From the Confucian Tradition to the Digital Era: The Case of Mathematics Teaching in Hong Kong

Abstract

Recent years have witnessed an increase in research focused on studying on perspectives of Chinese mathematics instructions. The sustained interest is partly due to the outstanding performances of Chinese students in international studies such as the Trends in International Mathematics and Science Study (TIMSS) (Mullis, et al., 2012) and the Programme for International Student Assessment (PISA) (Organization for Economic Co-operation and Development (OECD), 2010); and partly due to the shared interest in comparative studies of instructional practice across different cultural systems. What have we learned about in mathematics classrooms from international studies? Findings of international studies suggest that how the teacher used the tasks so that the cognitive demand of the learning tasks could be sustained is very important. Despite the good performances of Hong Kong students in international studies, there is a gap between traditional classroom practice and the long established goals for promoting generic capacity in mathematics learning; most of the traditional learning tasks in Hong Kong classrooms are apparently routine and serving a demonstrative purpose as a result of the highly competitive systems. Based on triangulation of the findings of the analysis of the mathematics lessons at different levels, the study shown some robust features in the traditional mathematics teaching practice in Hong Kong classrooms in contrast to some innovative scenarios in a special mathematics lesson. Finally, the author re-examines the robust features in the context of curriculum reforms and the cultural context of Confucian tradition.

Keywords

international assessment Confucian heritage culture cultural conception of learning and teaching curriculum reforms digital technology in teaching mathematics education

1 Introduction

Students from places such as Singapore, Japan, Korea and Hong Kong that are under strong influence of the Chinese Confucian heritage culture have consistently shown outstanding mathematics performance in In international comparisons such as international studies such as the Trends in International Mathematics and Science Study (TIMSS, Mullis, et al., 2012) and the Programme for International Student Assessment (PISA, OECD, 2014). On the other hand, there has been a widely accepted view that Chinese classrooms (including those in Hong Kong) are characterized by rote learning, passive students and large class size, such attributes together constituting an environment, seemingly not to be conducive to learning. These observations sat in uneasy juxtaposition and led to the identification of the phenomenon known as the “Chinese Learners’ Paradox” (Biggs & Watkins, 1996; Watkins & Biggs, 2001). What is special about the instruction in Chinese mathematics classrooms? Recent years have witnessed a sustained interest in research focused on studying on perspectives of Chinese mathematics instructions, that is partly due to the Chinese Learners’ Paradox phenomenon and partly due to the shared interest in comparative studies of instructional practice across different cultural systems (Mok, 2015). What lessons have we learned from international studies? Findings of international studies suggest that how the teacher used the tasks so that the cognitive demand of the learning tasks could be sustained is very important (e.g., Cai, et al., 2016). Despite the good performances of Hong Kong students in international studies, there is a gap between traditional classroom practice and the long established goals for promoting students generic capacity in learning. The Hong Kong government has advocated series of curriculum reforms since the early 90’s. The mathematics teachers have consistently experienced a dilemma. On the one hand, Hong Kong students in the traditional practice have achieved a record of outstanding performance in international studies. On the other hand, the goals and directions in the series of curriculum reforms are indeed articulation of an orthodox for a better ideal. The phenomenon inevitably leads to some queries in some people’s mind: Why do we have to change? Will the change make students perform less in their examinations? Via efforts of the reforms how are the changes in line with the curriculum goals? What are the good features in the traditional heritage that should be kept while bringing about change?

Attempting to make a contribution to some of these questions, the author will draw upon the findings in the project, “Fundamental challenges in using digital technologies in secondary mathematics classrooms: a comparison between different paradigms, over time, and between places” (called DTMC in short) to capture some robust features in the traditional practice and the changing paradigms in the mathematics classroom in Hong Kong. In the following sections, the author will present a summary of the curriculum reforms in Hong Kong since the 90’s providing a contextualization for the study, followed with a methodology section of the study; then the result sections will present the changing paradigms in two parts: the traditional practice in the eyes of the beholders (the teachers and students), and a special lesson showing the confluence between the traditional practice and innovations. Finally in the discussion, the author attempts to examine some robust features in the different paradigms of mathematics lessons in the context of the curriculum reforms and the Confucian tradition.

2 Through Decades of Changes: Curriculum Reforms in Hong Kong

A non-favorable image in the 80’s: Classroom teaching in the East Asian region was predominantly described in terms of an American view of whole-class instructional style with a stern and demanding teacher who stressed mechanical learning (Stevenson & Lee, 1997). The portrayal of Hong Kong classrooms in the 80’s was a typical of such image, for example, the 1982 Llewellyn Committee reported:

The lessons we observed tended to be teacher-centred, with little use of aids beyond chalk and blackboard. In “non-exam” years, the atmosphere seemed fairly relaxed, but in the examination preparatory forms all was deadly earnest and pupils were seen taking notes, laboriously completing model answers and learning texts by rote.

Llewellyn et al., 1982; Morris, 1995: 112

The teacher-model in the 90’s: In Leung’s comparative studies of mathematics lessons in Beijing, Hong Kong and London (Leung, 1995), Hong Kong teachers in their mathematics lessons used most of the teacher-talk time for demonstrating solutions to mathematics problems and usually appeared to be rushing through the subject content. According to Leung (1995), Chinese teachers held a more rigid view of mathematics as a product rather than a process and perceived their main role as to explain the mathematics content clearly. This image is somewhat consistent with Paine’s virtuoso model of the Chinese teachers (Paine, 1990) in which textbooks were seen as the source of knowledge and the teacher was the presenter; hence, the role of the teacher was mainly demonstrating and pupils are expected to follow the teacher’s model.

The Target Oriented Curriculum ( TOC ): Since the 90’s, Hong Kong education authority has been well aware that such environment was not necessarily conducive to learning and has advocates curriculum reforms, the Target Oriented Curriculum (TOC) was developed in accordance with the recommendations of the Education Commission’s Report No. 4 (Education Commission, 1990). The goals of and rationale for TOC were derived essentially from Western precepts of what constitutes a ‘good’ teaching environment, which was effectively the converse of those characteristics associated with Asian classrooms. The TOC documents proposed changes in the core subjects (English, Chinese and mathematics) in the curriculum and a major impact was the direction of putting emphasis on the learning process (Clark, Scarino & Brownell, 1994), advocating the elements of the activity approach and communicative method (Morris & Adamson, 1997) with the Vygotsky’s theoretical origin (Vygotsky, 1978).

Some changes were reported but some features remained prevalent. Mok and Morris (2001) in their study of the TOC mathematics classrooms, they analyzed 197 mathematics lessons for the period from 1995 to 1998, the results showed that the objectives of direct teacher instruction were beyond exposition but serving various purposes such as, recalling the prerequisite knowledge, delivery of new knowledge, giving demonstration for subsequent activities, encouragement, providing diagnostic feedback and encouraging peer assessment. It was obvious that the classroom practice in Hong Kong was not stagnant and gradually changed in pace of the curriculum reforms. According to Mok and Morris (2001) with the infusion of the knowledge of learning theories in the west the Hong Kong classrooms starting with a teacher-centred image have changed into one with more recognition of students’ participation in the process of learning. Despite these changes, teachers remained the important person in charge of the events happened in the classroom and the teaching was still likely to be directive, and guided by the teacher and by the tasks in the lessons.

As far as mathematics was concerned, the TOC offered a set of learning targets for the teachers and schools to develop effective teaching and learning. The content area of mathematics was organized under five strands of number, measure, algebra, shape & space, data handling. In addition to the structuring of the content, the curriculum also stated explicitly an emphasis of the process abilities of mathematical conceptualization, inquiry, reasoning, communication, application and problem solving (Curriculum Development Council, 1995). This change was in line with the direction of mathematics curriculum in other places in the world, namely, the National Council of Teachers of Mathematics curriculum standard in U.S. (National Council of Teachers of Mathematics, 1989) and the national curriculum in U.K. (National Curriculum Council, 1991).

From Target Oriented Curriculum ( TOC ) to New Senior Secondary ( NSS ): After TOC, continuous reforms have been promoted by the Hong Kong government, and the direction of the education reforms and action plan was confirmed in the report document, “Learning for Life, learning through Life: Reform Proposal for the Education System in Hong Kong,” to promote “life-long learning” and “all-around development” as a part of the aims of education for 21st century. The reform proposal covered the curricula, the academic structure, the assessment mechanisms and the interface between different education stages (Education Commission, 2000). A new academic structure of senior secondary and higher education known as NSS launched in 2004 and has been implemented in 2009 (Education Commission, 2004). Under NSS, the senior secondary school level has been changed from “2 + 2” year (Secondary 4–5 & Secondary 6–7) to 3-year (Secondary 4–6) while the length of higher educations had changed from 3 years to 4 years. The two public examinations (Hong Kong Certificate of Examination [HKCEE] and Hong Kong Advanced Level Examination [HKALE]) in the senior secondary level were replaced by one examination, namely, Hong Kong Diploma of Secondary Education (HKDSE). With respect to goals and objectives, generic skills such as collaboration, communication, creativity and critical thinking are included in the document to provide a direction for facilitating learning in mathematics classrooms (Curriculum Development Council and Hong Kong Examinations and Assessment Authority, 2014).

Today, the position of the mathematics curriculum states in the official website:

Students require knowledge and skills that will help them live a full life in the society of the 21st century, which is an information age. Knowledge of Mathematics is a necessity for every individual if they are to contribute towards the prosperity of Hong Kong. Mathematics pervades all aspects of life. It is not possible to live in the modern world without making some use of mathematics. Many of the developments and decisions made in industry and commerce, the provision of social and community services as well as government policy and planning etc., rely to some extent on the use of mathematics.

Hong Kong Education Bureau (2019)

For the role of mathematics, it includes the four aspects: (a) a powerful means of communication; (b) a tool for studying other disciplines; (c) an intellectual endeavor and a mode of thinking; and (d) a discipline, through which students can “develop their ability to appreciate the beauty of nature, think logically and make sound judgments”; and with respect to attitude, it further states the direction of enabling students to “build up confidence and positive attitudes towards mathematics learning, to value mathematics and to appreciate the beauty of mathematics.” Hong Kong Education Bureau (2019).

For the direction of information technology, it states: “The appropriate use of information technology (IT) in mathematics learning should be emphasized. High technology items like computers and calculators have profoundly changed the world of mathematics education. Students need to master IT to adapt to the dynamically changing environment. With the help of IT tools, meaningless drilling and obsolete topics are no longer essential and relevant in mathematics learning.” Hong Kong Education Bureau (2019).

3 Methodology

The findings reported in this paper drawn upon the analysis of data for the DTMC project that aims to compare and capture the change of Hong Kong mathematics lessons in different paradigms, namely, the today’s lessons using digital technology with the traditional lessons in 2000. The design of the project built on that of the international Learner’s Perspective Study (LPS) (Clarke, 2006), which generated a rich data set of lesson videos, teacher interviews and student interviews, of eighth-grade Hong Kong mathematics lessons in the period 2000 to 2002. Aiming to compare the variation in paradigms with a re-focusing on digital technologies, 16 lessons of two teachers were collected anew in the period 2015 to 2016. The data collection employed three cameras—a “Teacher Camera”, a “Student Camera” and a “Whole Class Camera”. This unique technique has been developed and designed to meet complex classroom research needs, including production of a composite-image video for stimulated-recall post-lesson interviews. The teacher and two focused students of each lesson were interviewed after the lesson separately with the video-stimulated-recall method to obtain the teacher’s and students’ reconstructions of the lesson and the meanings that particular classroom events held for them personally (Clarke, 2006). In addition, all the lesson materials including student work were collected. The generation of the DTMC data set in 2015–2016 employed the similar data collection protocol to facilitate the feasible comparison between the different data sets generated in the two different periods. The research design produces a large data set giving opportunities for regeneration of accounts of the data lessons into sufficient details for analysis from different theoretical frameworks addressing various relevant issues in classroom practice.

In this paper, the analysis applied the post-positivist approach of triangulation (Mok & Clarke, 2016). The complex, interconnected dataset in which a variety of data types relating to the same situation or phenomenon are strategically generated for both qualitative and quantitative analysis, makes possible the description of a social situation from a variety of perspectives as well without incurring any obligation to converge on a consensus interpretation. Triangulation was enacted at the level of data type (video, interview, questionnaire and classroom artefacts), informant (teacher, students and observing researcher), analytical perspective (e.g. discourse analysis, variation theory, socio-cultural theory) and cultural perspective. In such an interpretivist paradigm, triangulation becomes the aspiration to more thorough portrayal, rather than the aspiration to more precise location (Mok & Clarke, 2016).

4 Paradigm 1: Traditional Practice in the Eyes of the Beholders (the Teachers and Students)

How do students view their mathematics lesson in traditional practice? In this section, some important results of the Hong Kong LPS lessons in the period of 2000 to 2002 were recapitulated here.

4.1 Triangulation at the Level of the Informants (Teacher, Students and Observational Researchers)

Mok (2009) analysed 18 Hong Kong consecutive lessons (HK1) and presented a juxtaposition of the researcher’s analysis of the lesson with the teacher’s and students’ perspectives of the lesson. In the post-lesson student interviews, a high proportion (31 out of 34) expressed explicitly that they liked their lessons. The researcher’s perspective applies the theory of variation to delineate how the teacher-led whole class discourse with the careful selection of examples delineated a potential learning space for the students to experience the critical aspect of the object of learning, such as, factorization in algebra. To argue for the advantage of the traditional practice, some persistent features were found that included: the teacher talk was a major input in teaching; the technique of variation was used in the design of the mathematical problems and a potential learning environment with class discourse directing students to pay attention to the critical features for the object of learning. In addition, in the teacher post-lesson interviews, the teacher openly shared his ideology of teaching showing his seriously concern for the student factor, such as, deep reflection and inclination to capitalized on students’ mistakes and difficulty for scaffolding the learning experience; and consistent effort in establishing an encouraging class atmosphere where students might answer the teacher’s questions and raised their questions. In the teacher’s explanation of his ideology, his own the delineation of how to draw the students’ attention to the critical aspects of the mathematics was coincidently in alignment with the potential learning space from the framework of the learning theory of “variation” (Marton, 2014).

4.2 Triangulation at the Vygotsky’s Perspective of Learning, How Might Imitation Work

According to Vygotsky’s perspectives, students’ learning may take place through the verbal interaction with the teacher, the observation of procedures used by the teacher and imitation of teacher’s procedures (Vygotsky, 1978). Based on this perspective, imitation could be an important aspect in the process of learning. Yau and Mok (2016) analysed 14 consecutive mathematics lessons of a Hong Kong mathematics teacher (HK3), with a focus on the students’ notebook, a total of 252 items in the written work in the students’ notebook were collected from 27 students and they were analysed according to the cognitive demand of the problems and the level of imitation of the students’ work as compared with the teacher’s examples. The lessons showed a clear pattern of traditional directive teaching of sequences of teacher example-student exercise (TE-SE) cycles. The teacher examples played an important role in explaining and demonstration, whereas, student exercises indicated a high degree of similarity with teacher-example and there were different levels of imitative work: some directly copied of the teacher’s examples shown on the board, some contained some incomplete items and unattempted items; resulting in high proportion of student work that were categorized into “complete imitation” and “partial imitation”. Furthermore, some students would work on extra problems on their own initiative if they worked faster than other students. What aspects in the mathematics lessons did they see important? Supplemented with the analysis of 26 post-lesson student interviews with the stimulated video recall method, analysis of the video segments that the students attached importance to showed that 42% of the video segments were TE (teacher examples) and 55% of the video segments were SE (student exercises), therefore, both TE and SE were important although SE appeared to be slightly more important than TE. Further categorization of students’ comments on the lesson video segments showed that students thought that the teacher’s demonstration of procedures was the most important (Yau & Mok, 2016).

4.3 Triangulation at a Cultural Level, Contrast between Hong Kong and Singapore

Hong Kong and Singapore shared many similarities in their cultural and education system: both have population with Chinese as the majority, both are places with high examination pressure, both have high student achievement in international comparative studies (e.g., TIMSS and PISA). Mok et al. (2013) compare the student perspectives for their lessons between one Singapore class (SG1) and one Hong Kong class (HK1) via the analysis of the student interview data. During the interviews, the students were asked to stop at episodes and explained why they thought that was important. The findings of Hong Kong students’ interview data (HK1, 18 lessons) were similar to that of Singapore students (SG1, 13 lessons) in many aspects. The students attached importance to segments of the lessons associated with the instructional practice of exposition, seatwork; review and feedback. The results in both classes HK1 and SG1 suggested that the strong leading role of the teacher was inevitable. The students liked their teacher and their mathematics lessons although a high proportion of the instructional practice is expository. The students valued the expository instruction that served the purpose of helping them to learn. The functions that they saw important were clear explanation, demonstration of methods, strategies helping them to learn something new. Their keenness to follow the teacher’s exposition to obtain better understanding of concepts or mastering methods was explicit. The next thing that they saw as important was the opportunity to do work in the class and learning from either the teacher or other students. They appreciated the teacher’s effort of corrective feedback for mistakes and peer support given either in seatwork or public sharing in the lessons. Mok et al. (2013) further pointed out that the results suggested that expository instructions with strong teacher guidance were not necessarily unwelcome by students in the system, for expository instruction could be good experience of learning something new when the students were ready to follow the demonstration and explanation given by competent teachers. This harmonious match between teacher’s competence and students’ expectation might suggest a possible explanation for why students in Asian classrooms may have good performances despite the unfavourable factors of large class size and high examination pressure.

5 Paradigm 2: the Confluence between the Traditional Practice and Innovations

The lesson is taken from a larger data set collected in the period 2015 to 2016. The teacher of the lesson had been video recorded for 11 lessons including: 4 traditional lessons without using any special digital technology, 6 lessons using the mathematics software Geogebra in mobile tablets, one lesson with an outdoor activity using apps in the mobile phones.

5.1 The Special Lesson

The lesson was an 8th Grade lesson with duration of 1 hour and 30 minutes. The class size was 33 students. The teacher explained in the post-lesson interview that this was a special lesson for consisting the activity of making authentic measurement. The lesson consisted of two parts: (1) a trigonometry problem of finding unknown lengths in the diagram, (2) an activity for estimating the height of a bell from the school playground.

The first part was about 20 minutes for which the teacher showed a trigonometry problem to demonstrating a procedure for finding unknown lengths in a diagram containing two right-angled triangles (figure 1). The teacher via an interactive discourse guided the students to write down two equations for the unknown h and k. Teacher then asked students to solve two equations. While students were trying to solve the equations, the teacher walked around the classroom to give individual feedback and support to students. Later the teacher showed the solution procedure for the two equations on the blackboard and concluded the relationships of tan, angle, height and distance, and how to estimate height from two positions.

This part of the lesson took about 20 minutes in the lesson, a calibration with the lesson video data in year 2000 showed that this part of the lesson was very similar to a traditional lesson: the problem was a typical routine problem with standardized method and the teaching was spreading over lesson segments of consisting of problem set up via teacher-led whole class discussion, and student attempt of solving the problem via individual seatwork, followed by resuming teacher-led whole class discussion for which the teacher demonstrated and explained the correct procedures for finding the answers. This was a typical Teacher-example (TE) segment like those found in the analysis by Yau and Mok (2016).

Figure 1

Extracts of the teacher example

The second part of the lesson was an activity for measuring the height of the bell “Aramis” from the school playground. The teacher had prepared a special worksheet for the activity (figure 2). The students were divided into groups and had spent about 30 minutes outdoor making measurement in the school playground. In the playground, the students worked on the task in with their own methods; some marked on the ground with chalk and made measurement by with their own tools. Then they tried to use their measured data to calculate the height of the bell on worksheet. In the last 20 minutes of the lesson, they returned to the classroom to complete the calculation in the classroom.

Figure 2

The special worksheet for the activity measuring the height of the bell “Aramis”

5.2 The Learners’ Perspectives: What the Students Saw as Important

In the post-lesson student interview, two students (S1 and S2) were interviewed individually. The video-stimulated recall method was used. During the interview, they were invited to play back the video, stop at instances at where they thought important and tell the interviewer what they were doing at that moment and why they thought it was important.

Two students with different performances and attitudes. When the students were also asked how they saw their performance in mathematics and how they liked the subject mathematics. The two students showed different attitudes according to their answers: (1) S1 saw his performance non-steady in mathematics; his score might range from 50% to over 70% depending on how he liked the topics and how hard he did the exercises. He had a very positive towards mathematics and mathematics lessons. He liked mathematics because he found it challenging and interesting, liking solving mystery, not learning by rote. (2) S2 saw his performance as weak and his score in the last test was a bare pass. He did not like mathematics because he was not successful in the subject. His feeling towards mathematics lessons was only average for he did not quite like mathematics.

Contrasting between the special lesson and other lessons by the students. Both S1 (who liked math) and S2 (who did not like math) liked the special lesson for the reason that it was more lively, comparing with other mathematics lessons. When they described the other mathematics lessons, they used the adjective “boring” and that they had to sit all the time and the classmates were not keen to answer questions.

Episodes that the students saw important. In the analysis, the interview audio was played several times with reference to the lesson video, notes were taken for the episodes that they said important. S1 made 10 stops for the lesson video and S2 made 8 stops. S1 tended to be more detailed and sometimes expressed his feeling when describing what happened. S2 was relatively brief when comparing with S1 but they were very similar in seeing what were important in the lessons. They both liked the special lessons and used the adjectives of “lively” and “not boring”. In addition, they both attached importance for the following events in the lesson important: the teacher demo example at the beginning of the lesson, the preparation discussion before going for the playground, the actual fieldwork in the playground and the final discussion after returning to the classroom. Their ideas were summarized below:

The teacher demonstration example episode:

According to S1, the problem was different from those in other lessons, he said that the problem was difficult and the teacher asked a question that nobody seemed to know the answer, that seldom happened. As a result, the class was silence and he thought this was important. Referring to a slightly later moment but still within the episode of explaining the example, he said that he was thinking and about to answer the teacher’s question but he was surprised that he knew his answer for he had not known earlier.

According to S2, the teacher taught a method so that they did not need to go under the bell to measure the height, and he thought the teaching of how to set up the equations were important.

The episode after the demo-example and before going for the playground: They discussed in group for planning how to carry out the activity, realizing the link between the demo example and the activity, deciding what tools to use.

The actual fieldwork in the school playground: Some details of the events for making the measurement were mentioned, for example, finding two points in a straight line so that they could make measurement, the moment when they had different opinions and discussed over the chalk drawing on the ground, and measuring the height of their eye-level.

The final discussion after returning to the classroom: After returning to the classroom, they completed the calculation with their measurement data. S1 pointed out that their answer was not very good for the measured value was not good and the teacher gave them another number so that they could get a better answer. Also, he mentioned the moment that they were very happy while getting the answer, so he raised his hand to ask the teacher to check whether they were correct.

A preliminary calibration of the 11 lessons in 2016 with the typical lessons in data-2000 showed that the traditional lessons were very similar to the lessons in 2000, showing typical teacher example and student exercise TE-SE patterns with class discourse carried out in the format of teacher-led whole class discussion and between-desk-support for individual seatwork. The lessons using the Geogebra showed activity pattern also in the forms of teacher-led whole class discussion and individual seatwork; they were different from the traditional lessons for screen-display replacing the chalk writing and the students were engaged in working with tablets. The special lesson consisted a TE segment and an outdoor activity for which students worked collaboratively in groups. Hence, the calibration results showed a spectrum varying between “traditional lessons”, “using Geogebra software with similar lesson patterns” and the “special lesson with collaboration activity” with different images of the lessons (Figure 3).

Figure 3

Images in the “traditional paradigm”, the “lessons using Geogebra software” and the “special lesson”

What was the two students’ overall evaluation of the use of technology in their mathematics lessons? After the review of the lesson, they were asked how they liked the use of technology in mathematics lessons and they were also asked to compare the traditional lessons, the lessons using the software Geogebra in earlier period and this special lesson that they used the apps in their mobile phone.

S1’s answer was that of welcoming technology with reservation. S1 saw that technology was something good to use in lessons because it was something new. However, he did not want technology to be used in every lesson. He thought that using tablets every lesson would be like using the computer at home and it was better to let the teacher teach with face-to-face opportunities. Hence, technology was not a bad thing, but it was not always good. Although he liked the Geogebra lessons and this lesson using technology to measurement more than traditional lessons, he thought that it was not good to use mobile tablets or phones for every lesson. Using them every lesson would not be so much fun for he thought that students needed interaction and talking to each other.

S2 showed a very explicit preference to the use of technology. S2 saw technology (Geogebra, apps) as good, not boring. Different from drawing on blackboard, he thought that technology was more convenient, made the subject easier to understand for it showing more information, and he needed to understand before he could use it. He described himself that he would be day-dreaming while sitting still, thus moving about helped him become attentive. He thought that the school should use more technology.

5.3 The Teacher’s Perspectives

In the post-lesson interview, the teacher explained the objective of the lesson was to let the students via some authentic tasks of measurement to learn trigonometry. He believed that mathematics was beyond numbers and authentic in the real world. The application of mathematics in the activity in the lesson created an experience of making authentic measurement, such as, the elevation angles, walking on the ground to find the distances and appropriate position to make the measurements, and these experiences enrich their mathematics sense, an authentic mathematics sense that go beyond numbers. He explained that the lesson was special because it consisted of a task that the students needed to make actual measurement to estimate the height of the bell in the school, in order to carry out the task, the students had opportunities for group work and collaboration, discussion, planning, measuring a specific height, recording, evaluation and reflection to seek for improvements. He told the students about the tasks in the day before the lesson so that they could search the internet for diagrams or methods and bring their own measuring tools, hence, some students chose to use the apps in the mobile phone.

When he reviewed the lesson video, he gave comments to the four sections of the lesson. Below summarizes what the teacher said.

Section 1: The first 15 minutes was devoted to explaining a “simplified survey problem”. It was simplified for he skipped the consideration of the height of eye-level and the use of the terms of elevation. Hence, the problem was presented as a guided detailed method of solving a slightly complex of problem, through the teacher’s guidance, the basic concepts and skills of the right-angled triangle were revised and applied. The teacher said that alternatively this could be the instruction of the task of measuring the bell for he believed that without this, the students might not think of a way to complete the task. In his description, he had taken into consideration the complexity of the activity, the students’ ability and their mathematics capacity in his planning for the lesson.

Section 2: The next about 20 minutes section, the teacher explained the special worksheet and gave the students to divide into groups, to discuss and plan what to do. It also gave the teacher an opportunity to “listen to the students’ discussion”. He said that this was especially important for by this he got feedback whether the students understood the earlier example and instruction, gave him a chance to clarify further. He also explained the remaining pages of the worksheet and told the students the rules when they got outside the classroom.

Section 3: For this section the students worked outside the classrooms in the school playground. The teacher observed “many interesting instances” (helped how to measure the angle of elevation, helped two weak students who needed more support, The students were very engaged in the task, willing to try and ask questions).

Section 4: Back to the classroom. He helped the students check whether the measured value not appropriate. He got the idea for the next lesson to use Geogebra to verify the answer with a different possibility.

6 Conclusion and Discussion

The study has been situated in a context of curriculum reforms and changes in Hong Kong. The story set off with the non-favorable image of the classrooms in the 80’s that called for an awakening of the detrimental effect of rote and mechanical learning as well as the stress as a result of the highly competitive examination system, leading to reform directions much influenced by Western learning theories, world trends of increasing emphasis of the process of learning, and outlook for developing learners’ capacity for the 21st century including the disposition of using information technology in the recent decades. Interestingly international comparatives studies, such as, TIMSS and PISA, present a paradoxical phenomenon, known as the Chinese learner’s paradox or the Asian learner’s paradox: on the one hand, the students in the Asian region have a record of outperforming their counterparts in the west in mathematics; on the other hand, the reported image of classroom did not provide a learning environment favorable to learning and also not in line with the emphasis of the process of learning. The phenomenon unavoidably leads queries on how to change and the struggle for keeping good features in the traditional practice. In the aforementioned, a paradigm of traditional teaching practice is juxtaposed with a special lessons and the research design has made a special contribution of capturing the learners’ perspectives.

What are the robust features in the traditional paradigm? The traditional practice paradigm (Mok, 2009) demonstrates some robust features. The lessons demonstrated a consistent teacher-example-and-student-exercise (TE-SE) pattern providing evidence that imitation played a significant role in the students’ process of learning. What to imitate is a major concern? The students indicated an expectation of quality demonstration and clear explanation while they attached importance to the teacher’s expository segments in the lessons. Triangulation with the teacher’s perspectives further unfolds two important reasons supporting the teacher as an authority figure receiving high respect: (1) the teacher’s emphasis on taking into the students’ needs and ability consideration resulting in a design of lessons tailored for his perception of empowering students’ understanding of the subject matter. (2) The teachers’ demo and explanation supports learning and understanding, making a reference to the variation theory applicable in both the work of Chinese and northern European scholars (Gu, Huang & Marton, 2004; Runesson & Mok, 2005; Marton, 2014; Mok & Yau (2006) also argued that the variations embedded in the teacher’s examples and students’ exercises in fact helped the students to experience the object of learning in a deep sense leading to an understanding of the mathematical concepts and procedures from multiple perspectives (Gu, Huang & Marton, 2004; Huang, et al., 2006). In addition, the private work in the notebook and the explicit sharing of their thinking in the post-lesson interviews suggested evidence for some students, the traditional practice of TE-SE examples might help developing a confidence and motivation in the work and possibly a “deep” approach that might bring about understanding beyond memorization (Biggs, 1998).

What are the robust features in the special lesson? The lesson is “special” for several reasons. It is special for it is very different from the lesson of the traditional practice. It is a confluence of the traditional elements and a unique outdoor activity applying of the trigonometric example they learned inside the classroom. The two students expressed a conflicting sentiment to the traditional practice. They saw the importance of learning from the teacher’s demo example but they used the adjective “boring” to describe the lesson pattern of the traditional practice. Their open sharing of their feelings suggests that in addition to good quality demonstration, attributes such as, lively atmosphere, opportunities for collaborative work and discussion, hands-on activity such as making measurement, links between mathematics examples inside the classroom and authentic work, experience of overcoming difficulty and achieving success and suitable use of technology. The lesson in a sense was innovative for its alignment with several features advocated in the curriculum reforms, such as, collaborative and communicative work, application of problem solving skills and the use of technology. Similar to the traditional practice, the teacher played a directive role in the design of the lesson including the choice of the example and activity. In addition, his reflection and evaluation showed many elements of learning, he valued instances of listening to the students’ discussion, helping weaker students engage in the tasks and making better mathematics sense, observing the students working, getting new ideas of the next lesson.

What happens if “Confucius” were in a classroom experiencing the curriculum reforms in the 21st century? Teaching is often seen as a cultural activity (Stigler & Hiebert, 2004). Some scholars sought for answers for the Asian learning paradox in the cultural perspectives, or more the specifically, the Confucian heritage culture (CHC). For example, Asian students are often associated with diligence, high achievement motivation and having a high regard in education (Cleverly, 1985; Ho, 1986; Yang, 1986). Lee (1996) discussed the conceptions of learning in the Confucian tradition in relation to such beliefs as human perfectibility and educability, and the emphasis on effort and will power. In the special lesson paradigm discussed in this paper, the change in pedagogical design and the use of technology inevitably may create novel experience and learning trajectory, hence reflected in the enactment is the reverse role of the teacher as a learner learning more about his students, such as, listening to the students’ discussion, providing clues to support their discovery of methods, and getting inspiration for follow up teaching. In the study of the traditional paradigm, the students’ inclination to imitation of the teacher’s methods were attributed in significant part to the students’ Confucian heritage cultural background, such as, the traditional Chinese beliefs of “practice makes perfect” (Li, 2006) and memorization could come before understanding (Cai & Wang, 2010). Nonetheless, the word “imitation” also has the danger of associated to “surface learning” (Biggs, 1991). Hence, one cannot take it for granted that all kinds of practice will make perfect, nor all kinds of memorization may lead to understanding. It is worthy to revisit the Confucian conception of learning in the context of the two contrasting paradigms. According to Biggs (1991) and Lee (2006), “Confucius saw learning as deep: ‘Seeing knowledge without thinking is labour lost; thinking without seeking knowledge is perilous’ [Analects II. 15].” Hence, seeking for knowledge in inseparable with thinking in the conception of learning. Effort and stress is significant in the process of learning in the Confucian tradition that is indeed a process of “studying extensively, enquiring carefully, pondering thoroughly, sifting clearly, and practicing earnestly (The Mean, XX.19)” (Lee, 2006: 35). This interpretation of learning unfolds the multiple facets in the process of learning in the Confucian conception. Using these multiple facets as the Confucian lens for meaningful learning, some robust elements are found in the learning environment in the changing paradigms, and they are:

the creation of a learning space of variations enabling in-depth understanding of the subject matter,

the lively and interactive class atmosphere for making enquiry feasible, and

the experience and opportunities of realizing the values and overcoming difficulty in the process of learning.

Footnotes

Acknowledgements

The project is funded by General Research Fund, Research Grants Council, Hong Kong SAR, China.

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