4.10 Data analysis procedures
4.10.1 The LFUP questionnaire analysis
The LFUP questionnaire was designed such that learners’ understanding of the functions of proof was represented by numbers for quantitative analysis. Although numbers were assigned to learners’ demographic data (for example, gender, home language, and, grade class), they were merely labels to indicate the differences between these categories of learners. Thus, they required numeric measures of analysis. There exists no consensus amongst scholars as to whether Likert data should be analysed with parametric statistics such as the t-test for dependent means or nonparametric statistics such as the Wilcoxon Signed Ranks test (Carifio & Perla, 2008). In this
Research methodology Data analysis procedures
study, the Likert scale was as treated as eliciting interval data and therefore amenable to parametric statistical measures.
On the one hand, if the Likert items were treated as individual items, the data were to be analysed as ordinal; therefore nonparametric measures applied. On the other hand, when multiple Likert items were summed together to describe an attribute (and therefore data considered to be measured on interval scale) parametric measures were appropriate. Another reason for using parametric statistical measures in the LFUP scale was the assumption that, as in psychology research, distributions in education research often approximate a normal curve (Aron, Aron, &
Coups, 2014). In addition, the sample was regarded as normally distributed because of the large number of learners who participated in the investigation. Further, ‘the Likert scale (“strongly agree” to “strongly disagree”) illustrates a scale with theoretically equal intervals among responses’ (Creswell, 2012, p. 167).
According to Clason and Dormondy (1994), numbers in Likert scales presumed the existence of underlying continuous variables. Thus, the Likert-type interval scales on LFUP were treated as ratio scale (Austin, 2007). Thus, the five-point LFUP questionnaire responses with five subscales (factors or dimensions) of three to seven items each were be treated as Likert scales where: 1=strongly disagree; 2=disagree; 3=undecided; 4=agree; and, 5=strongly agree. The
“undecided” option was included on the basis that a respondent may truly hold no particular view about an item and if this option is absent, they may choose to respond to the question thus introducing bias in the data. Positively worded items signified agreement with the mathematical community and negatively worded items represented disagreement. Thus, the scoring of the LFUP scale was conducted according to the way in which the response reasonably reflected views in the mathematical community. Also, two items with “Leave this item blank” were added to the LFUP instrument to check on participants’ attentiveness while completing the questionnaire (Schommer- Aikins, Duell, & Barker, 2003). As already mentioned, LFUP was linked to the five-factor model (verification, explanation, communication, discovery, and systematisation) whose items were derived from research literature about proof functions.
Research methodology Data analysis procedures
The data were screened to test the presence of outliers and also assessed for linearity, normality and homoscedasticity through scatter plot matrix and boxplot. If outliers were found, the case(s) associated with them were eliminated if they only accounted for less than 5% of the total sample. If elimination were inappropriate, I minimised their effect through data transformation techniques such as square root transformation or logarithmic transformation.
Whether the data approximated a normal distribution was verified by using three tests: skewness and kurtosis z-value (ratio with standard error) which must lie between – 1.96 and +1.96 if data distribution is normal and also used the Shapiro-Wilk test for p > .05 (Wilson & MacLean, 2011).
The research by Shongwe and Mudaly (2017) was useful in determining and assessing the degree to which the LFUP instrument is unidimensional. Unidimensionality reflects that a scale taps a single composite construct (Streiner, 2003). Having obtained a factor structure that confirmed homogeneity – the existence of unidimensionality in the sample of items – I then proceeded to determine Cronbach’s alpha coefficient. The internal consistency reliability, Cronbach alpha, was calculated to determine the degree to which each item on the LFUP scale measured the same construct. Alpha is the mean inter-item correlation measuring internal reliability; determining how closely related a set of items measure the same construct when they are considered as a group. However, since there were five subscales in the LFUP questionnaire, the internal consistency was tested on each subscale rather than on the whole instrument only.
The rationale for determining alpha is that it is the only measure of reliability that can be determined with much less effort because it does not require test-retest (Streiner, 2003; Tavakol &
Dennick, 2011). Test–retest reliability involves the administration of a measure to the same group a second time and comparing the two scores (Kline, 2011). In the final analysis, the data were subjected to PAF to test the key assumption that there is one unique factor for each item which affects that item but does not affect any other items. In the factorial ANOVA where the means of three groups of learners (gender and resources), the homogeneity of variances (equal amount of variability of the scores of three groups of schools) assumption could not be assumed because the p-value associated with Levene’s statistic was lower than .05. However, I proceeded to perform independent factorial ANOVA on the groups because it turns out that in practice the test gives
Research methodology Data analysis procedures
almost accurate results even when there are fairly large differences in the population variances, particularly when there are equal or near numbers of scores in the groups (Aron et al., 2014, p.
321).