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
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

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
                                                       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.

ZDM 2006 Vol. 38 (5)                                                                                                                         Analyses

                   Yes Response (%)

                                                                                                                Answer Types:
                                       60%                                                                      A. Checking worksheet answers
                                                                                                                B. Checking homework answers
                                       40%                                                                      C. Checking quiz answers


                                                      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


                         80%                                                                               Answer Types:
Yes Response (%)

                                                             66.7%                                         A. Calling inattentive students
                                                                                                           B. Drawing lots
                                                                     49.3%                                 C. Volunteering
                                                                                                           D. Calling on students who
                         40%                                                                                  know the answers
                                                                                               26.6%       E. Taking-turns


                                              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.

Analyses                                                                                            ZDM 2006 Vol. 38 (5)

                                           78.9%                                         Answer Types:
 Yes Response (%)

                     80%                                                                 A. Making sure students
                                                                                         B. Preventing making the
                     60%                                                                    same mistakes
                                                                                         C. Practicing different types
                                                                                            of problems
                     40%                                                                 D. Providing standard
                                                                             27.6%          answer
                                                                                         E. Saving time

                               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,

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

Analyses                                                                                ZDM 2006 Vol. 38 (5)

what was taught. Sending them to the blackboard         Dweck, C. S., Chiu, C. Y., & Hong, Y. Y. (1995).
to display math procedures and answers is one            Implicit theories: Elaboration and extension of
effective way to check if students can get the right     the model. Psychological Inquiry, 6 (4), 322-
answers. That’s why our survey found that                333.
“checking if students learn” is one of the reasons      Dweck, C. S. & Legget, E. L. (1988). A social-
for teachers asking students to practice on the          cognitive approach to motivation and person-
blackboard.                                              ality. Psychological Review, 95, 256-273.
                                                        Fuson, K. C., Stigler, J. W., & Bartsch, K. (1988).
Moreover, since the teacher is regarded as an
                                                         Grade placement of addition and subtraction
authority figure, students are expected to show
                                                         topics in Mainland China, Japan, the Soviet
deference by being constantly alert to what the
                                                         Union, Taiwan, and the United States. Journal
teacher delivers in class. As the teacher is the
                                                         for Research in Mathematics Education, 19 (5),
source of knowledge, learning is viewed as a
serious process in which students are expected to
                                                        Fwu, B. J., & Wang H. H. (2002). The social
fully concentrate. To keep students on alert, the
                                                         status of teachers in Taiwan. Comparative
teacher tends to use tactics that may help students
                                                         Education, 38(2), 211-224.
concentrate on tasks. That’s why our survey
                                                        Gao, M. S. (1999). The history of Chinese edu-
showed that “drawing lots” and “calling upon
                                                         cational system (Zhong Guo Jiao Yu Shi Lun).
inattentive students” are frequently used methods
                                                         Taipei: Lian-jing Bookstore. [in Chinese]
of selecting students to the board.
                                                        Hau K. T. & Salili, F. (1989). Attribution of
As mentioned earlier, teaching is a cultural             examination results: Chinese primary students
activity in which cultural scripts of teaching/          in Hong Kong. Psychologia, 32, 163-171.
learning and the roles of teacher/student are           Hein, S. J., Lehman, D. R., Markus, H. R. &
learned implicitly through observation and               Kitayama S. (1999). Is there a universal need for
informal participation over a long period of time        positive self-regard? Psychological Review,
growing up in a culture. This study provides a           106(4), 766-794.
case in point that the Taiwanese teachers, growing      Hess, R. D., & Azuma, M. (1991). Cultural
up in the Chinese cultural context, have learned         support for schooling: Contrasts between Japan
and demonstrated a distinct math instructional           and the United States. Educational Researcher,
pattern. This pattern highlights the Chinese deep-       20 (9), 2-8.
rooted belief that “practice makes perfect,” and        Hiebert, J. et al. (2003). Teaching mathematics in
that learning is refined through constant effort in a    seven countries: Results from the TIMSS 1999
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                                                        Holloway, S. D. (1988). Concepts of ability and
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supported by a grant from the National Science          Hong, Y. Y., Chiu, C. Y., & Dweck, C. S. (1995).
Council, Taiwan, The Republic of China (NSC              Implicit theories of intelligence: Reconsidering
90-2511-S-002 -023).                                     the role of confidence in achievement moti-
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