning compared to simply explaining again.

1. Misconceptions in mathematics and diagnostic teaching

2. Introduction Section 1

3. A pupil does not passively receive knowledge from the environment - it is problematic for knowledge to be transferred faithfully from one person to another . A pupil is an active participant in the construction of his/her own mathematical knowledge. The construction activity involves the reception of new ideas and the interaction of these with the pupils extant ideas.

4. Misconceptions arise frequently because a pupil is an active participant in the construction of his/her own mathematical knowledge via the reception and the interaction of new ideas within the pupils extant ideas. New Concept: Decimal numbers and fractions Extant idea: If I multiply two whole numbers I get a bigger number Accommodation Misconception: If I multiply two fractions I will always get a bigger number

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

7. He knew that He also knew the concept of ordinal numbers. So he (mis) applied this to all unitary fractions to obtain: Source: J.F.Scott, The mathematical work of John Wallis, Chelsea, New York, 1981 The Renaissance mathematician John Wallis used a naïve method of induction as follows . Misconceptions in mathematics are not new; accomplished mathematicians also have misconceptions.

8. In the example below a one-off mistake is made by a pupil due to lack of care or attention to the procedure for decimal addition: Source: Pearson Publishing

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

14. Mistakes : The pupil understands an algorithm but there is a computational error due to carelessness. A mistake is normally a one-off phenomenon. Misconceptions : The pupil has m isleading ideas or misapplies concepts or algorithms . A misconception is frequently observed.

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

16. Multiplication is repeated addition. 3 x 6 = 3 + 3 + 3 + 3 + 3 + 3 What problems can this specialization lead to later?

17. Children construct meaning internally by accommodating new concepts within their existing mental frameworks. Thus, unless there is intervention, there is likelihood that the pupil’s conception may deviate from the intended one. Pupils are known to misapply algorithms and rules in domains where they are inapplicable. A surprisingly large proportion of pupils share the same misconceptions.

18. A pupil says this is the rule to divide two fractions: This is a misconception This is a perfectly good rule

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

25. This misconception is not just related to school children but also with adults. The first scheme is chosen because of the persuasive misconception that 20% on a larger amount in year 2 will yield more money. This misconception is countered by demonstrating that multiplication is commutative: 1.1  1.2 = 1.2  1.1 Q. A bank offers two savings schemes that last 2 years: 10% interest in year 1, followed by 20% in year 2. 20% interest in year 1, followed by 10% in year 2. Which scheme is better? A. Scheme i)

26. Children construct meaning internally by accommodating new concepts within their existing mental frameworks. Thus, unless there is intervention, there is likelihood that the pupil’s conception may deviate from the intended one. Pupils are known to misapply algorithms and rules in domains where they are inapplicable. A surprisingly large proportion of pupils share the same misconceptions.

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.

29. The unknown x is in the ratio 1: 2 and the pupil misapplies simplifying ratio into this domain: s/he divides the coefficients of x by 3 This misconception is also evident in this ‘simplification’. Beware: The pupil could argue that it is correct and with justification! Q 1. Solve 3x + 3 = 6x + 1 3x + 3 = 6x + 1 x + 3 = 2x + 1 2 = x

30. The two arithmetic operations are: First: Multiply by 3 Second : Add 6 The pupil inverts both to obtain the answer: First: Divide by 3 Second : Subtract 6 This misconception can be countered by the ‘socks and shoes’ analogy. Q. I multiply a number x by 3 and add 6 to obtain 27? What is the number x ? 27 divided by 3 = 9 And 9 take away 6 = 3 So x = 3

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.

32. The pupil was initially introduced to quadratic equations by investigating equation such as x 2 - 4 x + 3 = 0 Solvable in this manner: x 2 - 4 x + 3 = 0  ( x – 3)( x – 1) = 0 ........(1)  ( x – 3) = 0 or ( x – 1) = 0 ..........(2) So x = 3 or x = 1. The pupil misapplies the method to quadratic equations not equal to zero. The reasons why (1) leads to (2) needs to be clearly understood to avoid this misconception. Q. Solve x 2 - 4 x + 3 = 12 x 2 - 4 x + 3 = 12 ( x – 3)( x – 1) = 12 ( x – 3) = 12 or ( x – 1) = 12 x = 15 or x = 13

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.

37. Invariably pupils are conditioned by the standard triangle presented to them when the area of a triangle algorithm is presented: one with horizontal base and height ‘upwards’ from the base. The pupil assumes the base must be 13 cm and the height ‘appears’ to naturally be 5 cm, thereby leading to the incorrect answer. Q. Find the area of the right angled triangle 5 cm 12cm 13cm Area = ½ x 5 x 13 = 32.5 sq cm

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

40. A pupil does not passively receive knowledge from the environment - it is not possible for knowledge to be transferred holistically and faithfully from one person to another . A pupil is an active participant in the construction of his/her own mathematical knowledge. The construction activity involves the reception of new ideas and the interaction of these with the pupils extant ideas.

41. Because a pupil is an active participant in the construction of his/her own mathematical knowledge via the reception and the interaction of new ideas with the pupils extant ideas, misconceptions arise frequently. New Concept: Decimal numbers and fractions Extant idea: If I multiply two whole numbers I get a bigger number Accommodation Misconception: If I multiply two fractions I will always get a bigger number

42. Because misconceptions arise frequently when a pupil’s intellect engages in the interaction of new ideas with his/her extant ideas, the analysis of misconceptions is crucially important to teaching and learning. Unless a misconception is identified and ameliorated there is the risk of cognitive conflict and/or further misconceptions: Misconception: If I multiply two fractions I will always get a bigger number Cognitive conflict: Find what number multiplied by ½ makes ¼

43. A teacher cannot correct a misconception unless s/he understands the reasons behind it. That is, the teacher has to diagnose the faulty interaction between the pupils extant ideas and the new concept by discussion . Once the diagnosis is made then the teacher can challenge or contrast the misconception with the faithful conception. Research evidence shows that a pupil is more likely to not only adopt the faithful conception but also retain the correct understanding by this diagnostic approach. Misconception: To add two fractions, I add the numerators and the denominators. Challenge: If the rule to add two fractions is to add the numerators and the denominators then

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

47. • Lessons focus on known, specific difficulties. Rather than posing many questions in one session, it is more effective to focus on a challenging situation or context and encourage a variety of interpretations to emerge, so that students can compare and evaluate them. • Questions or stimuli are posed or juxtaposed in ways that create a tension or conflict that needs resolving. Contradictions arising from conflicting methods or opinions create awareness that something needs to be reconsidered, and understandings clarified.

48. • Activities provide opportunities for meaningful feedback to the student on his or her interpretations. This does not mean providing superficial information, such as the number of correct or incorrect answers. Feedback is provided by students using and comparing results obtained from alternative methods. This usually involves some form of small group discussion. • Lessons include time for whole class discussion in which new ideas and concepts are allowed to emerge. This can be a complex business and requires non-judgmental sensitivity on the part of the teacher so that students are encouraged to share tentative ideas in a non-threatening environment.

49. A pupil says this is the rule to divide two fractions: Would this pupil conception be a good example for diagnostic teaching?

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.