4.7
The researcher originally proposed to interview all mathematics teachers teaching in the FET phase at each of the selected schools, but this was not possible, as some of them could not avail themselves for the interview due to work commitments. The researcher also resorted to focus group interviews with the learners since the time allocated for research activities was not adequate for individual interviews. Language problems were also encountered despite thorough revisions of the instrument after the pilot survey. Learners at times failed to interpret the questions or struggled to express themselves clearly in their responses. Nevertheless, participants managed to provide adequate data for purposes of the study. Thus, the results were not affected.
182 Response rate
4.8
The response rate for learners’ questionnaire was 70.63% since participants completed the questionnaire in the presence of the researcher. Fifteen, 15(83%) out of a possible 18 teachers completed questionnaires which is greater than the threshold recommended by Fincham and Draugalis (2013).
Demographic data for learner participants Table 4-2: Demographic Data
Variable Frequency(N) Percentage (%)
Gender Male 245 43.4 Female 320 56.6 Age (Years) 11-15 137 24.3 16-20 428 75.7 Home Language Sepedi 499 88.3 Sesotho 10 1.8 Tshivenda 11 1.9 Xitsonga 20 3.5 Other Languages 25 4.4 Grade 10 200 35.4 11 215 38.1 12 150 28.5 Residential Area Urban 180 31.0 Semi-urban 199 35.2 Rural 161 33.8 Deep Rural 14 2.5 Participants
Demographic data about learner respondents shows that 245 (43, 4%) were males and 320(56.6%) were females. Only two age groups were observed. The majority 428 (75%) of the participants were in the 16-20 years category while the 11-15 years age category had 137(24, 3%). Sepedi was the dominant home language with 499(88.3%) while 66(11.7%) speak other local languages. The other languages had an insignificant combined representation. The initial plan was to draw an equal number (200) of learners from each grade level; however, it was not possible to obtain 200 learners from the sampled schools due to the dwindling number of learners taking mathematics at Grade 12. Thus 200(35.4%) learners were drawn from Grade 10, 215(38.1%) were drawn from
183
Grade 11 while 150(28.5%) were drawn from Grade 12. The participants were drawn from rural schools 175(31%, semi urban 199(35.2%) and urban 195(33.8%). Most participants (435(76.9%) were drawn from families with household sizes ranging from 4 to 5 people.
Cluster Analysis 4.9
Cluster analysis is a segmentation or taxonomy technique that is used to identify homogenous groups of participants based on their responses or experiences of a given phenomenon (Gopichandran & Chetlapalli, 2013). Cluster analysis methods provide means for classifying a given population into groups (clusters), based on similarity or closeness measures (Ragno, De Luca & Loele, 2007). A cluster analysis identifies what homogeneous groups exist among learners (for example learners can be classified according to their challenges with mathematical symbols as mild, moderate and severe difficulties). Cluster analysis is a grouping technique that identifies cases when the groups cannot be determined in advance. Cluster analysis is also interpreted as a multivariate analysis that divides data into groups or "clusters" of objects (sample plots) that are "similar" to each other (Lance & Williams, 1966). Two-step cluster analysis was preferred in this study since it is quick and automatically selects the number of clusters and groups or clusters based on their experience of the phenomenon.
4.9.1 Demographic variables
To check if demographic variables can be used as predictors of learners’ competency with mathematical symbols, an SPSS two-step cluster analysis procedure was used to analyse the importance of the each of the variables. The results of the analysis are show in figure 4.1 below.
184 Figure 4-1: Predictor importance indicators
SPSS Predictor Importance view shows the relative importance of each demographic variable in explaining how these variables affect learners’ level of competence with mathematical symbolism. This indicates how well the variable can differentiate different clusters. The view shows that variables such as grade, gender, residential area and age have a significant effect on the learners’ understanding of mathematical concepts together with their symbols.
0 0.2 0.4 0.6 0.8 1 1.2
Grade Gender Residentail Age Home language
Im p o rt a n ce Va lu es Demographic variables
185
Figure 4-2: Model summary
Figure 4.2 shows a summary of the cluster model and a positive Silhouette measure of cluster cohesion and separation. This measure lies in the ‘Fair’ category, which implies that the model is unbiased cluster. Eight demographic variables were clustered into three clusters. This summary also shows that the cluster quality was fair. The assessment of the quality of clusters was based on the criteria suggested by Kaufman and Rousseeuw (1990). In the model summary view, a good result equates to data that reflects Kaufman and Rousseeuw’s (1990) rating as either reasonable or strong evidence of cluster structure, fair reflects their rating of weak evidence, and poor reflects their rating of no significant evidence.
A silhouette coefficient of 1 means that all cases are located directly on their cluster centres. A silhouette coefficient of −1 means all cases are located on the cluster centres of some other cluster. A silhouette coefficient of 0 means, on average, cases are equidistant between their own cluster centres and the nearest other cluster. In this data set, the Silhouette coefficient is approximately 0.35 suggesting that the structure is weak and the researcher’s prior classification can be allowed in clustering the demographic variables.
186 Demographic Clusters
Table 4-3: Demographic clusters
Table 4-3 above shows three clusters of demographic variables:
Grade level and gender were the main inputs (predictor) importance variables while ethnicity and home size were the least inputs (predictor) importance variables. This means that ethnicity and home size can be dropped from the list of demographic variables.
Cluster 1 consists of 214(37.9%) learners, cluster 2 consists of 184(32.6%) learners and cluster 3 was made up of 167(29.6%) learners.