Hong Kong teachers’ views of effective mathematics teaching and learning

ORIGINAL ARTICLE

Hong Kong teachers’ views of effective mathematicsteaching and learning

Ngai-Ying Wong

Accepted: 2 May 2007 / Published online: 25 May 2007

� FIZ Karlsruhe 2007

Abstract Twelve experienced mathematics teachers in

Hong Kong were invited to face-to-face semi-structured

interviews to express their views about mathematics, about

mathematics learning and about the teacher and teaching.

Mathematics was generally regarded as a subject that is

practical, logical, useful and involves thinking. In view of

the abstract nature of the subject, the teachers took abstract

thinking as the goal of mathematics learning. They re-

flected that it is not just a matter of ‘‘how’’ and ‘‘when’’,

but one should build a path so that students can proceed

from the concrete to the abstract. Their conceptions of

mathematics understanding were tapped. Furthermore, the

roles of memorisation, practices and concrete experiences

were discussed, in relation with understanding. Teaching

for understanding is unanimously supported and along this

line, the characteristics of an effective mathematics lesson

and of an effective mathematics teacher were discussed.

Though many of the participants realize that there is no

fixed rule for good practices, some of the indicators were

put forth. To arrive at an effective mathematics lesson,

good preparation, basic teaching skills and good relation-

ship with the students are prerequisite.

1 A brief description of mathematics education in Hong

Kong

The first elementary mathematics curriculum was issued by

the Education Department in 1967, which was influenced

by the UK Nuffield mathematics. While the ‘‘activity ap-

proach’’ was introduced in the 1970s, the ‘‘Target Oriented

Curriculum’’ was put forth in the early 1990s. The inten-

tion of the latter was ‘‘to provide clear learning targets to

help teachers and schools develop more lively and effective

approaches to teaching, learning and assessment’’ (Edu-

cation Department 1994, p. 26). ‘‘The five dimensions of

number, measure, algebra, shape and space, and data

handling were identified. These five strands were incor-

porated with process abilities of mathematical conceptu-

alization, inquiry, reasoning, communication, application

and problem solving’’ (Curriculum Development Council

1992, p. 12). The Target Oriented Curriculum brought

about heated debates and the mathematics curriculum

underwent a holistic review in the late 1990s. As a result,

the new elementary mathematics curriculum was published

in 1999. The government launched for the holistic educa-

tional reform at the turn of the millennium, in which

‘‘learning to learn’’ and ‘‘higher ordered thinking skills’’

were emphasized (Curriculum Development Council

2001). It does not change the contents of the elementary

and junior secondary mathematics curricula, while the

mathematics curriculum at the senior secondary level will

be restructured according to the new ‘‘6–3–3–4’’ system

(Curriculum Development Council and the Hong Kong

Examinations and Assessment Authority 2006). New ini-

tiatives like projects, generic skills, standard-based

assessment, and school-based assessment are introduced

too.

Previously, most of the elementary school teachers are

non-degree holders. On the other hand, those who obtain a

first degree (in mathematics or related topics) are eligible

to teach (mainly secondary) school mathematics. The sit-

uation is greatly improved starting from the mid-1990s. At

present, around 90% of school teachers are degree holders

N.-Y. Wong (&)

Department of Curriculum and Instruction, Faculty of Education,

The Chinese University of Hong Kong, Shatin 852, Hong Kong

e-mail: [email protected]

123

ZDM Mathematics Education (2007) 39:301–314

DOI 10.1007/s11858-007-0033-4

and most of them went through teacher education programs

(i.e., Teacher Cert., PGDE or PDCE1). The idea of ‘‘having

(mathematics) subject specialists teaching the subject

(mathematics)’’ was also put forth. It is envisaged that all

teachers teaching mathematics would have a considerable

qualification in mathematics in the future.

2 The participants

Our target informants are experienced elementary mathe-

matics teachers because novice teachers may not have

formulated a stable conception of effective mathematics

learning and teaching. Twelve such teachers were invited

to participate in the face-to-face semi-structured interview.

Table 1 shows the brief descriptions of them. To prevent

their identity from disclosure, pseudonyms were used in

this chapter.

3 Results

The interviews were then transcribed and analysed. The

results were then categorised into the teachers’ views about

mathematics, about mathematics learning and about the

teacher and teaching. The results are presented as follows.

3.1 Teachers’ view about mathematics

The participants expressed their opinions on ‘‘what is

mathematics’’. It was generally agreed that mathematics is

practical, logical, useful and involves thinking. Mathematics

is language and a set of rules. In particular, much delibera-

tions were made on the abstract nature of mathematics as

they unanimously took abstract thinking as the goal of

mathematics learning. They reflected that it is not just a

matter of ‘‘how’’ and ‘‘when’’, but one should build a path

so that students can proceed from the concrete to the abstract.

3.1.1 What is mathematics?

The practical significance of mathematics constituted a

salient theme in the teachers’ responses. Seven out of the

Table 1 The participants of the

studyName Gender Academic

qualification

Teaching

experience

(years)

Other qualification

HK1 Male Teacher Cert., B.Ed. 17 Member of the editorial board of a local

mathematics education periodical

HK2 Female Teacher Cert., B.Ed. 6 Council member of a local mathematics

education professional body; served at the

government Education Department as

seconded teacher for a year

HK3 Female Teacher Cert., B.Ed. 10+ Council member of a local mathematics

education professional body

HK4 Female Teacher Cert., B.Ed., M.Ed. 10 Council member of a local mathematics

education professional body

HK5 Female Teacher Cert. 25 Teaching practice supervisor of a university

HK6 Male B.Sc., PGDE 5 Head mathematics teacher in school

HK7 Female Teacher Cert. 22 Curriculum leader in school; team member of

a university project on students’

motivation of learning

HK8 Male Teacher Cert., B.Ed. 15 Principal; member of the government’s

Curriculum Development council

(mathematics)

HK9 Female Teacher Cert., B.Ed. 15 Head mathematics teacher in school; team

member of a university project on

students’ motivation of learning

HK10 Female Teacher Cert., B.Ed., M.Ed. 17 Member of the government’s Curriculum

development council (mathematics)

HK11 Male B.Sc., PGDE 10+ Senior teacher in school

HK12 Female B.Ed., M.Ed. 12 Member of the editorial board of a local

mathematics education periodical;

teaching practice supervisor of a university

1 Teacher Certs. stands for Teacher Certificate, PGDE stands for

Postgraduate Diploma in Education, and PDCE stands for Post-

graduate Certificate in Education.

302 N.-Y. Wong

123

twelve participants explicitly mentioned that mathematics is

practical. ‘‘Help solving daily life problem’’ is one: ‘‘we

apply mathematics when we solve problems in our real life’’

(HK10; similar response: HK5). Another teacher elaborated:

For instance, you could use number to solve prob-

lems. In daily life, child may face problems in books.

When they grow old, they use it in buying house. I

think that we learned some skills and method of

calculation, then apply them in life to solve problems

continually (HK11).

One teacher saw the application of mathematics in other

disciplines, something that helps ‘‘explain natural phe-

nomena’’ (HK10). Another made a fuller account:

Mathematics is the beginning of science. After you

understand mathematics, you can apply the mathe-

matics knowledge. Mathematics also has logic and it

has its pattern. After you understand the logic and

pattern, you can solve other problems (HK7).

‘‘Mathematics is logic or is logical’’ constituted another

salient theme. Five teachers expressed such a view.

Mathematics being a science of pattern was seen to have

close relationship with the logical nature of mathematics.

As one teacher said: ‘‘[mathematics is] something that has

logic and pattern’’ (HK7), and another said: ‘‘[mathemat-

ics] includes logical thinking and proofs’’ (HK2). Mathe-

matics is seen as a logical language that helps us

communicate patterns and phenomena in nature. This is

clear from the following response: ‘‘we have to use the

numbers, symbols to communicate, to construct with logic,

to create with reasons’’ (HK5).

So, mathematics is seen to be useful, applicable to other

disciplines (science in particular), and is a logical language

that explains natural phenomena. Though whether there is a

causal relationship between the two was not explicit from the

interview, many did mention that mathematics is ‘‘a kind of

language, a worldwide language’’ (HK4) and ‘‘a language,

but different from Chinese and English [language]’’ (HK2).

Not only that, mathematics is perceived as a subject that

is highly related with thinking, including logical thinking.

We had similar responses like ‘‘Mathematics is a thinking

method, in which you could follow’’ (HK11); ‘‘Mathe-

matics is a way to nurture student’s intelligence. Students

not only learn the mathematics knowledge, but also get an

opportunity to think’’ (HK8; HK12).

We also found responses that describe mathematics by its

content or terminologies concerned: ‘‘[Mathematics as a

discipline is] arithmetic, algebra, measurement and statis-

tics’’2 (HK3). We got similar responses like ‘‘[Mathematics

concerns] relationship between measurements, characteris-

tics of shapes, and many others’’ (HK1; HK12).

Mathematics is also seen as something universal, ‘‘not

affected by the regions [countries]’’ (HK4); it is something

abstract (HK1), ‘‘that can use simple number or symbol to

represent our ideas’’ (HK4). A teacher even viewed it as a

game: ‘‘We can manipulate numbers like playing games’’

(HK3). In fact, David Hilbert once said that ‘‘Mathematics

is a game played according to certain simple rules with

meaningless marks on paper’’ (Rose 1988). ‘‘Mathematics

is some rules discovered or created by human beings’’

(HK10) is yet another response. Another teacher saw the

artistic nature of mathematics. She said: ‘‘[as] I also teach

art [in school] and learned design [myself]. They [art and

design] also involve a lot of mathematics’’ (HK3).

When participants were confronted with the statement

‘‘Some people believe: A lot of things in mathematics must

simply be accepted as true and remembered and there

really isn’t any explanation for them’’, most of them made

no comments on whether mathematics is really a set of

truths. However, a majority of them did not agree that just

remembering such facts is a good way of learning. Most of

the responses argued that the process of learning is of vital

importance. Inevitably, their arguments were very much in

line with their perceptions of mathematics and their per-

ceptions of the value of learning mathematics. For instance,

mathematics facts should not be just remembered since

‘‘one of the goals of Mathematics is to train one’s think-

ing’’ (HK4). Also, ‘‘We expect the student to get the

knowledge from the process of learning. They are not only

memorizing the thing, but also discovering some new thing

or think about the mathematics concept’’ (HK8). Another

pointed out that ‘‘This [the arrival of mathematics facts] is

only the final step but students may encounter many

problems in the process. So, I think we cannot just ask

them to memorize the result but we should let them

experience’’ (HK2).

The above findings basically are in concordance with

those found in previous studies on student and teacher

conceptions of mathematics conducted in the Chinese

mainland and in Hong Kong. Previous research reviewed

that students associated mathematics with its terminologies

and content, and that mathematics was often perceived as a

set of rules. Wider aspects of mathematics such as visual

sense and decision making were only seen as tangential to

mathematics. In particular, mathematics is seen as a subject

of ‘‘calculable’’. Students also recognized mathematics as

closely related to thinking. We obtained similar results

with teachers from Hong Kong though teachers’ concep-

tions were more refined (Wong 2000, 2002; Wong, Marton,

Wong, & Lam 2002). As found in the present study,

seemingly contradictory themes emerged: mathematics is

both abstract and practical; it is artistic and logical.

2 These are actually the classification (learning dimensions) found in

the curriculum document.

Hong Kong teachers’ views of effective mathematics teaching and learning 303

123

Besides, it is a game and it involves thinking. However,

these facets of teacher’s conception of mathematics may

not necessarily be conflicting. Mathematics is often said to

be widely applicable since it has a high level of abstraction

and the ‘‘game’’ of logical reasoning is coupled with

inspection and induction in the course of mathematical

invention. As DeMorgan has said, ‘‘The moving power of

mathematical invention is not reasoning but imagination’’

(Moritz, 1914).

3.1.2 Abstract and abstraction in mathematics learning

When the participants were asked whether we should teach

students abstract thinking through the study of mathemat-

ics, most of them responded that this is one of the goals of

mathematics education (HK6; HK3). However, many of

them pointed out that it should be done gradually according

to students’ ability, developmental stage and age (HK2;

HK5; HK4). So it is not a matter of ‘‘should’’ or ‘‘should

not’’, but a matter of ‘‘when’’ and ‘‘how’’.

One of them illustrated with the topic of ‘‘speed’’: one

can progress from the idea of ‘‘fast and slow’’, which is

concrete and easy to understand, to ‘‘speed’’, which is an

abstract notion. Finally, we end up with the measurement

of speed (HK1). It was repeatedly pointed out that the

teaching of abstract thinking should be built upon experi-

ence with concrete objects:

For example, introducing a cube through 2D [ob-

jects]. First, you can give them a pile of papers. Then,

you increase the number of piles. Finally, you can get

a 3D object. In fact, many Primary 1 and 2 students

have already learned the idea of 3D objects, but I

would like to introduce the relationship between the

2D and 3D objects (HK4).

Another elaborated:

I think students have to see, try, touch, and then their

abstract thinking can be enhanced.... At present, I am

taking a [certain] course... on the teaching of algebra

and students think it is really abstract. They [the

lecturers] are using real objects... and the original

question [can be transformed into one for what is] left

are just blanks for the student to fill in.... Or this can

even be a story [the abstract algebra question is

presented as a story).... Those which are abstract are

made concrete as they have been embedded in the

story line (HK5).

By ‘‘establishing [such] a habit [of linking the abstract with

the concrete], they [the students] could learn abstract

thinking’’ (HK1). The notion of ‘‘concrete’’ is, however,

not confined to real objects. Less abstract concepts can help

building up more abstract ones (HK6). Abstract notions

should be developed using students’ prior knowledge as the

foundation (HK7).

While some reflected that primary level (lower primary

in particular) is too early for abstract notions, some others

felt that abstraction, or the teaching of abstract thinking, is

a gradual path (though the two views may not be con-

flicting). For instance, one of the participants opined that:

It [the teaching of abstract thinking] is one of the

[curriculum] aims and it is true for higher grade

levels. But I am afraid that this may not be the case in

lower grade levels. It depends on the age group. We

have to consider what the students should learn at

each age group.... I think it begins from ages of 12 to

13. It is sure that there was already some training

since they were 10. However, if it is for them to learn

more, get more and imagine more, I think the [suit-

able] time is when they are in Secondary 2 or Sec-

ondary 33 (HK2).

Others also said that there are fewer (or scarcely any) ab-

stract mathematics in junior classes and there are more in

senior ones (HK7; HK3). Another held the view that ‘‘I

think most of the things they learnt are concrete and in-

volve daily examples’’ (HK6). Yet some others even

mentioned that:

According to educational psychology, if one has not

reached certain stage, early learning may not be good

to him/her and she/he would only memorize [the facts

involved]. As they grow older, we start to teach some

abstract idea. In fact, even in the senior classes, not

all the students can understand (HK4; Betty).

Not only that, concrete objects are often deliberately used

to scale down abstract notions, if there are, in primary

mathematics: ‘‘We make use of some concrete matter to

explain some abstract Mathematics’’; in a sense, not much

abstract notions would the student come across in primary

level (HK11; HK5).

Though many of the teachers interviewed held the view

that there is hardly anything abstract at junior levels and we

should not start the teaching of abstract thinking too early,

they saw the need of a gradual development of it:

I believe there are levels of abstract mind [thinking].

There may be some basic level of abstract mind at the

junior level. I think the level of abstract mind will rise

as the children grow (HK8).

When they are promoted to a higher form, they get

more experience, they will then develop a mathe-

matical sense in their minds and they can then accept

3 Secondary 2 is equivalent to Grade 8.

304 N.-Y. Wong

123

the abstract things more easily, they can think of that

[understand] (HK12).

Nevertheless, the abstract and the concrete are sometimes

relative notions:

I think my explanation is concrete. The students may

not think so, although they may not use the word

‘‘abstract’’. For instance if I say ‘‘folding from left to

right’’ without pointing the picture and its motion, to

the students, probably almost all of them would be

unable to distinguish left and right (HK11)!

For instance, one of the teachers said ‘‘[what we] have to

teach is that 1 km equals to 1,000 m, but it’s hard to make

them grasp the idea of 1,000 m.... One kilometre is very

concrete and down to earth, but it is very abstract in the

mind of children’’ (HK9).

Most of the teachers in the study agreed that abstract

thinking/abstraction is one of the central goals of mathe-

matics learning. Again this coincides with what was found

in previous studies (Wong 2000, 2002; Wong, Marton et al.

2002). Yet when such an abstraction should be started

varies among these informants. However, most of them see

that the road of abstraction is a long path and mathemati-

zation is basically a path from the concrete to the abstract.

3.2 Teachers’ view about mathematics learning

During the interviews, participants expressed their con-

ceptions of mathematics understanding. The roles of

memorisation, practices and concrete experiences were also

discussed. As for mathematics understanding, knowing how

and the ability to transferred knowledge to a new area are

some of the criteria. However, memorisation was not seen

as necessarily an obstacle to the enhancement of under-

standing. In fact, rote-memorisation and memorisation with

understanding (keep something in mind) could refer to quite

different things. Yet it was generally agreed that if one

possesses a deep understanding, rote-memorisation is not

necessary. Practices are valued among participants. In

particular, to them, practices help memorisation and fa-

miliarisation. Certainly, both quality and quantity count.

Likewise, though concrete experience and manipulatives

could be helpful, it was pointed out that the effect really

depends if they are used wisely.

3.2.1 What is mathematics understanding?

The participants almost unanimously agreed that under-

standing is of utmost importance. It helps recalling the

facts (HK9) and problems can be worked out more

smoothly. Understanding is not something difficult but is in

fact a source of interest (HK7). It builds a firm foundation

for future studies so that ‘‘Later when s/he [the student]

faces some more complicated cases, when they are in a

higher grade levels, s/he can still manage’’ (HK5).

There were various criteria to judge whether students

understand. Students can explain what they have learned

and know why is a major one: ‘‘The most traditional one

[to test if one understands] in school is test and examina-

tion. In fact, we can know whether they understand or not

through their oral presentations’’ (HK4; HK8). Creating a

variety of contexts and requiring the students to explain is

one good way:

We should create more situations and ask students to

explain, and we can see whether they can do it. For

instance, if we have given them the area and ask them

for the height, we can then ask them to explain why

we can find the height in this case (HK2; HK12;

HK10; HK3).

Though getting the answer correctly is still a dominating

requirement (HK6; HK11), certainly teachers are looking

for more to see if students really understand:

Finishing some exercises is one of the determinants,

but according to my experience, that is not sufficient

to say that they understood. That they could get right

answer does not mean they understood.... My stan-

dard is that they will not have great difficulties in

doing homework. Or they could accurately express

their learning that I have expected, during class dis-

cussion (HK11).

Knowing the rationale of every step certainly involves

meta-cognition among the students:

If that is addition and subtraction of fraction, s/he [the

student] should be able to tell why we have to convert

the fractions [involved] into the common denomina-

tor, and [in solving word problems] s/he should be

able to explain why that is a problem of addition and

subtraction of fraction, but not any other topics in

Mathematics (HK1).

The ability of transferring one’s knowledge and skill to

other problems is another important indication of genuine

understanding (HK6; HK3). This can be tested by ‘‘using

questions that they [the students] don’t frequently come

across’’ (HK10). ‘‘If a student understands one thing, s/he

should know the different sides of it’’ (HK1), which is very

much in line with Confucius’ famous saying of

‘‘responding to the other three corner when only one corner

is shown’’. Another participant gave a yet clearer expla-

nation:

For understanding, I think the first is they can accept

the rule as a fact.... Second, when the rule appears in


123

another format, s/he can still think in the reverse

manner.4 And I think this is understanding. On the

other hand, if s/he can only apply the rule in one way

or use it in just one direction, I do not think s/he

[really] understands (HK2).

It is generally believed that the use of daily life examples

and various activities would enhance understanding

(HK10; HK3; HK6). Here is a concrete example:

For measuring capacity, we can do some practical

activities. For instance, students usually confuse

about 1 L with 1 mL. When we drink, do we drink

200 L, or 200 mL? We can just take one real drink

out [and everything becomes clear].... To teach the

concept that [vessels of] different shapes can have the

same capacity, I would buy different drinks. Then I

asked them to estimate the capacity of the drinks. But

some of them have the same capacity. At first, they

may think the capacity of the taller drink is larger.

But they finally find that taller in shape may not mean

larger in capacity, since it may be smaller in other

dimensions.... Therefore, for some topics, we should

make teaching as concrete as we can (HK11).

Other means include letting students know the thinking

process, guiding students to observe (HK8), and question-

ing (HK9). However, it was also pointed out that a good

foundation is prerequisite to understanding (HK11). The

issue may be related to the issue of ‘‘content’’ (or

‘‘product’’) and ‘‘process abilities’’ (Wong, Han, & Lee

2004). Preparatory knowledge is important in understand-

ing subsequent contents:

For example, in the teaching of fractions, they [the

students] must understand integer. Decimal comes

from fraction and there are many levels [of com-

plexity of these numbers]. Another example is that

triangle comes from rectangle or parallelogram

(HK7).

Teaching mathematics for understanding is one of the

hallmarks of current mathematics education. While pre-

vious research on the conception of mathematical under-

standing among Hong Kong students revealed that

‘‘getting the correct answer’’ is still the most salient

indicator of understanding in the mind of the students

(Wong & Watkins 2001), teachers in the present study

possess a wider perspective of mathematics understanding.

Knowing why, transferal of knowledge, and flexible use

are stressed. In fact, though students look high upon

‘‘getting the correct answer’’, they do realize that under-

standing mathematics may mean the ability to solve

problems, the knowledge of underlying principles, the

clarification of concepts, and the flexible use of formulas

(Wong & Watkins 2001). Not only that, even if one is

confined to getting the correct answer, understanding is the

‘‘royal path’’. This is also realized by the students as

found in previous studies (Wong, Lam, Leung, Mok, &

Wong 1999). Probably this cannot be attained through

rote-learning and it was repeatedly found that CHC5 stu-

dents have stronger preference and deeper approaches to

learning, which is the opposite to rote learning, than do

Western students (Wong 2004).

3.2.2 The role of memorization in mathematics learning

Probably to the astonishment of many, memorization was

not very highly regarded by the participants (e.g., HK3).

One of them even said that ‘‘someone can also learn

Mathematics even they have poor memory’’ (HK6). It is

less important in the learning of languages (HK7). Mem-

orization does not help students’ learning if they don’t

understand (HK3). In brief, memorization may have some

effect on mathematics learning, but it is not an important

component (HK4; HK12). One of the participants said that

‘‘I will not ask students to memorize anything’’ (HK12),

while another pointed out that we simply don’t have too

many things that need to be memorized in the curriculum

‘‘[only up to] 20–30%, I guess’’ (HK1). To these teachers,

rote memorization is just a ‘‘better-than-nothing’’ sub-

stitute when understanding is not readily at hand (HK2).

Smart students need only to memorize a few facts and

those who do not understand have to rote-memorize

everything. ‘‘If there is something we really cannot

understand, we should memorize it first as to tackle the

examination’’ (HK9). Here are two examples that help

explain their opinions:

When a student can get hold of the meaning, even

though s/he does not memorize the formula, s/he will

have that in mind.... Take an example, when we are

calculating the area of parallelogram and triangle, it

does not matter if s/he cannot remember the formula,

if s/he knows where the formula come from [was

derived... since] s/he actually experienced how the

area of parallelograms was derived from the area of

rectangles; though s/he could not get the formula

directly and quickly, but s/he really understands

(HK12).

4 Just like what the same participant mentioned: given the area (of a

rectangle, together with the width), one is able to find the height,

though the original formula is to find the area with the width and

height given.

5 CHC is Confucian Heritage Culture, and generally refers to regions

like the Chinese mainland China, Taiwan, Hong Kong, Japan, and

Korea.

306 N.-Y. Wong

123

If students understand the meaning of speed, it is

not necessary to have the three formulas,6 but [what

one needs is] just applying the concept.... Why cannot

we just use one of the formulas and derive the other

two from it (HK9)?

So what was indicated in the second quotation is in fact a

flexible use of the formulas.

Though many teachers opined that memorization should

not play a central role, it is yet indispensable, especially

when one needs to work out problems quickly (HK8). After

all, deriving the formulas from conceptual understanding

each time is too slow, for instance the identification of

prime numbers within 100 (HK6; HK7). This could be in

line with what Kerkman and Siegel (1997) spoke of ‘‘fastest

strategy’’ and ‘‘backup strategy’’. Inevitably, the multipli-

cation table is a prototype of the need for memorization.

‘‘Something like multiplication table is something that we

must memorizes.... We cannot just explain to them that 2

times 6 is the sum of six 2’s adding together by not asking

them to memorize’’ (HK2; HK1). As the speed of com-

pleting mathematics problems is one of the major concern

in the Hong Kong mathematics classroom, memorization

(and quick retrieval) of formulas become important:

For example, sometimes mathematics formulas or

rules are involved, like square numbers. If you list out

the square of the number within 20 [certainly] they

understand what a square number is. However, if the

student can remember all these square numbers, the

speed will be increased when they were asked to

calculated questions involving square number. Apart

from motivation, confidence is also important in

mathematics learning. If they can remember the

things they learned, their confidence in calculation

will also be increased (HK8).

In that sense, memorization could be seen as more

important for the abler students when there is a high

expectation that they should solve problems quickly and in

automation. The fault does not lie in memorization, but in

memorization without understanding (HK1). It is also

perceived that the need is stronger at junior levels in which

conceptual understanding cannot go too far (HK11).

There had been some opinions that memorization should

go after understanding (HK8); it is a kind of consolidation

(HK10). As one said: ‘‘There is something you must

memorize after you understand. For instance, I think it is

not a must to memorize the formulas as long as you get the

concept and you can write it out [again]’’ (HK5). A teacher

made it more explicit: ‘‘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).

Yet there are some who suggested a route of the re-

versed direction: remembering the facts first, applying with

them, and through such applications achieving under-

standing. It would cultivate students’ interest (for deeper

understanding) when students find themselves successful in

solving problem through these memorized facts (HK11).

Before discussing teachers’ conceptions of memoriza-

tion and its relationship with understanding, we would like

to draw the attention that there could be at least two facets

of memorization: keeping something in mind (being

memorized) and memorizing something, which includes

recitation. Obviously, while the latter refers to rote-mem-

orization, the former is closely related to understanding,

without which something cannot be efficiently stored and

retrieved. The fact that students regarded understanding

and memorization as something not segregated but under-

standing helps ‘‘memorizing’’ facts better was actually

revealed in previous studies (Wong & Watkins 2001).

Research also supported the hypothesis that excellent

academic performance of CHC learners may be due to a

synthesis of memorizing and understanding which is not

commonly found in Western students (Dahlin & Watkins

2000; Marton, Tse, & dall’Alba 1996; Marton, Watkins, &

Tang 1997). We obtained similar findings among teachers

in the present study. Once one understands, there are not

too many things one needs to (take the effort to) memorize

(they are automatically kept in the mind: ‘‘memorized’’!).

There had been discussions among our informants on

whether one should memorize rules first and then try to

understand the principals behind later or vise versa. If one

looks at the dual nature of mathematics (Sfard 1991), the

two could be enhanced reciprocally: one starts off with a

brief understanding, and through practices and other

means, one begins to memorize the rules behind and arrive

at a deeper understanding; with such a deeper under-

standing, the rules are more firmly memorized. The situa-

tion is made more complicated when student performance

is stressed (in high-stake assessments) in which automation

and speed are required. That could be the major cause of

requesting students to memorize facts and rules first (so

that one can implement such rules in the tests), leaving

aside genuine understanding.

3.2.3 The role of practices in learning mathematics

The importance of practices varies among the participants.

Some took it as something not only important but indis-

pensable. ‘‘Students have to do exercises’’ (HK10) consti-

tutes the most straightforward response (also HK8; HK5;

HK2). ‘‘Practice makes perfect’’ (HK3) was also quoted as a

ground for practices. However, there are some who thought

6 Speed = distance/time; distance = speed · time; time = distance/

speed.


123

that they are ‘‘important, but not very important’’ (HK7).

Another said: ‘‘There is a need for the exercises ... [but] I

don’t have a very high regard on them’’ (HK4). Some also

put forth the argument that one only needs enough practices

for understanding: ‘‘They do not 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 do not need to do a

lot’’ (HK7). Another even said: ‘‘If someone understand, but

s/he still do exercises that are at the same level as his/her

level, I think s/he is wasting his/her time and s/he should do

some challenging exercises’’ (HK3). Certainly it is not easy

to judge how much is enough and what is too much. For

instance, some teacher asked for fluency and accuracy in

problem solving and not just conceptual understanding:

‘‘The criterion is that the student has to calculate accurately

and fast’’ (HK10), and ‘‘too much’’ refers to monotonous

mechanical calculation rather than actual quantity. This is

clear from the following response:

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 worst case

is that students do not think over the questions after

doing massive exercises. I do not agree with

mechanical training (HK5).

One of them pointed out that the role of exercises may vary

with topics: ‘‘[exercises are essential] in the topics which

involve arithmetic more. I think it would be relatively less

important in topics like statistics’’ (HK2). It varies with

grade levels too: ‘‘For Primary One or Two students, I

suggest a number of 8–10 exercises are already enough’’

(HK9).

Despite the difference in opinion on the importance of

exercises, a number of learning effects of doing exercises

was put forth. Helping understanding, memorization,

familiarization and consolidation of what is learned

(‘‘deepening the imprint’’, so to speak) are some (HK7;

HK6; HK8; HK4; HK12; HK10; HK2). It can also serve as

interactive and diagnostic means, ‘‘I will teach and let the

students understand. Then, test the students how much they

know and want to consolidate the students’ learning by

asking them to do exercises’’ (HK3; HK2). Exercises can

also serve dual purposes: ‘‘For the average students, we

can give them more drilling exercise, hoping they can

familiarize themselves [with the underlying concepts or

skills]. For smart students, we train them how to avoid

making mistakes’’ (HK11).

The above results reveal that teachers’ views on mem-

orization and on practices have a lot in common. As for

memorization, practices are indispensable but too much

could hamper students’ interest in learning. However, well-

designed practices enhance understanding. In fact, problem

sets with systematic variations has a long tradition in the

Chinese mainland (Sun, Wong, & Lam 2005; Wong 2007;

Wong, Lam, & Sun 2006). Such a curriculum design (based

on problem sets with variations) offers repetition with

variations and has a much deeper pedagogical meaning than

mere drillings.

3.2.4 How concrete experience and manipulatives help

in mathematics learning

Most of the participants took high regard on student

activities and the use of manipulates: ‘‘After they experi-

ence the process, most of them can learn’’ (HK3); it makes

learning student-centered (HK4). ‘‘Learning by doing’’ is

deep-rooted in the teachers’ mind: ‘‘Yes, students have to

do, and [learning] should be actually done by themselves

[and not just be told by the teacher]’’ (HK9). Another said:

‘‘For example when teaching pyramid, it is impossible that

you only talk without showing them how to make a pyra-

mid’’ (HK6). Making learning pleasurable is another rea-

son for use: ‘‘Moreover, they may find it interesting [We

let students realize that]. There is not only arithmetic in the

lesson. I think that having something interesting in class

would make students happy’’ (HK9). Some of them actu-

ally related some activities they conducted:

To demonstrate, I made a 10-m paper strip. How long

is 10 km? May be they even do not know how long is

10 m. I stretched the paper strip and place around the

frame of the classroom. May be it surrounded half of

the classroom. Then I used this to teach them how

long 1 km is. I asked them how many 10-m paper

strips can add up to 1 km? Also, I asked them to think

if there is a 1,000-m paper strip, how many class-

rooms can it surround? Let them imagine how long

1 km is [by] just sitting inside their classroom (HK9).

However, there are arguments that the use depends on

grade-levels and time. Yet there were conflicting views on

whether lower or upper grade levels need more activities.

On the one hand, some teachers thought that the use is

more essential in junior grade levels: ‘‘I agree we should

provide lower primary school students with more of these

activities’’ (HK2; HK7). Some others opined that these

activities are most useful when a topic is first introduced:

Yes and this also depends on what grade level they

[students] are in.... Ah, no, it is not the grade level but

whether it is their first time to learn the topic. For

example, they learn the place value in Primary One; it

is meaningless for you to tell them which is the tenth

value, etc.... However, it would be much better if you

teach them by asking them to pack ten things

together. It is better for them to learn through expe-

rience (HK10).

308 N.-Y. Wong

123

Another elaborated:

I think for those sell-and-buy practices, having

experience is better. For example, they can have the

experience of giving changes.... I think it is those

involving more calculations. Students gain experi-

ence after they practiced with the calculations (HK5).

On the other hand, some said that the use is more important

in upper grade levels:

Very important in the senior classes. In general, it is

especially important when talking about the concepts,

like 3D objects.... But the most important thing is

how to incorporate these teaching aids into the

teachers plan and teaching strategies (HK8; HK12).

Some other said that it really depends on topics:

‘‘Teaching aids have different significance in different

topics and concepts. They are very important in some

topics, but in some other topics, they are not useful at

all’’ (HK1). For some topics like algebraic symbols, it is

not easy to have concrete representations (HK6). How-

ever, we got counter-arguments: just because these topics

are so abstract, we need to create manipulatives so that

students can comprehend better (HK7). Though many of

them agreed that activities are helpful, whether it is

indispensable is another matter: ‘‘Some people say there

should be many activities and adequate teaching aid, but

I think it’s not a must’’ (HK1). There are some others

who expressed the exogenous constraints like the exten-

sive time in preparation and the limited time in each

class period.

As pointed out earlier, since teachers see learning

mathematics as a path that goes from the concrete to the

abstract, concrete objects or manipulatives can offer a lot

of learning opportunities, especially at the elementary

school level. With the curriculum reform that takes place in

recent years, hands-on experience is repeatedly stressed,

though the notion of ‘‘activity approach’’ was already

introduced into the Hong Kong mathematics curriculum

early in the 1960s (Tang, Wong, Fok, Ngan, & Wong

2006).

3.3 Teachers’ view about the teacher and teaching

The characteristics of an effective mathematics lesson and

of an effective mathematics teacher were expressed.

Though many of the participants realize that there is no

fixed rule for good practices, they pointed to some indi-

cators like having the goal achieved, understanding the

students, enhancing student participation and provoking

thinking among the students. To achieve these, good

preparation, basic teaching skills and good relationship

with the students are prerequisite.

3.3.1 Characteristics of an effective mathematics lesson

The responses ranged from ‘‘outcome’’ to ‘‘process’’,

from ‘‘objective’’ to ‘‘interest’’. First of all, as for meeting

the pre-set teaching objectives (curriculum goals), at least

some of the objectives were deemed importance as said by

one of the teachers: ‘‘May be we can only achieve one out

of the three teaching objectives, but I still think this is a

good lesson’’ (HK3). Five other teachers regarded the

meeting of teaching objectives as one of the criteria of a

good lesson. It seems that, by doing so, they know they

have performed their (teaching) duties.

Certainly, students’ demonstration of understanding is a

key indicator of whether the teaching objectives are

achieved. The same teacher said: ‘‘If you can achieve all

the teaching goals and are able to make the students

understand, then this scenario is the best’’ (HK3). Another

said: ‘‘Students should get some new knowledge after your

lesson’’ (HK6). In such a case, ‘‘We [can] see that stu-

dent’s response change from not understanding to having

understood’’ (HK1). Yet another commented:

For me, a good lesson is one in which students can

master the teaching aims, the theme [of the lesson],...

students should feel that they can grasp what I have

taught in that lesson. If the students are able to do

that, then I think that lesson is successful (HK5).

That understanding could be actualized by performance in

class and yet that performance may also mean a long-term

one, as argued by one of the respondents: ‘‘If the students

have a deep understanding, they can handle and tackle

[mathematics problems] better, especially when they were

promoted to higher grade levels’’ (HK12).

Students’ participation is another key area to judge an

effective lesson. Students should be deeply engaged and

pay attention (HK8). ‘‘It [a mathematics lesson] is suc-

cessful if the students participate fully in the lesson and the

teaching aim is also achieved’’ (HK10). In such a case,

‘‘they [the students] should realize they have learned a lot

and feel very happy’’ (HK6). From these responses, we see

that, in the view of the respondents, participation and

involvement are the keys for understanding (and for

achieving the teaching objectives) and also the source of

satisfaction in learning. Interest and academic success also

demonstrate a reciprocal relationship. In fact, ‘‘interest’’

was repeatedly mentioned. It is both a means and an end to

a good mathematics lesson (e.g., HK3; HK5). For instance,

one participant talked about the criteria of good lessons:

‘‘[It is one that] I can achieve the teaching goals. Also, I

can stimulate the students and arouse their interest. In

addition, students feel this is an interesting lesson and

enjoy this lesson. Then, I think this is a successful lesson’’

(HK4). Another said: ‘‘If students learned and learned


123

interestingly, it is considered to be good’’ (HK1). Not only

that, ‘‘once the student has interest, all else become easy’’

(HK4).

Very much in line with teachers’ conception that

mathematics involves thinking, whether student thinking is

provoked is another major criterion for a good mathematics

lesson. It should be one in which ‘‘students will think

continuously’’ (HK8) and there are ‘‘some good questions

to stimulate the students to think’’ (HK7). Not only that,

‘‘In the class, they [the students] can generate some new

idea in their mind and after discussion, the problem can

[thus] be solved’’ (HK7).

However, there is no fixed script of a good lesson. It all

depends on the situation. ‘‘So I have to change my method

[to test which is appropriate for that particular group of

students]’’ (HK12). One can test by observing students’

responses: ‘‘If students can answer your questions well,

then I know they understand’’ and ‘‘sometimes I will ask

students to do rather than answering my questions’’ (HK5).

Like many Asian countries, there is a long tradition that

Hong Kong follows a central curriculum and it is cus-

tomary for teachers to follow the curriculum closely,

especially when the central curriculum leads to public

examinations, which have high regard among the public

(Wong, Han, & Lee 2004). Thus, one of the prominent

factors of effective mathematics lesson, in the eyes of the

teachers, is to have the objectives of the lesson (which is

closely coherent with the curriculum objectives) attained.

An effective lesson is also one that students understood the

teaching contents. Student participation and interest are

also looked high upon. This resembles of what was found

in previous studies on students, in which they look for

teaching liveliness, student involvement, and rapport in the

mathematics classroom (Wong 1993). Again, similar to the

students’ preference, an effective mathematics lesson is

one in which thought-provoking tasks and exercises are

provided (Wong, Lam, Leung, et al. 1999). Obviously,

deep learning is looked for. It is interesting to note that the

teachers reflected that there is ‘‘no fixed rule’’ in con-

ducting an effective mathematics lesson, which is probably

a common belief in Chinese culture (Wong 1998).

3.3.2 What lead to an effective mathematics lesson?

It seems that a teacher-led ‘‘run-down’’ is a typical script

in the CHC mathematics classroom, particularly in Hong

Kong. Having clear objectives is the prerequisite and

having the lessons well-prepared is thus indispensable

(HK4; HK1; HK2). ‘‘One should think about what one is

going to teach before a lesson, but should not pack too

many objectives into a single lesson’’ (HK5). So, ‘‘another

important point is the flow of the lesson [well-designed]’’

(HK5). As another teacher put it: ‘‘I think an effective

lesson usually has good preparation, having thought about

the flow of it, and also with my teaching pointing to chil-

dren’s weaknesses’’ (HK11).

Having everything well-prepared, the contents should be

systematically delivered and clearly explained, as said by

one of the participants: ‘‘Our teaching plan must start from

the basic idea, then proceed to the difficult ones’’ (HK3).

Teaching skills, as a mode of delivery, is also seen as of

vital importance (HK12). For instance, one should teach

the contents step by step (HK5) and ‘‘[you] should have a

little pause after you have mentioned a key point’’ (HK5).

One should also make a good start because: ‘‘For the

students, the first impression is very important. If they get

the wrong message at the beginning, they need long time to

correct’’ (HK9). A right control of time (Effective means

finishing teaching in presumed time) (HK11) and the

teaching pace (HK1) are important too, the latter of which

will be discussed in the next section.

Very much in line with Ausubel’s (1963, 1968) notion

of ‘‘teacher-led yet student-centred’’, though teachers

stressed the importance of a well-designed lesson with its

objectives clearly laid down, the awareness of students’

situation and the consideration of their responses were once

again emphasized. The use of pauses (as raised in the last

paragraph) was further elaborated: ‘‘For those teachers

with no much experience, I would advise them to pause to

see students’ responses’’ (HK5). In general, students’ re-

sponse should be considered and the pace of teaching

should be so adjusted (HK2). The particular participant

(HK2) made it clear:

The teaching aim of that lesson should at least be

clear. The teacher should teach according to what the

students should learn. The teacher should not let other

things override the main theme. Secondly, it is the

explanation of the teacher. The teacher should ex-

plain clearly, not to use ambiguous words like ‘‘this’’

and ‘‘that’’, and the students do not know what

‘‘this’’ and ‘‘that’’ are. 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 consid-

ering students’ response (HK2).

A good understanding of students is thus crucial: ‘‘They

should know students’ need and inclination’’ (HK12; HK6;

HK8). ‘‘The teacher should know what students need when

they learn Mathematics’’ (HK9) and one should consider

‘‘the difficulties that may be encountered by students’’

when planning the lesson (HK1). One of the participants

elaborated: ‘‘I know which common problems would be

found by most students when I become more and more

experienced. It helps me to take preventive measures and I

know how to remind students in the next academic year’’

310 N.-Y. Wong

123

(HK9). ‘‘One should understand students’ learning differ-

ences and gear the teaching pace to them’’ (HK5; HK4;

HK8). A good teacher-student relationship through which

one can understand well the characteristics of students is

extremely useful to gear one’s teaching to their needs

(HK1). To this end, ‘‘the teacher should be patient, willing

to listen to students’ learning difficulties and help solve

them’’ (HK5; HK3). ‘‘S/he should encourage the students,

especially those who lag behind’’ (HK8).

As students’ participation and interest were seen to be so

important for an effective mathematics lesson, naturally,

cultivation of interest becomes a prominent element lead-

ing to an effective lesson. ‘‘Teaching aids, games, real-life

examples, introducing them various activities and outside

readers’’ (HK8) were various means suggested by some of

the participants. The connection with real-life situations

was seen as a major source of interest (HK5). The use of

cartoon is yet another means: ‘‘For example I framed the

question with a cartoon character, Digimon [Digital Mon-

sters], the kids like the questions more’’ (HK6). Connect-

ing to other concepts within the syllabus would arouse

students’ interest too: ‘‘They would feel surprised and this

would initiate their thinking [too]’’ (HK7). However, one

of the participants pointed out that teacher-student rela-

tionship is the key to all these: ‘‘First, it is the students’

impressions of the teacher [that counts]. This affects their

interests towards learning.... If students are interested in the

teacher, then no matter what teaching method the teacher

uses, students will all be interested to learn’’ (HK12). This

echoes the findings of previous studies that most students

identify a mathematics class with the mathematics teacher

(Wong 1993). This will be discussed further in the next

section.

Whether a lesson could lead to students’ involvement in

thinking is another criterion of effective mathematics les-

sons. Inevitably, teachers’ skill in initiating that to happen

is also a major factor leading to such an effective lesson.

This is attained, as seen by a participant, by good ques-

tioning skills ‘‘[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 about before’’ (HK1).

The qualities of a good mathematics teacher as per-

ceived by students were repeatedly investigated in previous

studies in Hong Kong. In one of them, it was found that:

A good mathematics teacher as perceived by the

students is one who explains clearly, shows concern

towards the students, treats them as friends, makes

sure that they understand, teaches in a lively way, is

conscientious, well-prepared and answers students’

queries.... A good mathematics teacher should also

provide more exercises and should generate a lively

atmosphere but keep good order; and a good learning

environment is one which is not boring, quiet, with

classmates engaged in learning, where order is ob-

served but discussion with classmates after lesson is

possible (Wong 1993, p. 304).

Similar findings were obtained in Wong, Lam, Leung, et al.

(1999) and are echoed among the teachers as found in the

present study. Clear explanation via a well-prepared and

organized lesson comes first. On the other hand, one needs

to know the students so that one can gear one’s teaching

according to the needs of students. Inevitably, this demands

good teaching skills. If we refer to the students’ preference,

they too prefer the teacher to treat them as friends and

shows concern toward the students. Also, it is a high de-

mand on teaching skills for a teacher who can generate a

lively atmosphere but at the same time keeping good order,

and can get the students involved yet keeping them quiet.

From the above discussions, one should get a scenario of

the typical CHC classroom: teacher-led yet student-cen-

tered (Leung 2004). While Huang and Leung (2006) de-

scribed the CHC mathematics classroom as having

‘‘teacher dominance and student active engagement’’, Gao

and Watkins (2001) put it more precisely as ‘‘learning

centred’’. In fact, in the eyes of the students, teachers are

seen as the crucial factor of the mathematics classroom; the

mathematics class is often identified by the teacher (Wong

1993).

3.3.3 What makes a good teacher

Naturally, fluency in mathematics is repeatedly taken as a

criterion of a good mathematics teacher. Five of the par-

ticipants explicitly mentioned that. We heard similar fac-

tors like mathematical sensitivity (HK12) and confidence

in solving mathematics problems (HK8). A considerably

number of them even raised the concern that the teachers

themselves should have an interest in mathematics (HK12;

HK3; HK4), which is more important than fluency (HK9).

Does it reflect that many mathematics teachers whom they

met are not keen in mathematics? This could be related to

the particular situation in Hong Kong that a considerable

portion of primary mathematics teachers are not brought up

with a strong mathematics background. Mathematics

teaching could just be considered as a ‘‘secondary duty’’ in

some schools (Tang et al. 2006). However, some raised the

point that fluency in mathematics is just one factor (HK9);

deep understanding of the curriculum content (HK1) and of

the curriculum structure (HK5) are also mentioned by

others.

Besides, a good mathematics teacher, as perceived by

the teachers themselves, is one who knows how to use


123

appropriate/effective teaching methods (HK5; HK8). The

mathematics teacher should make clear and systematic

explanation, using concrete example to explain abstract

concepts (HK3), and let students progress gradually in the

course of learning, ‘‘from the simple to the hard’’ (HK8).

The teacher should be able to conduct questioning that

provokes thinking (HK9; HK6), rather than presenting

everything to the students (HK6). However, it does not

only mean conforming to a ‘‘best way’’ of teaching, so to

speak, but also processing a variety of teaching methods

(HK4). S/he should be creative in teaching too (HK6;

HK8).

The participants in this study talked a lot on the personal

caliber a good mathematics teacher should possess. S/he is

portrayed as one who loves teaching (HK9), who like to

explore mathematics problems (HK12; HK8), who pos-

sesses a loving heart (HK8), and who has a sense of humor

(HK12). S/he should be objective, fair, looking into prob-

lems from multiple perspectives (HK3); be flexible in

handling problems (HK3), and accept different points of

views from the students (HK3). Also, s/he is attractive

to students, in terms of both physical appearance and

mathematical fluency (HK6). As for his/her relationship

with students, s/he should have a sharp mind but can tol-

erate students’ failures at the same time (HK9, HK6); s/he

should be strict to students, yet reasonable (HK6), and

impose high expectations on the students (HK8). If we look

other factors of preferred classroom environment as per-

ceived by the students, the above portrayal is not at all

surprising.

The participated teachers repeatedly pointed to the cri-

terion that a good mathematics teacher should be one who

always strives to improve oneself. S/he should reflect

preferably after every lesson (HK4), willing to learn

(HK12), adventurous, and receptive to new things (HK3;

HK9). S/he is seen to be one who always takes part in

continuous learning, through reading books and journals,

attending seminar and courses, and surfing through the

Internet (HK5; HK4; HK9). S/he is willing to take part in

collegiate exchange within and outside the school (HK5;

HK4; HK8), and knows the current trend of mathematics

(HK5; HK8) and of mathematics teaching (HK4).

With the ideal effective mathematics lesson we por-

trayed above from our findings, the image of a good

mathematics teacher is more than obvious. Strong profes-

sional knowledge, including subject and pedagogical

knowledge, is fundamental. The personality of the teacher

is another key element since it is expected that teachers

should show personal concern toward the students. Not

only that, setting a role model so as to conduct moral

education has always been taken as one of the duties of the

teacher in CHC regions (Leu & Wu 2006; Ho 2001).

4 Discussions

The above results reveal that, to our participants, mathe-

matics was generally regarded as a subject that is practical,

logical, useful and involves thinking. Abstract thinking is

regarded as one of the goals of mathematics learning and

teachers should build a path so that students can proceed

from the concrete to the abstract. Learning and teaching for

understanding is treasured but the participants find a role of

memorisation, practices and concrete experiences in the

enhancement of mathematics understanding rather than

seeing them as something solely good or bad. Good

teaching practices are exactly precisely those that make full

use of these means and let leaning for understanding to

happen. Though many of the participants realize that there

is no fixed rule to this, good practices, basic teaching skills

and good relationship with the students are major indica-

tors.

This echo with the results in previous research studies

on students’ conception of mathematics, in which students,

too, find mathematics as a subject of ‘‘calculables’’, in-

volves thinking and is useful (Wong 2002). This is not at

all surprising since they were all brought up from the same

culture! Very much in line with the students’ conceptions,

teachers see the ‘‘dual nature’’ of mathematics—it is useful

and involves thinking. Though the conception that mathe-

matics as a subject of calculable is not that salient as

among the students, some teachers did identify mathe-

matics by its content and terminologies (Wong 2002).

Being abstract is not only seen as the nature of mathe-

matics, abstraction is also perceived as one of the major

goals of learning mathematics. Thus the path of mathe-

matization of going from the concrete to the abstract is the

course of mathematics learning, which could be continued

from the elementary up to the secondary levels. Though

there are not too many abstract notions at the elementary

level, the route of abstraction has already been started. To

this end, the use of concrete objects has its particular role

as it helps grasping the abstract.

Learning for understanding is unanimously agreed. To

most of the informants, understanding means flexible use

of rules. This is in concordance with previous research on

Hong Kong students. There had been discussions on

whether understanding or memorization should go first.

Yet we can look at the issue from another viewpoint that

there could be at least two aspects of memorization: rote

memorization (recitation, so to speak) and having some-

thing memorized (kept in mind). If something (whether it is

a rule or a concept) is being remembered with rich con-

nections with other rules or concepts via a variation of

learning activities, in a sense, isn’t it precisely that some-

thing is being understood (Hiebert & Carpenter 1992)?

312 N.-Y. Wong

123

One of the possibilities is the careful arrangement of

tasks or practices with variation systematically introduced.

In this way, practices may not necessarily be seen as a

bunch of drilling exercises or meaningless repetitions but a

scaffolding that leads from the basics to higher-order

thinking skills (Wong, Lam, & Sun 2006). In fact, both the

teachers and students (in previous research) reflected the

importance of thought-provoking exercises. Of course, too

much emphasis on the speed of calculation (especially as

required by high-stake assessments) makes it difficult to

turn the above into reality.

The above discussions have painted a picture that the

teacher needs to be the central figure of the mathematics

classroom. It is the teacher who designs such learning

activities, which are the key to understanding. The need for

the teacher to lead the classroom and its learning activities

was reflected in the present research and in previous

studies. ‘‘Teacher-led, yet student-centred’’ (or ‘‘learning-

centred’’) is seen as the characteristics of the CHC class-

room (Leung 2004; Wong 2006).

So, having a series of learning activities, including

practices, carefully arranged (by the teacher) so that stu-

dents can move from the concrete to the abstract and have

the rules/concepts firmly put to heart (memorized) could be

the image of a effective mathematics lesson as perceived

by the teachers in Hong Kong. ‘‘Teacher-led, yet student-

centred’’, a scaffolding from the basics to higher-order

thinking is constructed in such a way. To have this realized,

teacher professionalism comes first place. Teachers must

have a strong professional knowledge, including the mas-

tery of teaching skills and the ability to understand the

students.

5 Conclusions and implications

With such an in depth investigation with experienced

mathematics teachers in Hong Kong, we came up with a

portrait of teachers’ views of effective mathematics

teaching and learning. The different aspects of mathemat-

ics, that it is practical, logical, useful and involves thinking,

precisely reflect the ‘‘dual nature’’ of mathematics. This

leads to the complexity of its acquisition. In particular, the

participants did not see memorisation, practices and con-

crete experiences in the enhancement of mathematics

understanding. It is not just a matter of using these means

or avoiding these means. It is how they are used that

counts. Good teaching practices are exactly precisely those

that these means are used to their full potential so as to

make learning for understanding to happen. All these have

their cultural assumptions. So, first of all, teachers have to

create a suitable (psycho-social and cultural) classroom

environment for the above to take place; and secondly

teacher professionalism is of utmost importance so that

teachers could have a professional judgment in every sec-

tion of classroom learning and teaching. Thus, teacher

professionalism should be the heart of all educational re-

forms.

References

Ausubel, D. P. (1963). The psychology of meaningful verbal learning.

New York: Grune & Stratton.

Ausubel, D. P. (1968). Educational psychology, a cognitive view.

New York: Holt, Rinehart & Winston.

Curriculum Development Council, Hong Kong. (1992). Learningtargets for mathematics (Primary 1 to Secondary 5). Hong Kong:

Government Printer.

Curriculum Development Council, Hong Kong. (2001). Learning tolearn: The way forward to curriculum development. Hong Kong:

Education Department.

Curriculum Development Council and the Hong Kong Examinations

and Assessment Authority, Hong Kong. (2006). New seniorsecondary curriculum and assessment guide (Secondary 4–

6)—Mathematics (Provisional final draft). Hong Kong: Educa-

tion Department. Retrieved November 1, 2006, from http://

www.emb.gov.hk/FileManager/EN/Content_5630/math-

s_e_060630a.pdf.

Dahlin, B., & Watkins, D. A. (2000). The role of repetition in the

processes of memorising and understanding: a comparison of the

views of Western and Chinese school students in Hong Kong.

British Journal of Educational Psychology 70, 65–84.

Education Department, Hong Kong. (1994). General introduction totarget oriented curriculum. Hong Kong: Government Printer.

Gao, L., & Watkins, D. A. (2001). Towards a model of teaching

conceptions of Chinese secondary school teachers of physics. In

D. A. Watkins, & J. B. Biggs (Eds.), Teaching the Chineselearner: psychological and pedagogical perspectives (pp. 27–

45). Hong Kong: Comparative Education Research Centre, The

University of Hong Kong.

Hiebert J., & Carpenter T. P. (1992). Learning and teaching with

understanding. In D. A. Grouws (Ed.), Handbook of research onmathematics teaching and learning (pp. 65–97). New York:

Macmillan.

Ho, I. T. (2001). Are Chinese teachers authoritarian? In D. A.

Watkins, & J. B. Biggs (Eds.), Teaching the Chinese learner:psychological and pedagogical perspectives (pp. 99–114). Hong

Kong: Comparative Education Research Centre, The University

of Hong Kong.

Huang, R., & Leung, K. S. F. (2006). Cracking the paradox of

Chinese learners: looking into the mathematics classrooms in

Hong Kong and Shanghai. In L. Fan, N. Y. Wong, J. Cai, & S. Li

(Eds.), How Chinese learn mathematics: perspectives frominsiders (pp. 348–381). Singapore: World Scientific.

Kerkman, D. D., & Siegel, R. S. (1997). Measuring individual

differences in children’s addition strategy choices. Learning andIndividual Differences 9(1), 1–18.

Leu, Y. C., & Wu, C. J. (2006). The origins of pupils’ awareness of

teachers’ mathematics pedagogical values: Confucianism and

Buddhism-driven. In F. K. S. Leung, G.-D. Graf, & F. J. Lopez-

Real (Eds.), Mathematics education in different cultural tradi-tions: the 13th ICMI study (pp. 139–152). Heidelberg: Springer.

Leung, F. K. S. (2004). The implications of the third international

mathematics and science study for mathematics curriculum

reforms in Chinese communities. In D. Pei (Ed.), Proceedings ofthe conference on the curriculum and educational reform in the


123

primary and secondary mathematics in the four regions acrossthe strait (pp. 122–138). Macau: Direccao dos Servicos de

Educacao e Juventude.

Marton, F., Tse, L. K., & dall’Alba, G. (1996). Memorizing and

understanding: the keys to the paradox? In D. A. Watkins, & J.

B. Biggs (Eds.), The Chinese learner: cultural, psychologicaland contextual influences. Hong Kong: comparative EducationResearch Centre, The University of Hong Kong; Melbourne (pp.

69–83). Australia: The Australian Council for Educational

Research.

Marton, F., Watkins, D. A., & Tang, C. (1997). Discontinuities and

continuities in the experience of learning: an interview study of

high-school students in Hong Kong. Learning and instruction 7,

21–48.

Moritz, R. E. (1914). On Mathematics and Mathematicians (p. 31).

New York: Dover Publications.

Rose, N. (1988). Mathematical maxims and minims. North Carolina:

Rome Press.

Sfard, A. (1991). On the dual nature of mathematical conceptions:

reflections on processes and objects as different sides of the same

coin. Educational Studies in Mathematics 22, 1–36.

Sun, X. H., Wong, N. Y., & Lam, C. C. (2005). Bianshi problem as

the bridge from ‘‘entering the way’’ to ‘‘transcending the way’’:

the cultural characteristic of Bianshi problem in Chinese math

education. Journal of the Korea Society of MathematicalEducation Series D: Research in Mathematical Education 9(2),

153–172.

Tang, K. C., Wong, N. Y., Fok, P. K., Ngan, M. Y., & Wong, K. L.

(2006). The long and winding road of Hong Kong primarymathematics curriculum development: modernisation, localisa-tion, popularisation, standardisation and professionalisation (inChinese). Hong Kong: Hong Kong Association for Mathematics

Education.

Wong, N. Y. (1993). The psychosocial environment in the Hong

Kong mathematics classroom. The Journal of MathematicalBehavior 12, 303–309.

Wong, N. Y. (1995). Discrepancies between preferred and actual

mathematics classroom environment as perceived by students

and teachers in Hong Kong. Psychologia 38, 124–131.

Wong, N. Y. (1998). The gradual and sudden paths of Tibetan and

Chan Buddhism: a pedagogical perspective. Journal of Thought33(2), 9–23.

Wong, N. Y. (2000). The conception of mathematics among Hong

Kong students and teachers. In S. Gotz., & G. Torner. (Eds.),

Proceedings of the MAVI-9 European workshop (pp. 103–108).

Duisburg: Gerhard Mercator Universitat Duisburg.

Wong, N. Y. (2002). Conceptions of doing and learning mathematics

among Chinese. Journal of Intercultural Studies 23(2), 211–229.

Wong, N. Y. (2004). The CHC learner’s phenomenon: Its implica-

tions on mathematics education. In L. Fan, N. Y. Wong, J. Cai,

& S. Li (Eds.), How Chinese learn mathematics: perspectivesfrom insiders (pp. 503–534). Singapore: World Scientific.

Wong, N. Y. (2006). From ‘‘entering the way’’ to ‘‘exiting the way’’:

In search of a bridge to span ‘‘basic skills’’ and ‘‘process

abilities’’. In F. K. S. Leung, G-D. Graf, & F. J. Lopez-Real

(Eds.), Mathematics education in different cultural traditions:the 13th ICMI study (pp. 111–128). New York: Springer.

Wong, N. Y. (2007). Confucian heritage cultural learner’s phenom-enon: from ‘‘exploring the middle zone’’ to ‘‘constructing abridge’’. Plenary lecture delivered at the 4th East Asia Regional

Conference on Mathematics Education, June 18–22, Penang,

Malaysia.

Wong, N. Y., Han, J. W., & Lee, P. Y. (2004). The mathematics

curriculum: towards globalisation or Westernisation? In L. Fan,

N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learnmathematics: Perspectives from insiders (pp. 27–70). Singapore:

World Scientific.

Wong, N. Y.; Lam, C. C.; Leung, F. K. S.; Mok, I. A. C.; Wong, K.

M. (1999): An analysis of the views of various sectors on themathematics curriculum. Final report of a research commis-sioned by the Education Department, Hong Kong. Retrieved

November 1, 2006, from http://www.emb.gov.hk/index.aspx?no-

deID = 4424&langno=1.

Wong, N. Y., Lam, C. C., & Sun, X. (2006): The basic principles ofdesigning bianshi mathematics teaching: a possible alternativeto mathematics curriculum reform in Hong Kong (in Chinese).

Hong Kong: Faculty of Education and Hong Kong Institute of

Educational Research, The Chinese University of Hong Kong.

Wong, N. Y., Marton, F., Wong, K. M., & Lam, C. C. (2002). The

lived space of mathematics learning. Journal of MathematicalBehavior 21, 25–47.

Wong, N. Y., Watkins, D. (2001). Mathematics understanding:

students’ perception. The Asia-Pacific Education Researcher10(1), 41–59.

314 N.-Y. Wong

123

Documents

Hong Kong teachers’ views of effective mathematics teaching and learning