one model (Green & Thompson, 2006). The major criteria for determining model goodness of fit included (1) a root mean square error of approximation (RMSEA) lower than .10, (2) comparative fit index (CFI) higher than .90, and (3) nonnormed fit index (NNFI) higher than .90 (Hair, Black, Babin,
& Anderson, 2010). Because of the large sample size in this study, the conventional criterion, a nonsignificant chi-square (χ2), would be easily violated (Bollen & Long, 1993). Thus, χ2 did not serve as the major criterion in this study.
The MANOVA results showed that the four profiles differed in some e/g-teaching behaviors (Wilks
= .22; F(4,3973) = 3386.60, p < .0005, η2 = .77). The ANOVA and TukeyHSD post hoc test results indicated significant differences between the four profiles in the four e/g-teaching behaviors (Table 3).
As shown by the last column in Table 3 for ICT use, students in Profiles C and D experienced more e-teaching than those in Profiles A and B (CD > AB); students in Profile D experienced more ICT use in teaching than those in Profile C (D > C). The same interpretation methods applied to formative assessment (BD > A; B > CD; D > C), student orientation (BD > A; B > CD; D > C), and teacher direction (BCD > A; B > CD). The differences between the profiles in the four e/g-teaching behaviors had medium to large effect sizes (η2 = .08 for student orientation to .79 for ICT use). On the basis of these results, the profiles are designated and interpreted as follows.
Parsimony e/g-teaching (Profile A). The parsimony approach to e/g-teaching involves low e-teaching (ICT use) (M = –.77) and medium, below-average g-e-teaching (Ms = –.22, –.11, and –.25 for formative assessment, student orientation, and teacher direction, respectively). In other words, parsimony teachers do not intensively use either e-teaching or g-teaching strategies in mathematics classrooms. Most students (75% = 2980/3978) experienced parsimony e/g-teaching.
Conservation e/g-teaching (Profile B). The major characteristic of conservation e/g-teaching is high degrees of g-teaching behaviors, with extremely high teacher direction (M = 2.55) and frequent use of formative assessment (M = 1.34) and student orientation (M = 1.10). However, conservation teachers seldom use ICT (M = -.76). Approximately 4% (=163/3978) of the students experienced conservation e/g-teaching.
Moderation teaching (Profile C). The moderation profile revealed medium degrees of e/g-teaching in all four behaviors, with ICT use as the highest (M = .52), followed by teacher direction, formative assessment, and student orientation (M = .11, –.09, and –.10, respectively). Approximately 9% (= 348/3978) of students experienced moderation e/g-teaching.
Liberal e/g-teaching (Profile D). The major characteristic of liberal e/g-teaching is intensive ICT use (M = 1.07) with emphasis on student orientation (M = .41) supplemented by formative assessment and teacher direction (Ms = .18 and .03, respectively). Approximately 12% (= 487/3978) of the students experienced liberal e/g-teaching.
Profile Differences in Explicit Elements of Cognition, Affect, and Condition
The MANOVA results revealed that profile differences occurred in some cognitive elements (Wilks = .99; F(3,3974) = 14.00, p < .0005, η2 = .01). In addition, the ANOVA results showed significant
differences between the four profiles in all the three cognitive elements (Table 3). The TukeyHSD post hoc test results indicated that students who experienced Profiles A and C exhibited higher employing, formulating, and interpreting abilities in mathematics than did those who experienced Profiles B and D.
The profile differences in the three cognitive elements had small effect sizes (η2 = .02 for all three cognitive elements).
The MANOVA results showed some profile differences in the three affects (Wilks = .98; F(3,1968)
= 14.55, p < .0005, η2 = .02). Furthermore, the ANOVA and TukeyHSD post hoc test results revealed no profile difference in self-efficacy (η2 = .00) but significant differences in interest (Profiles B, C, and D > Profile A; η2 = .02) and engagement (Profile D > Profile A; η2 = .01).
The MANOVA results showed some profile differences in the conditions (Wilks = .99; F(3,3951) = 18.03, p < .0005, η2 = .01). Moreover, the ANOVA and TukeyHSD post hoc tests revealed no profile differences in SES and home ICT availability (η2 = .00 for both) but a significant difference in school ICT availability (Profiles B, C, and D > Profile A; η2 = .01). The results imply that profile differences may only be conditioned by school ICT availability, a result suggested in previous research (Cuckle &
Clarke, 2002). Thus, the subsequent SEM analyses included only school ICT availability as the conditioning variable in the models.
Profile Differences Predicting Latent Cognition and Affect
SEM was applied to analyze six models (configured as Figure 2), with every two profiles being dummy coded to examine their differences in the latent constructs of cognition and affect; the two constructs were set to be correlated, a general phenomenon in mathematics education (Chiu, 2012a).
The SEM results showed that the six models were acceptable, as indicated by all the NNFI and CFI values being higher than .90 and RMSEA values being equal to .10, except for the RMSEA value (= .11) of Model 5 (Table 4).
Descriptive Statistics and Results of ANOVA and TukeyHSD Post Hoc Tests for the 4 Identified E/G-Teaching Profiles
Profile A: Parsimony e/g-teaching
Profile B: Conservation e/g-teaching
Profile C: Moderation e/g-teaching
Profile D: Liberal e/g-teaching
ANOVA TukeyHSD
post hoc test N1 Mean SD N2 Mean SD N3 Mean SD N4 Mean SD F(df1,df2) p η2 p < .05
E/g-teaching behaviors
ICT use 2980 -.77 .05 163 -.76 .09 348 .52 .35 487 1.07 .93 4960.00 <.0005 .79 CD>AB;D>C* Formative assessment 2980 -.22 .89 163 1.34 .96 348 -.09 1.01 487 .18 .83 175.90 <.0005 .12 BD>A;B>CD;D>C Student orientation 2980 -.11 .91 163 1.10 1.41 348 -.10 1.00 487 .41 .89 120.80 <.0005 .08 BD>A;B>CD;D>C Teacher direction 2980 -.25 .89 163 2.55 .12 348 .11 1.05 487 .03 .98 507.50 <.0005 .28 BCD>A;B>CD Cognitions
Employing 2980 552.42 105.11 163 517.32 102.19 348 562.08 105.33 487 516.08 114.77 22.89 <.0005 .02 AC>BD Formulating 2980 584.24 131.57 163 536.08 129.05 348 592.20 135.36 487 537.58 142.22 23.80 <.0005 .02 AC>BD Interpreting 2980 553.17 99.60 163 512.22 97.07 348 560.97 103.24 487 515.58 108.78 28.05 <.0005 .02 AC>BD Affects
Self-efficacy 1480 .16 1.20 97 .36 1.28 172 .30 1.05 231 .13 1.07 1.70 1.70 .00 NS Interest 1481 -.06 .94 98 .34 1.15 171 .20 .93 231 .26 .94 14.03 <.0005 .02 BCD>A Engagement 1484 .01 .97 98 .17 1.10 172 .21 .94 231 .27 1.02 .01 .0002 .01 D>A Conditions
SES 2972 -.38 .82 162 -.39 .87 348 -.37 .87 486 -.48 .87 2.17 .09 .00 NS Home ICT availability 2980 -.37 .90 162 -.22 .99 347 -.39 .85 487 -.26 1.03 3.092 .03 .00 NS School ICT availability 2971 -.27 .81 163 -.10 .83 348 -.07 .77 485 -.05 .90 15.69 <.0005 .01 BCD>A Note. *D > C = Profiles D > Profiles C (same interpretation methods applying to the others). Small effect size: .01 < η2 < .06; medium effect size: .06 < η2 < .14;
large effect size: η2 > .14 (Cohen, 1988, p. 283). F(df1,df2) = F(3,N1+N2+N3+N4-4); df = degree of freedom. NS = not significant.
Figure 2 Structural model for the effects of profile differences on latent cognition and affect. Model 1 (Table 4) served as an example with Profile A coded as 0 and Profile B as 1. All the parameter estimates presented are significant at p = .05.
Table 4
Parameter Estimates Obtained by SEM Model
Relation
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Profile A(0)
Profile B(1)
Profile A(0) Profile C(1)
Profile A(0) Profile D(1)
Profile B(0) Profile C(1)
Profile B(0) Profile D(1)
Profile C(0) Profile D(1) school ICT -> profiles .05 .08 .09 .01 .02 .01 profiles -> cognition -.08 .03 -.12 .21 .00 -.21 profiles -> affect .07 .07 .06 -.03 -.07 -.04 cognition <-> affect .75 .74 .75 .76 .74 .75 cognition ->employing .98 .98 .98 .96 .98 .96 cognition ->formulating .97 .97 .96 .95 .97 .95 cognition ->interpreting .95 .95 .94 .93 .95 .93 affect -> self-efficacy .82 .82 .81 .83 .83 .81 affect -> interest .55 .55 .56 .54 .54 .55 affect -> engagement .63 .63 .64 .62 .62 .64 Fit indexes
χ2 780.80 778.84 798.29 802.97 847.99 792.99
df 18 18 18 18 18 18
RMSEA .10 .10 .10 .10 .11 .10
NNFI .95 .95 .95 .95 .94 .95
CFI .97 .97 .97 .97 .96 .97
Note. The underlined figuresare significant at p = .05.
employing
Cognition
.98
formulating .97
Profile A (0) Profile B (1)
interpreting -.08 .95
SchoolICT availability
.75
self-efficacy .07
interest .82
Affect
.63
engagement .55
.05
The factor loadings for cognition leading to employing, formulating, and interpreting (.93–.98) were large, and those for affect leading to self-efficacy, interest, and engagement (.54–.83) were acceptable (above .30; Costello & Osborne, 2005, p. 3). The two constructs (cognition and affect) were highly correlated (.74–.76), as suggested by previous research (Chiu, 2012b). These results suggest that SEM is suitable for examining profile differences because the six measures of cognition and affect have underlying constructs. SEM also allows for including school ICT availability as a condition. School ICT availability plays significant roles for models including Profile A (i.e., Models 1–3 in Table 4), with Profile A having less school ICT availability than Profiles B, C, and D (parameter estimates = .05, .08, and .09 respectively).
Both RQs 2 and 3 focused on the differences between the profiles in learning outcomes of cognition and affect, but RQ 2 focused on those in explicit elements and RQ 3 focused on those in latent elements.
Table 5 presents a comparison of the answers to RQs 2 and 3. The answers to RQs 2 and 3 were the same in explicit and latent cognition (Profiles A and C > Profiles B and D), interest, latent affect (Profiles B, C, D > Profile A for both), and school ICT availability (Profiles B, C, and D > Profile A). The answers to RQs 2 and 3 differed only in affects, with Profile D having more engagement than Profile A but having less latent affect than Profiles B and C. One reason for the slightly unstable answers about affects may be that the factor loadings of the three affective elements were not as large as those of the three cognitive elements (Table 4).
Table 5 presents a comprehensive description of profile differences. The results were stable for cognition. Profiles A and C were determined to benefit cognitive learning outcomes more than Profiles B and D did. In affective learning outcomes, the profile differences were relatively unstable, which means that the four profiles performed slightly differently between different observed measures and the latent measure. Nevertheless, a general trend still occurred: Profile D was determined to benefit affect most, followed by Profiles B and C, and then Profile A. Profile differences in conditions were stable:
the only difference occurred in school ICT availability. Detailed interpretations of the four profiles and their differences in learning outcomes are presented in the Discussion section.
Table 5
Test Results of Profile Differences in Cognition, Affect, and Condition Obtained by MANOVA and SEM
MANOVA and related post hoc tests (Table 3) SEM(Table 4)
Cognitions
na AC>BDEmploying AC>BD* na
Formulating AC>BD na
Interpreting AC>BD na
Affects
na BCD>A; BC>DSelf-efficacy NS na
Interest BCD>A na
Engagement D>A na
Conditions
na naSES NS na
Home ICT availability NS na
School ICT availability BCD>A BCD>A Note. *AC > BD = Profiles A and C > Profiles B and D (same interpretation methods applying to the others);
NS = not significant; na = not available.