In choosing the school within which my study was conducted, I looked at those schools that were part of the WMC-S project with a large number of learners who had written the test so that I would have an adequate sample to analyse. The other reason for choosing VSS was that it is in close proximity to where I live and work.
My study consists of three sets of data i.e. 29 test scripts for Grade 9 learners, 29 test scripts for the same learners who were in Grade 10 in 2012, as well as interviews with 7 Grade 11 learners. The 7 learners from Grade 11 were selected from the sample of the 29 learners. In 2011 the entire enrolment of Grade 9 learners in the school who were willing to participate wrote the test. In Grade 10, there were Mathematics as well as Mathematical Literacy learners who wrote the tests.
My analysis consists of two phases. The first phase consists of analysing the WMC-S test scripts in an attempt to address the first research question.
The second phase of the study encompasses the analyses of interviews with learners (who were in Grade 11) who were selected from the original sample, in order to address the second, third and fourth research questions which are:
What errors are made by learners?
Which errors continue from Grade 9 to Grade 11? Which errors fade away by Grade 11?
All 7 of the selected learners continued with Mathematics in Grades 10 and 11.
3.5.1. Selection of Sample for Phase One
Researchers from the WMC-S coded the test scripts according to a uniform coding system (see appendix C). Each of the questions in the algebra component was coded by the
members of the WMC-S unit, using codes for correct response, incomplete response,
incorrect response and unanswered. These results were entered on a spread sheet. I added
on a column to reflect the total number of correct responses. Each learner’s total was then divided by the total number of questions in the algebra section (33) to convert the mark to a percentage. Those learners in Grade 10 who had obtained a mark above 40% were then selected for script analysis.
The decision to choose learners who obtained above 40% was made on the basis of having enough answers to analyse. A number of learners left answers blank or incomplete. I therefore felt that those with marks above 40% would provide me with sufficient information from their answers to investigate their errors. The other motivation in choosing those above 40% in Grade 10 was so that I could see if individual learners had progressed through the levels from Grade 9 to Grade 10. Only 30 of the 212 learners in Grade 9 obtained a mark greater than 40% whereas this number was 53 in Grade 10. The marks of those learners who were selected in Grade 10 ranged from 40% to 93%. This wide range of selected learners in terms of their performance in the test would provide me with the necessary data to ascertain which errors were persistent and which were not. It was therefore necessary to consider those learners in Grade 10 who had made some improvement but who still made errors. This would make it possible for me to investigate those errors that are persistent as opposed to those that have disappeared.
3.5.2. Selection of Sample for Phase Two
After analysing the results of the written work of learners in the ICCAMS part of the WMC-S test, it was found that most learners (19 out of a total of 29 i.e. 66% of learners) progressed from a lower level to a higher level from Grade 9 to Grade 10. This is not surprising as the learners had written the same test the year before and had learned more algebra during the course of the year. I decided to focus on problems experienced by the learners with level 2 questions during the interviews in the second phase of my study. My justification for this is that it was unexpected that 9 learners progressed through more than 1 level. For example, there were 2 learners who were on level 0 in Grade 9 but who progressed to level 3 in Grade 10. Even more surprising was that 3 of these 9 learners progressed from level 1 to level 3 but were unable to answer the minimum number of questions at level 2 to be on level 2. In other words, these learners experienced problems with some of the level 2 questions and were not able to correctly answer at least 4 out of
the 6 questions at level 2. This is unexpected as, according to Hart (1981), success at a lower level is a pre-requisite for success on a higher level. In addition, there were another 2 learners who moved down a level. Both of them moved from level 2 in Grade 9 to level 1 in Grade 10. This is also unexpected. The other interesting observation is that there were more learners on level 1 and 3 than on level 2 in Grade 9, while in Grade 10 there was the same number of learners on level 1 and 2 but more than this number on level 3. This is outlined in the results for phase 1 of the study.
I have selected particular learners for the interviews using criterion sampling by incorporating the criteria outlined below:
One learner who moved down a level i.e. from level 2 to level 1
Two learners who progressed from level 1 to level 3 without first reaching level 2 Two learners who remained on level 1 in Grade 9 and 10
One learner who remained on level 2 in both Grades 9 and 10
All of the above learners were selected since 5 of the learners experienced some form of problem with level 2 questions as is evidenced above. The last learner was selected as she was on level 2 and her answers could be used for comparison purposes between her and those who were unable to answer the minimum number of questions correctly to be on level 2.
According to Sandelowski (2000) criterion sampling is “a type of purposeful samplingof cases based on preconceived criteria such as scores on an instrument”. She maintains that purposeful sampling is used in order to collect more data, using instruments such as interviews or observations, from the participants in order to enhance understanding of “the information rich case” as well as to elaborate on or clarify the results. Criterion sampling is therefore an appropriate method of sampling in my study as the sample selected not only allowed me to investigate the errors made by learners but more importantly to address the problems centred on the answering of level 2 questions.