Educational Studies ISSN: 0305-5698 (Print) 1465-3400 (Online) Journal homepage: http://www.tandfonline.com/loi/ceds20 Exploring profiles of ideal high school mathematical teaching behaviours: perceptions of in-service and pre-service teachers in Taiwan Feng-Jui Hsieh, Ting-Ying Wang & Qian Chen To cite this article: Feng-Jui Hsieh, Ting-Ying Wang & Qian Chen (2017): Exploring profiles of ideal high school mathematical teaching behaviours: perceptions of in-service and pre-service teachers in Taiwan, Educational Studies, DOI: 10.1080/03055698.2017.1382325 To link to this article: http://dx.doi.org/10.1080/03055698.2017.1382325 Published online: 07 Oct 2017. Submit your article to this journal Article views: 13 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=ceds20 Download by: [UAE University] Date: 25 October 2017, At: 08:47 Educational Studies, 2017 https://doi.org/10.1080/03055698.2017.1382325 Exploring profiles of ideal high school mathematical teaching behaviours: perceptions of in-service and pre-service teachers in Taiwan Feng-Jui Hsieha, Ting-Ying Wanga and Qian Chenb Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan; bSchool of Mathematics and Software Science, Sichuan Normal University, Chengdu, China Downloaded by [UAE University] at 08:47 25 October 2017 a ABSTRACT This study explored and compared the perspectives of Taiwanese inservice and pre-service high school mathematics teachers regarding ideal teaching behaviours; the perspectives of a nationwide sample of students were taken as the baseline. Fourteen factors contributing to ideal teaching behaviours were identified through exploratory factor analyses. Nine factors, including idea explanation and speedy lecture, were rooted in traditional Chinese culture; five factors, including concrete representation and student activities, were influenced by Western cultures. Three teacher profiles were identified through k-means clustering analysis. The perspectives of in-service teachers were dominated by a painless meaning-emphasised profile; these teachers emphasised meaningful learning for students and avoided the fast pace and demanding requirements that can cause distress in students, whereas pre-service teachers were dominated by an allround profile, revealing their openness to all factors. Compared with pre-service teachers, in-service teachers’ perspectives were more similar to those of students. ARTICLE HISTORY Received 15 August 2016 Accepted 16 September 2017 KEYWORDS High school teacher; ideal mathematics instruction; teaching behaviour profile; teaching material; teaching method Introduction Characteristics of ideal mathematics instruction Numerous studies have sought to determine the characteristics of high-quality mathematics instruction to facilitate student learning. Diverse aspects of ideal mathematics teaching have been depicted using various research approaches on different targets at various school levels. For example, in the West, Martinez-Sierra (2014) employed questionnaires with openended items to investigate Mexican high school students’ perspectives on good mathematics teaching; Perry (2007) and Murray (2011) explored the views of Australian elementary school teachers and high school students, respectively, through interviews. In East Asia, Pang (2009) identified key characteristics of good mathematics teaching by observing elementary classroom practice in South Korea; Kaur (2009) combined classroom observation and interviews with high school students to investigate good mathematics teaching in Singapore; and Li CONTACT Ting-Ying Wang email@example.com © 2017 Informa UK Limited, trading as Taylor & Francis Group Downloaded by [UAE University] at 08:47 25 October 2017 2 F.-J. HSIEH AND T.-Y. WANG (2011) explored Chinese elementary school teachers’ perspectives by requiring them to write essays to answer questions. Overall, student-centred and process-oriented features characterise Western perspectives of ideal mathematics instruction (Kaiser and Vollstedt 2007; Pang 2009). Regarding teaching methods, several frequently promoted behaviours are pertinent to adequate illustration of mathematical ideas; such as teaching concepts and procedures with clear, simple, and stepby-step explanations that are appropriate to students’ prerequisite knowledge of mathematics and their learning paces (Martinez-Sierra 2014; Murray 2011; National Council of Teachers of Mathematics [NCTM] 2000). Teachers are also expected to provide students with opportunities to discover mathematical ideas through exploratory or hands-on activities, group discussions, and technology (Wang and Cai 2007a; Wilson, Cooney, and Stinson 2005). Regarding handling teaching materials, establishing connections within and between mathematics and real-life examples is deemed ideal teaching behaviour; establishing such connections has been rooted in Western educational norms for decades (Ausubel 1961; Sigurdson and Olson 1992; Skemp 1989). Likewise, providing visual, manipulative, simple, or concrete representations and examples is regarded as ideal teaching behaviour (Martinez-Sierra 2014; Murray 2011; Stigler and Perry 1988; Wang and Cai 2007a; Wilson, Cooney, and Stinson 2005). Regarding assessment, ideal teachers must allow students to complete exercises and should implement frequent evaluations with multiple methods to inform students about their comprehension and difficulties, as well as provide feedback and help every student (MartinezSierra 2014; Murray 2011; Wilson, Cooney, and Stinson 2005). Ideal mathematics instruction in East Asia Mathematics instruction in East Asia, including Taiwan, is often considered as typically lecture-oriented and examination-driven, and is characterised by passive and rote student learning, procedural teaching, and large classes; none of these things are considered desirable in the West (Li 2011; Park and Leung 2006). However, recent studies have argued that the misinterpretation of mathematics classes in East Asia is caused by the simple dichotomy of teacher-centred versus student-centred (Leung 2001; Pang 2009). Through investigations of actual practices in mathematics classes and the delineation of ideal mathematics classes, researchers have revealed that under the aforementioned undesirable characteristics, students in East Asia are actually provided with opportunities for meaningful learning in well-organised academic learning activities thoughtfully designed by their teachers (Park and Leung 2006; Stevenson and Lee 1997). In East Asia, mathematics is generally viewed as a system of interwoven concepts, facts, procedures, and relationships, reflecting the structural perspective of Ernest (1989). The mathematics curriculum is demanding, and students are expected to grasp and master the knowledge structure (Cai 2007; Leung 2001). Thus, teachers strive to teach mathematics in a coherent way that emphasises the connections among various mathematical ideas; this practice aligns with what is stressed in Western educational systems (Cai 2007). Teachers in East Asia also review past knowledge and organise new and complex mathematics content in sequences appropriate for the abilities and specific conditions of the students (An 2004; Liu 2007). To achieve curricular expectations, these teachers endorse and employ clear lecturing (Leung 2001), discuss mathematically challenging problems and provide or encourage multiple solutions (Lin and Li 2009; Pang 2009; Stigler and Stevenson 1991), guide students in the use of generalised Downloaded by [UAE University] at 08:47 25 October 2017 EDUCATIONAL STUDIES 3 problem-solving strategies and abstract representations (Cai 2004), and provide sufficient exercises to prepare students for examinations (Kaur 2009). These characteristics are strongly rooted in traditional East Asian and Chinese culture (Leung, Graf, and Lopez-Real 2006). In addition, the notion of ideal mathematics instruction in East Asia also includes characteristics influenced by Western culture, such as exposing students to exploratory or conjectural activities, generating small-group or whole-class discussion, using physical and manipulative materials, providing real-life examples, using multimedia, and delivering timely assessments and effective feedback (Kaur 2009; Li 2011; Lin and Li 2009; Lin and Tsai 2007; Liu 2007; Pang 2009; Zhao and Ma 2007). To further understand the viewpoints of ideal mathematics instruction among various countries and cultures, Kaiser and Vollstedt (2007) created a spectrum by combining knowledge from several empirical studies, for ideal instruction from the perspective of teachers. At one end, where China and Hong Kong stood, were approaches involving teachers as explainers to present and elaborate mathematical ideas, and classes with teacher-dominated “chalk and talk”. At the other end, where the United States and the United Kingdom stood, were approaches involving teachers as facilitators to engage students in problem solving with long-term tasks, and classes which were student-centred and process-oriented. The desired teaching behaviours of Western cultures reveal that the dominant ideology is focused on the process of doing mathematics, rather than grasping the body of mathematics knowledge itself. Western ideology also stresses classroom activities, such as student exploration and hands-on work, rather than teacher-driven lectures and student practice problems (Leung 2001). However, it is unclear whether these Western ideals of mathematics teaching behaviours extend to East Asian cultures. As Clarke (2013) noted, these instructive approaches largely pertain to Western educational philosophy and pedagogical practice, which may not be feasible in East Asian countries. Indeed, because students in East Asian countries usually outperform their Western counterparts in international comparison studies of mathematics achievement (e.g. Mullis, Martin, and Foy 2008; OECD 2013), the ideal instruction of the West may not fulfil the academic pursuits of East Asia. Bypassing Western criteria, both Kaur (2009) and Bryan et al. (2007) argued that mathematics instruction characterised as “teacher-centred but student-focused” or “teacher-led, yet student-centred” could be ideal in East Asian countries. Background of the present study In the 1970s, traditional mathematics instruction in Taiwan was teacher-directed, contentoriented, practice emphasising, and examination-driven, focusing on memorisation with little emphasis on the interaction between teachers and students (Lin and Li 2009). However, with the introduction of new mathematics curriculum standards in the 1990s, innovative instruction techniques have been adopted from Western cultures, which emphasise learner-centred and constructivist-based approaches (Hsieh 1997). Hsieh claimed that the views of not only teachers but also students regarding ideal mathematics instruction should be respected (Hsieh, Shy, and Wang 2015). She and her colleagues conducted a 4-year project to develop a framework composed of critical mathematical teaching dimensions (e.g. mathematics problem solving) and teaching behaviours specific to these dimensions (Hsieh 2012, 2013).1 Questionnaires addressing these teaching behaviours were developed and distributed to a random sample of 4514 students from nationwide Downloaded by [UAE University] at 08:47 25 October 2017 4 F.-J. HSIEH AND T.-Y. WANG large-scale senior and junior high schools, to obtain their opinions on which mathematical teaching behaviours were ideal. Their results showed that the opinions of Taiwanese high school students corresponded with some of the expectations of Taiwan’s mathematics curriculum standards that had been adopted from Western views, and also retained some views rooted in Chinese tradition (e.g. Hsieh, Shy, and Wang 2015; Hsieh and Wang 2014; Wang and Hsieh 2015). However, few studies have empirically investigated the perspectives of senior and junior high school mathematics teachers, regardless of in-service or pre-service status, on ideal mathematical teaching behaviours; moreover, comparisons of students’ with teachers’ opinions are also scant. Teacher perspectives (such as those in Taiwan) may become the most influential factor of their teaching practices when their teaching knowledge is sufficient (Hsieh et al. 2011). Aligned with Hsieh’s student-centred view, this study explored the perspectives of in-service and pre-service mathematics teachers on ideal teaching behaviours in Taiwan, providing student perspectives as the base. In particular, the present study answered the following research questions: (1) What are the factors that contribute to the perspectives of ideal mathematical teaching behaviours for in-service and pre-service mathematics teachers? (2) What are the commonalities and the differences among in-service teachers, preservice teachers, and students on the perspectives of ideal mathematical teaching behaviours? (3) What are the profiles that portray the high school in-service and pre-service mathematics teachers’ perspectives of ideal mathematical teaching behaviours? Research methods Measures The present study surveyed teachers using the same questionnaires that Hsieh had used earlier (2012, 2013). The candidate items of ideal mathematical teaching behaviours for the questionnaires of the current study were obtained through a survey of constructive-item questions that collected the opinions of 238 students, and a series of processes that consulted expert opinion. The students were asked to list what an ideal mathematics teacher should and should not do to facilitate their mathematics learning in various teaching tasks, such as introducing new concepts and demonstrating how to solve a problem, and providing an explanation. A content analysis was then conducted by experts, including a mathematics teacher educator, three mathematics education researchers, and six high school mathematics teachers, on the questionnaire responses to produce the mathematical teaching dimensions and their corresponding teaching behaviours. Ten focus group discussions were conducted to identify the expert views regarding what and how ideal mathematics teachers should teach. The first three group discussions involved four Ph.D. students and six master’s students. Among them, one Ph.D. student and all six master’s students were high school mathematics teachers with an average of 11.1 years of teaching experience. In the fourth discussion group were 43 high school mathematics conveners from 43 schools, each of whom was the official representative of the entire body of mathematics teachers in the corresponding school. The remaining six group discussions involved 22 high school mathematics teachers in compulsory education advisory groups EDUCATIONAL STUDIES 5 Downloaded by [UAE University] at 08:47 25 October 2017 operated by the government in Taiwan, who guided other teachers in teaching mathematics. The views of both students and teachers on ideal mathematical teaching behaviours were integrated into 18 dimensions (such as teaching method or problem solving); two were not suitable for student questionnaires (such as the mathematics knowledge of teachers), and the rest were presented as dichotomous items in the questionnaires. Eight dimensions involving seven facets were included in the present study (Figure 1): handling teaching material (also including the teaching concepts dimension), representation, problem solving, teaching process, teaching method, mathematical media and teaching aids, and evaluation. The dimensions had 11, 11, 14, 7, 14, 3, and 14 items, respectively. An example of the problem stems that prompted the questions was as follows: “Regarding teaching behaviours of problem solving in class, a great high school mathematics teacher should …” Participants The sample consisted of 125 in-service and 113 pre-service high school mathematics teachers. The in-service teachers comprised 65 randomly selected senior high school teachers from 36 schools and 60 junior high school teachers from 31 schools, in 20 of Taiwan’s 22 cities. These participants were the mathematics teachers of the previously randomly sampled students studied by Hsieh and her colleagues. The pre-service teachers consisted of two cohorts: 75 college sophomores, who were at the entry point of mathematics teacher preparation programmes, and 38 college seniors, who were at the exit point and were about to start internships in high schools. Both cohorts were from the same normal university, which was intentionally chosen for its representative department of teachers. It was the first university-level teacher education institution in Taiwan, and the most experienced institution to prepare high school level teachers (Ministry of Education 1996). Before the Taiwanese government opened access to teacher preparation in 1994 (Hsieh et al. 2013), this normal university was the biggest, and one of the only three universities that prepared high school mathematics teachers. After the Taiwanese government allowed all universities to educate teachers, this Ideal teaching behaviours Evaluation Instruction Concepts and ideas Handling teaching material Representation Teaching process Teaching method Mathematical media and teaching aids Figure 1. Framework of the present study. Note: The bold labels are the facets investigated in this study. Problem solving 6 F.-J. HSIEH AND T.-Y. WANG normal university trained one-sixth to one-fifth of high school mathematics teachers in Taiwan each year. The pre-service teacher sample of the present study included 100% of the students at the entry point and 95% of the students at the exit point in the university. Downloaded by [UAE University] at 08:47 25 October 2017 Data analysis To determine the factors contributing to ideal mathematical teaching behaviours from teacher perspectives, exploratory factor analysis (EFA) was performed on the responses of teachers to questionnaire items. The EFA on the aforementioned seven facets was performed separately with Mplus (Version 6.12). A robust weighted least squares estimator was applied because it is typically considered robust for nonnormal data (Flora and Curran 2004). The fit indices employed to determine the number of factors were a (a) comparative fit index (CFI), (b) Tucker–Lewis index (TLI), and (c) root mean square error of approximation (RMSEA). A factor model was adopted in this study when the values of CFI and TLI ≥ .9 which indicated a good fit; the value of RMSEA ≤ .08, which indicated a reasonable approximate fit, and; all factor loadings ≥ .3, which indicated an adequate level (Diamantopoulos and Siguaw 2000; Hu and Bentler 1999; Kline 2011; Wang and Wang 2012; Williams, Brown, and Onsman 2010). For each factor obtained from the EFA, an item response theory (IRT) scaling model, the Rasch logistic model was used with Winsteps (Version 3.72.3; Linacre 2011) to create interval logit scores of the endorsement of ideal mathematical teaching behaviours for every teacher. For each factor, to determine the fit of the data to the model, the weighted fit mean square (infit) and the unweighted fit mean square (outfit) statistics were applied. The acceptable values for both statistics were between 0.7 and 1.3 (Bond and Fox 2007). Misfit items were removed, and then the EFA and Rasch logistic model procedures were rerun on the remaining items until a good fit of the data was obtained. To facilitate reading the results, the final logit scores were transformed to a range of 0–10 points, with teachers who had endorsed no items receiving 0 points and those who had endorsed all items receiving 10 points. Each logit score for a particular cohort was calculated by averaging the logit scores of all the individuals in that cohort. To make sense of the logit scores of the teachers, the average logit scores of the previously randomly sampled students studied by Hsieh and her colleagues were calculated for each factor obtained from the EFA on teacher data. K-means clustering analysis was performed using SPSS to identify relatively homogeneous groups of teachers, according to the patterns of their perspectives on ideal mathematical teaching behaviours (Kaufman and Rousseeuw 2005). Clusters were determined by computing Euclidean distances among the samples in multidimensional space. For each factor obtained from EFA, a one-way analysis of variance combined with post hoc analysis was conducted to obtain the between-group differences of the average logit scores of the teacher groups, as well as the differences between the scores of the teacher groups and those of the students. Results Factor models The factor models of the seven facets from EFA and IRT analysis are presented in Table 1. According to the fit statistics, the models fit the data well (CFI ≥ .9, TLI ≥ .9, RMSEA ≤ .08, and all factor loadings ≥ .3). EDUCATIONAL STUDIES 7 Table 1. Factor models of the seven facets. Downloaded by [UAE University] at 08:47 25 October 2017 Facet Handling teaching material Abbreviation HM Model Two-factor model Representation R Two-factor model Problem solving PS Two-factor model Teaching process TP Two-factor model Teaching method TM Three-factor model Mathematical media and teaching aids Evaluation MA E One-factor model Two-factor model Factor Connection Simplification Concrete representation Formal representation Skill preparation Heuristic guidance Eliciting Outlining Student activities Idea explanation Speedy lecture Media use Performance demand Comprehension diagnoses Handling teaching material was captured by two factors, connection and simplification, with seven and four items, respectively. Connection represented a group of teaching behaviours that provide meaning to mathematical concepts by connecting new concepts to previously learned ones, both in mathematics and in the real world. Examples of the items were “Formulate connections and comparisons among new mathematical concepts and other already learned concepts” and “Clearly illustrate the meaning of mathematical concepts”. Simplification represented a group of teaching behaviours that starts with basic ideas, focuses on textbooks rather than handouts with challenging problems, and provides numerous examples that are more concrete than mathematical concepts alone. Examples of the items were “Start from basic concepts so students can develop a full picture of the concepts” and “Rather than heavily illustrate the meanings of concepts, provide many examples”. Representation was captured by two factors, concrete representation and formal representation, with six and five items, respectively. Concrete representation comprised representations with characteristic concreteness, such as actual numbers, graphs, metaphors and real objects in life. An example of the item was “Use things in real life to present new concepts and ideas as much as possible”. Formal representation comprised representations involving formal and structural means, such as abstract symbols, definitions, formulas and proofs. An example of the item was “Use a formal approach (for example, definitions) to present new concepts and ideas as much as possible”. Problem solving was captured by two factors, skill preparation and heuristic guidance, each with seven items. Skill preparation referred to the approaches that prepare students’ problem-solving skills economically, such as omitting nonessential steps in solutions or demonstrating multiple problem-solving methods simultaneously by solving a challenging problem to save instruction time for teaching more problems, or making comparisons between and synthesising problems to help students succeed in exams. Some examples of items were “Only go over important steps rather than every single one when solving a problem” and “Compare and synthesise various problems and problem types”. Heuristic guidance involved teachers elaborating on and representing their own thought processes in detail to guide students heuristically, thereby encouraging students to think ahead, leading students to analyse and solve problems step-by-step, and accepting students’ individual problem-solving methods. Some examples of items were “Let students think about how to solve a problem Downloaded by [UAE University] at 08:47 25 October 2017 8 F.-J. HSIEH AND T.-Y. WANG by themselves before explaining it” and “Elaborate on the teacher’s own train of problem-solving thoughts in detail”. Teaching process was captured by two factors, eliciting and outlining, with four and three items, respectively. The eliciting teaching process involved helping students elicit mathematics ideas by requiring them to study new concepts before they are taught them or by reviewing previously taught concepts before teaching new ones, collocating corresponding examples for student understanding of concepts, and synthesising and comparing concepts after students are taught them to facilitate the construction of conceptual networks. The exemplified items were “Explain the correct concept to be used before a new lesson or before discussing math problems” and “Teach concepts alongside problems”. The outlining teaching process involved outlining main ideas and then going over them one by one, providing students with time to organise the ideas and practise with in-class exercises which associated with the ideas. The exemplified items were “Determining the key points and then explaining them respectively” and “Arranging the students to do some exercises in class after the concepts or the example questions were taught”. Teaching method was captured by three factors, student activities, idea explanation and speedy lecture, with six, five and three items, respectively. Student activities included teaching methods that require students to actively engage in small group work or discussions, games, hands-on activities and exploration. These activities often require a considerable amount time for students to complete. The exemplified items were “Ask students to explore new concepts or ideas before providing instruction” and “Employ small group learning when applicable”. Idea explanation involved teaching methods that used clear illustrations and explanations, and guidance in students’ conjecturing, observation, and induction to teach new ideas or clarify student confusion and doubts. The exemplified items were “Introduce new concepts from easy to difficult levels” and “Guide students in observation and induction to develop their concepts”. Speedy lecture referred to fast-paced teaching methods in which the teacher serves as the primary source of knowledge and takes responsibility for determining and presenting content to students, as indicated by the items “Lecture mainly to avoid unnecessarily wasting time” and “Emphasise critical ideas repeatedly in class”. Mathematical media and teaching aids was formed with one factor, media use. The exemplified items were “Use teaching aids such as cards, models, and surrounding objects, to instruct when appropriate”, and “Use multimedia such as PowerPoint and GSP to assist instruction when appropriate”. Evaluation was captured by two factors, performance demand and comprehension diagnoses, with six and eight items, respectively. Performance demand referred to assessments that provide students with substantial practice time and assign several exercises and quizzes; this can include preparing a broad range of problems from basic to challenging ones, and even expediting the teaching schedule to ensure that students have sufficient time to practise before examinations. Notably, these approaches aim to prepare students for high academic performance. The examples were “Arranging substantial quantities of homework for students to practise on” and “Meet the teaching schedule two weeks ahead, and start to request that students practise with problems to prepare for midterm or final exams”. Comprehension diagnoses involved assessments that diagnose student comprehension through multiple approaches. The examples were “In addition to written exams, use various methods such as having students answer questions when called on to assess students” and “Thoroughly discuss assignments and tests, and require students to correct the errors”. EDUCATIONAL STUDIES 9 Downloaded by [UAE University] at 08:47 25 October 2017 Comparison of the views of ideal mathematical teaching behaviours among teachers and students Figure 2 depicts the profiles of the in-service teachers, pre-service teachers and students. The factors from Table 1 were categorised into two sets. The set on the left consisted of those influenced by Western cultures (Martinez-Sierra 2014; Murray 2011; Wang and Cai 2007a; Wilson, Cooney, and Stinson 2005), which were promoted in Taiwanese teacher preparation programmes subsequent to the mathematics curriculum reform (Hsieh 1997; Lin and Li 2009). The set on the right consisted of the factors rooted in traditional Chinese culture (Cai 2004; Kaur 2009; Li 2011; Liu 2007; Pang 2009; Park and Leung 2006; Stevenson and Lee 1997), although some of them were ideas shared with Western cultures. Examining the factors influenced by Western cultures indicated that neither the profile of the in-service teachers nor that of the pre-service teachers shared the same patterns as the profile of the students. The factors were endorsed by both in-service teachers and students to similar levels with one exception, media use of mathematical media and teaching aids. Notably, the pre-service teachers endorsed all the factors to a higher degree than did the students, with score differences between 1.15 and 2.72. Both cohorts of the teachers significantly endorsed media use, which the students did not share. The pre-service teachers overestimated the importance of student activities of teaching method and simplification of handling teaching material when the levels of student endorsement were used as the baseline. This may have resulted from the emphasis placed on using student activities to provide high school students with opportunities to engage in and think about mathematics, and from simplifying Figure 2. Mathematical teaching behaviour profiles of in-service teachers, pre-service teachers, and students. Notes: To the right and the left of the underscores are the names of the factors and the abbreviations of the facets the factors belong to, respectively. E = Evaluation; HM = Handling teaching material; MA = Mathematical media and teaching aids; PS = Problem-solving; R = Representation; TM = Teaching method; TP = Teaching process. Downloaded by [UAE University] at 08:47 25 October 2017 10 F.-J. HSIEH AND T.-Y. WANG and concretising mathematics content so that students can comprehend ideas in the teacher preparation programme. The profiles of the two cohorts of teachers were not as deviated from the profile of the students on the factors rooted in traditional Chinese culture. The profile of the pre-service teachers shared a similar pattern with that of the students considering the relative amounts of emphasis on the factors, except formal representation. This exception may have resulted from all the pre-service teachers’ having majored in mathematics; thus, they valued the use of formal representations. However, the pattern of the in-service teacher profile was distinct from that of the pre-service teachers and students regarding the focus of mathematics teaching processes. Although in-service teachers shared similar opinions on instructional concepts, such as eliciting, they placed less emphasis on outlining than the students expected. These teachers also placed less emphasis on speedy lecture of teaching method, performance demand of evaluation, and formal representation. Concrete representations may facilitate comprehension of newly learned mathematical ideas, but formal representations are unavoidable in mathematical materials, actual mathematics classes, and assessments in Taiwan, and speedy lecture and performance demand are particularly pertinent to the examination-driven ideas in traditional Chinese culture. The higher endorsement of these factors by the students than the teachers suggested that high school students in Taiwan tend to value academic performance (Tan and Yates 2011), which may include both the development of mathematical abilities and achieving high grades (Hsieh and Wang 2014); they were more willing to learn formal representations that were difficult, as well as go through instruction with a quick pace and endure intensive practice, than the teachers thought they were. Profiles of ideal mathematical teaching behaviours among the teacher cohorts Figure 3 illustrates the results of the k-means clustering analysis on Rasch logit scores of the 14 factors of the in-service and pre-service teachers. Each profile represented the preferred type of a cluster of mathematics that the teachers regarded as ideal teaching behaviours. The teachers in Class 1 had high scores for all ideal mathematical teaching factors from both traditional Chinese culture and Western culture, scoring higher than 8 points on all except three factors (whose scores were a minimum of 7.53 points). The teachers in this class either highly or moderately highly endorsed all factors in the seven facets, and considered a mathematics teacher capable of implementing all of the embedded teaching behaviours. Thus, the profile was called the “all-round mathematical teaching behaviour profile” (all-round MTB profile). In total, 38% of the teachers fit this profile. In Class 2, a similar pattern to Class 1 was produced regarding Western-influenced factors, with simplification of handing teaching material being less desired (the difference in scores was 1.24). The teachers in Class 2 also had similar opinions with those in Class 1 on five factors rooted in traditional Chinese culture; however, skill preparation in problem solving, speedy lecture in teaching method, performance demand in evaluation, and formal representations in representation were less desired in Class 2. The scores of these four factors in Class 2 were between 3.92 and 7 points, with the biggest difference from Class 1 located in formal representation (Class 1 was 8.72 and Class 2 was 3.92). The teachers in Class 2 instead endorsed concrete representations, which were more helpful for students to construct meanings of mathematical ideas (Marzano, Pickering, and Pollock 2001). This class also endorsed various concepts of teaching behaviours, including teacher-leading behaviours such as heuristic Downloaded by [UAE University] at 08:47 25 October 2017 EDUCATIONAL STUDIES 11 Figure 3. Mathematical teaching behaviour profiles of three classes of teachers identified using k-means clustering analysis. Notes: To the right side and the left side of the underscores are the names of the factors and the abbreviations of the facets the factors belong to, respectively. E = Evaluation; HM = Handling teaching material; MA = Mathematical media and teaching aids; PS = Problem-solving; R = Representation; TM = Teaching method; TP = Teaching process. guidance, idea explanation and elicitation, and media use, as well as student-centred behaviours such as long-term student activities, student practising, and multiple comprehension diagnoses. The features of this class reflected Kaur’s (2009) commentary on ideal mathematics instruction in Singapore, which was teacher-centred but student-focused. The factors that the Class 2 teachers did not endorse were considered characteristic of traditional mathematics instruction in East Asia but disfavoured in the West because of a lack of meaningful learning and an emphasis on heavy practice and examinations (Huang and Leung 2004; Park and Leung 2006). Overall, the teachers in Class 2 endorsed the factors that comprehensively promoted students’ meaningful learning, but avoided the fast pace and demanding requirements that can cause distress in students. The profile of this class was thus named “painless meaning-emphasised mathematical teaching behaviour profile” (painless meaning-emphasised MTB profile). A total of 44% of the teachers fit this profile. The Class 3 teachers did not endorse the factors which were not endorsed by the teachers in Class 2, either. In contrast with the first two classes, Class 3 endorsed the teaching behaviours involving teacher guidance rather than those related to student activities, although they also applied various approaches in their teacher-guided instruction, including heuristics and media use. Another discrepancy among the classes was that Class 1 and Class 2 both emphasised eliciting concepts, outlining the main ideas, and student practising in class with respect to the teaching process facet, whereas Class 3 favoured only the process of eliciting concepts. Class 3 teachers also endorsed connection and meaning when handling teaching material with the use of concrete representations; the profile was thus named “concept-focused teacher-guidance mathematical teaching behaviour profile” (concept-focused teacher-guidance MTB profile) and comprised 17% of the teachers. 12 F.-J. HSIEH AND T.-Y. WANG Table 2. Distribution of teachers in the three classes. In-service teacher Pre-service teacher College senior College sophomore Teacher Class 1: All-round (%) 9 70 84 63 Teacher Class 2: Painless meaning-emphasised (%) 57 30 16 37 Teacher Class 3: Concept-focused teacher-guidance (%) 34 0 0 0 Downloaded by [UAE University] at 08:47 25 October 2017 Comparison of the profiles of ideal mathematical teaching behaviours among cohorts of teachers Table 2 shows the distribution of the three mathematical teaching behaviour profiles in the two teacher cohorts. The cohort of in-service teachers was dominated by the painless meaning-emphasised MTB profile (57%). The remaining in-service teachers were primarily of the concept-focused teacher-guidance MTB profile (34%), and only 9% of the in-service teachers matched the all-round MTB profile. Contrarily, the cohort of pre-service teachers had dominant all-round MTB profiles (70%), and none matched the concept-focused teacherguidance MTB profile. A higher percentage of the pre-service teachers than the in-service teachers were open to various mathematical teaching factors. None of the pre-service teachers could accept giving up the use of long-term student activities and neither could they accept that only concepts without problem solving were emphasized in the instruction. Prior research revealed that teachers’ notions of ideal mathematics instruction were profoundly affected by their teaching experiences in classes (Wilson, Cooney, and Stinson 2005), which might be the reason for the distinct percentage distributions of the three profiles between the cohorts of in-service and pre-service teachers; the in-service teachers likely adapted their concepts of instruction once in the classroom. Additionally, in comparison with college seniors (16%), a higher percentage of college-sophomore pre-service teachers fit the painless meaning-emphasised MTB profile (37%), which was the profile dominating the cohort of in-service teachers. This might have resulted from the sophomores’ limited experience in teacher preparation programmes, and their tendency to accept the approaches they experienced from their high school teachers (Wilson, Cooney, and Stinson 2005). Conclusion and discussion What constitutes effective teaching is a profound question and a topic worthy of investigation. Comparing student and teacher views enables teachers and teacher educators to determine whether the strategies and behaviours used by teachers coincide with what students admire and apply adjustments accordingly. Thus, the present study built on earlier research (Hsieh 2012, 2013) and used a teaching behaviour list to survey in-service and pre-service teachers’ perspectives on which teaching behaviours a great mathematics teacher should employ in class. Fourteen teaching behaviour factors (Table 1) in seven mathematical teaching facets and three ideal mathematical teaching profiles from the perspective of teachers (all-round, painless meaning-emphasised, and concept-focused teacher-guidance) were identified. This study also found that the profiles of teachers, both in-service and pre-service teachers, were not identical to those of the students. However, the perspectives of the inservice teachers resembled those of the students more than those of the pre-service teachers. EDUCATIONAL STUDIES 13 A discussion on lenses of culture and educational practice is included below to broaden the interpretations of the findings in this study. Downloaded by [UAE University] at 08:47 25 October 2017 Perspectives of ideal mathematical teaching behaviours and cultures Focusing on students and providing them with opportunities to learn meaningfully are internationally prevailing concepts on ideal mathematics instruction (Kaiser and Vollstedt 2007; Kaur 2009; Murray 2011; Pang 2009; Wang and Cai 2007a, 2007b). Aside from these common concepts, studies have revealed that teaching is profoundly entrenched in culture (Pang 2009). Both East Asian and Western cultures have endorsed specific teaching behaviours that build meaning for new mathematical concepts by connecting them to previously learned concepts and using concrete or life-related representations (Kaur 2009; Li 2011; Murray 2011; Wang and Cai 2007a). Our results support this conclusion, because Taiwanese high school in-service and pre-service mathematics teachers considered the factors connection (in handling teaching material) and concrete representations to be ideal teaching behaviours. Connection is a factor rooted in traditional Chinese culture, reflecting the long-standing view of mathematics as a systematic and coherent structure (Schmidt et al. 2007). However, concrete representations is a factor adopted from Western culture; the teachers’ endorsement of this factor may result from beliefs in the process- and application-oriented views of mathematics nature that prevail in the West (Schmidt et al. 2007) and are consistent with the ideas promoted in Taiwan’s 1990s mathematics curriculum reform. Another possible reason is that educational goals shifted during the curriculum reform, from elite education to education for all, with the notion that at least 80% of students should have opportunities to learn in class. Thus, Taiwanese teachers employ this type of representation as a feasible start to instruction to facilitate student understanding (Hall 1998; Ministry of Education 2000). Formal representations, although not always considered ideal, were also unavoidable in mathematics classes and assessments in Taiwan. Notably, some teachers from Western cultures have stated that they have reluctantly used concrete examples to facilitate student comprehension, despite an awareness that mathematical knowledge is abstract (e.g. Wang and Cai 2007a). By contrast, Taiwanese teachers are required to use representations, including symbols, formulas, definitions, and proofs in their classes, despite this strategy receiving low endorsement from most in-service teachers, particularly when teaching new ideas. This may reflect their frustrating experiences of teaching students with abstract representations and their desire to use this type of representation less, especially when teaching new ideas. Although the views of Taiwanese teachers were aligned with Western views for these factors, there were other Western-based teaching behaviours that Taiwan did not endorse. For example, curricula supporting the “no student left behind” policy are found throughout the United States and many other Western countries, which typically employ simpler and more concrete materials as a result. However, this study determined that Taiwanese teachers disapproved of teaching behaviours embedded in the simplification factor when handling teaching material, likely because this factor did not match the demanding mathematics curriculum in Taiwan. Two factors identified in the teaching process facet were rooted in traditional Chinese culture, but also had a substantial role in Western teaching. First is the eliciting factor, which involves organising mathematics classes to help students gradually comprehend Downloaded by [UAE University] at 08:47 25 October 2017 14 F.-J. HSIEH AND T.-Y. WANG mathematical concepts. The idea underlying this factor has been emphasised in Western teaching behaviours for many years, and is practised through meaningful learning (Ausubel 2000) and relational comprehension (Skemp 1989). Second, outlining, which involves organising mathematics classes in a structured manner to cultivate problem-solving competence among students, has also been a critical approach in the West (e.g. Mesiti and Clarke 2006; NCTM 2016). Providing outlines to inform students of the learning objectives of a topic is a common arrangement in mathematics classes and textbooks in Western cultures (Cunningham 2009; Molix-Bailey, Day, and Frey 2008). Both factors were also endorsed by the teachers in Taiwan. This may result from the long-standing two basics that are emphasised in East Asian mathematics classrooms: mathematics content and the procedures or skills with which content is applied (Zhang, Li, and Tang 2004). Of the six factors singled out from the facets teaching methods, problem solving, and mathematical media and teaching aids, Taiwanese teachers’ perspectives were similar to those of German teachers, who stood in the middle of the spectrum described by Kaiser and Vollstedt (2007). Specifically, Taiwanese teachers approved of both the factors resulting from the influence of Western culture (student activities and media use) and the factor rooted in Chinese culture (idea explanation). The teachers also endorsed heuristic guidance when teaching problem-solving, which combines characteristics from both Western and East Asian cultures. This factor allows students to think on their own and develop their own strategies to solve problems; it also involves elaborating on teachers’ thoughts during problem-solving, and presenting all of these details for the students. However, teachers in Taiwan did not endorse two factors rooted in Chinese culture, speedy lecture and skill preparation, which aim to quickly and economically cultivate students’ problem-solving skills to attain academic achievement. To help students successfully learn the demanding mathematics curriculum, Taiwanese teachers noted that employing idea explanation to convey mathematical ideas is inevitable. However, for the same purpose, they did not endorse pure teacher-directed approaches; instead, they supported student activities, heuristic guidance, and media use to provide students with some control over the learning process, to proceed according to the students’ learning pace, and to demonstrate ideas with various appropriate media. These ideas are consistent with the innovative instruction promoted in Taiwan’s mathematics curriculum reform. Although superior academic achievements are still highly valued, Taiwanese teachers did not approve of achieving it through speedy lecture and skill preparation, which are criticised as causing rote learning in the West (Leung 2001). Two factors were identified under the evaluation facet: comprehension diagnoses and performance demand. The teachers in Taiwan were aligned with Western culture on these factors, attaching great importance to frequent and quick assessments through various methods that diagnose student comprehension in classes (Wilson, Cooney, and Stinson 2005). Performance demand emphasised repeated practice with the use of many exercises and quizzes, which also remains endorsed by teachers in China (Wang and Cai 2007b). However, teachers in Taiwan have deviated from this factor rooted in traditional Chinese culture. One possible reason is that Taiwanese teachers did not consider students’ substantial and repetitive practice to build understanding; rather, they suggested that students’ classroom engagement would cause effective learning instead. This phenomenon is again consistent with the ideas promoted in Taiwan’s 1990s curriculum reform. EDUCATIONAL STUDIES 15 Overall, the perspectives about ideal mathematics teaching behaviours among Taiwanese high school teachers are shaped by the ideas rooted in traditional Chinese and the ideas originated in Western cultures and then adopted as well as promoted in mathematics curriculum reforms in 1990s. Downloaded by [UAE University] at 08:47 25 October 2017 Profiles of ideal mathematical teaching behaviours and educational practice Educational research often applies Western criteria to evaluate perspective and practices; however, its validity may be problematic if the underlying cultural norms are very different (Clarke 2013). Thus, the present study used Taiwanese students’ perspectives as a baseline for investigation and comparison of in-service and pre-service teachers’ perspectives to stand for student-centred view promoted in Taiwan’s mathematics curriculum since the 1990s. The dominant ideal mathematical teaching behaviour profile among the in-service teachers was the painless meaning-emphasised MTB profile. The concept-focused teacher-guidance MTB profile applied to approximately one-third of the teachers, and the all-round MTB profile applied to less than 10% of the teachers. However, among pre-service teachers, the dominant ideal mathematical teaching behaviour profile was the all-round MTB profile (70%), with a few of them fitting the painless meaning-emphasised MTB profile, and none of them fitting the concept-focused teacher-guidance MTB profile. The dispersion of MTB profiles between the two teacher cohorts may result from their various experiences in actual mathematics classrooms, because the in-service teachers had already adapted to their teaching experiences; this is consistent with what has been reported elsewhere (Skott 2001; Wilson, Cooney, and Stinson 2005). Between the cohorts of pre-service teachers, college seniors were dominated by the all-round MTB profile, whereas college sophomores were divided between the all-round MTB profile and the painless meaning-emphasised MTB profile at a ratio of approximately 6 to 4. Notably, because college sophomores were closer to their high school student experiences, they had tendency to accept their teachers’ instructions; this group also had less experience in the teacher preparation programmes than college seniors did, which might explain why a higher percentage of sophomores were classified under the painless meaning-emphasised MTB profile, which dominated the cohort of in-service teachers, when compared with college seniors. Conversely, a majority of college seniors belonged to the all-round MTB profile, which may reflect their open mind to the mathematical teaching behaviours embedded in all the factors; it is also possible that college seniors struggled with whether certain teaching behaviours would be feasible in real classrooms when they started as intern teachers. More research is necessary to thoroughly investigate this issue. Compared with the pre-service teachers, in-service teachers held perspectives that were more similar to the students. Nevertheless, in-service teachers’ perspectives still deviated from those of the students on several factors. Since the mathematics curriculum was reformed, teacher educators in Taiwan have focused on developing the competence of in-service and pre-service teachers to implement innovative, learner-centred, and constructivist-based instruction (Hsieh 1997; Lin and Li 2009), where the use of teaching aids and multimedia were advocated. The higher endorsement of media use in mathematical media and teaching aids among the teachers than the students may indicate the successful promotion of innovative instructional methods. However, the lack of endorsement from students remains an unanswered question; further research is necessary to determine whether this Downloaded by [UAE University] at 08:47 25 October 2017 16 F.-J. HSIEH AND T.-Y. WANG relates to teachers’ being capable of efficiently using multimedia during student learning processes or to their use of multimedia to help students attain their learning goals. The pre-service teachers overestimated the emphasis on student activities in teaching method compared with the level of student endorsement. Although the aforementioned indicates successful training in the teacher preparation programme, the amounts of emphasis that should be placed on media use and student activities in teacher preparation programmes and junior high school mathematics classes require additional investigation. Student endorsement of speedy lecture in teaching method, performance demand in evaluation, and formal representations was higher than that of in-service teachers, a phenomenon suggesting that, to improve academic performance, the students accepted teachers’ use of abstract representations, fast-paced instruction, and heavy practice. This confirmed the influence of the deep-rooted Chinese cultural value that high academic achievement is a priority among students in Taiwan (Tan and Yates 2011). The profiles of the in-service and pre-service teachers regarding ideal mathematical teaching behaviours revealed their malleable features in adapting to teaching experiences in schools and incorporating the learning experiences in teacher preparation programmes. This phenomenon supported the results of international comparison studies (Hsieh et al. 2011; Schmidt et al. 2011), thus indicating high teacher quality. The present study employed a quantitative approach to investigate various mathematical teaching facets. The findings provide broad pictures of ideal mathematical teaching behaviours from the divergent perspectives of teachers and students in Taiwan. Qualitative studies are necessary to derive additional rich and deep information on the teaching behaviour factors endorsed by teachers and the factors that received varying endorsements from the teachers and students for applicable purposes. Note 1. A master’s degree thesis by Hsieh’ student, entitled, “The instructive characteristics secondary school teachers should possess – The perspectives of students”, was completed with this study. Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported by the Ministry of Science and Technology [grant number 6510]. Notes on contributors Feng-Jui Hsieh is a professor at National Taiwan Normal University. She is also incumbent Chair of Taiwan Association for Mathematics Education. Her research has focused on international studies of mathematics teacher education and the assessment of mathematics teaching competence, awareness, and reflection of secondary school mathematics teachers. She is currently cooperating with researchers from Mainland China to explore ideal mathematics teaching behaviours from students’ perspectives and professional mathematics teaching indicators for secondary school teachers in Taiwan and China and to make comparisons of related issues between the two countries. EDUCATIONAL STUDIES 17 Ting-Ying Wang is a project assistant professor at National Taiwan Normal University. Her research has focused on international studies of mathematics teacher education and secondary school mathematics teachers’ teaching competences. She is currently cooperating with researchers from Mainland China to investigate and compare pre-service and in-service teachers’ perspectives of ideal mathematics teaching behaviours in Taiwan and China. Qian Chen is currently an associate professor at School of Mathematics and Software Science, Sichuan Normal University. She obtained her BSc and MSc from Southwest Normal University, and her PhD from the University of Hong Kong. The author has broad research interests, including mathematics teacher education, etc. She has been involved in various research projects and has published work internationally. She has been teaching courses at both undergraduate and post-graduate level, and supervising masters students. Downloaded by [UAE University] at 08:47 25 October 2017 References An, S. 2004. “Capturing the Chinese Way of Teaching: The Learning-Questioning and Learning-Reviewing Instructional Model.” In How Chinese Learn Mathematics—Perspectives from Insiders, edited by L. Fan, N.-Y. Wong, J. Cai and S. Li, 462–482. Singapore: World Scientific. Ausubel, D. P. 1961. “In Defense of Verbal Learning.” Educational Theory 11 (1): 15–25. Ausubel, D. P. 2000. The Acquisition and Retention of Knowledge: A Cognitive View. Dordrect: Kluwer Academic Publishers. Bond, T. G., and C. M. Fox. 2007. 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