CHAPTER 7 CHARACTERISTICS OF THE SCHOOL AND THE CLASSROOM DATA
7.4.1 Exploring and Confirming a Structure for Teacher Beliefs
Belief statements in the teacher survey questionnaire were formulated on the basis of findings from the Askew et al. (1997) study. Therefore, the basis for belief statements in the teacher questionnaire was empirical rather than theoretical. Consequently, the validity of instructional constructs relevant to belief statements required exploration. A sample of 89 teachers is rather small for factor analysis (Comrey & Lee, 1992). Yet, the author proceeded because the sample achieved the minimum 1:5 subject to item ratio (Gorsuch, 1983). More recently, Ko and Sammons (2010) found that a small sample of 79 teachers could produce a six-factor model using confirmatory factor analysis with 30 items (from a scale of 45 items). In the current study, alpha factoring techniques with varimax rotation were used to explore the possibility that items would group around three factors (transmission, discovery, connectionist). This solution failed to converge. During the next round, items were not constrained. This resulted in a six-factor solution. Table 7.8 gives factor loadings from this solution for items with a loading of .40 and over.
Table 7.8 – Exploring a Structure for Teacher Beliefs
Skills (item) 1 2 3 4 5 6
Pupil misconceptions must be remedied by reinforcing the correct method (19)
.782 Pupils must be taught standard methods
and procedures (23)
.425 Pupils learn mathematics by working
sums out on paper (42)
.845 Pupils do not need to be able to
read/write/speak English well to learn mathematics (46)
-.803
Table 7.8 – Exploring a Solution for Teacher Beliefs (continued)
Routines and Methods 1 2 3 4 5 6
Pupil misunderstandings need to be made explicit and improved upon (34)
.777 Teachers must help pupils refine their
problem-solving methods (35)
.785 Talk, Readiness and Ability
Engaging pupils in meaningful talk is the best way to teach mathematics (8)
.600 Teaching is best based on verbal
explanations (16)
.431
.435
Pupils make mistakes because they are not ready to learn mathematics (24)
.487 All pupils are able to learn mathematics
(36)
.525 Understanding
Pupils learn mathematics by reasoning (44)
.730 Pupils need to learn to understand the
mathematics context to solve a problem (45)
.855 Connections/Materials and Methods
Pupils need to be taught how topics link (38)
.648 Pupils need to learn to solve problems by
using concrete materials (47)
.409 Pupils may be taught any method as long
as efficient (48)
.549 Other Routines/Methods
Teaching is best based on practical activities so that pupils discover methods for themselves (14)
.871
Pupils must be taught how to decode a word problem (11)
.909
The Kaiser-Meyer-Olkin (KMO) statistic describes the adequacy of the sample (as cited in Dziuban and Shirkey, 1974:359). Kaiser-Meyer-Olkin, refined an index for the interpretation of this statistic. He recommended that anything in the: .90‘s was
―marvelous‖, .80‘s ―meritorious‖, .70‘s ―middling‖, .60‘s ―mediocre‖ and .50‘s
―miserable‖. The six factors in this solution have a KMO of .748. Internal reliability, as indicated by the alpha statistic, is acceptable for each of the six factors in the above solution: ―Skills‖ (α = .735), ―Routines and Methods‖ (α = .876), ―Talk/Readiness and Ability‖ (α = .781), ―Understanding‖ (α = .754), ―Connections/Materials and Methods‖
(α = .779) and ―Other Routines/Methods‖ (α = .750). An item with a split loading was included with the factor upon which it next loaded the highest. Names given for each of the six factors describe, as much as possible, the reconfigured nature of items. The correlation matrix in Table 7.9 shows associations as generally weak (r is below .40).
Table 7.9 – Correlation Matrix for Teacher Beliefs
B8 B11 B14 B16 B19 B23 B24 B34 B35 B36 B38 B42 B44 B45 B46 B47 B48
B8 1.000
B11 .211 1.000
B14 .093 .112 1.000
B16 .416 .177 .002 1.000 B19 .132 .023 .066 .211 1.000 B23 .249 .020 .031 .141 .284 1.000 B24 .334 .116 .318 .095 .025 .258 1.000 B34 .047 .217 .384 .028 .057 .014 .316 1.000 B35 .075 .036 .292 .138 .029 .077 .242 .766 1.000 B36 .167 .186 .084 .135 .080 .266 .242 .200 .195 1.000 B38 .210 .138 .275 .123 .106 .237 .252 .194 .120 .005 1.000
B42 .216 .137 .226 .295 .172 .070 .241 .335 .167 .129 .236 1.000
B44 .196 .276 .104 .032 .149 .133 .098 .122 .009 .006 .023 .048 1.000 B45 .093 .148 .012 .250 .176 .263 .176 .073 .050 .106 .139 .233 .622 1.000 B46 -.151 -.101 -.035 -.186 -.547 -.258 .017 .002 -.209 .110 .111 -.322 .088 .051 1.000
B47 .084 .203 .095 .163 .251 .054 .053 .110 .043 .241 .185 .065 .013 .059 .001 1.000 B48 .056 .035 .243 .018 .028 .006 .177 .331 .226 .065 .332 .177 .210 .081 .117 .208 1.000
Cells in white mean that the coefficient r is not significant. Cells in orange mean that the coefficient r is significant at p < .001. Cells in yellow mean that the coefficient r is significant at p < .01. Cells in light blue mean that the coefficient r is significant p < .05
Structural equation modelling is more rigorous than exploratory factor analysis.
Confirmatory factor analyses, using the software AMOS, explored the structure associated with constructs underpinning the belief responses of Year 2 teachers.
Minimum sample size requirements are vexing in structural equation modelling (Brown, 2006). A sample of 89 teachers is below a critical n of 100 to 150 subjects (Ding, Velicer & Harlow, 1995). However, a ratio of one subject to five variables usually suffices for normal distributions (Bentler & Chou, 1987). Here, the model (for testing) postulates that there are six correlated factors: Skills Needed, Routines and Methods, Talk/Readiness and Ability, Understanding, Connection/Materials and Methods/Other Routines/Methods. The root mean square error of approximation (RMSEA) and the comparative fit index (CFI) describe fit. RMSEA values of less than .05 indicate good fit and values less than .08 represent reasonable errors of approximation (Browne & Cudeck, 1993). MacCallum et al. (1996) extend these cut- off points. Values between .08 and .10 indicate poor but acceptable fit. Browne and Cudeck (1993) and MacCallum et al. (1996) argue that this is more realistic than an exact fit of RMSEA = 0.00. The CFI index ranges from 0 to 1 and is a measure of the complete co-variation in the data (Byrne, 2001) and is not as affected by small sample sizes (Iacobucci, 2010). A CFI value >.90 is indicative of a well-fitting model but this was later revised to <.95 (Hu & Bentler, 1999).
The hypothesized solution did not fit as well with the structure of the local data (RMSEA = .098, CFI = .930, χ2 = 218.10, df = 152, p < .001). Three of the six factors:
―skills needed‖ (RMSEA = .020, CFI = .980, χ2 = 14.5, df = 5, p < .05), ―other routines/methods‖ (RMSEA = .046, CFI = .970, χ2 = 8.80, df = 3, p < .05) and
―routines/methods‖ (RMSEA = .046, CFI = .970, χ2 = 8.80, df = 3, p < .05) separately approached or achieved acceptability. Further attention was given to the items: ―pupils must be taught how to decode a word problem‖ (item 11) and ―teaching is best based on practical activities so that pupils discover methods for themselves‖ (item 14). Fit improved when item 11 was included with the factor ―skills needed‖ (RMSEA = .063, CFI = .973, χ2 = 22.20, df = 9, p < .01). Fit also improved when item 14 was included with the factor ―routines/methods‖. (RMSEA = .058, CFI = .950, χ2 = 66.5, df = 34, p
< .05). Figure 7.7 presents a valid model with items 11 and 14 included (RMSEA = .057, CFI = .960, χ2 = 66.5, df = 34, p < .001) in Figure 7.7.
Figure 7.2 – A Confirmed Structure for Teacher Beliefs Key: S = skills and U = understanding.
.64
S
B46
.52
46 1
B42
.33
42
B23
.72
23
B11
.46
11
.42
U
B35
.14
35
B34
.18
34
B48
.66
48 1
.50
B8
.79
8 1
.27 -.98
.98 1.00
.80 .55 1
1
1
1.00 .52
B16
.60
16
1 1 .39
1
7.4.1.1 Teacher Responses for Skills and Understanding
Figures 7.3 and 7.4, give percentage figures for teacher responses to belief statements from the validated factors of Skills and Understanding.
Figure 7.3 – Percent Responses of Teacher Beliefs from the Factor Skills
Most teachers agreed that: ―pupils must be taught how to decode a word problem‖
(item 11), ―pupil misconceptions must be remedied by reinforcing the correct method‖
(item 19), ―pupils learn mathematics by working sums out on paper‖ (item 42) and
―pupils may be taught any method as long as efficient‖ (item 48). Teachers tend to disagree that: ―pupils must be taught standard methods and procedures‖ (item 23) and
―pupils do not need to be able to read/write/speak English well to learn mathematics‖
(item 46). No teacher exhibited uncertainty for: ―pupils may be taught any method as long as efficient‖ (item 48).
In Figure 7.4 below, most teachers agreed that: ―engaging pupils in meaningful talk is the best way to teach mathematics‖ (item 8), ―pupil misunderstandings need to be made explicit and improved upon‖ (item 34) and teachers ―must help pupils to refine their problem-solving methods‖ (item 35). Most teachers disagreed that: ―teaching is best based on practical activities‖ (item 14).
Figure 7.4 – Percent Responses of Teacher Beliefs from the Factor Understanding