1 Introduction

In the previous four papers (Perry, 2007; Wang & Cai, 2007a, b; Wong 2007), we presented the teachers’ views of mathematics and the teaching and learning of mathematics from four regions. In this paper, we provided a comparison and contrast to highlight the similarities and differences among teachers from Australia, Mainland China, Hong Kong SAR, and the United States. Though we fully understand that there is no single educational region that can represent the East (likewise the West), and we cannot draw a line on the globe to divide the East from the West, these four regions were chosen to make cultural contrasts. For this purpose, the US can be considered “Western” and Mainland China “Eastern”. Hong Kong is clearly influenced by Chinese culture, but it is also influenced by the more than a century of Western (British) colonization. Australia is also strongly influenced by British colonization but, over the last 60 years, has grown into one of the world’s most multicultural nations and represents a highly diverse cultural melting pot. To some extent, Hong Kong and Australia can be considered to be between the Eastern and Western extremes exemplified by Mainland China and the US respectively.

2 Teachers’ views about mathematics

What is the nature of mathematics? Of the three fundamental questions investigated in this study, this one received the most varied responses among teachers from the four regions. Overall teachers from Australia and US both hold more to the “functional view” of mathematics, which focuses on its usage in the physical world. Teachers from Mainland China and Hong Kong are of the “Platonist view”, meaning that they focus more on the internal structure of mathematical knowledge (Ernest, 1989). We discuss the similarities and differences on a few major themes cross-culturally.

2.1 Mathematics is practical

Of all the common themes among the four regions, the utilitarian aspect of mathematics was dominant. There is an agreement among teachers from Australia, Mainland China, Hong Kong, and the US that mathematics is applicable to real life problems and that it is a necessary skill for living. All of the teachers interviewed from four regions felt that mathematics has many utilitarian aspects.

For Australian teachers, mathematics is one of those essential subjects that allows us to function in the world. (AU1)

For Mainland Chinese teachers, mathematics is practical in daily life and can help people solve real life problems in an efficient way. It is a science as well as a necessary tool for life. (CH8)

For teachers from Hong Kong, the practical significance of mathematics constituted a salient theme in the teachers’ response. In daily life, [a] child may face problems in books. When they grow old, they use it in buying [a] house. I think that we learned some skills and method of calculation, then apply them in life to solve problems continually. (HK11)

For U.S. teachers, mathematics could provide a new perspective for looking at the world: “I see it as a tool in order to solve problems... But it’s a tool that enables people to do things or to reach goals that they have. The substance of mathematics would be things like a set of rules, a set of methods that allow me to achieve goals or achieve things I’m trying to do or other people are trying to do.” (US5)

2.2 Mathematics is a language

Mathematics as a “language” was the second most common theme in regards to the nature of mathematics across the four groups of teachers. Though this aspect was held more prevalently in some regions than others, what is meant by mathematics having the nature of a “language” is the same: it is a system of knowledge that provides the means of description and explanation of natural phenomenon. The formal language of mathematics is a logical framework of rules and terms that can be used effectively to solve problems of many kinds and communicate the procedures with others in a somewhat universal dialect.

The emphasis of mathematics being a language decreased as it passed through Mainland China, Hong Kong, the US, and Australia. It is likely that this language description is held more strongly in Mainland China and Hong Kong because of the language’s relation to the Platonist view of mathematics (i.e., language, being a structure itself, is related to the structural view), and then held less by Australia and the US because of their emphasis on the functionality of mathematics.

2.3 Mathematics is derived from real life

Only the teachers from Mainland China explicitly argue that mathematics is derived from real life, though it is implicit in some of responses of Hong Kong teachers. In fact, all of the nine Mainland Chinese teachers believe that mathematics is an abstract and generalized knowledge system refined from real life problems.

Mathematics stems from real life...but it is the knowledge refined [tilian] from real life. Once our ancestors help us get the knowledge, we can directly apply the general knowledge without considering some unnecessary features of each specific real life problems (CH3)

Perhaps this view relates to the abstract nature of mathematics that Mainland Chinese teachers have stressed. In any case, the view extends the more practical orientation held by the majority of teachers from all four regions.

2.4 Mathematical knowledge is abstract

This particular characteristic of mathematics drew out a sharp distinction among the four groups of teachers; specifically between the Eastern and Western regions. Teachers from Mainland China and Hong Kong had considerably more to say (both quantitatively and qualitatively) about the abstract nature of mathematics than teachers from Australia and the US. There was a decreasing emphasis on the abstract nature of mathematics, in the following order: Mainland China, Hong Kong, Australia, and the US.

All of the nine Mainland Chinese teachers differentiate mathematics knowledge from real life problems in that mathematics is an abstract and generalized knowledge system refined from real life problems. The real life problems provide the raw materials that can be purified and abstracted as mathematics knowledge.

Though they did not have much to say about the abstract structure itself of mathematics, the majority of the interviewed teachers from Hong Kong said that developing abstract thinking in students is one of the objectives of teaching mathematics. Unlike teachers from Mainland China, they did not give deep descriptions of what they thought abstraction is. However, they spoke mainly of the process of developing abstract, logical thinking in the classroom.

In Australia and the US, few teachers explicitly say that mathematics is abstract. The reluctance to teach and encourage students to learn abstract principles is evident, especially in the US. For the most part, that which is concrete is the focus in the classroom and lessons of Australia and the US.

This distinction between the Eastern and Western cultures in regards to the abstract nature of math and how it affects mathematics education is predictable in light of the previously discussed views of the usefulness of mathematics. It follows from Mainland China and Hong Kong’s structural or “Platonic” view that the teachers place greater importance on the abstraction of mathematics than the countries that hold more to a functional view.

3 Teachers’ views about mathematics learning

To discuss teachers’ views about mathematics learning, we focus on the three themes (See Cai, 2007 in this special issue for details):

  1. 1.

    The nature of understanding—this includes the teacher’s belief of what “understanding” is and how the teacher should help the students gain it.

  2. 2.

    Memorization and understanding—what role, if any, does memorization play in a student’s development of understanding? With this is the question of whether or not memorization should come before or after understanding.

  3. 3.

    The role of practice—what role does practice play and how much should be conducted? What kind of practice develops understanding?

3.1 The nature of understanding

In terms of the importance of understanding, teachers from the four regions have little disagreement. They all agree that understanding is the ultimate goal of learning mathematics and that using real-life problems and concrete experiences can facilitate mathematical understanding.

3.1.1 Understanding means being able to apply knowledge flexibly

Teachers from the four regions agree that an indicator of mathematical understanding is flexible application of what has been learned to various problem situations; that is, problems that require the student to use what they have learned in different ways.

For understanding, I think the first step is they can accept the rule as a fact... Second, when the rule appears in another format, s/he can still think in the reverse manner. And I think this is understanding. (HK2)

[Understanding] is being able to use what you are able to apply in many different situations rather than just applying a skill or a piece of knowledge in one situation repeatedly. (AU12)

Another common theme in terms of the nature of understanding is that the student is able to communicate what they have learned. When a student is able to communicate with others using mathematical language, this means the student is able to give an explanation about that which was explained to him. Teacher AU11 gives a thorough account of this point:

Understanding is achieved when they are able to explain the ‘why’, the ‘how’ and the ‘do’ in a situation using mathematical language to support their explanation.

3.1.2 Understanding at concrete and abstract levels

Teachers from the four regions agree that mathematics understanding should start from students’ concrete experience. However, they have different views on functions of concrete examples in mathematical learning. While the US teachers put emphasis on helping students realize the relatedness between mathematics and real life problem, Mainland Chinese teachers tend to encourage students to reach abstract concepts and thinking from concrete examples.

All Mainland Chinese teachers argue that the ultimate goal of introducing concrete examples is to help them derive abstract mathematics concepts. Once the students’ understanding reaches this abstract level, they can be freed from the constraints of concrete representations. After the students have established some abstract mathematical concepts, the teachers should emphasize the importance of connecting different concepts and integrating them into a systematical knowledge system. While most Mainland Chinese teachers emphasize the importance of helping students master abstract and connected mathematics concepts, only one US teacher (US7) explicitly mention that helping students connect abstract concepts is important. Instead, most US teachers express their reluctance to encourage students to learn mathematics on an abstract level (e.g., deriving formulae) especially in 6 or 7th grade classrooms. It seems that most of the US teachers think their students at this stage are cognitively not ready to think abstractly. That is not the case for the Mainland Chinese teachers.

While Mainland Chinese and US teachers seem to hold quite opposite views about understanding at the concrete and abstract levels, teachers from Australia and Hong Kong hold views between these extremes. For Australia and Hong Kong teachers, while concrete experiences offer great opportunities for fostering mathematical understanding, the individual characteristics of the learner need to be taken into account before offering particular materials. They see that the road to abstraction as a long process. Teachers from Australia and Hong Kong explicitly point out that concrete materials are particularly relevant for certain groups of children and not for others. The level of abstraction teachers can reach depends on the characteristics of students and nature of the mathematics to be learned.

3.2 Memorization and understanding

Though they all agree that memorization plays an important role in mathematical understanding, teachers from the four regions do not fully agree on what that role is or to what degree memorization is important. Furthermore, there was a re-occurring concern of whether or not memorization should come before or after understanding. For teachers from Mainland China and Hong Kong, memorization can come before or after understanding. However, for Australia and US teachers, memorization can only come after understanding. Nevertheless, memorization after understanding is held in higher regard than memorization before understanding (or “rote” memorization), though some of the teachers from Mainland China say that perhaps the latter could be an intermediate or transitional step towards understanding the mathematics.

While students start with rote memorization (without understanding), they should be able to gradually come to understanding by practicing. (CH8)

While Mainland Chinese teachers express the value of memorization, they also give a distinction between what they call “live knowledge” and “dead knowledge.” “Live” knowledge is easy to transfer for solving new problems. In particular, CH1 uses the Chinese idiom “juyifansan” (knowing one concept and applying it into three situationsFootnote 1) to describe the “live knowledge.” In contrast, “dead knowledge” cannot last long and is difficult to be applied and transferred into a new situation. Therefore, it seems the Mainland Chinese teachers believe that even that knowledge which is memorized before understanding (“rote” memorization) must eventually be converted to knowledge which is understood.

In general, teachers from Hong Kong do not believe that memorization has a central role in mathematics learning.

Memorization may have some effect on mathematics learning, but it is not an important component. (HK4, HK12)

However, when asked what kind of memorization (if and when used) is best, a higher regard for memorization after understanding than for rote memorization was expressed. The majority of teachers here view rote memorization as a final alternative for the student when he/she does not have understanding of that knowledge:

If there is something we really cannot understand, we should memorize it first as to tackle the examination. (HK9)

Though there is a general inclination among the teachers from Hong Kong not to make memorization (whether before or after understanding) an imperative, the teachers value the type of memorization that follows understanding and therefore makes the knowledge mentally available for application:

After the students understand, then memorization is important. It would be useful if s/he has a good memory. In fact, if s/he understands, memory would be useful to future application. (HK8)

Australian teachers give high regard to memorization as the recall of pertinent information. They support the process of memorization as a key factor in learning almost as strongly as teachers from Mainland China.

Memory is very important. They have to start off with a core amount of information. (AU2)

I think there’s a place for memorization. I’m glad to see that the new syllabus puts some emphasis back on learning times tables. I think that is very important. Along with that comes understanding. (AU1)

However, Australian teachers do not necessarily share the idea that rote memorization can serve as a transitional step to understanding. The majority sees rote memorization—retaining facts without understanding—as something to be avoided. When the Australian teachers speak of memorization, they tend to conjunct words like “reinforce”, “connections”, and “understanding” to the idea. So, in general, the Australian teachers believe memorization should follow understanding.

The US teachers are all in agreement that memorization after understanding is the type of memorization that is valuable. They believe it is necessary for retaining knowledge, applying the knowledge to solve problems, and learning new knowledge. However, when it comes to rote memorization, the teachers have various views. Some believe it has little to no use, while others see it has something that is necessary.

I think that if they encounter something enough times, they’re just going to remember it anyway. That rote memory is not something they are going to remember. (US8)

In contrast, US9 suggests:

I think some people that follow the NCTM standards very closely would disagree with me, but I still think there’s a place for memorization and rote memorization of basic facts.

In summary, the idea of understanding before memorization seems to be the most prominent trend. All of the teachers interviewed from the four regions, except for those from the US, explicitly affirm that rote memorization is only useful in making knowledge something that can be recalled quickly and as a last resort for examination when understanding is not fully developed. Only teachers from Mainland China expressed the idea that this rote memorization could be useful as a transition to understanding.

3.3 Role of practice

All teachers from the four regions view practice as essential, but to varying degrees. Teachers from Mainland China place as much value on it as they did on memorization. Teachers from Hong Kong, much like teachers from Mainland China, view practice as a means to facilitate understanding, but some also present a minimalist view:

They don’t need a lot of exercises, but [just] one in each type. If you understand, you just have to have a quick glance [to understand] and don’t need to do a lot. (HK7)

This minimalist view is also observed in US responses:

I don’t do a lot of practice...students in my classroom don’t necessarily get a lot of practice repeatedly on [a] particular concept. They may only be exposed one or two times to a particular concept, and then we move on. (US11)

One significant difference among teachers from the four regions is that teachers from Mainland China were the only group who did not mention the risk of over-burdening the students with “drill and kill” practice. Hong Kong, Australia, and the US all shared concern that practice can be over done and student’s interest in the subject can be dulled.

Yes they [exercises] are important but I do not agree on letting students doing the same kind of exercises too much unless there are variations. The worse case is that students do not think over the question after doing massive exercises. I do not agree with mechanical training. (HK5)

Of the other three regions, teachers from Australia and the US shared similar concerns related to practice:

I think that you only need to go so far as realizing that when most people are confident with the idea and not flogging it to death.” (AU13)

I find that if you give too much, it’s like they’ll just turn off from it...especially with word problems, you know, you don’t want to turn them of either. (US10)

4 Teachers’ views about the teacher and teaching

The focus of this study is to understand the teachers’ views about effective teaching. Understanding teachers’ views about mathematics and learning of mathematics provide a context to understand teachers’ views about effective teaching. Of all of the questions used in this cross-cultural study, this one revealed most about the beliefs of all the teachers involved regarding effective mathematics teaching. How a teacher decides to run a classroom is, to a great extent, the reflection of their goals for maximizing students’ learning and their beliefs and values in relation to the subject being taught.

4.1 Characteristics of an effective teacher of mathematics

The topic of what characterizes an effective teacher also accentuated the differences of mathematics education between the Eastern and Western cultures. The teachers from Australia and the United States had much more to say about the teacher’s enthusiasm and rapport with the students than teachers from Mainland China and Hong Kong. Teachers from Mainland China and Hong Kong focused on how well the teacher prepares and presents a lesson and the ability to provide clear explanations of the points to be covered in the lesson.

4.1.1 Strong background in mathematics

Nearly all the teachers from Australia, Mainland China, and Hong Kong make a strong point of this characteristic of effective teachers of mathematics. According to their statements, well-grounded knowledge and understanding of the subject is a crucial element in being able to effectively teach mathematics. In addition, teachers from Mainland China place an extremely strong emphasis on understanding of the curriculum and texts being used. According to Mainland Chinese teachers, it is clear that an effective mathematics teacher should explore and study textbooks intensively and carefully and should precisely predict the possibly difficult concepts for their students so that they can devise instructional strategies to overcome the difficulties. Teachers from Australia and Hong Kong also emphasize deep understanding of the curriculum content and structure.

They have to have an understanding of the syllabus to start with and what they should be teaching. (AU5)

4.1.2 Adept in instructional skills

All teachers from the four regions agreed that an effective teacher should possess the skills needed to instruct properly. For teachers from Australia, this point was made implicitly through their discussion about making the lessons relevant to current society and balancing humor and authority in lessons. In particular, the following three specific instructional skills were mentioned consistently by teachers from the four regions:

  1. 1.

    Clear communication and explanation of the topic and goals (which requires their own knowledge of math and of the curriculum).

  2. 2.

    Being able to use a variety of methods of instruction (manipulatives, thought-invoking lecture, etc.) according to the students’ needs.

  3. 3.

    Building interest and maintaining it by varying methods and/or by making the topic, when possible, relevant to the students’ experiences.

Although teachers from all four regions agree that the personal magnetism and solid mathematical understanding are both important traits of an effective math teacher, there is again a general difference between the teachers from Mainland China and Hong Kong, and teachers from Australia and the US. While teachers from Mainland China and Hong Kong highlight the need for the teacher’s ability to provide insightful explanation and stimulate thinking, teachers from Australia and the US focus more on how well the teacher can listen to their students and get them to interact with teachers and one another.

4.1.3 Knowing and caring for the students

Teachers from Australia, Mainland China, and the US all explicitly agree on the necessity of this characteristic; Hong Kong teachers did not mention this point as a characteristic of an effective mathematics teacher. The general theme among the teachers from Australia, Mainland China, and the US is that a teacher should understand the needs of their students and have the desire to understand their needs. For example, CH1 argues that a good teacher is always passionate in caring about students both in and out of the classroom. CH2 further argues that this kind of passion not only builds a positive rapport between the teacher and students, but also could directly impact on students’ learning.

Overall, however, the teachers from Australia and the US had more to say about building this positive rapport with the students than teachers from Mainland China did.

Once there is a level of empathy with the student so that you know this person reasonably well, at least in terms of their interests, you can start to get somewhere. (AU9)

The [effective teachers] are caring. They relate to the students as far as I show them respect and they show me respect...(US10)

4.1.4 Classroom management

Classroom management seemed to be much more important to the teachers interviewed from the US than for any of the other three regions. One teacher asserted:

Well, first of all, I think the classroom management is truly important. If that’s not there, we’re not going to...nothing will be accomplished. (US2)

US5 also heavily emphasized this need:

In the public school, the number one thing that you have to have is you have to control the classroom. From my experience of teachers that appeared not to be as effective it’s more of a discipline issue and a control issue.

The teachers interviewed from Mainland China did not mention anything about classroom management being a concern, reflecting that, for most Mainland Chinese educational environments, classroom management is simply not an issue. Neither does this appear to be an issue with teachers from Hong Kong or Australia, as nothing was said by the teachers from these two regions in regards to classroom management. It must be remembered that the interviewed teachers in all four regions were chosen for the study because they were effective mathematics teachers. Consequently, they would be expected to have solved issues around classroom management in their own teaching.

According to Ernest (1989), there are three basic teaching models related to the responsibilities that a teacher has in the classroom: facilitator model, explainer model, and instructor model. The first model, the facilitator, has the goal of having students develop confidence in establishing and solving problems. The explainer’s aim is toward the students gaining conceptual understanding with cohesive knowledge. Finally, the instructor’s intention is that the student masters the skills necessary for proper performance. There are some teachers that fit more into one model than another, and this can be observed in this study as well. The description of an effective teacher according to teachers from Mainland China and Hong Kong seems to fit more into the “instructor model.” For teachers from Australia, it would seem that their description of an effective teacher is somewhat between the explainer and facilitator: one who helps the students connect the knowledge mentally, yet encourages them to confront and solve mathematical problems themselves. The US teachers’ descriptions of an effective teacher generally seem to fit the facilitator model.

4.2 Characteristics of an effective mathematics lesson

As might be expected, some of the aspects that were mentioned in regards to what makes a teacher effective were reiterated in the responses to the characteristics of an effective lesson, albeit from a different angle. There is a tendency for the teachers from the Eastern regions (Mainland China and Hong Kong) to emphasize the “teacher-led” aspect of the mathematics education in the classroom, while the Western region teachers (Australia and US) emphasize the “student-centered” aspect (Leung, 2004).

4.2.1 Active engagement of students

All 11 US teachers agree that active student engagement in the classroom is necessary to keep the students interested. Therefore, concrete examples are often implemented into the lesson:

I usually start off talking about why we’re going to learn this topic, why we need this topic. Let’s say with percent. And I have the kids say, well we need it for sales tax, to leave a tip at a restaurant…so we talk about why we’re dealing with this topic. Then I try to go into what does it really mean. (US7)

For the majority of the US teachers, active student engagement also involves hands-on manipulative activities for the purpose of student exploration:

I think that investigation by students and allowing them to find rules, allowing them to find the way things behave is very effective compared to just always lecturing and giving formulas and telling them how things behave. (US5)

The teachers from Australia emphasize the students’ verbal involvement more than the physical involvement as a characteristic of an effective lesson. There was little said specifically about the use of hands-on manipulatives. To them, active student engagement can arise by tapping the students’ curiosity:

I think curiosity is a big thing with kids...and active student involvement. (AU2)

[An effective mathematics lesson is] one where all the conversation is about the maths, the students are engaged and there is not too much teacher talk. (AU13)

Such verbal engagement is also expressed as a respectful exchange between the student and teacher on what is being taught:

You try to make sure that children have the opportunity to question, to discuss, to answer and that there’s an atmosphere where the children and teacher respect each other’s views and that those are listened to. (AU1)

For teachers from Hong Kong, student participation and involvement are the keys for understanding as well as achieving the learning objectives. Student participation is also the source of satisfaction in learning. For teachers from Hong Kong, participation mainly refers to the vocalized interactions in classrooms.

Mainland Chinese teachers acknowledged that there was a necessity for a “lively and comfortable learning environment”. CH1 asserts that:

A terrible lesson is the teacher lecturing the whole lesson without student participation because you would have no idea whether your students understand the material.

In following this point made by CH1, all the teachers seem to agree that concrete types of examples serve a purpose in helping the students understand mathematical concepts. However, there are varying opinions about how this should be practiced. For some Mainland Chinese teachers, in order to understand a concept clearly, students should physically operate the concrete examples and tools. However, due to constraints of time and class size, other teachers argue that, in real teaching, a teacher often just demonstrates the process without having students manipulate tools.

It was also expressed by Mainland Chinese teachers that the hands-on manipulatives should be used only towards the objective of bringing about understanding, and therefore the teachers need to have the students contribute mentally and verbally to what they have just done physically.

4.2.2 Group activities/in-class student collaboration

Significantly more was said about group activities and in-class student collaboration by the US teachers than those from Australia, Mainland China, and Hong Kong. Teachers from Mainland China and Hong Kong made no mention of this being a characteristic of an effective lesson. The teachers from Australia held a more middle-road view: small group activity was neither a necessity nor an impediment to an effective lesson, though the majority of Australian teachers interviewed utilize this educational technique in their classrooms.

At times you need to have groupings. So you might have a core lesson and some work that the children who have obtained or [understood] that knowledge can go on with. Then you can spend some intense time with the ones that don’t. (AU7)

For U.S. teachers, in-class peer interaction is essential to a lesson being effective.

And to have a problem like that where the kids are, you know, four of them together, are communicating mathematically, trying to solve a problem. Maybe not one of them individually could solve it, but all of them could solve it. (US3)

In general, it seems that the US teachers are more comfortable with having group activities and discussion during the class time than the teachers from the other three groups.

4.2.3 Coherence

Of the four groups of teachers interviewed in this study, only teachers from Mainland China and Hong Kong explicitly addressed the issue of having a well-structured, coherent lesson for the class. All nine Mainland Chinese teachers maintain that an effective lesson should coherently develop well-planned content. The following statement is typical for Mainland Chinese teachers:

An effective lesson should have all the steps [of instruction] closely serve for the essential points…so that students can actively participate in each step. (CH3)

About half of the Hong Kong teachers interviewed made mention that an effective math lesson is one which is well structured. For example:

One should think about what one is going to teach before a lesson, but should not pack too many objectives into a single lesson...another important point is the flow of the lesson [well-designed]. (HK5)

The teachers from Australia did not give as many specific responses that emphasized the need for coherence of a lesson as the teachers from Mainland China and Hong Kong. However, there was mention that lesson objectives should be made clear and that there needs to be a structured routine in the classroom:

I have clear goals to be reached, they know where the journey is going, it’s very clear and they have to be focused. (AU4)

There were no comments about the coherence of lessons from the US teachers interviewed.

4.2.4 Flexibility of teaching fits individual students’ needs

Teachers from Australia and the US addressed the issue of flexibility with significantly more emphasis than teachers from Mainland China and Hong Kong.

For the US teachers, flexibility is a prominent characteristic of an effective mathematics lesson. However, it is primarily addressed through the characteristics of an effective math teacher.

Being able to observe, and judge, and evaluate each student and meeting their individual needs probably is the most difficult and probably one of the most crucial parts [for an effective teacher]. (US4)

In regards to the math lesson itself, there is acknowledgment that the lesson needs to be appropriate to the students’ stages of development:

[I]f you can gear a lesson so that it’s just at the right spot for where the students are developmentally, where it stretches them just enough so they’re not frustrated but it challenges them at the same time. (US9)

Australia’s teachers were specific about how the lessons themselves should be both planned and yet flexible.

Most of my lessons are planned... For example, you would have to have the resources you needed there and if there is a child who needs concrete material then you have to have it available. There has to be an ability to change. (AU7)

When new ideas have been discovered, when perhaps what I had planned is not what we’ve done at all, which is what happened to us this week because somebody came up with something and we’ve gone off on a tangent and discovered something totally new, then that is an excellent lesson. (AU3)

Some of the Mainland Chinese teachers acknowledged the need for teaching flexibly in order to address students’ needs. They agreed to the necessity of flexibility in the lesson, though that the flexibility is constrained both by the large number of students in the class and the amount of content that is required to be taught in a lesson. One teacher argues:

In terms of how to unfold a planned lesson, the teacher should always flexibly adjust his path according to student status. After a student answered a question, [I can find] what is still not understood by him. Then I will continue [to] explain it carefully. Therefore, I cannot just rigorously follow the plan. (CH2)

Only one teacher from Hong Kong explicitly commented on the need for flexibility in order to be sensitive to the developmental pace of the students:

The teaching pace should be adjusted with the response of the students in the lesson. The teacher should not just blindly follow the lesson plan and let the lesson go on without considering students’ response. (HK2)

4.2.5 Cultivating students’ interests

This characteristic of an effective lesson was prominent for all four groups of the teachers.

Six teachers from Hong Kong comment on the cultivation of student interest, giving various ideas on how this can be done:

Teaching aids, games, real-life examples, introducing them various activities and outside readers. (HK8)

In regards to the teacher drawing connections between the mathematical ideas on the syllabus, HK7 suggests:

They would feel surprised and this would initiate their thinking [too].

HK1 comments on the importance of a teacher using a good question and answer technique:

[One has to] ask questions, from which can inspire students to further imagine...[One should ask] how can we make use of questions to guide students to think something new, deeper, and those things they have never thought before.

Many of Australia’s teachers hold the characteristic of cultivating interest in students in high regard. Some of the teachers point out that if one is able to begin the lesson in an interest-capturing way, then the student’s interest is more likely to be maintained for the rest of the lesson.

When it is time for math groups it should be “Yes! Off we go to maths” and they should be coming into the classroom excited. For all sort of reasons, the mathematics classroom should be a place where they feel really good about themselves, where they’re feeling really enthused to be there... Not everybody feels like that all the time but there are times when the recess bell has gone and I am shooing them out the door and they’re still not going. (AU3)

I like to start the lesson off with something that makes the children think. It doesn’t have to be anything to do with the particular topic that you’re learning but it just means that you are trying to get the answer to something. (AU13)

Though there is no specific comment made by teachers from the US on the cultivation of student interest being a characteristic of an effective lesson, the study showed that all of the 11 US teachers see the importance of cultivating students’ interest. The majority of teachers from Mainland China agree.

At the beginning, they can learn some mathematics, and then they are willing to learn more mathematics. Finally, the enjoy mathematics. (CH8)

Some Mainland Chinese teachers also commented on the technique of having thought-provoking question, which is similar to what was found from the Hong Kong teachers.

5 Reflections

Over centuries of mathematical progress new knowledge has been acquired that needs to be passed on to following generations. With this imperative there arises the need to have willing and capable teachers to pass on this knowledge for the benefit of global posterity. In every country, mathematics impacts the way we understand our environment, control our finances, construct enterprises, and conduct businesses. Therefore, to have the opportunity to take into account the global status of mathematics education and to hear the perspectives of teachers from a variety of regions about the effective teaching is invaluable.

This study provided a cross-cultural perspective about teachers’ views of effective mathematics teaching, with a focus on the comparisons of East and West. Australia, Mainland China, Hong Kong SAR, and the United States were selected for the study because they represent a spectrum of Eastern and Western cultures.

5.1 Nature of mathematics

Some of the beliefs of teachers from Australia, Mainland China, Hong Kong SAR, and the US about the nature of mathematics, learning and teaching of mathematics showed an East and West cultural dichotomy while others resulted in much more of an East/West cultural continuum. For example, the teachers from Mainland China and Hong Kong SAR view the nature of mathematics from a “Platonic view”; that is, their focus is on the internal, logical structure of mathematics, which reflects mathematics as an abstract body of knowledge. In contrast, the teachers from Australia and the US place much emphasis on the “functional view” of mathematics, that is, mathematics is a useful tool that is used everyday to solve real-life problems. Teachers from Australia and the United States have more emphasis on the aspect of mathematics being a language by which physical phenomenon can be described and explained. This does not mean that there is no acknowledgment by teachers from Mainland China and Hong Kong SAR about the usefulness of mathematics in real life problems. Rather, there is not as much emphasis placed on its functionality by teachers from Mainland China and Hong Kong SAR as teachers from Australia and the US.

5.2 Understanding, memorization, and practice

In regards to the nature of understanding, there was not a great deal of variance among the four groups of teachers. Teachers from the four regions by and large agreed that the goal of mathematics education is that the students gain understanding of the material. They all also agree that both the student’s ability to apply the mathematics to various problems and his/her ability to mathematically communicate the learned material to the teacher or other students indicates the presence of understanding. However, there is a remarkable difference in terms of the teachers’ descriptions of the relationship between understanding and memorization.

Two types of memorization were identified: memorization before understanding (sometimes understood as “rote” memorization) and memorization after understanding. For teachers from Mainland China and Hong Kong, memorization can come before or after understanding. However, for Australia and US teachers, memorization can only come after understanding. For teachers from Mainland China, memorization before understanding could serve as an intermediate step towards understanding; in other words, as long as this type of memorization leads to understanding, then the memorized knowledge is not simply “dead knowledge”. In general, however, teachers from all four regions agreed that understanding after memorization is ideal, though it is also acknowledged by those teachers that this is not always the case in the classroom. In both cases, it seems that memorization is regarded as a means and understanding as the goal, though Automation (getting something memorized by heart) is also regarded as important, especially when one needs to solve mathematics problems fluently (Kerkman & Siegel, 1997; Wong, 2006). Other research supports the hypothesis that excellent academic performance of “Asian” learners on international mathematical comparison programs may be due to a synthesis of memorizing and understanding which is not commonly found in Western students (Marton, Tse, & dall’Alba, 1996; Marton, Watkins, & Tang, 1997; Watkins, 1996). Recitation is often used to bring about sharp focus and better understanding (Dahlin & Watkins, 2000).

In Mainland China and Hong Kong SAR, “knowing how” is seen as equally important as “knowing why”. While Sfard (1991) pointed out the dual nature of mathematics, the recent discussions of Star (2005) and Baroody, Reil, & Johnson (2007) also open up the possibility of “deep procedural understanding”. The seminal work of Skemp (1972) on procedural and relational understanding is also relevant here.

In terms of the role of practice, it seems that the teachers from Mainland China are the most comfortable with having students practice since there was no concern expressed about over-doing it; there were only expressions of its value. Hong Kong teachers place nearly as high a value on practice as teachers from Mainland China, but only if that practice is constituted as exercises with variations. Australian and US teachers, in general, are not as keen as Mainland China and Hong Kong on the value of practice. Teachers from Australia and the US shared a common concern that practice can be overdone and therefore dull student interest. None of the teachers from Mainland China, however, expressed this concern.

5.3 Characteristics of an effective teacher

Teachers from Mainland China, Hong Kong SAR, and Australia agree that competence in mathematics is a necessary characteristic of an effective teacher. It was also mentioned (especially by teachers from Mainland China) that a teacher’s in-depth understanding of the curriculum and textbooks being used is key for an effective teacher. The US teachers did not note this as an important point in their responses.

Teachers from all regions concurred that a teacher should both understand the needs of his/her students and have interests in understanding their educational needs. Teachers from the US expressed concern for a teacher to have appropriate classroom management skills, particularly in terms of discipline and control. Teachers in the remaining three groups did not address this issue, possibly because classroom discipline does not seem to be as major an issue with them, particularly for teachers from Mainland China and Hong Kong. The general difference here was that the teachers from Mainland China and Hong Kong emphasize the ability of a teacher to provide the information with clarity and to stimulate thinking, while the teachers from Australia and the US emphasize the ability of the teachers to listen to the students and to get them to respond with interest.

5.4 Characteristics of an effective lesson

Based on the responses of the teachers from the four regions, it appears that the teachers from Australia and the US are more comfortable with frequent use of hands-on manipulatives than teachers from Mainland China and Hong Kong. Hong Kong teachers, in particular, suggested that if physical manipulatives are used, they are commonly used by the teacher for purposes of demonstration and not by the students, mainly because of time restraints. The teachers from Mainland China tend to stress verbal engagement over physical engagement on the part of the students. The teachers from Mainland China and Hong Kong did not mention in-class group activities, yet this is stated as a characteristic of an effective lesson by the US. In fact, group activities are usually included in US teachers’ lesson plans, but not on those from Mainland Chinese teachers (Cai, 2005; Cai & Wang, 2006). Teachers from Australia saw group activity neither as a necessity nor an impediment to mathematical understanding.

While the US teachers focus more on the students’ engagement and interaction during the lesson, the teachers from Mainland China and Hong Kong emphasize the importance of the coherence of a lesson. In summary of what characterizes an effective lesson, teachers from the East have more of a teacher-led view of classroom instruction than the teachers from the West, who hold more to a student-centered view.

It is impractical to look for a “national/regional script” of mathematics teaching. Yet classroom practices are often shaped by cultural, environmental and societal assumptions. Watkins and Biggs (2001) have warned that teaching and learning traditions that appear to work well in a certain culture may not necessarily work in another. For instance, when high-stake examination is the “ticket to the future”, fast and accurate solutions to mathematics problems are needed. The ramifications of such high-stake examination are often manifested through general parental expectations. When there are large class sizes, hands-on explorations can become difficult and individual care is often left to after-class hours (Gao & Watkins, 2001; Wong, 2004).

In this study, we see a broad-stroke linkage between teachers’ beliefs in mathematics, their image of an effective mathematics lesson and that of the effective teacher. For instance, for the two “Eastern” regions (Mainland China and Hong Kong SAR), since the mathematics teachers general hold a Platonic view, the mathematics knowledge structure is very much stressed in teaching. It is important to let the student understand the “truth” and generalization (Cai, 2004). That is why practice plays a central role. Though these teachers fully understand the importance of individual guidance, this can only be done after class. The major task of classroom teaching lies in the transmission of knowledge, the teacher must get oneself well prepared and have the lesson well structured, so as to run a “teacher led, yet student centered” mathematics lesson. Two things are pre-requisite: thorough understanding of the curriculum and textbook on the teacher’s side and the establishment of a classroom routine on the students’ side. That explains partially why classroom management is not the major concern among Eastern mathematics teachers since students are accustomed to the various routines in flow of classroom teaching: when to talk, when to do seat work, when to open one’s book, when to look at the chalk-board (or computer projection), and so on early at their early age (Wong, 2004). Classroom transmission is, however, only the first step to learning (“entering the Way”). Further steps have to be taken for “transcending the Way” (Wong, 2006).

In the two “Western” regions, there is a much stronger emphasis on student-centered approaches to mathematics teaching and learning and the need for the mathematics being learned to be practical and relevant to the learners. While teachers’ understanding of the mathematics being taught is seen as important, reliance on planning, knowledge of the syllabus and textbooks is less so. Both US and Australian teachers see that part of their being effective teachers relies on their knowledge of the students and their understanding of the students’ needs. Consequently, the “functional” understanding of mathematics leads to less structured lessons that are more able to reflect flexibly the needs of the students and the teachers.

In this study, we have confined our investigations to teachers’ perspectives on the effectiveness of mathematics teaching. Many similarities and some differences have been discerned across the four regions considered. Further investigation is needed to see if these similarities and differences are sustainable across the populations concerned.

There are many other international studies that take different starting points and have found results that both contrast and compare with those presented in this volume (Clarke & Keitel, 2006; Leung, Graf, & Lopez, 2006; Ma, 1999; Stigler & Hiebert, 1999). It is possible that asking different questions of different people in different ways might result in different conclusions. What is notable are the similarities that have resulted from these different approaches.

However, more needs to be, and will be, done. Detailed observations of lessons need to be continued. The perspectives of student and novice teachers and school students need to be canvassed further in a consistent, valid cross-cultural methodology. We need to strive for as full a picture as possible of effectiveness in mathematics teaching and learning so that future generations of students in all regions can benefit. In particular, we need to investigate teachers’ perspective of effective teaching in mathematics from broader national and international contexts (Cai, Gabriele, Perry, & Wong, 2007).