How teachers respond to students’ mistakes in lessons: A cross-cultural analysis of a mathematics lesson

Abstract

Purpose

The purpose of this paper is to capitalize on the advantages of an evidence-based lesson analysis while proposing a method of research on teaching that offers opportunities for deeper reflections. The objective is to examine how well a transnational learning project such as this one can determine the cultural script of a mathematics lesson in Malaysia through the perspective of Japanese educators well trained in the lesson study approach. Emphasis here is on a cross-cultural analysis to view in depth the cultural script of teaching mathematics in Malaysia with particular focus on how teachers respond to students’ mistakes in a mathematics lesson.

Design/methodology/approach

This paper draws on data collected by the authors in a lesson study in Malaysia that aimed to provide a cross-cultural analysis of a Malaysian mathematics lesson (grade 10) through the eyes of Japanese educators. Data retrieved should determine the cultural script of a mathematics class in Malaysia with an emphasis on Malaysian teachers’ responses to students’ mistakes in class. The cross-cultural analysis of a lesson is a comparative method that reveals the hidden factors at play by increasing awareness of characteristics in classroom situations that are self-evident to all involved members.

Findings

The findings are intended to the cultural script of Malaysia in the context of “classroom culture regarding mistakes” and “mistake management behavior.” The impact on the quality of teaching and learning also discussed in relation to how it can be improved in practice from the following perspectives: the teacher’s attitudes toward student mistakes; how mistakes are treated and dealt with in class; and how learning from mistakes is managed. The data in Table II provide a meta-analysis of evidences of “classroom culture regarding mistakes” and “mistake management behavior” of the teacher from the Malaysian researchers and practitioners’ perspective as well as from the lens of the Japanese educators.

Research limitations/implications

This study realizes that both sets of research studies value the importance of mistakes. It is important to identify the source of students’ mistakes and further learn from them. In order to reveal the overall structure of the cultural script of lessons, we need to realize that various cultural scripts are at work in the production of any given lesson. In the future, the authors hope to develop the potential of this view of culture script of teaching through cross-cultural analysis for lesson study and curriculum research and development.

Practical implications

This study aims to capitalize on the advantages of evidence-based lesson analysis through the lesson study process while proposing a method of research on teaching that offers opportunities for deeper reflections. The objective is to examine how well a transnational learning project such as this one can determine the cultural script of a mathematics lesson in Malaysia through the perspective of Japanese educators well trained in the lesson study methodology.

Social implications

The authors need to obtain reflective feedback based on concrete facts, and for this reason “lesson study,” a pedagogical approach with its origins in Japan, is attracting global attention from around the world. This study focuses on the discrete nature, the progression, significance, and the context of lessons. That is, by avoiding excessive abstraction and generalization, reflection based on concrete facts and dialogue retrieved from class observations can be beneficial in the process. The mutual and transnational learning between teachers that occurs during the lesson study process can foster the building and sharing of knowledge in teaching practice.

Originality/value

There is currently little empirical research addressing “classroom culture regarding mistakes” which mostly represents how teachers and students learn from mistakes in the classroom. This study focuses on a cross-cultural analysis to view in depth the cultural script of teaching mathematics in Malaysia with particular focus on how teachers respond to students’ mistakes in a mathematics lesson. The following perspectives are examined: the teacher’s attitudes toward student mistakes; how mistakes are treated and dealt with in class; and how learning from mistakes is managed.

Citation

Sarkar Arani, M.R., Shibata, Y., Sakamoto, M., Iksan, Z., Amirullah, A.H. and Lander, B. (2017), "How teachers respond to students’ mistakes in lessons: A cross-cultural analysis of a mathematics lesson", International Journal for Lesson and Learning Studies, Vol. 6 No. 3, pp. 249-267. https://doi.org/10.1108/IJLLS-12-2016-0058

1. Introduction

Building teaching ability of teachers has been a significant issue across the ages. However, teaching ability is difficult to acquire outside of teaching practice (e.g. Stenhouse, 1975). It is said that teachers changing their awareness (view of lessons) or learning pedagogical approaches through training seminars and published literature often does not lead to actual change in teaching practice (e.g. Hiebert and Morris, 2012).

Specifically, changing teachers does not necessarily improve quality of teaching. It is difficult to improve lessons by simply understanding new theories, or acquiring new methods and tools. The key is in applying approaches that link theory with method, and integrating cultural known factors that allow those approaches to work effectively (e.g. Morris and Hiebert, 2011). However, such approaches and cultural factors are often hidden or difficult to recognize. Stigler and Hiebert (2009, p. 12) point out that “much of what happens in the classroom is determined by a cultural code that functions, in some way, like the DNA of teaching. This could be why simply changing teachers will not automatically produce a change in results.” This cultural code is referred to as teaching script. The teaching script, in other words the cultural script of a lesson, is the imagery, values, and customary ways of acting in lessons where people share one culture, and the interrelated aggregate of these (Sarkar Arani, 2017; Stevenson and Stigler, 1992).

Although the teaching script (the cultural script of the lesson) is a determining factor of teaching practice, for the most part we are not aware of it. The authors believe that the teaching script cannot be extracted from practitioner reflection and researcher observations alone. The script within each culture is self-evident to people within that culture. Reflection cannot tell of things that are seen as self-evident. Knowledge gained from reflection is only relevant to particular individuals and local communities (e.g. particular schools, study groups), and is not open to external use.

The process of discovering possible points of improvement through reflection, implementing these, and subsequently iterating further evaluation and improvement is crucial. To this end, we need to obtain reflective feedback based on concrete facts, and for this reason “lesson study,” a pedagogical approach with its origins in Japan, is attracting global attention from around the world (Elliott, 2016; Wood, 2017). Lesson study focuses on the discrete nature, the progression, significance, and the context of lessons. That is, by avoiding excessive abstraction and generalization, reflection based on concrete facts, and dialogue retrieved from class observations can be beneficial in the process. The mutual learning between teachers that occurs during the lesson study process can foster the building and sharing of knowledge in teaching practice.

1.1 Purpose of this study

This study aims to capitalize on the advantages of evidence-based lesson analysis through the lesson study process while proposing a method of research on teaching that offers opportunities for deeper reflections. The objective is to examine how well a transnational learning project such as this one can determine the cultural script of a mathematics lesson in Malaysia through the perspective of Japanese educators well trained in the lesson study methodology. Emphasis here is on a cross-cultural analysis to view in depth the cultural script of teaching mathematics in Malaysia with particular focus on how teachers respond to students’ mistakes in a mathematics lesson. The following perspectives will be examined: the teacher’s attitudes toward mistakes; how mistakes are treated and dealt with in class; and how learning from mistakes is managed.

1.2 Classroom culture regarding mistakes and mistake management behavior

There is currently little empirical research addressing “classroom culture regarding mistakes” which mostly represents how teachers and students learn from mistakes in the classroom (Santagata, 2004, 2005). Although cross-cultural studies are expanded to uncover the culture of teaching in the last few decades (e.g. Alexander, 2001; Hayhoe, 2015; Li, 2012; Moore, 2000; Stigler and Hiebert, 2009, 2016), relatively a few empirical research studies have focused on “mistake management behavior” which mostly represents how students and teachers react to the mistakes, especially how teacher responses to students’ mistakes (Santagata, 2005; Tulis, 2013).

According to Santagata (2004) a comparative study between Italian and American teachers’ mistake-handling strategies, “mistake-handling activities may be understood as culturally specific interplays of beliefs and practices” (p. 141). Her research shows that handling of students’ mistakes is an important activity and should be regarded as a “cultural logic of practice.” She asserts that “mistakes are a fundamental aspect of learning, and teacher-student interactions surrounding mistakes are frequent; the unfolding of these interactions has important effects not only on learning outcomes, but also on students’ motivation” (p. 143). One of her research findings shows that “Italian teachers may respond to students’ mistakes by openly showing their disappointment. Whereas in the U.S., teachers show a positive disposition toward students who make mistakes” (p. 161).

Ten years later, a study by Tulis (2013) revealed that “only a few studies have focused on how teachers deal with mistakes in the actual classroom setting” (p. 56). Tulis claims that “only very few teacher-student-interactions included reinforcement of error risk taking and emphasizing mistakes as learning opportunities” (p. 60). In the case of Japanese culture of teaching, Tanaka’s (2017) historical research on “practices of leading educators” shows, “[w]hen a child makes a mistake during class, he [the teacher] did not pin the blame on that child, but called the mistake “xx’s type of mistake,” and classified the same type of mistakes into patterns so that the whole class could share and learn from them” (p. 75).

Current studies contribute to existing attempts for a theoretical framework of this study which tries to examine “mistake management behavior” and the “classroom culture regarding mistakes” in actual classroom settings. Additionally, this study focuses more on teachers’ response to students’ mistakes. The theoretical framework for this study was adopted based on the literature by Van Dyck et al. (2005), Heimbeck et al. (2003), Santagata (2004, 2005), Schleppenbach et al. (2007), Stevenson and Stigler (1992), Stigler et al. (1995), Stigler and Hiebert (2009), and Tulis, 2013. The coding format of comparative research by Tulis (2013, pp. 58-59) and Santagata (2004, 2005) of teacher beliefs and practices between Italy and the USA was of especial interest (see Figure 1).

2. Research design

2.1 Methodology

The cross-cultural analysis of a lesson is a comparative method that reveals the hidden factors at play by increasing awareness of characteristics in classroom situations that are self-evident to all involved members. The concept of comparison employed in our proposed “cross-cultural lesson analysis” is different to the traditional idea of comparison.

The first main difference is we extract multiple cultural codes through the practice of analysis. We then interpret the meaning of situations, being very careful not to fall into the trap of arbitrary interpretation. Subsequently, we group these meanings into cultural codes before focusing on uniformity and format present in the actions that appeared within. Finally, we use an explorative approach to reveal the structural factors thought to be at work across multiple cultural codes, and identify these as cultural script. This study refers to the situations interpreted that have arisen in practice from the implicit and explicit effects of culture as a “cultural code.” The “cultural script” is used to refer to the structure of meaning behind each cultural code when decisions are made in the classroom.

2.2 Comparison

The concept of comparison used in this research differs from that traditionally used. That is, this research does not center on highlighting features shared by a particular country or culture. In short, we did not conduct comparative research seeking a standard or typical model. We look at comparison as a lens from both the Malaysian and Japanese researchers’ perspective. As Hayhoe (2015) noted “comparative education became a lens through which I tried to reflect on and interpret the educational developments of a changing China” (p. 1). For instance, Stigler and Hiebert (2009) have already conducted a comparison of lessons in Japan, the USA, and Germany. In their study they adopted the method of extracting features of teaching common to multiple countries and highlighting the differences between lessons in each one. However, ultimately the problem is that the concrete actions of teachers and learners are made abstract. This calls for a new research method (Thomas, 2012), which is required to uncover the cultural teaching script of each particular country (Sarkar Arani, 2017; Stigler and Hiebert, 2016).

The “cross-cultural analysis” proposed in this study has its origins around this point. Thus, “comparative” does not mean “competitive.” It is not a matter of ranking countries as better than each other. This cross-cultural analysis is not a comparative approach based on generalization and standardization, but rather it reveals valuable differences through exchange. Looking through the lens of people with different cultural backgrounds enables us to extract a certain cultural code. Further discussion with researchers can then reveal a cultural script. On the other hand, it enables researchers to notice the existence of their own “lenses,” a perspective which to them is self-evident. In this way, cross-cultural analysis is not a comparison for the sake of highlighting differences, but rather a lens and a way of mutual understanding of individual differences and similarities. In fact, in this current study, the analysis of lessons conducted in different countries and cultures is considered as an opportunity for cross-cultural learning. Accordingly, certain values and qualities come to light when they are viewed through a different lens.

2.3 The cross-cultural analysis procedure

A cross-cultural analysis of lessons seeks to elicit the structure of the cultural meaning hidden in each teaching practice by analyzing local and foreign practice through the eyes of education researchers and teachers with different cultural backgrounds. It not only reveals the unique advantages of lessons in each country but also further initiates a deeper understanding of lessons and the way they are conducted in one’s own country. The cross-cultural analysis procedure for this study is as follows:

1. Observe and record the Malaysian mathematics lesson with video and audio recorders. This was conducted by Japanese and Malaysian researchers. The investigation also relies on evidence based on the observations and recordings. A word-for-word transcript was created for the lesson which transcribed all dialogue within the lesson.

2. Teachers and researchers of Malaysia where which the lesson was conducted, analyzed the transcript. The analysis is not limited to an evaluation of the events of each lesson (e.g. children/students’ speech, teachers’ actions); rather, the basic attitude toward analysis is to learn and understand the aim and pedagogy of the lesson.

3. Lesson transcripts are exchanged and examined between teachers and researchers from different countries, in this case Japan and Malaysia. Lesson analysis based on the transcript is primarily conducted qualitatively through reading and exchanging of lesson transcripts. However, there is a focus on the analysts’ own concerns, perception, and viewpoints.

4. The results of the analysis of the mathematics lesson through the eyes of Japanese and Malaysian researchers are compared.

5. In order to encourage active and constructive comments, lesson analysis meetings use a group work and workshop style format allowing all participants to discuss together and provide feedback collaboratively. By doing this, it helps to test a limited hypothesis. This approach is similar to that of generating an exploratory hypothesis.

Using this procedure brings to the fore determining the unique qualities of the mathematics lesson in each other’s countries while simultaneously helping to deepen our understanding of the way a lesson is taught in our own country. It should be noted that the above procedure is a base format and does not necessarily need to be followed in its entirety. Where it is difficult to use lessons that teach the same curriculum and content units, useful knowledge can also be garnered from merely having teachers and researchers from different countries/cultures analyze the same lesson. Here, more specifically, exploring the structure of meaning hidden in lesson practice takes an approach not of typifying or standardizing but of “comparison as a lens” by educators from differing countries. Furthermore, this helps to illuminate issues through intricate exchanges, to better understand the style, cultural script, process, and quality of teaching and the pedagogical reasoning of teachers and their teaching designs and methods in each country.

2.4 Data collection

This study employs the cross-cultural analysis procedure to empirically investigate the cultural script of a mathematics lesson in Malaysia from the perspective of Japanese educators well trained in the lesson study approach with particular focus on how teachers respond to students’ mistakes. This involved peer observation of a mathematics lesson by the authors, the school principal, and teachers of the school. The transcript of the lesson, post-lesson discussions in Malaysia, ethnography field notes, and semi-structured interviews with the teacher were also used.

The lesson was conducted on March 1, 2013, in a secondary boarding school in the city of Seremban, Malaysia. A post-lesson discussion was held in the school immediately after the lesson observation with all participants (Japanese and Malaysian researchers, the school principal, teachers, and three advisors from Seremban City Board of Education). Further discussion meetings on the lesson were held in Malaysia with the participation of nine teachers who are also post-graduate students on August 30, 2013, in the town of Bangi. More detail discussions were also held between four teachers from the school and four researchers from the University of Kebangsaan Malaysia on September 19, 2013, again in Seremban.

Based on the data collected in the local context (Malaysia), the transcripts of the lesson and the discussion meetings were translated from Malay (Bahasa Melayu) into English and Japanese. Further discussion meetings on this mathematics lesson were held in Japan with the participation of 37 Japanese educators on December 21, 2013 and again with 22 educators on January 7, 2014, at Nagoya University, Aichi. Ideas and opinion raised in these meetings were analyzed from a range of perspectives and through different socio-cultural lenses. The authors also conducted semi-structured interviews with some Malaysian and Japanese participants, especially mathematics educators to hear their general opinion about the mathematics lesson in depth.

The results of these discussion meetings and interviews revealed that some issues raised were viewed differently through the lens of Japanese educators in regard to lesson structure and the quality of teaching mathematics in Malaysia. Therefore, again a further discussion meeting on the lesson was held with the collaborative participation of eight Japanese educators and four Malaysian educators including the actual teacher of the class, on August 8, 2015 at Miyakomesse Public Hall, Kyoto, Japan. By adopting this approach, it was predicted that the apparent cultural structure of meaning hidden in lesson practice could be discovered. In addition, various comment dealings to each other revealed the structure of the cultural script of the Malaysian mathematics lesson from the eyes of the Japanese educator.

The findings were categorized into two main aspects of how teachers respond to students’ mistakes in classrooms, “classroom culture regarding mistakes” and “mistake management behavior,” as shown in Table II. Then it presented to a range of educators in Malaysia and Japan for reconfirming reliability of the comparison. Finally, the research team asked 27 research experts (14 Japanese on September 3, 2015, 13 Malaysian on September 5, 2015) to recognize the understanding in detail and approve the validity of the comparison presented in Table II. The finding is based on a meta-analysis of all data collected in Malaysia and Japan as cross-cultural analysis and discussion to provide evidence in detail.

3. Case

The research lesson that is analyzed and reported in this paper was implemented by a veteran female teacher of mathematics with 27 years of teaching experience. The lesson was a math lesson at the tenth grade level in a secondary boarding school in Seremban, Malaysia. There were 20 students present in this class. The boy’s school where this lesson was conducted had 541 students divided into two grades (grade 10 and grade 11). Each grade had 10 classes comprising of between 20 and 30 students each at age 16 and 17.

The lesson was planned by the teacher in collaboration with her three colleagues in the school. All four educators met several times to discuss the lesson plan during January and February before implementing it on March 1, 2013.

In this particular lesson, the teacher was following national curriculum guidelines. The curriculum unit in this lesson was numerical “sets.” Students were to learn about unions, intersections, complements, and a combination of these. The relationships between their symbolic descriptions and graphic area were also discussed (see Table I).

The aim of the school’s “open doors” policy was to redesign the way mathematics is taught. By agreeing to the lesson study class observation format, the school hoped that a so-called “active learning” environment can be introduced, which encourages students to participate more in classroom activities and support teachers to apply various strategies to further help motivate students. If successful, this would maintain a conceptual understanding in mathematics and develop “active learners.” For the school and teachers, its defining characteristics are that students are dynamic participants in their learning and that they are reflecting on and monitoring both the processes and the results of their learning. As Barkley (2010) noted, “‘Active learning’ is an umbrella term that now refers to several models of instruction, including cooperative and collaborative learning, discovery learning, experiential learning, problem-based learning, and inquiry-based learning” (p. 16).

Most of the participants in this case study had a positive view of the Japanese lesson study as an effective approach for redesigning teaching and refreshing teachers’ perception enabling students’ academic achievement and enhancing learning. It was predicted that through a lesson study like this, students could deepen their understanding of mathematics through active learning.

3.1 Process of lesson

Table I provides an online description of the process of the lesson with the boundaries of each part indicated by numbers from the lesson transcript.

3.2 Cultural code and cultural script

This study refers to the situations interpreted that have arisen in practice from the implicit and explicit effects of cultural artifacts – referred to here in this paper as a “cultural code.” The “cultural script” is used to refer to the structure of meaning behind each cultural artifact when decisions are made in the classroom during the teaching and learning process.

We extract multiple cultural codes through the practice of analysis of the mathematics lesson. We then interpret the meaning of situations, being very careful not to fall into the trap of arbitrary interpretation. Subsequently, we group these meanings into cultural codes before focusing on uniformity and format present in the actions that appeared within. Finally, we use an explorative approach to reveal the structural factors thought to be at work across multiple cultural codes, and identify these as the cultural script.

Figure 2 shows the extracted cultural code and cultural script of this case study. Five scripts exist as a conglomerate in the lesson. The authors focused not on which scripts had the greatest effect in this conglomerate, but rather the integral nature of the conglomerate, that is the way in which each script was incorporated into it. In other words, the fact that various views of lessons and of teaching materials were historically and culturally embodied by the teacher, and existed alongside each other even when partially in contradiction with each other, is a fundamental characteristic of the cultural practice of this lesson.

In revealing the cultural script, as we were aiming to understand the situations in the lesson as a cultural conglomerate, we did not concern ourselves with a hierarchy between the scripts or with the strength of the effect of each. The fact that the scripts exist alongside each other has already contributed to our understanding of the layered structure of the lesson. However, in the following example, the two scripts do not simply co-exist, but are actually observed in a symbiotic relationship with each other. This is when one situation (cultural code) is affected by various background factors (cultural scripts).

For instance, in the lesson, the teacher has students work on a learning task individually and awards a prize of chocolate to the best five students. If we consider the chocolate to be a reward for a successful learning outcome, then a “competition script” is observed. However, if we ask ourselves why the teacher decided to award students in this way and how students reacted, we can further understand decisions adopted by both parties. Perhaps the teacher had predicted the initial reaction of the students to share their award with their group. This further indicates a homelike “family script” based on the evaluator and evaluated teacher to student relationship. Figure 2 provides further details on this point. This may be an unexpected reception for an outside observer to see that the 5 best students who receive the chocolate normally share it with their friends in their group rather than keeping it to themselves.

4. Findings

This section will mainly examine the content of the cross-cultural lesson analysis meetings in Japan from the perspective of both Malaysian and Japanese educators. Emphasis is placed here on the cultural script of both Malaysia and Japan in the context of “classroom culture regarding mistakes” and “mistake management behavior.” The impact on the quality of teaching and learning will also be discussed in relation to how it can be improved in practice from the following perspectives:

1. allowing students to make mistakes;

2. understanding and evaluation of the mistakes; and

3. learning from mistakes.

The features of the cultural script of teaching a mathematics lesson observed in Malaysia are categorized from these three perspectives and “a comparison as a lens” of these from the perspective of Japanese educators is summarized in Table II. Emphasis here is on a cross-cultural analysis to view in depth the cultural script of teaching mathematics in Malaysia with particular focus on how teachers respond to students’ mistakes in a mathematics lesson.

It represents evidence on how teachers respond to students’ mistakes in classrooms in Malaysia in comparison to Japanese teachers’ point of view in a similar situation. The main objective in the cross-cultural analysis meetings of the mathematics lesson in Malaysia and Japan was to discover the inconsistencies involving “classroom culture regarding mistakes.” There is an anomaly in how teachers and students learn from mistakes in the classroom, and the differences between “mistake management behaviors” in the two cultural contexts. This also involves how students and teachers react to mistakes, especially how teachers respond to students’ mistakes.

The findings in Table II are based on a meta-analysis of results to provide features of “classroom culture regarding mistakes” and “mistake management behavior” of a lesson from the Malaysian perspective and through the lens of Japanese researchers. These results were retrieved from discussion between 27 experts in the field as previously mentioned in the research methods section of this paper.

4.1 Allowing students to make mistakes

In the lesson, the teacher says multiple times to the class that “it is okay to make mistakes,” subsequently some students do present wrong answers to the rest of the class. We also observed cases where students actively raised their own hands, only to give the wrong answer. However, from a mistake management culture point of view the classroom climate was negative in comparison with that of Japan where teachers try to face mistakes specifically such as classified the same type of mistakes into patterns so that the whole class could share and learn from the mistakes (Tanaka, 2017). In some cases in Japan, teachers try to link the mistake with misconceptions and false understanding of the learning task at hand (discussion meeting, Japan).

From the Japanese perspective, one participant, Hiro, commented that, “we observed students raising their hands in an encouraging manner, volunteering themselves to give answers. This is something we would rarely see in Japan. I also noticed that the teacher said ‘my sons’ over and over to her students. This was amazing to see, and it made us wonder about the relationship Malaysian teachers have with their students. This relationship provides an environment where students try to be more active” (Hiro, discussion meeting, Japan). Furthermore, after the discussion meeting, Hiroyuki, a mathematics teacher, mentioned that “Malaysian students saying ‘pick me, pick me,’ was quite unique. He remarked that this was something that would not be seen in Japan” (interview, Hiroyuki, Japan). Although Malaysian educators viewed this remark favorably, they also agreed that the teacher tried to use many strategies to motivate student at the beginning of the lesson by using colorful teaching aids in instruction set, recalling, and asking questions. It was also noted that students call their teacher “mother” and the teacher calls her students “my son.” From the environment and relationship of teacher and students it can be claimed that students did not appear to be afraid of the mathematics teaching style. However, the thinking process is still limited because no proactive discussion from the students was observed (discussion meeting, Japan). Conversely, one of the Malaysian researchers commented that “teachers have try to make students more active, but the students themselves did not reciprocate. Another teacher disclosed that students looked very serious” (interview, Zoheir, Malaysia).

Furthermore, one Malaysian educator claimed that according to the process of this lesson, the teacher’s perspective is one thing, and the students’ perspective is another (discussion meeting, Malaysia). However, from one Japanese researcher’s point of view, the students don’t say much about the activities themselves, but in T102 (Teacher, can the numbers be put in any order and area that we like?) students actually ask themselves. Maybe there aren’t many scenes like this in the lesson. In T115 (The wrong answer, look at the part that has been mistaken. 4, where does this 4 come from? 4 is R. It says (P∩Q)’, so there should not be the number 4 there. So, this is where it is mistaken.), the teacher says that’s the wrong answer, and then in the right-most column is S10’s mistake (discussion meeting, Japan).

Nowadays the view of mathematics education is changing to be more about students thinking together and helping each other, a Japanese mathematics teacher said (interview, Masao, Japan). The teacher usually ends up explaining it himself. If we look at it with this in mind, the most interesting part of this lesson is enjoyed by the teacher; it is understood through him explaining it (discussion meeting, Japan). One Japanese researcher also commented, “I feel like by explaining away that process where students discover and understand for themselves, the teacher gobbles up the most interesting part of the lesson” (Komatsu, discussion meeting, Japan).

From the Malaysian researchers’ lens, the lesson suffers from a lack of creative thinking, a mathematics educator said (discussion meeting, Malaysia). Malaysian educators viewed the teacher’s role critically. When the teacher feels that a misconception has occurred, the teacher must immediately take action to explain the right concept (discussion meeting, Malaysia). From the view of a veteran teacher who is teaching science in the same school level, “the lesson is still exam-oriented. We are under pressure to teach toward the test and not be more creative by introducing ideas and concepts that differ slightly from the contents of the textbooks” (interview, Zohaier, Malaysia).

However, as can be seen in this case, this represents the effect of the “family script” where there is a diminished risk of failure compared with Japanese high school lessons. Furthermore, the value placed on autonomy to think for oneself and express opinions means that a “behaviorism script” is symbiotically at work.

In comparison, in Japan’s case, when Japanese people want to create outstanding students, they do so very seriously, a Japanese researcher said. This is an aspect where Japanese people have a serious side. The teacher’s ulterior motive for offering chocolate in the Malaysian lesson is an attempt to get students to make effort. Students also won chocolate and after the lesson everyone shared it. So if this were Japan, students would end up only making effort for the sake of getting chocolate. (discussion meeting, Japan).

From the view of a veteran mathematics teacher, “Japanese classrooms may looks idealist, sometimes with a samurai mentality I think, and this is reflected, meaning we would label such lessons low level, so I felt like in some respect we are determining this for ourselves. I feel like our national character is reflected in this way in our classrooms, and this sort of aspect, this aspect of giving out chocolate has a lot hidden behind it” (interview, Nakayama, Japan).

4.2 Understanding and evaluation of mistakes

As described above, although the teacher welcomes mistakes in the classroom, it seems she does not expressively aim to solidify or strengthen understanding through those mistakes (discussion meeting, Malaysia). However, this does not mean that she does not place importance on understanding and high order mathematics learning (discussion meeting, Japan). In this lesson, “the teacher utilizes various strategies such as ICT tools to encourage understanding by relating the symbolic description of groups, and graphic representation of their respective areas,” a Japanese educational technologist said (interview, Yoshiaki, Japan). Following the lesson the teacher herself claimed that even though she valued students’ understanding, she fell short of being able reassure understanding in all students while still making sure the lesson is kept on schedule. She also declared that a teachers’ mental notion toward a student’s mistake in class may be interpreted as a student mistake and the responsibility of the student and not the teacher or other students (discussion meeting, Japan).

In the lesson analysis meeting in Japan, a Japanese mathematics teacher commented that “after reading the transcript of the lesson, I felt like when at T115 (The wrong answer, look at the part that has been mistaken. 4, where does this 4 come from? 4 is R. It says (P∩Q)’ so there should not be the number 4 there. So, this is where it is mistaken.) the teacher outright said that the answer was wrong, I realized that normally, as a teacher we take wrong answers and discuss them together to get to the right answer in class. When the teacher said the answer was wrong I automatically stopped” (Suzuki, discussion meeting, Japan). However, from the Malaysian perspective, the teacher deliberately chose a weak student in mathematics with hope it could aid her lesson by confirming the student’s understanding from the contents of her lesson. In this case it was still OK if the weak student got the wrong answer and was mistaken. The mistake could be used for reference-learning (discussion meeting, Malaysia). Here it can be seen the competing relationship between an “institutional script,” which requires the lesson to progress as planned, and an “individualism script” that values each students’ individual understanding.

According to the process of this lesson and from classroom culture regarding mistakes, a Japanese High School teacher commented that “We’ve spoken about mistakes, but in the middle of a lesson when mistakes arise, normally if it came out that students don’t understand that part, if the teacher would continue by encouraging students to think together about this part that students haven’t understood, then he might be able to deepen students’ understanding” (Tamura, discussion meeting, Japan). However, from the Malaysian educator’s eyes, students have a good attitude and try to be more actively involved in class, and give appropriate responses to teacher questions. Although they do not have enough opportunity to discuss and contribute to the dialogue and mathematics communication for more deep understanding (discussion meeting, Malaysia). Refereeing by this comments, a Japanese high school teacher who is teaching mathematics mentioned that from transcript-based analysis of this lesson, it seems to me that more students like mathematics here than in Japan, although as Table III shows, a few of them reached all of the problems mentioned by the teacher at the end of the lesson and awarded the best one (discussion meeting, Japan).

4.3 Learning from mistakes

Transcript-based analysis of the lesson shows that the teacher does not attempt to explain the reason “why” the answers to some questions were incorrect, or to have the students think about their response in more detail. Rather, she evaluates the students’ learning situation and merely gives the correct answer to them individually. From the Japanese educators’ perspective, this represented a missed opportunity to deepen students’ mathematical thought process and solidify their understanding. Most of the learners’ wrong answers were not simply careless errors, but in fact were underpinned by certain or likely reasons (discussion meeting, Japan). By interpreting incorrect student responses as new problems and investigating them as a class, “the teacher could deepen individual student understanding of the learning content, and promote mutual collaborative learning between students,” a Japanese researcher said (interview, Harada, Japan). That is, from the perspective of the student who made the mistake, as well as for their peers, a situation like this would provide a further learning opportunity. By further examining this opportunity, the teacher can help the student focus their attention on the reasons for the correct and wrong answers and help them understand and accept these reasons better. It can be said that, not enough collaborative learning from mistakes took place in this lesson in comparison with Japanese lessons (discussion meeting, Malaysia).

In the eyes of Japanese educators, “it seems like the teacher is deliberately creating an atmosphere that allows mistakes. Listening to the discussion just now, it’s like she’s not taking advantage of mistakes, like they’re happening all over the place. This is the kind of thing we would be worried about in Japan-there are a lot of teachers who do believe that wrong answers are assets in the classroom, but this idea is relatively ignored” (Yoshiaki, discussion meeting, Japan). Malaysian teachers and researchers also viewed this point critically. One of the Malaysian educators said “the teacher expects some mistakes made by the students in order to learn from them. Mistakes can be useful and should be further learning opportunities” (Zakariya, discussion meeting, Malaysia).

In this sense, the effects of the “individualism script” and the “collaborative script” are considered weak. According to the process of this lesson, some of us were saying that the teacher is not capitalizing on students’ comments right there and then to build the lesson. Nonetheless, if we look at the end of the lesson, the teacher does say she will make sure to check the worksheets, and I think this means she is able to plan the fruition of the one-on-one relationship between herself and the students. It seems that the points to take note of in this lesson are around T68 (OK, here comes the second example that involves 2 operations (A∩B)∪C. In this case, this question is a little different from the previous one as this involves a universal set outside the circles. Therefore, we have to label the area outside the circles as well because it is part of the universal set. So, the areas are 1, 2, 3, 4, 5, 6. Based on the operation, A is…) and T140 (My dear sons, look here please. I have tips, tips on how to shade the areas. Firstly, shade the whole area, not just part of it. Secondly, do not shade the areas with different patterns – some diagonal to the left, some diagonal to the right). The answer is correct, but you wanted to be creative, so you shade the areas with different patterns. Please do not do that. I repeat, first, shade the whole area and second, do not use different patterns of lines (discussion meeting, Japan). From the Malaysian lens, the teacher prepares the entire lesson to make sure that students can answer the test at the end of the class. The students’ outcome can be realized in detail through data as shown in Table III. So, teacher did not relate this topic to the real context. They suppose could realize the knowledge with the reality (discussion meeting, Malaysia). Additionally, one Malaysian participant, Al-ahram, believes that the lesson has a good competitive stance, by providing the chance for other students to work competitively together. This participant also commented that the teacher may need another assistant to help her in terms of weak learner in mathematics (interview, Al-ahram, Malaysia).

5. Discussion and conclusion

The aim of this study is to determine the cultural script of a mathematics lesson in Malaysia through lens of Japanese focusing on how teachers’ responses to students’ mistakes. The results of this cross-cultural analysis of a mathematics lesson showed that the lesson was formed of multiple cultural scripts existing alongside one another, at times symbiotically and at times in competition. It also highlighted the ability of cross-cultural analysis of a lesson to contrast the hidden cultural characteristics of teaching within a particular culture.

As the evidence in Figure 2 shows, a variety of views of education and teaching are involved in a lesson. It was discovered that such views co-exist alongside each other, even though they may be partially contradictory. That is, when we talk of a cultural script of teaching, not one but various cultural artifacts are at work in the production of a lesson. We found an integrated view of culture behind the lesson and the teacher’s mental pedagogical model. It appears that the collaborative script and the competitive script are in opposition to one another (see Figure 2). Cross-cultural analysis thus far has focused on which factor to give weight to, yet here we focus on the fact that a blended approach incorporates both. That is, various beliefs of what lessons are and how they should be taught with specific teaching materials, here mathematics, have been historically and culturally input into teachers depending on the country. Even these beliefs may partially be in contradiction, they may also co-exist alongside each other. This is a fundamental characteristic of lessons as a cultural practice and artifacts.

Table II demonstrates a variation in the collective perception of how teachers respond to student mistakes. By close examination of this table, it can be seen that the Malaysian communication style regarding mistakes is more verbal as opposed to the Japanese style which adopts a more non-verbal way of communicating perhaps through facial expressions. In addition, teachers in Japan typically introduce the mistake to the whole class and encourage students to discuss and correct the mistake during the lesson. However, in Malaysia, teachers may make self-judgments about the mistake and respond accordingly sometimes during and mostly after the lesson. This may be because students have more responsibility regarding mistakes in Malaysia, while the teacher has more responsibility in Japan (see Table II).

In case of the “classroom culture regarding mistakes” which represents how teachers and students learn from mistakes in the classroom, students’ and teachers’ attitude toward learning from mistakes is low in Malaysia. This may be because of the low cost of mistakes there. Thus, the possibility of mistakes being repeated is high while the emotional feeling toward making mistakes is low. From the Japanese perspective, the possibility of mistakes being repeated is low. It is maybe because the cost of the mistake in Japan is higher as the emotional feeling toward making mistakes is high. As such, teachers may hesitate to encourage students to make mistakes. However, when a mistake occurs teachers and students try to look at mistakes as an opportunity for learning from each other (see Table II). Stigler and Hiebert (1999) research also showed that “seeing why they are mistakes are believed to be essential parts of the learning process in Japan” (p. 91).

As a finding which leads further research on the cultural script of teaching, the Malaysian teacher’s way of communication and sharp directions are little bit harsh for the Japanese to listen to and react. In the same situation as the Malaysian mathematics lesson, Japanese would avoid making a mistake. However, for Malaysian students, saying that an answer is wrong actually is not criticism but rather is just telling what is right and what is wrong, but in Japan students may tend to read into this too much, trying to interpret what is behind the words being said “wrong.” Japanese teachers do not encourage students to make mistake. However, mistakes that occur or are identified in class provide a valuable learning opportunity for the whole class and help to provide a deeper understanding of the learning task and why such mistakes are made.

In the future, we hope to develop the potential of this view of research on the culture script of teaching through cross-cultural analysis research methods for raising the quality of teaching and understanding culture artifacts of teaching for enhancing learning and curriculum development. We have realized from this study that both sets of research value the importance of mistakes. It is important to identify the source of students’ mistakes and further learn from them. In order to reveal the overall structure of the cultural script of teaching, we need to realize that various cultural scripts and social states such as the cost and price of the mistake are at work in the production of any given lesson.

It may be interesting for educators to see how findings of this cross-cultural analysis of a mathematics lesson support the Malaysian teacher and school to make explicit the values that underpin and shape their teaching. Through such findings we can further understand the cultural script and how this can improve pedagogical practices. The emphasis here is on learning from mistakes, such as recovering the mistakes and linking them to students’ misconceptions, which in turn may help to encourage students to discuss the rationale behind each mistake and learn more. Furthermore, learning through mistakes can be applied as a lens for the deeper understanding of learning tasks and outcomes. This understanding reiterates how actions in class can be interpreted by society as a whole and how pedagogical decisions are made in the classroom.

This study highlights what can be learnt at the micro-level learning community, in this case a 50-minute mathematics lesson, and links that to the cost and price of that mistake at the macro-level, in this case Malaysian society as a whole. This invites the question for further future research as to how research on topics like this at the micro-level can be influenced by cultural traits at the macro-level and vice versa. Similarly, research like this provides us with the opportunity to learn more at the national socio-cultural level.

It may also be interesting for researchers to look at mistake management behavior from the students’ perspectives. This raises a further question of what the role of the students is in cross-cultural analysis. The students are an invaluable resource in this research. They can provide a different theoretical lens and hold a unique perception of the quality of teaching compared to teachers. This provides great potential for further transnational learning in future studies. For instance, it can be illustrated from this cross-cultural analysis that the mathematics teacher tried to be a risk taker to provide a homelike atmosphere for students to do their best and be encouraged as a creative learner. However, this teacher-student relationship may look at the lesson and its outcomes through the lens of teacher authority. These are very important research questions which the research team hope to cover in future manuscripts.

Figures

Figure 1

Mistake management behavior framework

Figure 2

The cultural code and cultural script of the mathematics lesson

Table I

The process of the mathematics lesson

Table II

How teachers respond to students’ mistakes

Categories	Japan	Malaysia
(1) Classroom culture regarding mistakes	High	Low
Cost of mistake	High	Low
Relationship between the kind of mistake (content, procedure, concept) and the way they are handled	Sometimes	Usually
Assessment of mistakes and the impact they have on students’ feelings toward them	High	High
Students’ and teachers’ attitudes toward learning from mistakes	High	Low
Impact of the subject matter (mathematics, science) on teachers’ reaction to mistakes	Weak	Strong
Relationship between teachers’ reaction and students’ attitude toward mistakes	High	High
Use of mistakes as a method of evaluating students	Not consider	Consider
Mistakes as a lens for the deeper understanding of learning tasks	Usually/strong	Sometimes/weak
Mistakes as opportunities for students to learn from each other (collaborative learning)	Provided	Not provided
Possibility of repeating mistakes	Low	High
Learning from mistakes (mistakes as learning opportunities)	Provided	Not provided
Students’ fear toward mistakes	High	Low
(2) Mistake management behavior	Positive	Negative
Teachers’ communication styles regarding mistakes	Non-verbal	Verbal
Who typically corrects the mistake	Students	Teacher
Indirectness of teacher response to students’ mistakes	Ordinary	Unusually
Tolerance of the teacher when mistakes are made	High	Low
Teacher’s supportive comments on how to correct the mistake	Sometimes and prudently	Frequently and frankly
Introduction of the mistake to whole class	Normal	Unusual
Returning of the mistake to students who made them for self-reflection	Unusually	Ordinary
Teacher response regarding mistakes	Slow	Quick
Teachers’ reaction to other students in class in order to expand further discussion	Encouraging	Judgmental
Final responsibility of the mistake	Teacher	Student
Recovering the mistake and linking it with students’ misconceptions	Provided	Not provided
Which type of response is most important to the teacher	Wrong	Right
Encouraging students to discuss the rationale behind the mistake	Encouraged	Not encouraged

Table III

The test results of students

Number of questions	Number of questions students reached
6	3
5	2
4	7
3	1
2	2
1	2
0	3
	20