Corresponding author：Mei-Shiu Chiu，e-mail：meishiuchiu@gmail.com Received：9 January 2017;

Accepted：18 October 2017.

Chiu, M. S. (2017).

Identifying effective e-teaching and general mathematical teaching profiles to predict student mathematical cognition and affect.

*Taiwan Journal of Mathematics Education, 4(2), 69-94.*

doi: 10.6278/tjme.20171018.001

**Identifying Effective E-Teaching and General Mathematical **

通訊作者：邱美秀，e-mail：meishiuchiu@gmail.com 收稿：2017 年 1 月 9 日；

接受刊登：2017 年 10 月 18 日。

邱美秀（2017）。

辨識可預測學生數學認知和情意的有效 E 化和一般數學教學法組合：潛在剖面分析法。

**臺灣數學教育期刊，4（2），69-94。 **

doi: 10.6278/tjme.20171018.001

### 辨識可預測學生數學認知和情意的有效 **E 化和 ** 一般數學教學法組合

邱美秀 國立政治大學教育學系

本研究旨在辨識 E 化和一般數學教學法的組合類型，並探討所辨識出的教學組合類型在學生數學認知和 情意上的差異情形。以潛在剖面分析（LPA）方法分析 3,978 名臺灣 15 歲學生在數學教室中的 E 化數學 教學和三項一般性的數學教學法（形成性評量、學生導向和教師指導）。LPA 的結果辨識出四種 E 化與 一般數學教學法組合：節約、保守、協調和自由使用 E 化與一般數學教學法的組合。接著，使用多變量 變異數分析（MANOVA）和事後檢驗，來考驗四種教學法組合在學生各數學認知和情意細項內容上的差 異，並且使用結構方程模式（SEM）考驗四種教學法組合在認知和情意二潛在構念上的差異。MANOVA 和 SEM 的分析結果顯示：協調的 E 化與一般數學教學法組合同時有益於學生認知和情意，節約的 E 化 與一般數學教學組合有利認知但犧牲情意，保守和自由的 E 化與一般數學教學組合有利於情意。

關鍵詞：E 化教學、潛在剖面分析、數學情意、數學認知、數學教學法

**Introduction **

Information and communication technology (ICT) has gradually been incorporated into general teaching (teaching), resulting in “e-teaching” (González, 2012). The relationships between e- and g-teaching, however, remain vague. Both e- and g-teaching may include pedagogies or practices for traditional, lifelong, and connected learning (Blignaut, Hinostroza, Els, & Brun, 2010). Pedagogical knowledge can fully comprise technological pedagogical knowledge or technological pedagogical content knowledge (Chai, Koh, & Tsai, 2013). The term “blended teaching” (González, 2012) explicitly indicates the diverse practices of integrating e- and g-teaching in real educational settings.

Researchers have identified different approaches to integrating e- and g-teaching (termed “e/g-teaching” in this study). Innovative e/g-teaching cases (e.g., Tan & Tan, 2015) and survey studies (e.g., Sang, Valcke, van Braak, & Tondeur, 2010) have integrated e- and g-teaching on the basis of ideas or theories. Qualitative studies have categorized methods of using ICT in teaching (González, 2012). These studies have tended to define or group e/g-teaching in a predetermined categorical manner. The aim of the current study was to identify e/g-teaching profiles on the basis of student perceptions of real mathematics teaching by using latent profile analysis (LPA) (Fraley & Raftery, 2007). LPA is a person-centered clustering method that aims to maximize the most likely profiles of distinct meanings on the basis of empirical data. LPA can exceed the linear relationship between e/g-teaching and learning outcomes to identify nonlinear profiles likely to address differences in learning outcomes. Examples of nonlinear relationships between e- and g-teaching include teachers using both e- and g-teaching intensively (e.g., the constructivist approach; Park et al., 2015), using e- and g-teaching moderately (e.g., the balance approach; Tan & Tan, 2015), and using e-teaching for g-teaching (e.g., the traditional approach; Lan, Chang, & Chen, 2012).

After the patterns of mathematics teachers’ e/g-teaching are identified, assessing how the patterns relate to students’ cognitive and affective learning outcomes is essential. Mathematics teaching and national curricula both relate to and emphasize learning outcomes in both cognitive and affective aspects (Chiu, 2009; Chiu & Whitebread, 2011). Competent mathematics learners require both cognitive and affective dispositions, such as domain knowledge, meta-knowledge, heuristics methods, self-regulatory skills, and beliefs about self and mathematical learning (De Corte, 2004). The current study also devoted partial attention to ICT availability and socioeconomic status (SES), which may condition e/g-teaching profiles (Cuckle & Clarke, 2002). In summary, the purpose of this study included identifying profiles or

patterns of mathematics teachers’ e/g-teaching and assessing how the identified profiles interact with students’ cognitive and affective learning outcomes, conditioned by students’ ICT availability and SES.

**Approaches to Integrating E- and G-Teaching **

E/g-teaching profiles can be implied by past studies on approaches to integrating e- and g-teaching.

Traditional and constructivist approaches are two extremes, and balance and theoretical approaches are mixed types of e/g-teaching, as shown in the following paragraphs.

**The traditional, activating, or teacher-centred approach. This approach entails using ICT to **
present concepts, explain ideas, and lead discussion (Lan et al., 2012). Most teachers appear to focus on
this traditional method of e-teaching (Blignaut et al., 2010; Louw, Brown, Muller, & Soudien, 2009;

Smeets, 2005).

**The constructivist, facilitating, or student-centred approach. This approach entails using ICT **
as a platform to transform teacher roles from dominant to parallel status (Park et al., 2015). Examples
of this approach include collaborative creative writing (Vass, 2007) and use of Web 2.0 tools (Chai,
Koh, Ho, & Tsai, 2012).

**The balance approach. This approach entails using ICT as a tool to compensate for conventional **
g-teaching methods for distinct blocks of teaching sessions (Tan & Tan, 2015). For example, g-teaching
(paper-and-pencil or concept development) is followed by e-teaching (ICT use for generalization or
application).

**The theoretical or pedagogical approach. E- and g-teaching can be fully integrated by existing **
higher-order conceptions of g-teaching. Examples of g-teaching conceptions include reflection (Leijen,
Admiraal, Wildschut, & Robert-Jan Simons, 2008), learning theories, teacher knowledge (Benson &

Brack, 2009), and learning models (e.g., cognition, action, and reflection) (Lan et al., 2012). This
approach fully integrates e- and g-teaching, through which e-teaching has actually transformed existing
**g-teaching conceptions into innovative ones. (Nachmias, Mioduser, & Forkosh-Baruch, 2010; Tømte, **
Enochsson, Buskqvist, & Kårstein, 2015).

**Relations between E/G-Teaching Profiles and Learning Outcomes **

Student learning outcomes can be effectively promoted by both e- and g-teaching, such as collaborative learning in both face-to-face and online settings (Solimeno, Mebane, Tomai, &

Francescato, 2008). Teachers tend to perceive ICT use as potentially benefiting student learning outcomes in the aspects or constructs of cognition (e.g., mathematics knowledge) and affect (e.g.,

motivation and collaborative skills) (Blignaut et al., 2010). If both e- and g-teaching can benefit students, the next question may be what e/g-teaching profiles are more effective than others in benefiting cognition and affect.

**Cognition. Mathematical cognition for learners can be defined as applying mathematical **
knowledge and reasoning to study patterns and relationships (Burton, 1994). Mathematics education
researchers have identified the cognitive activities involved in mathematical problem-solving. For
example, mathematical problem-solving may include the procedures of understanding, planning,
implementing, and reviewing (Polya, 1945, 1962) and addressing problems, thus reflecting on the
experience, studying the process of resolving problems, and noticing the interaction between the
experience and what is learned (Mason, Burton, & Stacey, 1996).

Successful mathematics cognitive activities can be measured as mathematics performance or achievements. Research has indicated that student achievements positively relate to teacher-centered g-teaching, such as reasoning orientation (Thorvaldsen, Vavik, & Salomon, 2012), direct instruction, and frequent test use to assess student learning, and negatively relate to rule orientation (Hinostroza, Labbé, Brun, & Matamala, 2011). The effects of e-teaching on achievement are perceived by teachers of low mathematics-ability classes but less often by those of high-ability ones (Thorvaldsen et al., 2012), who may nevertheless frequently employ e-teaching (Hinostroza et al., 2011). Positive relations between constructivist e-teaching and students’ meaningful learning perceptions, achievements, and course satisfaction may not be guaranteed (Wurst, Smarkola, & Gaffney, 2008) without formative feedback (Espasa & Meneses, 2010). Therefore, formative assessment in teaching and learning processes may play a role in the effect of e/g-teaching on learning outcomes.

**Affects. Mathematics affects are an indispensable part of mathematics cognitive activities **
(Gómez-Chaćon, 2000; Hannula, 2002). Mathematics affects include beliefs (e.g., I can competently solve
problems), attitudes (e.g., I enjoy problem-solving), and emotions (e.g., mathematics is beautiful)
(McLeod, 1992, 1994). Confidence-related mathematics beliefs (including self-efficacy) typically have
high correlations with mathematics achievement (Chiu, 2012b; Grootenboer & Hemmings, 2007).
E-teaching generally benefits student affects such as self-efficacy or confidence (Tan & Tan, 2015),
interest, and engagement. Constructivist e-teaching (e.g., real-world settings, collaboration, and
individual choices) can increase student interest in science (Wilson & Boldeman, 2012) and engagement
(Rappa, Yip, & Baey, 2009). College teachers having multiple g-teaching concerns and using ICT for

teaching tend to emphasize the roles of g-teaching in engaging students through ICT use (Webster &

Son, 2015).

**Relations Between E/G-Teaching Profiles and Conditions **

Naturally, e-teaching at school is conditioned by school ICT availability (Cuckle & Clarke, 2002).

Constructivist e-teaching further requires student ICT availability (Smeets, 2005). SES is potentially another condition, which generally has a positive relation with home ICT availability, ICT use quality, and achievement (Lee & Wu, 2012).

**Research Questions **

The literature review suggests that innovative and diverse e/g-teaching profiles may be identified on the basis of real context data by using nonlinear person-based modeling analysis. The identified profiles may address differences in learning outcomes in the explicit elements or latent constructs of cognition and affect through profile difference analysis partially considering conditions. Therefore, the objective of this study was to answer the following three research questions (RQs):

1. What are the profiles of mathematics e-teaching (ICT use) and g-teaching behaviors (formative
**assessment, student orientation, and teacher direction) perceived by students? **

2. What are the differences between the identified profiles in the explicit elements of cognition (e.g.,
employing, formulating, and interpreting), affects (e.g., self-efficacy, interest, and engagement), and
**conditions (e.g., SES, home ICT availability, and school ICT availability)? **

3. How do the identified profiles, conditioned by school ICT availability, predict differences in the latent constructs of cognition and affect?

**Method **

**Data Source and Sample **

This study used data on Taiwan from the main and ICT surveys of the Programme for International Student Assessment (PISA) in 2012 (Organisation for Economic Co-operation and Development [OECD], 2014b). The PISA started in 2000 and is a triennial international survey examining the achievement of 15-year-old students, principally in the fields of mathematics, science, and reading. PISA also collects self-report, contextual data from students, teachers, schools, parents, and national PISA administrators. PISA 2012 is the fifth survey focusing on mathematics.

The total Taiwan sample of PISA 2012 comprised 6,046 students. The four e/g-teaching measures

used in this study (cf. the Measures section) included approximately 33.6% missing data, which prevented the use of LPA to identify e/g-teaching profiles (cf. the Data Analysis section). To handle the problem of missing data, multiple imputation procedures were attempted by using the mix package in R. However, the several imputed data sets generated unstable profile solutions, implying that different imputed data sets generated different profile solutions. Therefore, listwise deletion was used for the four e/g-teaching measures, which resulted in a total sample of 3,978 students in this study. Sampling weights were not used in this study because of the considerable amount of missing data and the use of listwise deletion. Accordingly, the findings of this study can only be explained as a phenomenon of the specific sample and cannot be generalized to the population.

The exact sample sizes for the other measures, as derived after the aforementioned listwise deletion,
are presented in Table 1. Notably, the affective measures had small sample sizes because of missing data
(Table 1), implying that some participating students did not fully complete the related affective measures
in the survey. Medium correlations were observed between self-efficacy and mathematics cognitive
*measures (r = .63, .64, and .59), and these results are consistent with previous study findings revealing *
stable relationships between achievement- and confidence-related constructs (Chiu, 2012b; Grootenboer

& Hemmings, 2007). Moreover, medium correlations were observed between self-efficacy and the other
*two affective measures (r = .43 and .48, respectively), implying relatively high differences between *
efficacy and the other two affective measures. This may result in lower factor loadings for either
self-efficacy or the other two affective measures.

**Measures **

This study focused on 13 student measures obtained from the PISA 2012 database (OECD, 2014a, 2014b). These measures were grouped into four categories (e/g-teaching, cognition, affect, and condition). All 13 measures were derived from several items in the PISA data sets and rescaled using item response theory, with higher scores representing higher degrees in the meanings of these measures.

Table 2 presents detailed information on the 13 measures, including measure names in this study;

original labels in the PISA data set; item stems, sample items, and item numbers; measurement methods;

*OCED means, standard deviations (SDs), and internal reliability coefficients (Cronbach’s alpha (α)); *

*and Taiwan’s α. Table 1 presents the means and SDs of the present Taiwan sample. *

**Data Analysis **

The RQs were answered through statistical analysis using the software R Version 3.1.3 (R Core

**Descriptive Statistics and Correlations Between E/G-Teaching Behaviors, Cognition, Affects, and Conditions **

*N * *Mean * *SD * *r *

Measures 1 2 3 4 5 6 7 8 9 10 11 12

*E/g-teaching behaviors *

1. ICT use 3978 -0.43 0.75

2. Formative assessment 3978 -.10 .95 .09 3. Student orientation 3978 .01 .98 .13 .50 4. Teacher direction 3978 -.07 1.06 .06 .68 .42

*Cognition *

5. Employing 3978 547.38 107.12 -.12 -.01 -.28 .01

6. Formulating 3978 577.25 134.29 -.12 -.02 -.28 -.01 .95 7. Interpreting 3978 547.57 102.01 -.12 -.05 -.31 -.02 .94 .92

*Affects *

8. Self-efficacy 1980 .18 1.18 -.01 .14 -.07 .13 .63 .64 .59

9. Interest 1981 .02 .96 .10 .32 .13 .27 .32 .32 .26 .43 10. Engagement 1985 .07 .98 .09 .27 .10 .19 .45 .43 .37 .48 .53

*Conditions *

11. SES 3968 -.39 .84 -.04 .06 -.07 .05 .43 .40 .38 .32 .11 .25
12. Home ICT availability 3976 -.35 .92 .03 .09 .06 .04 .10 .10 .08 .15 .01 .11 .44
13. School ICT availability 3967 -.22 .82 .08 .15 .11 .12 .03 .03 .01 .04 .07 .09 .08 .22
**Note. The underlined correlations (rs) are significant at p = .05. **

**Detailed Descriptions of the 13 Measures **

Measure name PISA label

Item stem

sample items (item numbers) Measurement methods

OECD mean

OECD SD

OECD α

Taiwan mean

Taiwan SD

Taiwan
α
*E/g-teaching *

*behaviors *

1. ICT use Use of ICT in Mathematic Lessons

Within the last month, has a computer ever been used for the following purposes in your mathematics lessons?

Drawing the graph of a function (such as y = 4x+6). (7 items)

3-point Likert scale:

1 = yes, students did this, 2 = yes, but only the teacher demonstrated this, 3 = no. (reverse coded)

-1.57 1.57 .91 -.43 .75 .95

2. Formative assessment

Teacher Behavior:

Formative Assessment

How often do these things happen in your mathematics lessons?

The teacher gives me feedback on my strengths and weaknesses in mathematics. (4 items)

4-point Likert scale:

1 = every lesson ~ 4 = never or hardly ever (reverse coded)

-.28 1.35 .76 -.10 .95 .74

3. Student orientation

Teacher Behavior:

Student Orientation

(The same item stem as the above.) The teacher has us work in small groups to come up with joint solutions to a problem or task. (4 items)

(Same as the above) -.98 1.06 .68 .01 .98 .69

4. Teacher direction

Teacher Behavior:

Teacher-directed Instruction

(The same item stem as the above.) The teacher asks me or my classmates to present our thinking or reasoning at some length. (5 items)

(Same as the above) .54 1.14 .73 -.07 1.06 .78

*Cognitions *

5. Employing Plausible value 1 in process subscale of Maths - Employ

(PISA 2012 released mathematics problems) Cognitive performance test

493* na .91 547.38 107.12 .93

6. Formulating Plausible value 1 in process subscale of Maths - Formulate

(PISA 2012 released mathematics problems) Cognitive performance test

492* na .89 577.25 134.29 .93

7. Interpreting Plausible value 1 in process subscale of Maths - Interpret

(PISA 2012 released mathematics problems) Cognitive performance test

497* na .90 547.57 102.01 .90

(continued)

**Table 2 (continued) **

Measure name PISA label

Item stem

sample items (item numbers) Measurement methods

OECD mean

OECD SD

OECD α Taiwan

mean Taiwan

SD

Taiwan
α
*Affects *

8. efficacy Mathematics Self-Efficacy

How confident do you feel about having to do the following mathematics tasks?

Calculating how many square metres of tiles you need to cover a floor. (8 items)

4-point Likert scale:

1 = very confident ~ 4 = not at all confident (reverse coded)

1.15 1.50 .85 .18 1.18 .91

9. Interest Mathematics Interest Thinking about your views on mathematics: to what extent do you agree with the following statements?

I enjoy reading about mathematics. (4 items)

4-point Likert scale:

1= strongly agree ~ 4 = strongly disagree (reverse coded)

-.82 2.93 .89 .02 .96 .91

10. Engagement Mathematics Behavior

How often do you do the following things at school and outside of school?

I do mathematics more than 2 hours a day outside of school. (8 items)

4-point Likert scale:

1 =always or almost always ~ 4 =never or rarely (reverse coded)

-1.55 1.12 .72 .07 .98 .76

*Conditions *

11. SES Index of economic, social and cultural status

(1) home possessions, (2) the highest parental occupation, and (3) the highest parental education. (3 items)

3 derived items, each z-standardized

-.22 .94 .65 -.39 .84 .69

12. Home ICT availability

ICT Availability at Home

Are any of these devices available for you to use at home?

Desktop computer; portable laptop or notebook; Internet connection. (11 items)

3-point Likert scale:

1 = yes, and I use it, 2 = yes, but I don’t use it, 3 = no (reverse coded)

.59 .76 .53 -.35 .92 .63

13. School ICT availability

ICT Availability at School

Are any of these devices available for you to use at school?

Desktop computer; portable laptop or notebook; Internet connection. (7 items)

(Same as the above) -.21 1.15 .65 -.22 .82 .59

*Note. The OECD data with *^{*} were obtained from Figure I.2.37 in OECD (2014a) and the other OECD data and Taiwan’s α were obtained from OECD (2014b).

*Taiwan’s means and SDs were calculated on the basis of the final sample (n = 3978) used in this study. α = Cronbach’s alpha (internal reliability coefficient); na *

= not available

### .

Team, http://www.R-project.org/). This study focused on answering the three RQs, but descriptive statistics and correlations (obtained by the psych and stats packages in R) facilitated a basic understanding of the measures and data structures.

RQ 1 was investigated through LPA, because all the 13 measures were continuous variables (Muthén & Muthén, 2012); the analysis was conducted using the mclust package in R. LPA can identify latent profiles with distinct meanings such as different SES levels (Chittleborough, Mittinty, Lawlor, &

Lynch, 2014) and combinations of academic/cognitive, social/emotional, and behavioral risks (Wang &

Peck, 2013). LPA is more efficient than conventional cluster analysis (Chiu, Douglas, & Li, 2009). A simulation study indicated that the mclust package in R tends to outperform Latent Gold® and the poLCA package in R, particularly for continuous measures (Haughton, Legrand, & Woolford, 2009).

The mclust package applies a model-based clustering technique and uses higher Bayesian information criterion (BIC) values to represent more favorable profile number solutions. Notably, Mplus (Muthén &

Muthén, 2012), another software package widely used by researchers for LPA, uses lower BIC values to represent more effective profile number solutions, because Mplus and mclust use different formulae for the BIC. A priori theories may also be used to determine proper profile numbers (Marsh, Lüdtke, Trautwein, & Morin, 2009). This means that profile names and numbers must be determined by considering existing research findings and educational practices in a society. For example, direct teaching and liberal teaching may be one of the dominant mathematics teaching profiles in Taiwan (Chiu, 2009; Chiu & Whitebread, 2011).

RQ 2 was answered through multivariate analysis of variance (MANOVA) for the categories of
cognition, affect, and condition by using the base package in R. When MANOVA results showed
significant differences, each element in the category was subjected to analysis of variance (ANOVA),
followed by TukeyHSD post hoc tests using the base package. Subsequently, effect sizes (partial eta
*squared (η*^{2})) were obtained using the heplots, MASS, and car packages. According to Cohen (1988, p.

*283), .01 < η*^{2}* < .06 shows small effect sizes, .06 < η*^{2}* < .14 medium effect sizes, and η*^{2} > .14 large effect
sizes.

RQ 3 was answered through structural equation modeling (SEM) using the MASS, matrixcalc, and sem packages. The SEM technique used in this study focused on multiple-indicator/multiple-cause (MIMIC) analysis, because the models were aimed at examining profile differences (cf. Figure 2) (Hsu, Zhang, Kwok, Li, & Ju, 2011). Similar to MANOVA and ANOVA, MIMIC examines profile differences but additionally allows for measures with underlying latent constructs and conditions to be included in

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 (χ*^{2}*), would be easily violated (Bollen & Long, 1993). Thus, χ*^{2} did not serve
as the major criterion in this study.