Discussion

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

Employing AC>BD^* na

Formulating AC>BD na

Interpreting AC>BD na

Affects

Self-efficacy NS na

Interest BCD>A na

Engagement D>A na

Conditions

SES 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.

(Tan & Tan, 2015), and is similar to the theoretical or pedagogical approach, in which the higher-order conception of g-teaching is fully integrated with e-teaching (Tømte et al., 2015). The moderation profile appears to be the most effective e/g-teaching profile in terms of its positive effects on both student cognition and affect among the four identified profiles (Tables 3–5). E-teaching increases not only student interest but also noise because of novel materials/tasks and collaborative works (Watts & Lloyd, 2004). ICT use may increase difficulty in teaching, and the difficulty may be reduced by high-quality teacher direction in design and management.

Parsimony e/g-teaching (Profile A). This approach, experienced by most students, combines slight use of e-teaching and slightly medium but below-average use of g-teaching in Taiwanese mathematics classrooms (Table 2). The parsimony profile benefits student cognition but may be at the expense of student affect, having the least affect among the four profiles (Table 5). The parsimony profile depicts mathematics teaching and learning to be serious and boring. Parsimony teachers have medium, below-average degrees of g-teaching, in addition to having the highest degree of student orientation, followed by formative assessment and teacher direction. Parsimony e/g-teaching may reflect Confucianism (emphasizing respectable teachers’ serious roles) in Taiwanese society and recent constructivism (emphasizing student-centered teaching) in Taiwanese mathematics curricula (Chiu, 2011).

Conservation e/g-teaching (Profile B). Conservation teachers frequently engage in direction supplemented by formative assessment and student orientation. The conservation profile appears to partially reflect traditional/activating/teacher-centered approaches to e/g-teaching, in which e-teaching is seldom used or only for traditional purposes such as presenting materials (Lan et al., 2012). The conservation profile benefits affects but not cognition, a result different from previous research findings that high-quality g-teaching behaviors benefit cognition (Hinostroza et al., 2011; Thorvaldsen et al., 2012). One reason may be that effective g-teaching, which implies intense affective teacher–student relations, relates to affect (e.g., engagement) more than to cognition (e.g., achievements) (Roorda, Koomen, Spilt, & Oort, 2011).

Liberal e/g-teaching (Profile D). In a liberal classroom, class time is mostly allocated for teacher and student ICT use with high student orientation and medium, above-average formative assessment and teacher direction (Table 3). This profile appears to slightly reflect the constructivist approach to integrating e- and g-teaching (Park et al., 2015). The liberal profile benefits student interest, engagement, and latent affect (compared with Profile A), but its benefit to latent affect is slightly less than those of

the moderation or conservation profiles (Table 5). The slight benefit of the liberal profile to affect, but not to cognition, is unsatisfactory because constructivist approaches to e-teaching can transform educational practices, which is advocated by scholars (e.g., Chai et al., 2012). ICT use in mathematics teaching requires teachers to extend their expertise from g-teaching to e/g-teaching, especially when teachers aim for a liberal e/g-teaching profile. Teachers with a liberal profile may need more professional development and support than those with the other three profiles. How to transform the liberal profile from benefiting only affect to benefiting both affect and cognition remains a concern for educators and future research.

Few Profile Differences in Conditions

Profile differences occur in school ICT availability, not in SES or home ICT availability (Table 3).

Future research should consider other potential conditions that may play a role in e-teaching, such as digital learning materials, school management, and ICT technical support (Cuckle & Clarke, 2002;

Shohel & Kirkwood, 2012; Somyürek, Atasoy, & Ö zdemir, 2009).

Only the parsimony profile was observed to involve low school ICT availability, which may partially explain the low ICT use revealed by the parsimony profile (Table 3). However, after being conditioned by school ICT availability, the parsimony profile was still observed to involve high student cognition and low affect (cf. Table 5 for comparison between the MANOVA and SEM solutions). The results regarding the conditioning effects of school ICT availability on cognition and affect imply that g-teaching behaviors play more roles in student learning outcomes than simple ICT use. ICT use in teaching may need to be closely linked to g-teaching for achieving traditional learning objectives of subject matters. Future research should validate this speculation.

Limitation and Suggestions for Education and Future Research

A limitation of this study is that the three affective measures appeared to perform differently, which may reduce the model fit to data because of measurement errors. In particular, self-efficacy acted differently from the other two affective measures (i.e., interest and engagement). The results show that self-efficacy exhibited higher correlations with cognitive measures than the other two affective measures did (Table 1). Furthermore, the profiles differed in interest and engagement but not in self-efficacy (Table 3). However, self-efficacy had a higher factor loading than interest and engagement did (Figure 2). All the results imply that different affective measures of mathematics may represent dissimilar constructs such as different beliefs, attitudes, and emotions (McLeod, 1992, 1994). Future research can investigate the diversity and complexity of affective constructs and their interaction with diverse

cognitive measures and e- and g-teaching profiles.

The second limitation may be that the four e/g-teaching profiles were identified by statistical methods. Future research should investigate the validity of the four teaching profiles in real educational settings and interpret the identified four teaching profiles by using real cases in actual mathematics classrooms.

The four e/g-teaching profiles identified in this study and their interaction with student cognitive and affective learning outcomes may provide valuable suggestions for mathematics education practices.

Moderation e/g-teaching, which is moderately open to using ICT and diverse general teaching methods, appears to benefit students most in both cognitive and affective mathematics learning outcomes. The results suggest that the comprehensive but moderate use of diverse teaching methods, including e-teaching, may be one of the most favorable choices for developing effective teaching for student learning outcomes. Future research should validate whether moderate e/g-teaching is superior to parsimony teaching in terms of affective learning outcomes and whether it is superior to both conservation teaching and liberal teaching in terms of cognitive learning outcomes.

Conclusion

The major contribution of this study is the use of LPA to identify student-perceived e/g-teaching profiles (latent nonlinear relationships between e- and g-teaching behaviors) that successfully demonstrate the differences in students’ mathematics cognition and affect. First, the identified four e/g-teaching profiles contribute new knowledge to mathematics education research. The four e/g-e/g-teaching profiles identified in this study are outlined as follows: parsimony (low e-teaching and medium, below-average g-teaching), conservation (low e-teaching and high g-teaching, particularly in teacher direction), moderation (medium e-teaching and g-teaching), and liberal (high e-teaching and medium, above-average g-teaching of student orientation, formative assessment, and teacher direction, in descending order). The moderation profiles appear to be similar to the balance and pedagogical approaches and represent a thoughtful, considerate, and cautious use of e- and g-teaching. The conservation profile tends to reflect the traditional/activating/teacher-centered approaches to integrating e- and g-teaching, and the liberal profile reflects the constructivist/facilitating/student-centered approaches. The parsimony profile appears to be new in the literature and limited by school ICT availability.

Second, linking the identified teaching profiles with cognitive and affective learning outcomes provides practical implications for mathematics education. MANOVA and SEM were determined to generate similar results regarding the differences between the profiles in terms of learning outcomes;

however, MANOVA focused on elements and SEM focused on constructs of cognitive and affective outcomes. The moderation profile benefits both student cognition and affect. The parsimony profile benefits cognition but may harm affect. The two extreme profiles, conservation and liberal, benefit only affects. The literature tends to advocate constructivist e-teaching practices. However, the current study, based on data from a real educational setting, suggests that moderate ICT use with the merit of diverse g-teaching behaviors (in particular, teacher direction) may optimize student cognition and affect.

Finally, the successful use of LPA to identify distinct teaching profiles and the use of MANOVA and SEM to link teaching profiles with learning outcomes contribute a methodology to future research.

Future educational research can use similar statistical methods to find context-based, effective teaching profiles for predicting diverse learning outcomes.