Qualitative analysis of learners’ errors and misconceptions in Measurement:

In the following section, learners’ work is presented to analyse first-hand how their errors and misconceptions were expressed. 19 vignettes covering 5 items and sub- items were analysed in this section.

Item by item analysis of selected vignettes in Measurement

6.3.1.1 Item 21

Below is how item 21 reads from the ANA paper

Item 21 assessed learners’ skill to solve problems in the context of Measurement. As already outlined in figure 5.1, only 15 out of 50 learners (Thus 38% of the sample) were able to answer item 21 correctly. This analysis means that 62% of the sample was unable to answer item 21 correctly.

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Vignette 70 Learner 1’s response to item 21

The error shown in vignette 70 above may be conceptual error because as shown in the vignette, the learner added 9, 5 litres to 8, 7 litres. Such learner failed to realise that the calculation required is subtraction (Briggs, 2013). Lack of conceptual understanding of the concept involved in the problem resulted into the learner choosing the wrong operation (Heddens et al, 2009). The error of choosing addition operation instead of subtraction may be due to misinterpretation of verbal cues used in the word problem (Haylock & Cockburn, 2013). 11 out of 50 learners (thus 22% of the sample) wrote the same responses as the one shown on the vignette above.

Vignette 71 Learner’s response to item 21

The error shown in vignette 71 may be procedural error. Despite having an understanding of the concept behind the problem, the learner made errors arisen from failure to carry out algorithms. The learner understands that 8,7 litres must be subtracted from 9,5 litres, however, the has computed the decimals wrongly with the assumption that subtraction is commutative (Sadi, 2007). Using the vertical method of subtraction, the learner just subtracted 8 from 9 to get 1 before the comma and after the comma the learner subtracted 5 from 7 to get 2, as a result the learner’s response reads 1,2 litres as shown in the vignette.

Vignette 72 Learner’s response to item 21

The error shown in vignette 72 above may be conceptual errors because the response in the vignette demonstrates that the learner seemed to have no conceptual grasp of the problem. Owing to lack of understanding of the problem the learner just

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aligned the decimals numbers in column but resorted into writing response with no discernible link to the concept involved in the problem. The error may be due to difficulty in reading and comprehending the problem (Barbu, 2010). Most learners in this study responded by writing responses which turn to demonstrate lack conceptual of grasp to the problem they were required to solve.

Vignette 73 Learner’s response to item 21

The error shown in vignette 73 above may be conceptual error. Clearly, the response shown in the vignette shows that he learner intended to subtract 8,7 litres from 9,5 litres using the column method to subtract through decomposition, the vignette shows that the learner correctly exchanged 1 from the ones place for tens in the tens place. However, instead of subtracting the first raw digits with the digits in the second raw, the learner just erroneously added 8,7 to 9,5 . The learner’s procedures are flawed owing to lack of conceptual understanding.

Vignette 74 Learner’s response to item 21

The type of error shown in vignette 74 may be carelessness error because from the learner’s response in the vignette, such learner was able to identify that the calculation required is subtraction. However, an error ensued in carrying out the calculation. Instead of writing the difference between 9,7 litres and 8.7 litres as 0,8 litres the learner wrote 8 litres omitting the 0 before and comma. The error 8 litres instead of 0, 8 litres may be due to carelessness. 4 out of 50 learners (8% of the learners) have shown this type of error in their responses.

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What were the most common errors in solving word problem involving the concept of measurement?

The errors of choosing the wrong operation to solve the problem were common amongst 42% of the sample. The most commonly wrong operation was addition followed by multiplication. Some learners ignored the decimal point from the decimal numbers in a problem despite being able to choose the correct operation, such learners wrote 9, 5 as 95 and 8, 7 as 87 for example, this learners work;

and . Most of the learners who were unable to identify that the operation involve is subtraction only aligned the decimals in column and also made computation errors in addition to choosing the wrong operation. The error of writing responses with no discernible link to the problem was common amongst 20% of the sample. Such leaners did not even show the operations they used.

6.3.1.2 Item 22

Below is how item 22 reads from the ANA paper

The knowledge and skills assessed in item 22 involves calculation and problem in the context involving time based on different time zones. As already outlined in figure 5.3, only 2 out of 50 learners (thus 4% of the sample) was able to answer item 22.1 correctly and only 1 out of 50 learners (thus 2% of the sample) was able to answer item able to answer item 22.2 correctly. This analysis means that 96% of the sample

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was unable to give correct response for item 22.1 and 98% of the sample was unable to give correct response item 22.2.

Below are some of the learners’ responses for item 22.1

Vignette 75 Learner’s response to item 22.1

The error type shown in vignette 75 may be carelessness error because clearly the learner calculated the time difference between the two countries without taking into account the different zones shown on the clocks i.e. Rome time in the morning (a.m.) and Tokyo time in the afternoon (p.m.). Such error of calculating the time difference error of without considering the different time zones was common amongst 3 out of 50 leaners. Such learners wrote the same response as the one shown in vignette 74 above.

Vignette 76 Learner’s response to item 22.1

The error type shown in vignette 76 above may be conceptual error, because from the learner’s response, it is clearly that the learner intended to subtract Rome time from Tokyo time but failed to re-write the time for both countries as indicated from the clocks. Clearly, the learner was unable to identify hour hand and the minute from the clocks (Reys et al, 2012). The learner seems to lack an understanding of how to read the clock (Haddens et al, 2009)

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Vignette 77 Learner’s response to item 22.1

The error type shown in vignette 77 may be conceptual error because as in vignette 77 above, the learner erroneously wrote 21 hours 15 minutes as the time different between the two countries. Such response has no discernible link to the different time shown in the clocks. The learner made error in an attempt to find the length of time from one recorded time to the next and such error may be due inability to add-on (Haylock, 2006).

Sub item 22.2 and 22.1 are related in such a way that all learners who were unable to answer item 22.1 could not answer item 22.2

Below are some of the learners’ responses to item 22.2

Vignette 78 Learner’s response to item 22.2

The error type shown in vignette 78 may be conceptual error, because as shown in vignette 9 above, the learner erroneously considers 20:00 to be time in Rome given that the time in Tokyo is 17:00. Such error may be due to lack of understanding of the relation between Rome time and Tokyo time. 98% of the sample could not answer item 22.2 correctly owing to failure to calculate the time difference between the times shown on the clocks.

What were the most common errors in calculating the time differences?

Some learners erroneously calculated time differences directly without considering the different time zones indicated on the clock faces. Such learners seemed to read the clock faces in 12-hour only. Some of the learners confuse hour hand with minute hand.

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Most learners merely wrote responses with no discernible link to the time shown on the clock faces. Such learners seem to lack an understanding of concept to recorded time and the passage of time.

6.3.1.3 Item 23

Figure 4.9 below shows item 23 from the ANA paper

The knowledge and skills assessed on this item involves conversion between units (millilitres to litres). Only 8 out of 50 learners (thus 16% of learners) were able to answer item 23 correctly. This analysis means that 84 % of the sample was unable convert 600 millilitres indicated on the jug to litres.

Below are some of the learner’s responses to item 23

Vignette 79 Learner’s response to item 23

The error type shown in this vignette 79 may be conceptual error because the learner’s erroneous response shown in the vignette may be due to lack of conceptual knowledge of the equivalence or relation between litres and millilitres (Reys et al, 2012). Thus 1 litre = 1000 millilitres. This means that to convert litres to millilitres, one has to multiply the litres by 1000. And conversely, to convert the millilitres to litres, we divide the millilitres by 1000. The learner’s response in vignette 78 above may be resulting from dividing 600 millilitres by 2 to get 300 millilitres.

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Vignette 80 Learner’s response to item 23

The error type shown in vignette 80 above may be procedural error or conceptual error because as shown in the vignette, the learner erroneously converted 600 millilitres and wrote 0, 06 as a response. Here, Learner’s assumption may be that 1 litre = 10 000 millilitres, to convert the 600 ml indicated on the jug to litres, the response clearly shows that such learner divided 600 ml by 1000 to get the 0.06 shown in the vignette. Such error may also be due to lack of conceptual knowledge of equivalence or relation between litres and millilitres.

Vignette 81 Learner’s response to item 23

The error type shown in vignette 81 above may be procedural error or conceptual error because clearly the learner’s erroneous response shown in vignette 80 may be due to the wrong assumption that 1 litres = 100 millilitres, with this view the leaner divided the 600 ml indicated on the jug by 100 to get 6 litres. 5 out of 50 learners (that is 10% of the learners) have written the similar responses as the one shown in this vignette.

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Vignette 82 Learner’s response to item 23

The error shown in vignette 82 may be conceptual error or carelessness error because from vignette 82 above, it is apparent that the learner re-wrote the 600 millilitres as indicated on the jug as a response to item 23 without converting to litres as required. 19 out of 50 learners (thus 38% of the learners) wrote same responses as the one shown in vignette 81 above. The error of re-writing the 600 millilitres without converting may be due to difficulty in reading and comprehending what item 23 required (Haddens et al, 2009). The errors may also be due to lack of conceptual knowledge of the equivalence or relation between litres and millilitres (Reys et al, 2012).

What were the most common errors in converting millilitres to litres?

The error of re-writing the 600 millilitres as indicated on the jug without converting to litres as required was common among 19 learners. Some learners due to lack of conceptual knowledge of equivalence or relation between the two units measures merely wrote responses with no discernible link to the 600 millilitres they were required to be convert to litres. The error of writing 6 litres as a conversion of 600 millilitres to litres was also common amongst 10% of the sample.

6.3.1.4 Item 24

Figure 4.11 below shows how item 24 reads from the ANA paper;

Figure 4.11: Item 24

Item 24 assessed learners’ knowledge and skills in solving problem lengths given in different units i.e. metres and centimetres. As already outlined in figure12, only 15 out

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50 learners (30% of the sample) were able to answer item 24 correctly. This analysis means that 70 % of the sample was unable to answer item 24 correctly.

Below are some of the learners’ responses to item 24.

Vignette 83 Learner’s response to item 24

The error type shown in vignette 83 above may be conceptual error because the learner’s erroneous response shown in vignette 10 may be due to lack of conceptual knowledge of relationship between the metric measure of units measure of length i.e. mitres and the centimetres (Hylock & Cockburn, 2008). From the learner’s response, it is clear that such learner fails to recognise that 4,08 mitres is less than 429 centimetres (as 429 cm = 4,29 m) hence 1 centimetre is equal to 100 of a metres. 12 out of 50 learners (thus 24% of the learners) wrote same responses as the one shown in vignette … above.

Vignette 84 Learner’s response to item 24

The error shown in vignette 84 may be due to conceptual error or carelessness error because as shown in vignette 84 the learner erroneously considered 387 cm to be the furthest. The 387 cm written as a response for item 24 is in fact the shortest amongst the results indicated to a learner. The error of choosing the shortest throw instead of the furthest may be due to lack understanding of associated vocabulary or difficulty in reading and comprehending (Reys et al, 2012; Haddens et al, 2009). 13 out of 50 learners (thus 24% of the sample) wrote same response as the one shown in vignette 84 above.

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Vignette 85 Learner’s response to item 24

The error type shown in vignette 85 may be conceptual error. As shown in the vignette the learner erroneously considered Charles to be the one who threw the shot- put the furthest. Looking at the results, Charles is in fact in the third position. 6 out of 50 learners (thus 12% of the sample) wrote the same responses as the one shown in vignette 85 above.

Vignette 86 Learner’s response to item 24

The error type shown in the vignette may be conceptual error. Clearly the learner’s response shown in vignette 86 above has no discernible link to item 24. Such error may be due to difficulty in reading and comprehending what item 24 required (Haddens et al, 2009). 8% of the sample merely wrote responses with no discernible link to item 24.

What were the most common errors in solving problem involving lengths given in different units?

From the 70% of the sample who had difficulty in choosing the furthest throw from the results of the shot-put challenge, some learners (24% of the sample) chose the 4, 08 metres to be the furthest throw instead of 429 centimetres. Some learners chose the 387 centimetres to be the furthest throw. The error of choosing the shortest throw to be the furthest was common amongst 26% of the sample. Some learners chose 395 metres to be the furthest and such error was common amongst 12% of the learners.

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8% of the sample merely wrote numbers with no discernible link to the list of numbers they were required to compare and choose from.

6.3.1.5 Item 25

Figure 4.12 below shows item 25 from the ANA paper.

Figure 4.12: Item 25 and item 25.2

The knowledge and skills assessed on item 25 involves reading scales and conversion between kilograms and grams. 14 out of 50 learners (thus 28% of the learners) were unable to give the correct reading indicated on the scale. Quite disappointingly, only 6 out of 50 learners from the sample were able to convert 56, 8 grams shown indicated on the scale to kilograms. This analysis means that 88% of the sample was unable to convert the mass indicated on the scale to grams.

Below is one the learner’s response to item 25.1

Vignette 87 Learner’s response to item 25.1

The error type shown in vignette 87 may be conceptual error, because the learner’s response demonstrates the inability to read scale (Taylor & Harris, 2014). Learner’s inability to read scale may be due to lack of understanding of the use of scale (Tylor &

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Harris, 2014). 14 out of 50 learners (28% of the learners) were unable to give the correct reading indicated on the scale and 1 out of 50 learners (2% of the learners) did not respond to this item. Blank answers to the items imply that learners found the item difficult to answer to such an extent that they even fail to figure out how to start answering the item (Makonye, 2011).

Below is one the learner’s response to item 25.2

Vignette 88 Learner’s response to item 25.2

The error type shown in vignette 88 above may be Conceptual error because in an attempt to convert 56, 8 kilograms indicated on scale to grams, the vignette shows that the learner wrote 23. 4 grams. The 23, 4 grams written as a response has no discernible link to 56, 8 kilograms indicated on the scale. Such error may be due lack of conceptual knowledge of the equivalence or relation between the two units i.e. kilograms and grams (Reys et al, 2012). Kilo (k) means 1000, this means it takes 1 kg to make 1000 g. Therefore, to covert the 56,8 kilograms indicated on scale to grams, the learner was supposed to multiply 56,8 kg by 1000 and that will result into 56800 g. 42 out of 50 learners (thus 84% of the learners) have shown conceptual errors in their responses.

What were the most common errors in reading the scale and converting kilograms to grams?

With regard to reading scale, 28% of the sample demonstrated the inability to read scale and such learners merely wrote numbers which are different from the 56, 8 indicated on the scale. With regard to converting between kilograms to grams, most learners merely wrote responses with no discernible link to the mass indicated on the scale.

6.3.2 Conclusion

Numerous errors have been identified pertaining to the following main areas in Measurement: Problem solving involving decimal fractions within the context of Measurement, Time difference between clock faces in different time zones, reading

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scales, conversions from one unit to another, and comparison of lengths of different units.

6.4 Qualitative analysis of learners’ errors and misconceptions in Space and Shapes (Geometry): Vignettes of learners’ responses

In the following section, learners’ work on Space and Shapes (geometry) is presented to analyse how their errors and misconceptions were expressed. 13 vignettes covering 4 items and their sub-items are analysed.

6.4.1 Item by item analysis of the selected vignettes in Space and Shapes