During the period from December 2014 to January 2015, a total of 543 practising teachers volunteered for the online questionnaire producing 475 fully completed entries and 68 incomplete entries. Of the incomplete entries, 3 fulfilled all elements of the questionnaire with the exception of responding to the interview option and were subsequently retained, thus producing a total of 478 completed entries. All of the remaining entries were deleted.
The online data were exported directly from Survey Monkey to SPSS. Nevertheless, data cleaning procedures were applied in line with advice offered by Pallant (2013, p. 44), who warns that “it is important to spend the time checking for mistakes initially, rather than trying to repair the damage later”. To reduce response bias, all negatively worded statements (i.e. questionnaire items 18 to 34 respectively) were reversed to allow computation of an overall total mathematical beliefs score for the 39 items and for summation of each of the five sub- domain belief factors. A five point Likert scale was used to offer a choice of responses from ‘strongly agree’ to ‘strongly disagree’. Correspondingly, a score of 1 was assigned to the
‘strongly agree’ option and a score of 5 to the ‘strongly disagree’ option respectively. This numerical system produced a theoretical range of total mathematical beliefs scores from 39 (most favourable) to 195 (least favourable).
Internal consistency reliability
The Cronbach’s alpha coefficient was used to estimate how well the items that reflect the same construct yield similar results. This main study reports values for each of the five beliefs factors as 0.704, 0.759, 0.728, 0.789 and 0.699 respectively with an overall high coefficient value of 0.884. The 22 positive items (i.e. social-constructivist, problem-solving and collaborative orientation to the nature of mathematics, the learning of mathematics and the teaching of mathematics) measured 0.851. Likewise, the 17 negative items (i.e. static- transmission and mechanistic-transmission orientation to the nature of mathematics, the learning of mathematics and the teaching of mathematics) recorded a value of 0.817. Pallant (2013, p. 104) maintains that, “Values above .7 are considered acceptable; however, values above .8 are preferable”. Though, Field (2013) argues that it is more germane for a researcher to think about what obtained values mean within the context of their own research, opposed to applying any ‘general guidelines’. Accordingly, I identified similar studies of teachers’ mathematical beliefs with comparable overall coefficient measurements (e.g. Peterson et al., 1989; Van Zoest, Jones & Thornton, 1994) leading me to conclude acceptance of the computed result obtained in this study.
Parametric or non-parametric?
For a study of this nature, it may be natural to select from a range of non-parametric statistical techniques such as the Mann-Whitney U-test or the Wilcoxon signed rank test on the basis of utilising ordinal data, since Likert Scales are coded accordingly. Previous studies of teachers’ beliefs have employed these types of non-parametric tests (e.g. Jamieson-Proctor & Byrne, 2008; Rajabi, Kiany & Maftoon, 2011; Ampadu, 2014). Controversially, many authors promulgate conflicting statistical advice for researchers in this regard (e.g. Jamieson, 2004; Carifio & Perla, 2007; Norman, 2010; Brown, 2011). Though, what appears to be in harmony is that parametric tests are more powerful and exhibit additional applications than non- parametric tests (McCrum-Gardner, 2008; Field, 2013). Since the majority of previous studies involving teacher’s mathematical beliefs have been statistically analysed using parametric methods (e.g. Van Zoest, Jones & Thornton, 1994; Stipek et al., 2001; Barkatas & Malone, 2005; Yates, 2006; Yu, 2008; Depaepe, De Corte & Verschaffel, 2010; Memnun, Hart & Akkaya, 2012), it suggests that the belief construct is normally distributed.
6.1.1 Exploratory factor analysis
Factor analysis seeks to reduce or summarise a compilation of variables into a smaller set of dimensions termed factors or components. In this study, 39 items of the positive and negative scale were subjected to principle components analysis (PCA) in order to explore the nature of previously unknown variables to seek underlying patterns, clusterings or groupings. Prior to performing PCA, the suitability of data for factor analysis was assessed as follows:
Sample size
This is determined by considering a minimum sample size or a ratio of subjects to variables. Comfrey & Lee (1992, p. 317) suggest that “the adequacy of sample size might be evaluated very roughly on the following scale: 50 – very poor; 100 – poor; 200 – fair; 300 – good; 500 – very good; 1000 or more – excellent”. Whereas, Nunnally (1978) advises that the subject to item ratio should be at least 10:1, however this recommendation is not supported by published research. Irrespectively, exercising both distinctive approaches, the sample size of 478 is comfortably ‘good’ from a magnitude perspective and equally acceptable from an item ratio viewpoint.
Factorability of the correlation matrix
Inspection of the correlation matrix, as advocated by Tabachnick & Fidell (2014), revealed the presence of many coefficients of 0.3 and above. The Kaiser-Meyer-Olkin measure of sampling statistic was 0.903, generously exceeding the recommended minimum value of 0.6 (Kaiser, 1970, 1974) and Bartlett’s Test of Sphericity (Bartlett, 1954) reached statistical significance [χ2
(741) = 6057.958, p < 0.001], supporting the factorability of the correlation matrix.
Factor extraction
Factor extraction involves determining the smallest number of factors than can be used to best represent the interrelationships among a set of variables. Several techniques can be used to assist in this decision making process; Kaiser’s criterion, scree test and parallel analysis (Pallant, 2013).
Kaiser’s criterion
Kaiser (1960) recommended retaining all factors with eigenvalues greater than 1. According to Field (2013, p. 677), “This criterion is based on the idea that the eigenvalues represent the amount of variation explained by a factor and that an eigenvalue of 1 represents a substantial
amount of variation”. Principle component analysis revealed the presence of eight components with eigenvalues exceeding 1, explaining 20.338%, 11.429%, 7.078%, 3.536%, 3.311%, 2.957%, 2.657% and 2.606% of the variance respectively (Appendix G).
Cartell’s Scree Test
Cartell’s (1966) scree test is considered to be the best choice according to Field (2013) and involves plotting each of the eigenvalues of the factors and retaining all factors above the elbow. Conversely, Tabachnick & Fidell (2014, p. 697) caution that, “Unfortunately, the scree test is not exact; it involves judgment of where the discontinuity in eigenvalues occurs and researchers are not perfectly reliable judges”. An inspection of the scree plot obtained (Figure 6.1) revealed a clear break after the third component, and it was decided to retain three components for further investigation.
Figure 6.1 Scree plot
Parallel analysis
This involves comparing the magnitude of the eigenvalues with those obtained from a randomly generated data of the same size. For this analysis, I employed Monte Carlo PCA software which showed only three components with eigenvalues exceeding the corresponding criterion values (please refer to Appendix H and Table 6.1 respectively) for a randomly generated data matrix of the same size (39 variables
478 respondents). Therefore, the results of parallel analysis validate my decision from the scree plot to retain three factors for further investigation.Table 6.1 Comparison of eigenvalues from PCA and criterion values from parallel analysis Component
number
Actual eigenvalue from PCA
Criterion value from parallel analysis Decision 1 7.932 1.5790 Accept 2 4.457 1.5147 Accept 3 2.760 1.4693 Accept 4 1.379 1.4276 reject 5 1.291 1.3872 reject 6 1.157 1.3509 reject 7 1.036 1.3188 reject 8 1.016 1.2861 reject
Factor rotation and interpretation
To aid in the interpretation of these three components, direct oblimin rotation was performed. The rotated solution revealed the presence of a simple structure (Thurstone, 1947), with three components showing a number of fairly strong loadings and all variables loading substantially on only one component. This can be observed from the Pattern Matrix (Appendix I) and the Structure Matrix (Appendix J). To determine the strength of the relationship between the three factors, examination of the component correlation matrix was carried out. This revealed very weak positive affects between the three factors (r = 0.054, 0.147 and 0.140 respectively). The results of this analysis highlight the presence of three distinct mathematical belief systems as follows:
1. A social constructivist, problem solving and collaborative orientation; 2. A social constructivist, problem solving and static transmission orientation; 3. A static and mechanistic transmission orientation.
Further investigation will help to determine which belief system is associated with each sector and homogenous group.