Item by item analysis of the selected vignettes in Patterns, Functions and

Below is how item 5 reads from the ANA paper.

Item 5 assessed learners’ knowledge and skills in solving and completing the number sentence. As outlined in figure 11, only 18 out of 50 learners (thus 36% of the sample) were able to provide the correct answer for this item. This means that 32 out of 50 learners (thus 64% of the sample) were unable to complete the number sentence.

89

Vignette 52 Learner’s response to item 5

The error type shown in vignette 52 above may be conceptual error, because the learner incorrectly completed the number sentence by equating to 53 which is not equal to the expression on the left of the equal sign. Such learners’ errors may be due to lack of computation fluency which may be due to lack understanding of operations (Haddens et al, 2009; Reys et al, 2012). 24 out of 50 learners (thus 48% of the learners) completed the number sentence by just filling in the numbers not equals to the expression on the left of the sign.

Vignette 53 Learner’s response to item 5

The error shown in vignette 53 above may be conceptual error or carelessness error as the vignette shows that the learner incorrectly completed the number sentence by equating it with 3 which makes the number sentence not equivalent. Here, may be the learner just (51 48) and got 3 as the difference equated to the entire number sentence neglecting (14 ÷ 2) + being part of the number sentence. Such error may be due to carelessness. The error may also be due to lack of knowledge of equality (Reys, et al, 2012). The learner may be under the misconception of viewing “=” only as a command to carry out computation other than as a sign representing the equivalent relationship between quantities (Falkner, Levi & Carpenter, 1999). 2 out of 50 learners (thus 4% of the learners) wrote the same responses as the one shown in the vignette.

90

Vignette 54 Learner’s response to item 5

The error shown in vignette 54 above may be also conceptual error or carelessness error because of the following reasons; the vignette shows that the learner incorrectly wrote 31 as a number equal to the expression on the left side of the equal sign. Such error may be due to lack of computational fluency coupled with meaning of operations involved on the number sentence. On the other hand, starting from the left side of the number sentence, may be instead of dividing 14 by 2, the learner multiplied 14 by 2 and got 28 as a product and added 3 obtained as difference between 51 and 48 to get 31. 3 out of 50 learners (thus 6% of the learners) wrote responses similar to the one shown in the vignette.

What were the most common errors in completing the number sentence? Some learners seem to lack basic computation to handle number sentences. In completing the number sentence, 64% of the sample merely completed the number sentence by just filling in the numbers on the right side of the equal sign which are not equal to the expression on the left side of the equal sign making the number sentence unequal. Some learners (16% from the 64%) appeared to have basic computation skills but displayed some carelessness error in their responses.

5.5.1.2 Item 9

Below is how Item 9 reads from the ANA paper.

Item 9 assessed learners’ knowledge and skills in using multiplication and division as inverse operations and how they can use division problems to solve multiplication problems. As outlined in figure 11, 60% of the sample was able to answer Item 9 correctly. This analysis means that 40% of the sample had a challenge in responding to item 9.

91

Vignette 55 Learner’s response to item 9

The errors shown in vignette 55 above may be conceptual error because as shown in the vignette, given that 336 ÷ 14 = 24, the learner was unable to use the given division fact to compute 24 × 14. The learner’s erroneous response shown in the vignette may be due to lack of any kind of picture of what is going on when two numbers are multiplied together (Haylock & Cockburn, 2008). The error of equating 24 × 14 to 24 despite been shown that 336 ÷ 14 = 24 may also be due learner’s to failure to see division as having some sort of inverse relationship to multiplication (Haddens et al, 2009). 20 out of 50 learners’ (thus 40% of the learners) errors were due to knowledge gap of the inverse operation. Such learners errors of equating 24 × 14 to any number instead of 336 may also be due to knowledge gap of equality and multiplication facts (Reys et al, 2012).

What were the most common errors in using division problem to solve multiplication problem?

Learners merely wrote responses which made the number sentence unequal owing to failure to view multiplication and division as inverse operations. Apart from failure to view multiplication and division as inverse operations, learners wrote responses that demonstrates lack of sound knowledge and facts on multiplication of whole numbers.

5.5.1.3 Item 12

Below is how item 12 reads from the ANA paper.

Item 12 assessed learners’ skills solving word problems involving algebraic equations. Only 7 out of 50 learners (thus 14% of the sample) were able to answer item 12

92

correctly. This means that 86% of the sample was unable to answer item 12 correctly. Below are some of the learners’ responses to item 12.

Vignette 56 Learner’s response to item 12

The error type shown in vignette 56 above may be conceptual error. The learner in the vignette used the wrong operation for item 12. This may be due to the learner’s failure to grasp the concept involved in the problem (Seah, 2005). Owing to lack of conceptual understanding of a problem, the learner has multiplied Zonga’s payment of R240 which is said to be 12 times Peter’s payment instead of dividing Zonga’s payment by 12. Haylock & Cockburn, (2013) attributes the error of choosing wrong operation to misinterpretation of verbal cues used in the word problem. 36% of the sample erroneously chose multiplication operation instead of division operation with 18% of the sample writing the same response as the one shown in vignette 56 above.

Vignette 57 Learner’s response to item 12

The error shown in vignette 57 above may be conceptual error. Clearly, the learner’s response in the vignette shows that the learner added 12 to Zonga’s payment of R240 which is said to be 12 times Peter’s payment to have a sum of 252 shown in the vignette. Such learner’s error demonstrates lack of conceptual understanding of a problem which resulted into the learner choosing the wrong operation to solve the problem. i.e. the addition operation instead of division. 18% of the sample wrote the same as the one shown in the vignette.

93

Vignette 58 Learner’s response to item 12

The errors shown vignette 58 above may be procedural errors. The learner’s response in the vignette shows that the learner seemed to have a clear understanding of the concept involved in the problem. The learner was able to identify the correct operation to solve the problem. However, computation errors occur as shown in the vignette. Such errors according to Witzel et al, (2013) may be attributed to lack of requisite maths skill to perform accurately. 10% of the sample was able to realise that the operation involved in a problem is division, however leaners made errors when computing.

What were the most common errors in solving problems involving whole numbers in the financial context?

The error of choosing the wrong operation was common amongst most of the learners’ work. 36% of the sample erroneously chose multiplication operation, 18% erroneously chose addition operation and 4% of the sample erroneously chose subtraction operation. Some learners (12% of the sample) merely wrote numbers with no discernible link to the problem they were required to solve. Most learners despite choosing the wrong operations, did not succeed in computing.

5.5.1.4 Item 13

Below is how item 13 reads from the ANA paper.

Item 13 also assessed learners’ knowledge and skills in solving and completing number sentences. 25 out of 50 learners (thus 50% of the sample) were unable to answer this item correctly.

94

Vignette 59 Learner’s response to item 13

The error shown in vignette 59 above may be carelessness error or conceptual error because of the following reasons; as shown in vignette 59 above, may be the learner incorrectly considered 3 to be the missing value required to make the number sentence equivalent neglecting 8 that is multiplied by the 3 before the division sign resulting into simply considering the number that can divide into 3 and gives 1 as a quotient which equates to the 1 written on the right side of the equal side. The error may also be due to lack of computational fluency coupled with meaning of operations involved on the number sentence. 3 out of 50 learners (thus 6% of the sample) wrote the same responses as one shown in the in the vignette.

Vignette 60 Learner’s response to item 13

The error shown in the vignette 60 above may be conceptual error. The vignette shows that the learner wrongly considers 48 to be the number that makes the number sentence complete failing to see that 48 make the left side of the number sentence not equivalent to 1 written on the right side of the number sentence. May be the learner treated an empty space for the unknown on the number sentence more like an invitation to write an answer, overlooking the need for equivalence (Richards, 2013). 22 out of 50 learners (thus 44% of the learners) merely filled an empty space on the number sentence with the numbers without considering equivalence.

95

The same common errors committed when engaging with the number sentence containing addition and subtraction operations were also committed in number sentence containing multiplication and division operations. The error of completing the number sentences by merely writing numbers that only makes the expression on the left side of the equal sign not equal to the expression on the right side of the equal sign as in number sentence involving addition and subtraction was also found to be common. Although some errors seemed to be due to carelessness errors but most appeared to be conceptual errors.

5.5.1.5 Item 14

Below is how item 14 reads from the ANA paper

Item 14 assessed learners’ knowledge and skills in determining the input values, output values, and rules for the patterns and relationships using table. Only 10 out of 50 learners (thus 20% of the learners) were able complete the table correctly. This means that 40 out of 50 learners (thus 80% of the learners) struggled to complete the table.

Below are some of the learners’ responses to item 14.

Vignette 61 Learner’s response to item 14

The errors type shown in vignette 61 may be both procedural errors and conceptual errors because of the following reasons; as shown in the vignette, the learner erroneously completed the input raw by filling in 20. Furthermore, the vignette shows that the learner proceeded by erroneously completed the output raw by filling in 17 probably obtained from adding 3 into 14 in the output raw. Clearly, such learner failed to see links and connection between numbers in the input raw and numbers in the

96

output raw (Cooke, 2007). The learner treated the numbers in the input raw and numbers in the output raw as separate patterns. 5 out of 50 learners (thus 10% of learners) completed the input raw by filling in 20 as shown in the vignette and 12 out of 50 learners (thus 24% of the sample) completed the output raw by filling in 17.

Vignette 62 Learner’s response to item 14

The error type shown in vignette 62 may be procedural errors because the vignette shows that the learner correctly filled in 29 as the missing output value but in correctly filled the missing input value. Clearly, the learner was able to see the links and connection between numbers in the input raw and numbers in the output raw i.e thus input value × 3 – 1. Therefore, to obtain 29 as the missing output value, the learner multiplied 10 as the input value by 3 and subtracted 1. However, as shown in the vignette, the learner erroneously completed the input raw by filling in 20 which has no discernible link to 44 in the output value. The learner failed to apply the identified rule to work out out-put value (Haddens et al, 2009). 5 out of 50 learners (thus 10% of the learners) were able to determine the rule for the patterns in the table but struggled to manipulate the rule; input value × 3 – 1 = 44 to solve the unknown input value.

Vignette 63 Learner’s response to item 14

The errors type shown in vignette 63 may be conceptual errors because clearly, the responses shown in the vignette have no discernible link to numbers input and the output numbers in the table. The error of completing the table by merely filling numbers with no discernible link to the given input and output numbers may be due to failure to see links and connection between numbers in the input raw and numbers in the output raw (Cooke, 2007).

97

What were the most common errors in investigating patterns and determining the rule, input and output values?

The error of treating the numbers in the input and output raw as separate patterns was found to be common amongst a number of learners’ work. Such error makes it difficult for learners to recognise the rule used to link the input and output values. Errors of merely writing numbers with no discernible link to the input and corresponding output values were also found to be common in a number of learners’ work.

5.5.1.6 Item 15

Below is how item 15 reads from the ANA paper.

Item 15 assessed learners’ knowledge and skills in determining the input values when the rule is given as well as the corresponding output values in the flow diagram. 21 out of 50 (Thus 42% of the learners) were unable complete the correct input value for 15.1 and 20 out of 50 (40% of the learners) were unable to complete the complete the correct input value for 15.2.

Below are some of the learners’ responses to item 15.

Vignette 64 Learner’s response to item 15

The learner’s responses in vignette 64 above demonstrate lack of conceptual understanding in the use of two operations in flow diagrams. To complete the flow diagram, given × 3 + 2 as the rule that produces 11 as the given output value the learner just wrote 22 as the input value instead of 3. The learner fails to recognise that 22 × 3 + 2 is not equal to 11 given as the output value due to lack of conceptual

98

understanding. The learner further filled 46 as the input value instead of 7 failing to recognise that 46 × 3 + 2 does not produce the given output value of 23. 42% of the learners just filled the input values in the flow diagram that does not correspond with the given output values due to lack of conceptual understanding.

Vignette 65 Learner’s response to item 15

The learner’s error in vignette 65 may be carelessness error because from vignette 65 above, it is clear first input value that correspond with the second output value given in the flow diagram instead of filling in the input value that correspond with the first given output value. 5 × 3 + 2 gives an output value of 17 but not 11 given as the first output value for 15.1 in the flow diagram. However, the learner successfully filled in 7 as the third input value that corresponds with the third given output value.

Vignette 66 Learner’s response to item 15

The error shown in vignette 66 may also be carelessness error, as shown in vignette 65 above, correctly filled in the first input value for 15.1 in the flow diagram. However, error ensued when the learner filled 21 as the third input value, 21 × 3 + 2 ≠ 23. Clearly, due to carelessness the learner did not consider the rule in the flow diagram as “× 3 + 2”. May be the learner neglected “× 3” as part of the rule and regarded the rule as only “+ 2” resulting into such learner considering 21 as the input value for 15.1 in the flow diagram plus 2 giving 23 as the third given output value.

5.5.1.7 Item 16

99

Item 16 assess learners’ knowledge and skills in investigating and extending geometric patterns looking at the relationship or rule of the patterns. Only 14 out of 50 learners (thus 28% of the learners) were able to answer item 16 correctly. Meaning that 36 out of 50 sampled learners (thus 72% of the sample) were unable to answer item 16 correctly.

Below are some of the learners’ responses to item 16.

Vignette 67 Learner’s response to item 16

The error type shown in vignette 67 above may be conceptual error. The learner fails to extend the geometric patterns with respect to relationships and the rules of the patterns. The error of writing 40 to respond to item 16 may be due to learner’s failure to recognise and making sense of the diagram pattern (Taylor & Harris, 2014). The response shown in the vignette demonstrates failure to recognise that from the first figure formed with 4 match sticks, 3 more match sticks have been used to form the second figure totalling to 7 match sticks. And from the second figure formed with 7 match sticks, 3 more match sticks have also been added to form the third figure totalling to 10 match sticks. Therefore, the rule for the diagram patterns is “adding 3 match sticks to form the next diagram”. Meaning that if the diagram pattern is continued, 3 match sticks must be added to the third figure and that will result into 13 match sticks. 14 out of 50 sampled learners’ (thus 28% of the sampled learners) errors were due to failure to extend the diagram patterns with respect to the patterns rule.

100

Vignette 68 Learner’s response to item 16

The response in the vignette 67 above demonstrates that the learner was able to extended the diagram pattern with respect to the rule “Adding 3 match sticks to form the next diagram”, however, instead of counting the number of match sticks, the learner counted the number of squares which he/she refer as 4 blocks as shown in the vignette. This error may be due to carelessness or may be the learner had difficulty in comprehending what item 16 required (Haddens et al, 2009). 7 out of 50 learners (thus 14% of the learners) wrote the responses as the one shown in the vignette.

Vignette 69 Learner’s response to item 16

The vignette shows that the learner merely extended the diagram pattern with respect to the rule and did not count the number of match sticks used as required. This error may be due to difficulty in comprehending exactly what the problem required (Heddens et al, 2009). The error of drawing the diagram pattern instead of counting the number of sticks may also be due to carelessness. 12 out of 50 learners (thus 24% of the learners) extended the diagram pattern with respect to rule of the patterns, however, they responded by just drawing the diagram pattern instead of counting the number of match sticks used as shown in the vignette.

What were the most common errors in investigating and extending geometric pattern looking at the relationship or rule of the pattern?

101

The error of merely writing numbers with no discernible link to the geometric pattern was common amongst 28% of the sample. Some learners merely extended the geometric pattern but did not count the number of match sticks and such error was