Qualitative analysis of learners’ errors and misconceptions in Numbers,

Learners’ work is also presented in this section to analyse how their errors and misconceptions were expressed while responding to items under the content area Numbers, Operations and Relationships. The 51 vignettes covering 16 items and sub items were analysed in this section.

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5.4.1.1 Item by item analysis of the selected vignettes in Numbers, Operations and Relationships

5.4.1.1.1 Item 2

Below is how item 2 from the ANA paper

Item 2 assessed learners’ knowledge of place value and the skills to expand large numbers into their separate place values. Even though 35 learners (70% of the sample) scored on this item, 15 learners (that is 30% of the sample) were unable to provide correct response.

Vignette 1 below show one of the learner’s response to item 2.

Vignette 1: Learner’s response to item 2

The error type shown in vignette 1 above may be conceptualerror because the “eight hundred thousand” written as a response shown in vignette 1 above has no discernible link to the “seven million, three hundred and forty-two thousand, six hundred fifty-one”. The error may be due to lack of understanding of expanded notation, regrouping process, and place value (Heddens, William & Daniel, 2009). Due to lack of understanding of place value of digits, it is clear that the learner fails to recognise the value of 4 in the seven digits number 7 342 651 despite being given the other digits in expanded notation with respect to their place values. 15 out of 50 learners (30% of the sample) wrote responses with no discernible link to place value of 4 in their responses as in vignette 1 above.

What were the most common errors in expanding multidigit number in separate place values?

A number of learners wrote responses with no discernible link to the multidigit number 7 342 651 and the expanded numerals 7000 000; 3 × 100 000; 2 000; and 651. From the expanded numeral, such learners failed to recognise face value 4 which was

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missed out and could not recognise that 4 is in the forty thousand place in the seven- digit number 7 342 651.

5.4.1.1.2 Item 3

Below is how item 3 reads from the ANA paper

The knowledge and skills of rounding off the numbers to the nearest are assessed in this item. As already outlined in figure 10, 62% of the sample was able to answer item 3 correctly. This analysis means that 38% of the sample was unable to give the correct responses for item 3. Below are some of the learners’ responses to item 3.

Vignette 2: Learner’s response to item 3

The error type shown in vignette 2 could be procedural error as it is clear from the vignette that the learner failed to carry out the rounding off algorithms (Reys et al, 2012). The learner has chosen the correct place value required to rounding to but failed to increase it by 1 hence the digit to the right is greater than 5. This is called rounding up. The vignette shows that the learner has rounded down instead. 3 out of 50 learners (6% of the sample) wrote the same responses as the one shown in vignette 2 above.

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Vignette 3: Learner’s response to item 3

The error shown in vignette 3 may be procedural error or carelessness error because as shown in vignette 3 above, the learner has chosen the correct place value required to rounding to and have increased it by one hence the digit to the left is greater than 5. However, after rounding up, the learner wrote the rest of the digits as they appeared before rounding instead of representing every digit with zero. Such error may be due to carelessness or could be that the learner doesn’t know that procedurally after rounding up or down to the required place value the rest of the digits are represented by the zeros as place holders. 5 out of 50 learners (thus 10% of the sample) displayed this error type in their responses.

Vignette 4: Learner’s response to item 3

The error shown in vignette 4 above may be conceptual error because as shown in the vignette learner wrote response which shows no discernible link to rounding off the five digit number 56 673 to the nearest 10 000. Such learner merely interchanged 9 and 6 and thereafter reduced 7 to 6. The learner’s response demonstrates lack of fundamental concept of rounding off the whole numbers coupled with lack of knowledge of place value and naming numbers (Reys et al, 2012). 12 out of 50 learners (that is 22% of the sample) have shown this error type in their responses.

What were the most common errors in rounding off a multidigit number to the nearest require place?

Some learners (thus 22% of the sample) merely wrote responses with no discernible link to the multidigit number they were required to round off. Such learners seem to be lacking both conceptual and procedural knowledge of rounding off numbers to the required nearest. Some learners rounded up the required place value but erroneously re-wrote every digit in after rounding up. Some learners rounded down to the required place value instead of rounding up.

5.4.1.1.3 Item 4.1

Item 4.1 assessed learners’ knowledge and skills in addition of 5 digits whole numbers in columns. As already outlined in figure 10, 60% of the sample was able to answer item 4.1 correctly. This analysis means that 40% of the sample was unable to get the

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sum for the three multidigit whole numbers. Below are some of the learners’ responses to item 4.1.

Vignette 5 Learner’s response to item 4.1

The errors shown in vignette 5 above may be both procedural errors and conceptual errors because as shown in vignette 5 above, using vertical column method, the learner was only able to get the sum for the unit digits. Procedural error owing to confusion of standard algorithm for addition with standard algorithm for subtraction ensued when the exchanged one of tens in the tenth column for ten ones regrouped with the unit digit in the first raw. The source of such error lies in the learner retrieving the wrong schema and not recognising the retrieval error (Alwyn, 1989). Using the column method to add, the learner added the digits with respect to their place values, but erroneously changed the face values of all the digits in the first raw after applying the unnecessary regrouping algorithm i.e. the learner regarded 2(the unit digit in the 1st _{raw) as 12, 4 (the tenth digit resulting from erroneous exchange of one of tens for} ten ones regrouped in the 1st _{raw) as 14, 1 (the hundreds digit in the 1}st _{raw) as 21, 2} (the thousand digit in the first raw) as 32, and 4 (the ten thousand digit in the 1st _raw) as 44. These error patterns reflect the learner’s lack of conceptual understanding of procedures associated with addition of multidigit whole numbers (Haylock & Cockburn, 2013).

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Vignette 6 Learner’s response to item 4.1

The error shown in vignette 6 may be carelessness error because from the vignette 6 above, the learner seemed to know from cognitive perspective exactly how to get the correct answer for item 4.1. However, due to carelessness, the learner wrote the first 5 digits number on the first raw as 24 152 instead of 42 152 resulting such learner writing wrong response for item 4.1 despite having computed the digits numbers 24 152 + 28 945 + 76 361 correctly using the vertical column method to add. Two learners (thus 4% of the sample) have shown this error type.

Vignette 7 Learner’s response to item 4.1

The error shown in vignette 7 above may also be carelessness error because as shown in the vignette 7 above, using the vertical column method to add, the learner has correctly computed the units, tens, hundreds digit and the thousands digits. However, due to carelessness the learner added the 10th _{thousands digits and got 13} as the sum but learner erroneously ignored the regrouped digit from sum of thousands digits. 3 out of 50 learners’ (thus 6% of the sample) wrote the same responses as the one shown in vignette 7 above. In total, the error of adding the column ignoring the regrouped digit was common amongst 12% of the sample.

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The errors types shown in vignette 8 above may be both carelessness error and conceptual error. Due to carelessness, the vignette shows that the learner has written the second 5 digits whole number in the second raw as 28 942 instead of 28 945. 3 out of 50 learners (thus 6% of the sample) have shown this error type when writing the 5 digits whole numbers in columns. Furthermore, the vignette shows that the learner has attempted to add the three multidigit whole numbers using the vertical column method to add, however, the responses written in the 4th _{and 5}th _{raw shown} as sums in the vignette has no discernible link to the three multidigit whole numbers the learner was required to add. According to Hatfield et al. (2008) the learner committed errors owing to lack of mastery of addition facts. Knowledge of basic addition facts are considered to the cornerstones for success in addition (Heddens et al. 2009). The errors may also be due to lack of conceptual knowledge of place value (Haylock & Cockburn (2013). 14% of the sample decompose and align the multidigit numbers in column but failed to get the sum of each column. Instead such learners erroneously wrote the digits with no discernible link to the three multidigit whole numbers they were required to add as in vignette 8.

Vignette 9 Learner’s response to item 4.1

The error shown in vignette 9 may be conceptual errors because from vignette 9 above, using the vertical column method to add, the learner’s response shows that the learner correctly added the unit digits and got 8 as sum. However, the leaner erroneously computed the tens digits, hundreds digits, thousand digits and the tenth thousand digits. From the learner’s response it appears that the learner had challenge in adding the digits with the sum resulting to double digit (Haylock, 2006). The learner seems to have difficulty with the place value and regrouping algorithm (Heddens et al, 2009). 6 learners (thus 12% of the sample) was only able to get the partial sum for the unit digits which did not require regrouping but failed to get the partial sums for the rest of the digits as in vignette 9 above. Such learners wrote numbers with no discernible link as partial sums for the rest of the digits in columns.

What were the most common errors in adding multiple digit numbers?

From 40% of the learners who did not succeed in to adding the three multiple digit numbers, 14% of the sample made errors in writing the partial sums that involves regrouping. Such learners correctly added the digits in columns and regrouped to the

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next columns but ignored add the regrouped digit when adding the columns. Some learners (14% of the sample) aligned the digits vertically in column but could not find the partial sum of any digit. Instead, such learners wrote the digits with no discernible link the three multiple digit numbers they were required to add as partial sums. 12% of the sample correctly found the partial sum for the unit digits which did not involve regrouping but failed to get the partial sums for all the rest on the digits.

5.4.1.1.4 Item 4.2

The knowledge and skills assessed in item 4.2 involves subtraction of multiple digit numbers. As outlined in figure 10, 54% of the sample was able to answer item 4.2 correctly. This analysis means that 46% of the sample was unable to get the correct answer for item 4.2. Below are some of the learners’ responses to item 4.2

Vignette 10 Learner’s response to item 4.2

The type of error shown in vignette 10 above may be both procedural errors and conceptual errors. Clearly, starting from the unit digits, the vignette shows that the learner subtracted 6 from 8 to get 2 as the difference irrespective of the position the digits. Furthermore, the learner subtracted 4 from 6 in the tens place to get 2 irrespective of the digits’ position. The vignette further shows that the learner proceeded to the hundreds digits and subtracted 5 from 9 to get 4 as the difference. Here, the learner’s assumption in this vignette is that subtraction is commutative. The reason why the learner think subtraction is commutative stems directly from an outcome of their experience influenced by correct previous learning (Alwyn, 1989). For an example, knowing that 6 + 8 = 14 and also if one changes the position of the digits as 8 + 6 the answer is still 14, learners therefore are likely to conclude that 8 – 6 gives the same answer as 6 – 8. According to Alwyn (1989) & Sadi (2007), the errors shown in the vignette are due to over generalisation of subtraction to commutativity of addition and multiplication operations. 8 out of 50 learners’ (thus 16% of the sample) wrote the same responses as the one shown in this vignette. Such learners demonstrate lack of knowledge of correct algorithm i.e. exchanging one of the tens for units to have more units so that it become possible to make subtraction of 8 units.

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Vignette 11 Learner’s response to item 4.2

The error type shown in vignette 11 above may be carelessness error. From vignette above, it is clear that the learner seemed to have conceptual understanding in vertical column method to subtract. Applying the exchanging and regrouping algorithms, the learner correctly subtracted the unit digits to get 8 as the difference as shown in the vignette. The learner proceeded to the tens digits and exchanged one of the hundreds for tens to have more tens and make it possible to subtract 6. As shown in the vignette, the learner regrouped the tens. However, due to carelessness, the learner subtracted 6 from 13 and then wrote 9 as the difference instead of 7. This may be due to carelessness. From the learners’ responses, a total number of 7 learners’ errors were due to carelessness and 5 learners’ (thus 10% of the sample) errors were related to the error shown in vignette 11 above.

Vignette 12 Learner’s response to item 4.2

The error shown in vignette 12 above may be conceptual error because of the following reasons; clearly, the learner’s response in the 3rd _{raw shows no discernible} relationship with the digits in the 1st _{and the 2}nd _{raw shown in the vignette. The errors} shown in the vignette may be due to the learner’s lack of conceptual and procedural knowledge of multidigit subtraction coupled with lack of conceptual knowledge of place value (Fazio, Bailey, Thompson, Siegler, 2014; Mix, Prather, Smith & Stockton, 2014; Mooney et al. 2012). 8 out of 50 learners (thus 16% of the learners) have shown this error type in their responses.

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What were the most common errors in subtraction of multiple digit numbers? The error of subtracting the larger number from the smaller number was common amongst a 16% of the sample. Such learners turn to overgeneralise commutatively of addition to subtraction. Some errors were due to carelessness. Despite learners demonstrating conceptual grasp of the concept involved in subtraction. The errors of merely writing responses with no discernible link to the multiple digit numbers involved were common amongst 16% of the sample.

5.4.1.1.5 Item 4.3

Item 4.3 assessed leaners’ skills and knowledge in multiplication of 4-digits by 2-digit numbers. As already outlined in figure 10, only 38% of the sample was able to answer item 4.3 correctly. This analysis means that 62% of the sample was unable to answer item 4.3 correctly. Below are some of the learner’s responses to item 4.3

Vignette 13 Learner’s response to item 4.3

The errors shown in vignette 13 may be both procedural errors and conceptual errors; to multiply 3 107 by 35, the learner intended to use the long multiplication. As shown in vignette 13, the learner successfully multiplied the unit digit in the 2nd _{raw by} every single digit in the 1st _{raw and correctly wrote the products in the 3}rd _raw. However, the learner made errors in multiplying the tens digit from the 2nd _{raw by the} digits in the 1st _{raw. In multiplying the tens digit in the 2}nd _{raw by the digits in the first} raw, firstly, the vignette shows that the learner took cognisance of the place of 3 as the tens digit in the 2nd _{raw by placing 0 as the place holder before multiplying 3 as the} tens digit in the 2nd _{raw by every digit in the 1}st _{raw. Procedural error ensued when the} learner multiplied the 3 as the tens digit in the 2nd _{raw by 7 as the unit digit in the 1}st raw to get 21 as the product but failed to regroup the tens. The learner regrouped the unit digit to the next column instead of the tens digit. Such error may be linked to lack of conceptual grasp of place value (Hatfield et al. 2008). The learner proceeded by multiplying the tens digit in the 2nd _{raw by tens digit in the 1}st _{raw to get 0 and added} the wrongfully regrouped 1 and wrote the answer in the hundreds position in the 4th raw. Thereafter, the learner multiplied the tens digit in the 2nd _{raw by the hundreds digit} in the 1st _{raw and wrote 1 as the product which has no discernible link to the digits 1}st

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raw digits and 2nd _{raw digits. Such errors may be due to confused algorithm or lack of} mastery of multiplication facts (Wallace, 1984).

Vignette 14 Learner’s response to item 4.3

The errors type shown in vignette 14 above may be both procedural errors and conceptual errors, because locking at the leaner’s response it is clear that the learner multiplied the unit digits 3 × 7 and got 35. Using the knowledge of place value and regrouping the learner successfully regrouped 3. However, the learner proceeded by erroneously disregard 0 as tens digit in the first raw but considered the regrouped 3 and multiplied column by column to get 9. After all, the learner brought down 31. Such error may be due to overgeneralisation of addition and subtraction algorithm to multiplication (Hatfield et al, 2008). The learner fails to see that multiple digit multiplication is based on the distributive law for multiplication distributed over addition (Haylock, 2006).

Vignette 15 Learner’s response to item 4.3

The errors shown in vignette 15 above may be conceptual error because the response shown in the vignette shows that such learner has correctly multiplied the unit digit and correctly wrote 5 as the unit product. However, the procedural error ensued when the learner proceeded by multiplying the unit digit in the 2nd _{raw by the} tens digit in the 1st _{raw to get zero without adding 3 which was regrouped from units} product. The rest of the digits in the 3rd _{raw which are 9 and 2 written as the products} of multiplication of the unit digit in the 2nd _{raw by the digits in the 1}st _{raw show no} discernible link to the unit digit in 2nd _{raw and 1}st _{raw digits. The digits in the 4}th _raw also shows no discernible link to the tens digit in the 2nd _{raw and the digits in the 1}st raw. Such errors may be due to lack of conceptual understanding in long multiplication resulting from lack of mastery of multiplication facts and lack of understanding of place value (Hatfield et al, 2008; Maher & Muir, 2013). 16% of sample was only able to get

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the units product and correctly regrouped tens digit from the product but proceeded by writing responses with no discernible link to the whole numbers they were required to multiply as in vignette 15 above.