# Practice makes perfect on the blackboard: A cultural analysis of mathematics instructional patterns in Taiwan

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Analyses ZDM 2006 Vol. 38 (5) Practice makes perfect on the processes and patterns of math instruction applied by teachers in different countries need to be blackboard: A cultural analysis of investigated so that we may gather a deeper mathematics instructional patterns understanding of how teaching practices in in Taiwan different countries might have contributed to the differences in student performances. Bih-jen Fwu, National Taiwan University Stigler and Hiebert (1999) analyzed TIMSS (Taiwan) videotapes of math instruction by teachers from Hsiou-huai Wang, National Taiwan the U.S. and Japan and discovered an interesting University (Taiwan) “teaching gap” in math instruction. It was found that students in an American math classroom Abstract: Studies show a sharp difference in math spent most of their time acquiring isolated skills achievement between students in the U.S. and through repeated practice, whereas students in a students in several East Asian countries, amongst Japanese classroom devoted as much time to them Taiwan. It is suggested that the patterns of solving challenging problems and discussing math instruction applied by teachers may have mathematical concepts as they did practicing contributed to these differences. This study skills. This difference may be embedded in the intends to investigate the patterns of math different “cultural scripts” concerning the nature instruction applied by the Taiwanese teachers and of math, nature of learning and role of the teacher to delve into the cultural roots of these patterns. in the two countries. In the U.S., where math is Data source includes videotaping of instruction perceived more as a set of procedures for solving by three middle school math teachers and a problems, American teachers focus more on questionnaire survey of 297 eighth-graders. It was procedural skills and ask their students to practice found that the Taiwanese math instruction pattern many times in order to master the skills. Since the may be summarized as a cultural activity of practicing of the procedural skills is not very “practice makes perfect, on the blackboard.” The exciting, the teacher’s role is to keep students’ underlying cultural beliefs are further explored, attention by various non-mathematical techniques including the incremental view of human and provide relatively error-free practice with a intelligence, self-improvement through diligent high level of success to avoid frustration. In Japan effort, and the teacher’s role as an authority on the other hand, math is perceived more as a set figure. of relationships between concepts, facts and procedures; therefore, Japanese teachers appear to ZDM-Classification: D40, C20, C70 emphasize more on conceptual understanding, pose challenging problems and engage their students in exploring by developing new methods 1. Introduction for solving problems. Frustration is taken to be a natural part of this challenging process. A series of international comparative studies on student math performance (Fuson, Stigler, & From this perspective, teaching is perceived as a Bartsch, 1988; Stevenson, Lee, & Stigler, 1986; “cultural activity” in which cultural scripts of Stevenson, et al. 1990; Stigler, Lee, & Stevenson, teaching/learning and the roles of teacher/ student 1990; U.S. Department of Education, 1992; are learned implicitly through observation and Mullis et al., 2000; Mullis et al., 2004) shows that informal participation over a long period of time there is a sharp difference in math achievement growing up in a culture. As each society has its between students in the U.S. and students in own shared value system, teachers of different several East Asian countries, amongst them societies learn the implicit cultural scripts about Taiwan. Stevenson and Stigler (1992) found that teaching and learning and thus develop distinct the “learning gap” between West (American) and instructional patterns. East (Chinese and Japanese) lies in students’ and Students from East Asian countries including parents’ different beliefs, expectations, and Japan, Hong Kong and Taiwan ranked top in satisfaction with academic achievement. TIMSS 1999 and 2003 (Mullis et al., 2000; Mullis However, the beliefs alone cannot explain how et al., 2004). Hiebert et al. (2003) did the TIMSS student learning in the classroom might have 1999 video study of mathematical instruction in contributed to the sharp differences. The 368

ZDM 2006 Vol. 38 (5) Analyses seven countries including Japan and Hong Kong identified. The instructional patterns common to and found that there are strong differences in the the three teachers gradually emerged through lesson scripts of mathematical instruction between constant comparison and contrast among the three these two countries. The investigation of different cases. countries in East Asia such as Taiwan may 2.2. Research outcomes supplement the existing literature. This study, therefore, intends to examine the processes and It was found that the instructional pattern common patterns of math instruction applied by Taiwanese to the three teachers consisted of the following six teachers, who may have learned the cultural steps: scripts of teaching and learning in this (1) Review of previous materials: predominantly Chinese society with a Confucian cultural tradition. At the beginning of a class, the teacher usually starts with a check of the homework assignment 2. Mathematical instructional patterns or a quiz to review material taught in the previous In order to investigate the processes and patterns period. He/she usually calls on students to write of math instruction in the classroom, detailed and up the procedures and solutions on the blackboard reiterated analysis of the teaching flow is and then checks if the students give the right necessary. Videotaping of a complete unit of math answers. lesson by different teachers is the most effective (2) Presentation of the topic for the day: way to obtain such data. The teacher then moves on to present the new 2.1. Research design topic for the day by saying, “Today we are going Due to Taiwanese teachers’ prevailing reluctance to talk about .” At this point, the to being videotaped in the classroom, the students usually “automatically” take out the math researchers worked very hard to recruit teachers textbook and turn to the exact page from which for the study and finally obtained permission from the new topic begins. Most students appear to three math teachers in two middle schools in the show a “readiness to learn.” Taipei area. This sample of three teachers showed (3) Presentation of definitions of terms and rules: a variation in age and gender. While the two male teachers were new to the profession, the female The teacher presents the new terms and rules by teacher had more than 20 years of experience. contrasting them with the previously established ones, which cannot apply to the new situation. At Each teacher was videotaped on a unit of math this stage, the teacher usually asks some closed lesson lasting three to four periods (hours). A total questions to check if students get the point. The of 15 periods of lesson instruction was videotaped teacher seems to provide little context relevant to with the three teachers. The videotapes were then the students’ previous experiences and raises few reviewed and analyzed using both quantitative questions to arouse interest or curiosity on the and qualitative methods. A Teacher Observation topic. Schedule, adapted from the Stallings Observation System (Stallings, 1988) was used to quantify the (4) Demonstration with examples: number of instructional activities and teacher- After explaining a new term and a new rule, the student interactions in the classroom. Occurrences teacher then demonstrates how to apply the rule of different types of instructional activities (such and use the right skills to get the answers. He/she as demos, practices and quizzes) and teacher- usually picks two to four problems of different student interactions (such as instruction-related or types and with different degrees of difficulty to management-related interactions) were coded at a illustrate the procedures for solving the problems frequency of once every 2.5 minutes. Categories step by step. In the demo process, the teacher of most frequently occurring instructional usually highlights the “knack” for deriving correct activities and interactions were found and answers for different types of problems. highlighted. Moreover, a reiterated review of each videotape segment was conducted. The flow of (5) Practice on the blackboard and at the seat: different instructional activities in chronological To check if students have learned the rules and order was differentiated and aligned, and distinct mastered the skills, the teacher usually calls on patterns in teaching flow by each teacher were some students by drawing lots or selecting a 369

Analyses ZDM 2006 Vol. 38 (5) certain sequence of students to practice textbook 3. Practice on the blackboard problems on the blackboard while other students What stands out in the Taiwanese instructional do the same problems at their seats. Usually several students are called at a time to solve pattern derived from the video study of the three problems of various types and/or degrees of math teachers’ instruction is the student’s repetitive practice on the blackboard. To further difficulty. If students at the blackboard are stuck, understand when, how and why this repetitive the teacher will usually help by giving some hints. practice is implemented in regular math If the students get the wrong answers, the teacher classrooms, the researchers supplement the video will then correct the mistakes and remind the whole class to beware of making the same errors. study by conducting a questionnaire survey from In the case of “hopeless” students, the teacher will the perspective of students. solve the problem for the students. With the 3.1. Research design correct procedures and answers listed on the A questionnaire was developed to ask students to board, the teacher then asks all the students identify the following occurrences in their regular practicing at their seats to check their own math classes: (1) occasions when teachers ask answers against the “standard” ones on the students to practice on the board (such as blackboard. checking homework, practicing textbook (6) Assignment of homework: worksheets, or giving quiz answers); (2) ways by which teachers select students to the board (such At the end of the class, the teacher usually gives homework either from the textbooks or from self- as volunteering, drawing lots, taking turns, or produced worksheets. He/she may also announce calling on inattentive students); and (3) reasons why teachers ask students to practice on the board a quiz to be held in the next period on the topic (such as to save time, present the right answer, or just taught. make sure students have learned). The It was found that the cycle of teacher introducing questionnaire was administered to a sample of 297 new terms and rule (step 3), demonstration with eighth-grade students from four schools in the example (step 4), and student practice on the Taipei area. Descriptive statistics and Chi-square board and at seats (step 5) is usually repeated tests were used to analyze the data. several times, occupying the major block of an instruction period; thus, little time was spent on 3.2. Research outcomes managerial or non-instructional tasks. Teacher- The questionnaire survey showed the following student interaction focused more on whole-class results: instruction in which teachers explain rules and ask (1) When to practice on the board closed questions and students respond accordingly. Among the three occasions for practicing on the blackboard listed in the questionnaire, students’ Comparing our findings with those of Stigler and “yes” responses to “checking textbook worksheets Hiebert (1999), it was found that math teachers in answers” were significantly higher than “no” the U.S. and Taiwan all reviewed previous materials, presented the problems, and asked responses (χ2[1, N = 297] = 166.5, p < 0.001). students to practice problems at their seats. It is Figure 1 shows the percentage of students who the process of presenting the problem that reveals agree that the occasion for practicing on the board great differences between the two countries. was highest for “checking worksheet answers” Similar to American teachers, the Taiwanese (87.5%), followed by “checking homework teachers in our sample focus more on answers” (49.0%), and lastly “checking quiz demonstrating procedures rather than on math answers” (25.5%). In sum, students perceive that concepts, and ask students to practice procedural teachers are more likely to ask students to practice skills rather than understand the reasoning behind on the blackboard when teachers want to check if the procedures. However, American students tend students get correct answers on the textbook to practice procedures at their seats, while worksheets after listening to explanations and Taiwanese students practice both on the demonstrations. blackboard and at their seats. 370

ZDM 2006 Vol. 38 (5) Analyses 100% 87.5% Yes Response (%) 80% Answer Types: 60% A. Checking worksheet answers 49.0% B. Checking homework answers 40% C. Checking quiz answers 25.5% 20% 0% A B C Figure 1. When to Practice on the Board? (2) How to select students to the board students to the board was highest for “calling inattentive students” (79.9%), followed by Among the five ways for selecting students to the “drawing lots” (66.7%), “volunteering” (49.3%), board listed in the questionnaire, students’ “yes” “calling on students who knew the answers” responses to “calling inattentive students” and (46.7%), and finally, “taking turns” (26.6%). In “drawing lots” were significantly higher than “no” sum, students perceive that teachers are more responses (χ2[1, N = 297] = 104.52, p < 0.001; likely to call inattentive students or use lot- χ2[1, N = 297] = 32.67, p < 0.001, respectively). drawing when selecting students to practice on the Figure 2 shows that the percentage of students board. who agree that the ways teachers use to select 100% 79.9% 80% Answer Types: Yes Response (%) 66.7% A. Calling inattentive students 60% B. Drawing lots 49.3% C. Volunteering 46.7% D. Calling on students who 40% know the answers 26.6% E. Taking-turns 20% 0% A B C D E Figure 2. How to Select Students to the Board? (3) Why practice on the board reasons teachers ask students to practice on the board was highest for “making sure students Among the five reasons for asking students to learn” (90.5%), followed by “preventing the same practice on the board listed in the questionnaire, mistakes” (78.9%), “practicing different types of students’ “yes” responses to “making sure problems” (76.9%), “providing standard answer” students learn” and “preventing the same (45.1%) and finally, “saving time” (27.6%). In mistakes” and “providing different types of sum, students perceive that teachers ask to problems” were significantly higher than “no” students to practice on the board probably because responses (χ2[1, N = 297] = 194.60, p < 0.001; teachers want to make sure that students learn, to χ2[1, N = 297] = 98.30, p < 0.001; χ2[1, N = 297] prevent students from making the same mistakes, = 85.70, p < 0.001, respectively). Figure 3 shows and to provide students with different types of that the percentage of students who agree that the problems to practice. 371

Analyses ZDM 2006 Vol. 38 (5) 100% 90.5% 78.9% Answer Types: 76.9% Yes Response (%) 80% A. Making sure students learn B. Preventing making the 60% same mistakes C. Practicing different types 45.1% of problems 40% D. Providing standard 27.6% answer E. Saving time 20% 0% A B C D E Figure 3. Why to Practice on the Board? 4. Discussion on the problem can a student master the skills to solve the problem. In summary, from the three cases of math instruction videotaping and student questionnaire Underlying this emphasis on practice is a deep- survey, it was found that Taiwanese teachers rooted view of the “incremental” perception of focused more on demonstrating procedures and human intelligence. Dweck et al. (Dweck, 1999; asking students to practice at seats and on the Dweck, Chiu, & Hong, 1995; Dweck & Legget, blackboard. Moreover, teachers usually drew lots 1988; Hong, Chiu, & Dweck, 1995; Levy & or called on inattentive students to practice Dweck, 1998) found that some people hold an textbook worksheets on the board for the purpose “incremental” view while others hold an “entity” of making sure that students learn, preventing view of intelligence. Those who hold the them from repeating the same mistakes and incremental view tend to see intelligence as a providing them with different types of problems malleable quality that can be increased through for practice. effort, while those who hold the “entity” view believe human intelligence to be a fixed Concurring with Stigler and Hiebert’s (1999) permanent entity that cannot be changed. This argument that teaching is a cultural activity, it is belief may vary with cultures. Cross-cultural of interest to consider the cultural beliefs studies on people’s beliefs in intelligence found underlying the distinctive Taiwanese math that while Westerners tend to hold an entity view, instructional pattern. the Chinese tend to hold an incremental view 4.1. Practice makes perfect which emphasizes that human intelligence can be perfected (Chen & Uttal, 1988; Tong, Zhao, & The reason why repeated practice at seats and on Yang, 1985). the board plays such a major role in math instruction in the Taiwanese classroom may lie in Related to this incremental belief in human the deep-rooted conviction that “practice makes intelligence is the pattern of attribution to one’s perfect” (shou neng sheng qiao). Many Chinese effort, rather than innate ability. Weiner (1986, idioms express a concept that places “practice” in 2001) contend that people attribute their success the pivotal role in human learning, such as and failure to innate ability, effort, luck and task “diligence makes up for inadequacy” (qin neng bu difficulty. Many previous studies on student zhuo), “a gem unless polished forms no article of academic achievement found that while Western virtue” (yu bu zuo bu cheng qi), and students were more likely to attribute their “perseverance can grind rough iron into a delicate academic success or failure to ability, the Chinese needle” (tiechu mo cheng xiuhuazhen). It is tended to attribute their success to effort (Hau & believed that only through constant practice can Salili, 1989; Hess & Azuma, 1991; Holloway, learning be perfected. Therefore, in the context of 1988; Salili, Hwang, & Choi, 1989; Yang, 1986). a math classroom, only through repeated practice They believe that through constant effort-making, 372

ZDM 2006 Vol. 38 (5) Analyses their ability can be incrementally increased and tend to see errors as an opportunity improve the task of learning can be perfected. oneself. This is why Taiwanese teachers frequently send a student to the board to display Underlying this emphasis on malleability of his answers, even erroneous, in front of the whole ability through effort is the optimistic Confucian class. They do not seem too concerned about view of humanity that everyone is educable, and damaging the student’s self-esteem due to public perfection in human nature can be attainable by failure and making errors. Instead of seeing the all in spite of variance in people’s innate ability open embarrassment as punishment, they tend to (On, 1999). As expressed in the Analects (XVI.9), regard mistakes as an indication of what needs to “The smart can learn a task easily and quickly be learned for oneself and for others. For oneself, while the dull learn arduously through much through persistence and effort, a student can practice. But in the end, they all learned”. eliminate errors and eventually produce the Therefore, one’s diligence compensates for correct answer. For others, a math error made by a inadequacies in innate ability and constant particular student can be instructive because the practice can perfect a task of learning. This deep- teacher may use it as an example for correction rooted belief in the malleability of students’ and as a reminder to avoid that mistake. This is learning capacity through effort is revealed in the reflected in our student survey that “preventing Taiwanese math instructional pattern, in which the the same mistake” is one of the main reasons for teacher tends to ask students to practice repeatedly teachers to send students to the blackboard. In at the seat and on the board. fact, the Chinese character for “error” (tsuo) 4.2. Practice makes perfect on the blackboard contains the ideogram for "gold," reflecting the The distinct Taiwanese pattern focusing on idea that even a mistake can harbor golden practice “on the blackboard” may be related to, opportunities for learning. first, a tendency of self-improvement and second, Another possible explanation for why Taiwanese the role of the teacher as an authority figure in the teachers send students to the blackboard is related Chinese cultural context. to the teacher’s role as an authority figure in the Kitayama et al. (1997) and Hein et al. (1999) traditional Chinese culture. The respect for propose that while European Americans show a teachers was so high that the ancient Chinese tendency for self-enhancement, a general placed teachers on the same level as “heaven, sensitivity to positive self-relevant information, earth, emperor and parents” (tien, di, jun, qin, shi) and an orientation to enhancing one’s own in the temple of worship (Gao, 1999). Teachers uniqueness and self-esteem, East Asians tend to have always been highly respected because they display an inclination for self-improvement, a are perceived as “learned scholars” (jinshi) who general sensitivity to negative self-relevant transmit knowledge and skills essential for living information, and an orientation to making up for and also as “moral figures” (renshi) who set good one’s shortcomings and perfecting one’s actions examples for students to follow (Fwu & Wang, to meet standards of excellence shared in a social 2002; Wang, 2004). Not everyone can be a context. In U.S. where the self-enhancing teacher; only those with great knowledge and orientation is preferred, teachers are expected to virtues can assume this honorable role, implying provide practice relatively error-free with high that teachers are the source of knowledge to their levels of success to avoid frustration, and tend to students. conceive of errors as a possible precursor to This deep-seated view penetrates into the ultimate failure. This may explain why U.S. classroom where students are obliged to listen to teachers seldom call on students to practice on the the teacher and absorb the knowledge the teacher blackboard, because they fear if a student failed to delivers to them. In this teacher-centered answer correctly and his errors were witnessed by instruction, students usually take a low profile, the whole class, this public display of failure rarely asking questions or volunteering answers. might damage the student’s self-esteem and In fact, students may feel timid about expressing counter his self-enhancing tendency (Stevenson & themselves in front of the authority figure Stigler, 1992). (Duncan & Paulhus, 1998; Pratt & Wong, 1999; In a culture where self-improving orientation is Tweed & Lehman, 2002). Under this emphasized, the Taiwanese teachers and students circumstance, the teacher will need to employ tactics to see whether students actually learned 373

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