THE SUBJECT MATTER KNOWLEDGE OF HONG KONG PRIMARY
SCHOOL MATHEMATICS TEACHERS
Francis K W TSANG
Hong Kong Examinations and Assessment Authority
Tim ROWLAND
The University of Cambridge
Paper presented at the European Conference on Educational Research,
University College Dublin, 7 -10 September 2005
current legislation regarding the minimum Subject Matter Knowledge (SMK) of
teachers in Hong Kong (HK) is rather permissive, and research into analyzing the
SMK of HK mathematics teachers is virtually absent. This paper describes an
investigation into the SMK of HK primary school mathematics teachers. A
mathematics subject audit instrument used by researchers in England was adapted
for an initial exploration into HK teachers’ mathematics subject knowledge. The
collected data were analyzed and compared with the results of a mathematics subject
audit undertaken by a teacher training institute in England. The SMK of an
‘convenience’ sample of HK primary school mathematics teachers was found to be
relatively shallow des pite of the outstanding mathematics attainment of HK students
in recent international comparative studies like 2003 PISA and TIMMS.
The Content Knowledge and Pedagogical Knowledge of Teachers
Lee Shulman (1987) lists seven categories of knowledge constituting the ‘knowledge base’ for
teachers. These seven categories could be broadly grouped into two main kinds. One kind of
knowledge is different for different subjects taught in schools and is required by teachers in order to
teach their respective subject(s) effectively in classrooms. We refer to this kind as the content
knowledge of teachers. The other kind is not subject-specific, is generic in nature, and is required by
teachers irrespective of the subject(s) they teach in order to function professionally in the field of
education. We refer to it as the pedagogical knowledge of teachers.
Shulman (1986) identifies three kinds of content knowledge: (a) subject matter content knowledge
(which we call subject matter knowledge or SMK in this paper), (b) pedagogical content knowledge
and (c) curriculum knowledge. The SMK of teachers is “the amount and organization of knowledge
(of a subject) per se in the mind of the teacher” (ibid. p. 9). The structures of SMK of different subject
areas are different. For mathematics, the substantive and syntactic structures of a subject discipline as
proposed by Schwab (1978) are particularly relevant. To teach mathematics effectively, teachers
must have good mastery of the substantive and syntactic structures of mathematics. They must not
only be capable of telling students the accepted facts, concepts and principles of different branches of
mathematics, they must also be able to explain to students why a particular mathematical principle is
deemed warranted, why it is worth knowing and how it relates to other principles within the same
branch and across other branches of mathematics.
1
Content Knowledge versus Pedagogical Knowledge of Teachers – Which is more important?
Before the 1970s, the emphasis for including knowledge of subject matter into the curriculum for
teacher training was well articulated by Dewey (1904/1964). However, since the 1970s, the
pendulum swung to the other extremity where pedagogical knowledge, i.e. theories and methods of
teaching, was considered to be of paramount importance in the training and accreditation of teachers.
As there were no studies conducted within that period demonstrating an empirical link between
teachers’ content knowledge of a subject and the student learning which they hope will occur (Floden
& Buckmann, 1989), there had been a tendency around the 1970s and 1980s for research studies on
teaching to focus on generic skills and techniques rather than on the content of instruction. There is an
extensive body of research literature describing the results of process-product, teacher behaviour and
teaching effectiveness studies, which identify content-free teaching traits that account for promoting
student learning and improving their academic performance (Berliner & Rosenshine, 1977;
Rosenshine & Stevens, 1986). However, in such research studies, investigators usually ignored one
central aspect of classroom teaching: the subject matter. As observed by McNamara,
There has been a tendency to investigate and analyze teaching and learning as generic activities
without reference to the subject knowledge which provides the substantive content for most
lessons. (McNamara, 1991, p. 113)
Shulman (1986) reasserts the importance of the content knowledge of teachers. He uses the term
‘Missing Paradigm’ to refer to this blind spot with respect to content that characterizes most research
studies on teaching.
For mathematics teachers, besides the ‘Missing Paradigm’ syndrome as described by Shulman, there
is another reason for concern about the adequacy of their content knowledge. Through her study of
the mathematical understandings that prospective teachers bring to their teacher education, Ball
(1990) shows that the prospective teachers’ pre-university and university mathematics experiences
and understandings tended to be rule-bound and thin. Recently there has been research evidence
showing the inadequacy of content knowledge of mathematics teachers (Martin & Harel, 1989; Ma,
1999; Rowland et al., 2001; Goulding & Suggate, 2001).
Now different education systems around the world have put or intend to put in place measures to
guarantee a minimum level of content knowledge of teachers (DfEE, 1998; NCTM, 2000). Though
the pendulum of teacher training programmes had swung from one end to the other within the last
century, it seems that at present, the policies in many education systems are to promote the
importance of subject matter as the way to reform teacher education in order to enhance the quality of
teaching and learning in schools.
The Hong Kong Context
Apart from the Language Proficiency Requirements recommended by the Education Commission
(EC)1 (EC, 1996), the legislation regarding the minimum SMK of teachers teaching subjects other
2
than English Language and Putonghua (also known as Mandarin to westerners) is rather loose in
Hong Kong (HK). For mathematics, a teacher achieving a Grade E (the pass Grade) in Mathematics
in the Hong Kong Certificate of Education Examination (HKCEE), which is equivalent to the GCSE
in England, will be allowed to teach the subject up to Secondary 3 level. Hence, one can see that the
minimum qualifications governing the SMK of HK primary school mathematics teachers (HKPSM
teachers) are really minimal. To make the situation even less satisfactory is the fact that the academic
status of a Grade E in the HKCEE has often been challenged in recent years. For instance, EC admits
that
At present, the HKCEE follows the norm-referencing approach … An examination using this
approach reflects individual candidates’ performance in comparison with all other candidates.
Yet, it fails to indicate whether the candidates indeed possess the basic skills and knowledge
required of a Secondary 5 graduate. (EC, 2000, p. 105)
In HK, the majority of serving primary school teachers are Secondary 5 or 7 graduates taking the
initial teacher training programmes offered in the former Colleges of Education2 or the present Hong
Kong Institute of Education (HKIEd). Under the traditional practice of primary schools in HK,
Chinese, Mathematics and General Studies are considered as ‘general subjects’, which all primary
school teachers are supposed capable of teaching. Tsang (2004) explains why the
non-mathematics-elective teacher trainees would not have gained from their teacher training
programmes much insight into the subject matter underpinning the primary school mathematics
curriculum, and yet they are supposed to be academically and professionally prepared to teach
mathematics in primary schools. In fact, a recent study in HK shows that “the mathematical
knowledge of pre-service teachers is shaky” (Fung, 1999, p.124). That means it is reasonable to doubt
whether the majority of the HKPSM teachers are competent in their SMK to deliver effectively the
primary school mathematics curriculum. Despite this, HK students have attained notably good results
in international mathematics comparative studies. For instance, HK 15-year old students ranked first
in mathematics in the 2003 Programme For International Student Assessment (PISA) (OECD, 2004).
Furthermore, HK 9-year and 13-year old students both ranked fourth in the Third International
Mathematics and Science Study (TIMSS) (Law, 1997).
Ma (1999) suggests that the ‘learning gap’ in mathematics might not be just limited to students and
teachers’ knowledge could directly affect mathematics teaching and learning. Following this line of
argument, the SMK of HKPSM teachers should not be at risk given the high attainments of their
students. In fact, no adverse comments on teachers’ SMK have been made in the annual summary
reports of the Quality Assurance inspection, which is equivalent to the OFSTED inspection in
England, of mathematics in HK primary schools (QAD, 2000, 2001, 2002). So, have HKPSM
teachers mastered sufficient SMK after all?
According to Fung (1999), research into analyzing HK mathematics teachers’ SMK is virtually
absent. The present study constitutes an initial attempt to explore into HKPSM teachers’ subject
3
domain.
Researching the Subject Matter Knowledge of Hong Kong Primary School Mathematics
Teachers
Ma (1999) observes that earlier studies on teachers’ mathematics SMK often measured teachers’
knowledge by the number and type of mathematics courses taken or degrees obtained, and found little
correlation between these measures of teacher knowledge and various measures of student learning.
Since the late 1980s, the conception has changed to investigation of the knowledge that a teacher
needs to have or uses in the course of teaching a particular school-level curriculum in mathematics,
rather than the knowledge of advanced topics that mathematics teachers might have (Leinhardt et al.,
1991). It seems, therefore, that a more direct way of investigating the SMK of HKPSM teachers
would be to probe into their knowledge of mathematics in the context of the HK primary school
mathematics curriculum that they have to discuss and deliberate in their course of teaching. As the
school mathematics curriculum in HK and the Mathematics National Curriculum in England are very
similar (CDC, 2000; DfEE & QCA, 1999), both in terms of content and organization, the Subject
Audit Instrument (Instrument) used by Rowland et al. (2001) in their study on investigating the
mathematics SMK of pre-service primary school teachers in England was adapted as an initial
exploration into HKPSM teachers’ subject domain. Ten items from the Instrument were selected to
form the SMK Survey Questionnaire (Questionnaire) to be used in the present study. These items
related to three themes – basic arithmetic competence; mathematical exploration and justification;
and geometrical knowledge. These three themes were chosen because they are the basic elements of
the HK primary school mathematics curriculum and they address both substantive and syntactic
knowledge of mathematics. Besides the mathematical problem and the ‘Self Audit’3 already included
in each item, research participants were asked to assess the ‘Professional Importance of Mathematics’
of each of the ten items, i.e. how important they considered their mastery of the SMK tested by that
item for their teaching of primary school mathematics.
There are about 23,000 primary school teachers in HK (Statistics Section, 2003) and most of them
teach mathematics as a ‘general subject’ as discussed earlier. Since auditing teachers’ SMK is a very
sensitive issue in HK, it would be difficult to find a probability sample representing these 23,000
teachers willing to participate in the present study. However, we did secure the consent of 138
teachers from eight primary schools to form a ‘convenience’ sample. Since these eight schools have
their own management structures, recruit their teachers independently and nearly all the mathematics
teachers in these schools participated in the study, it would be likely that this sample of teachers is
heterogeneous enough to represent the population of HKPSM teachers as a whole. A small number of
pre-service primary school teacher trainees were also included so that their performance in the
Questionnaire could be compared with that of the participating teachers as well as with the teacher
trainees studying for the primary PGCE in the University of Cambridge. This sample of teacher
trainees consisted of six final year B.Ed.4 students from the HKIEd, 14 final year B.Ed.5 students
from the Chinese University of Hong Kong (CUHK) and 14 PGDE6 students also from CUHK.
4
The SMK survey in this study was conducted as a ‘test’ in which participants worked independently
on their own. For the teacher participants, the survey was mostly held after their normal school hours.
Participants were told that they could take as much time as they needed to complete the Questionnaire.
Specific scoring criteria for each item developed from the study of Rowland et al. (2001) were
adopted in the present study. Following a holistic approach, which underpins the scoring of the
participants’ solutions, Table 1 gives the general principles behind the scoring scheme:
Score
0
1
2
3
4
Mastery of relevant
General scoring principles
SMK
Insecure
Not attempted, no progress towards a solution
Insecure
Partial and incorrect solution
Insecure
Correct in parts, incorrect in parts
Correct solution with small errors, explanations acceptable but not
Secure
completely convincing
Full solution with convincing and rigorous explanations (not necessarily
Secure
using algebra)
Table 1: Scoring scheme for marking participants’ completed questionnaires
The range of coding used in participants’ assessment of ‘Self Audit’ is from 1 to 5, with 1 indicating
“have no confidence at all” and 5 indicating “have very strong confidence”. The range of coding used
in participants’ assessment of ‘Professional Importance of Mathematics’ is also from 1 to 5, with 1
indicating “of no importance” and 5 indicating “of very high importance”.
Some interviews were also undertaken with volunteers from the participants, so that the qualitative
data obtained from interviews could be used to supplement the quantitative data from the SMK
survey. The findings from these interviews with the volunteer teacher and teacher trainees are not
discussed in this paper.
The Subject Matter Knowledge of the Hong Kong Primary School Mathematics Teachers
The ten items in the Questionnaire can be grouped into three categories according to the nature of the
SMK being audited:
Category I: Items 1 to 4 – testing basic arithmetic competence
Category II: Items 5 to 8 – testing mathematical exploration and justification
Category III: Items 9 and 10 – testing geometrical knowledge
Table 2 summarises the percentages of serving teachers showing ‘secure’ mastery of the SMK being
audited by each item in the Questionnaire.
Item No.
% Secure
1
48.5
2
36.6
3
87.3
4
61.9
5
17.2
6
51.5
7
32.1
8
28.4
9
54.5
10
4.5
Table 2: The percentage of the participating teachers showing ‘secure’ mastery of the SMK being
audited by each item of the Questionnaire
It can be seen that fewer than half of the teachers demonstrated ‘secure’ mastery of the relevant SMK
in six of the ten items. Many teachers seemed not to understand what the items required them to do.
5
On the whole, they were relatively weak in those items or part(s) of an item that required them to
explain or justify their working or arguments. They were also weak in exploring patterns or
generalisations as well as in transformation geometry.
The performance of the teachers was not affected by the time that they spent completing the
Questionnaire (Pearson’s correlation coefficient between the time spent by the teachers and their total
scores = 0.159: not significant at  = 0.05 (two-tailed)).
The chart below shows the means of scores, ‘Self Audit’ and ‘Professional Importance of
Mathematics’ for each item in the Questionnaire.
Chart 1: The means of scores (S), 'Self Audit' (A) & 'Professional Importance
of Mathematics' (I) for each item in the Questionnaire
5
Mean S
Mean A
4.5
Mean I
4
Overall Mean A=3.726
3.5
Overall Mean I=3.620
Mean
3
2.5
Overall Mean S=2.241
2
1.5
1
0.5
0
Item 1
Item 2
Item 3
Item 4
Item 5
Item 6
Item Number
Item 7
Item 8
Item 9
Item 10
Pearson’s correlation coefficient between the mean of scores and the mean of ‘Self Audit’ = 0.539
(significant at  = 0.01 (two-tailed))
Pearson’s correlation coefficient between the mean of scores and the mean of ‘Professional
Importance of Mathematics’ = 0.224 (significant at  = 0.01 (two-tailed))
Pearson’s correlation coefficient between the mean of ‘Self Audit’ and the mean of ‘Professional
Importance of Mathematics’ = 0.165 (not significant at  = 0.05 (two-tailed))
It seems that the teachers’ perception of the SMK that is important in primary school mathematics
teaching is different from what they thought they are capable of doing. Although the correlation
between the mean of scores and the mean of ‘Professional Importance of Mathematics’ is significant,
many teachers performed quite poorly in items which they regarded as important SMK in their
mathematics teaching. There is a better match between the teachers’ actual performance and what
they thought they are capable of doing. However, it is interesting to note that some teachers claimed
high confidence in items which they gave ‘insecure’ solutions or did not attempt at all.
6
Pearson’s correlation coefficient between the teachers’ total scores and their number of years of
teaching experience = 0.256 (significant at  = 0.01 (two-tailed)).
This indicates that the teachers’ mastery of the SMK underpinning the primary school mathematics
curriculum is affected by their teaching experience. In fact, there is a tendency that their performance
deteriorates with their years of teaching experience.
An aggregate score for prior mathematics learning for each of the participating teachers was
calculated to reflect the ‘quality’ of their mathematics learning in secondary school and Sixth Form
education, both in terms of their achievement in public examinations and also their exposure to the
different levels of the mathematics curricula (for details see Tsang, 2004). It was found that the
attainment in public examinations and the level of mathematics learning correlate significantly with
the performance of the teachers in the Questionnaire (Spearman’s correlation coefficient between the
teachers’ aggregate scores for mathematics learning and their total scores attained in the
Questionnaire = 0.600: significant at  = 0.01 (two-tailed)).
To look at the same issue from another angle, a t-test was used to compare the total scores of those
teachers with the bare minimum mathematics qualifications in school leaving public examinations
(group size of 72) with the total scores of the rest of the sample (group size of 62). The t-value = 7.368
(significant at  = 0.01 (one-tailed)), i.e. the group of teachers with just a pass Grade in the HKCEE
Mathematics but with no further study of mathematics performed significantly poorer than the
remaining group of teachers in the sample.
Table 3 summarises the t-values for comparing the performance of the participating teachers in the
three categories of items (Category I – testing basic arithmetic competence; Category II – testing
mathematical exploration and justification; Category III – testing geometrical knowledge).
t-values
Category I Items compared with Category II Items
12.742*
Category I Items compared with Category III Items
16.217*
Category II Items compared with Category III Items
4.363*
Note: *significant at  = 0.01 (one-tailed)
Table 3: The t-values for comparing the performance of the participating teachers in the three
categories of items
These teachers performed best in items testing basic arithmetic competence, followed by items
testing mathematical exploration and justification; they performed poorest in items testing
geometrical knowledge. However, we need to emphasize that Item 10 has a high negative influence
on the teachers’ performance in Category III items since most of them had not learned transformation
geometry before.
7
Performance of the Participating Teachers on some Items of the Questionnaire
In this section, we will select one item from each of the three categories of items and describe some
representative/interesting responses of the participating teachers.
Category I – Item 2
Use any written method to multiply 63 and 37. Does your method use the distributive law? If so,
explain how.
Most teachers used the standard multiplication algorithm to arrive at the correct answer of 6337, but
almost half of them just stopped there without answering the part relating to the use of the distributive
law. This is why 44% of the teachers just attained the score of 2. Some teachers showed expansion
using the distributive law that did not correspond to the multiplication algorithm used,
e.g.
6 3

1
3 7
8 9
6337=63(403)=6340633=2520189=2331
4 4 1
2
3 3 1
A small proportion of teachers explicitly stated that they had not used the distributive law in the
standard multiplication algorithm and a few even said that using the distributive law in the expansion
of 6337 (actually showing the expansion) is a more complicated way of doing multiplication than
the standard algorithm. A few teachers used the distributive law twice with working such as
6337=(707)(403)=2800210280+21=2331. It seems that these teachers, even though they could
demonstrate the distributive law, did not recognise the use of the law in their chosen multiplication
process. A few teachers showed the working 6337=33737=11137= 3337=2331 and seemed
to have confused the distributive law with the associative law.
Category II – Item 8
In the figure, the number in each rectangle is the sum of the two numbers in the circles at either end of
the line segment through the rectangle.
(a) Calculate the sum of the numbers in the 3 rectangles.
3
(b) Calculate the sum of the numbers in the 3 circles.
(c) State the relationship between the sums in parts (a) and
(b). Will this relationship hold if there were different
numbers in the 3 circles? Justify your answer.
9
7
8
Many teachers got the correct answers of parts (a) and (b) and yet could not correctly state the
relationship between them. As a result many could not continue with the remaining part of the item.
Many teachers tried only one or two other sets of numbers in the three circles and then claimed that
the relationship would hold. These claims are in line with the results of the study by Martin and Harel
(1989) that many pre-service elementary school teachers in the US derived the truth of a general
mathematical statement basing only on a sequence of particular instances. Quite a number of teachers
stated, some with appropriate proof, that the relationship between the answers of parts (a) and (b)
would hold without stating what the relationship is; some of them did not answer parts (a) and (b).
Some teachers seemed not to understand the intended question as they changed the numbers in the
circles without correspondingly changing the numbers in the rectangles and so stated that the
relationship would not hold. Many teachers presented weak arguments when explaining why the
relationship would hold generally, for instance, just writing equations such as (1+2+3)2=3+4+5,
(3+6+7)2=9+10+13, etc. It is interesting to note that a few teachers explained in words the reason
why the relationship would hold by observing that each of the numbers in the three circles appears
twice in the three rectangles.
Category III – Item 9
Find the perimeter and area of the parallelogram drawn in the square grid below (each square
presents a square of length 1 cm). Explain your methods.
About half of the teachers remembered Pythagoras’ Theorem in finding the length of the slanting side
of the parallelogram and correctly calculated its perimeter, while the remaining teachers seemed not
to have recalled the Theorem. Quite a number of teachers used the formula for the area of a trapezium
in finding the correct area of the parallelogram. A few teachers confused the formula for area of a
triangle with that for parallelogram and gave 10 cm2 as the area of the parallelogram. Several teachers
arrived at the answers perimeter=18 cm and area=20 cm2 with strategies implying that by
transforming the parallelogram ABCD into the rectangle ABEF with the same base (5 cm) and same
height (4 cm) (see the diagram below), perimeter is preserved as well as area.
9
Interpretation and Implications of the Performance of the Hong Kong Primary School
Mathematics Teachers in the SMK Survey
It seems that the HKPSM teachers did not perform too well in the SMK survey. However, the
following factors should be taken into account when interpreting their performance:
1.
The SMK survey was conducted outside the teachers’ normal school hours and most of the
survey sessions (six out of eight) were held after the completion of their daily teaching duties.
The teachers were probably tired and in a hurry to leave school. Some had been working from
7.30 a.m. till 3.30 p.m. and took part in the survey around 4.00 p.m. It is reasonable to suppose
that the teachers were not in their best state of mind when working on the Questionnaire.
2.
The SMK survey was a low-stake and voluntary exercise for the teachers. They might not have
been very committed to completing the Questionnaire and tended to skip those items which they
at first sight did not clearly understand. In fact, a considerable number of teachers did not
attempt some of the items in the Questionnaire.
3.
It seems that the teachers were not well prepared, at least psychologically, for undergoing a
SMK survey. Many of them thought that they were only being asked to participate in an opinion
survey; therefore half of them had not brought along their calculators.
4.
The ten items in the Questionnaire cover only a small proportion of the SMK underpinning the
HK primary school mathematics curriculum. Therefore, the teachers’ performance in the
Questionnaire might not truly reflect the full extent of their SMK.
The participating teachers clearly performed much better in items auditing their substantive SMK of
mathematics. However, their relatively poor performance in the syntactic SMK of mathematics does
raise a concern about the adequacy of their overall SMK in delivering effectively the new HK primary
school mathematics curriculum, which requires teachers to strengthen students’ development of
higher order cognitive skills.
In their 1985 study, Steinberg et al. found that teachers who lack sufficient SMK adopt various coping
strategies in their teaching, such as relying heavily on textbooks as a source of information and
avoiding discussions and student questions (as cited by Huckstep et al., 2002). As mentioned earlier,
though there have been no adverse comments on teachers’ SMK, the annual summary reports of the
Quality Assurance inspection of mathematics in HK primary schools observe that teaching was
text-bound, teacher-centred and lacked teacher-student interactions. Teachers emphasized basic
computational skills, rarely used group activities and seldom asked open-ended questions to
encourage discussions (QAD, 2000, 2001, 2002). If the results of the present study reflect HKPSM
teachers in general, then the observations of the HK inspectors were in line with the findings of
Steinberg et al.
10
In his study, Fung (1999) gives a detailed analysis of the mathematics portrayed by a popular series of
HK primary school mathematics textbooks and concludes that “Any expectation to acquire the kind
of knowledge essential for teaching mathematics solely from reading school textbook is vulnerable”
(ibid. p. 54). The Mathematics Education Section in EMB runs a considerable number of in-service
teacher training courses every year to enhance the professional development of HKPSM teachers.
However, most of these courses are related to the pedagogy for mathematics teaching rather than
improving the SMK of teachers. Therefore, the finding that the performance of the participating
teachers in the Questionnaire tends to deteriorates with teaching experience is not surprising since
there are very few effective channels for teachers to improve their SMK after their initial teacher
training.
The results of this study also show that the participating teachers with the bare minimum of
mathematics attainment in school leaving public examinations performed significantly poorer than
the rest of the sample. This fact is worrying since EMB in HK takes this bare minimum of
mathematics attainment as the qualifying criterion for teachers teaching mathematics from Primary 1
to Secondary 3 levels. If the sample in this study is a fair representation of HKPSM teachers, then
about 54% of them are at such a minimum mathematics standard and the adequacy of their SMK is
therefore doubtful.
Conclusion
The present study is the first investigation of the mathematics SMK of HKPSM teachers. Although
the sample of teachers selected is not biased, it is nevertheless a ‘convenience’ sample and too small
for statistical purposes. Hence the sample might not be a good representation of all HKPSM teachers.
The scope of the Questionnaire is also not extensive enough to cover the whole spectrum of
mathematics SMK underpinning the HK primary school mathematics curriculum. Therefore, further
research is necessary to confirm the indications of this study. Furthermore, in addition to quantitative
data relating to the mathematics SMK of teachers, more qualitative information regarding their
thinking strategies and perceptions on mathematics teaching and learning would be necessary if
remedial interventions were to be taken. However, the present study does raise issues of concern
about the quality of the SMK possessed by HKPSM teachers with regard to their effective delivery of
the new primary school mathematics curriculum. It also appears that at present, the regulations
governing the minimum mathematics qualifications of mathematics teachers in HK are out of step
with new developments in teacher training and enhancement of teachers’ professionalism. This study
shows that teachers’ performance in the SMK survey is significantly correlated with their
mathematics achievement in public examinations and their exposure to different levels of
mathematics learning. The existing practice by schools of deploying as many teachers as possible in
sharing the total mathematics teaching load is not advisable. Since a high proportion of the sample of
teachers in this study do not possess satisfactory mathematics SMK, such regulations and practice
needs to be reviewed in the light of these results.
11
As a final remark, we understand that EMB is now implementing an improvement initiative to
provide additional resources for HK public sector primary schools to adopt specialized teaching in the
subjects Chinese, English and mathematics in the primary school curriculum (EMB, 2005). A similar
practice is being adopted in Israel, and this may be the right direction for schools to pursue in the
future. However, in addition to this initiative, EMB needs to raise the minimum qualifications for the
registration of mathematics teachers and strengthen their subject content training, both at the stages of
pre-service and in-service training.
NOTES:
1. Education Commission (EC) is a statutory advisory body under the Education & Manpower
Bureau (EMB) in Hong Kong. The former Hong Kong Education Department (ED) was
restructured as EMB in January 2003. EMB is now the policy bureau in the government of the
Hong Kong Special Administrative Region responsible for the formulation and implementation
of all education policies.
2. Like the situation in the UK before the 1970s, the former Colleges of Education in HK provided
initial and in-service teacher training for most of the primary and junior secondary school teachers.
The four Colleges of Education were re-structured, merged and upgraded as a degree-bestowing
teacher training institution called the Hong Kong Institute of Education (HKIEd) in 1994.
3. ‘Self Audit’ refers to the participants’ self assessment of their confidence in successfully solving
each item in the Questionnaire before they actually started attempting that item.
4. The HKIEd runs a Full-time Four-Year Bachelor of Education (Primary) Programme, which
supplies the main bulk of primary school teachers in HK. The six teacher trainees from the HKIEd
took very few learning modules on mathematics and did not have strong mathematics background
knowledge.
5. The CUHK runs a Full-time Four-Year Bachelor of Education Programme with mathematics as
an elective subject. All the teacher trainees of this Programme have Advanced-Level qualification
in Mathematics and have to undergo an in-depth study in mathematics SMK within the 4-years of
training.
6. The CUHK also runs a Full-time One-Year Postgraduate Diploma in Education (Primary)
Programme with mathematics as an elective subject. The trainees of this Programme have varied
background in mathematics and only a very small part of the Programme is related to
mathematics SMK.
REFERENCES:
Ball, D. L. (1990) The Mathematical Understandings that Prospective Teachers bring to Teacher
Education, The Elementary School Journal, Volume 90, No. 4, pp. 449-466
Berliner, D. C. & Rosenshine, B. (1977) ‘The acquisition of knowledge in the classroom’ in
Anderson, R. C. et al. (Eds.) Schooling and the Acquisition of Knowledge, pp. 375-396, New
Jersey, US: Erlbaum
Curriculum Development Council (CDC) (2000) Mathematics Education Key Learning Area –
Mathematics Curriculum Guide (P1-P6), Hong Kong: Hong Kong Education Department
Department of Education & Employment (DfEE) (1998) Circular 4/98 – Teaching: High status, High
Standards – Requirements for Courses of Initial Teacher Training, London, UK: Her Majesty
Stationery Office
12
Department of Education & Employment (DfEE) and Qualifications & Curriculum Authority (QCA)
(1999) Mathematics – The National Curriculum for England, Key Stages 1-4, London, UK: Her
Majesty Stationery Office
Dewey, J. (1904/1964) The Relation of Theory to Practice in Education, National Society for the
Scientific Study of Education, Third Yearbook, Part I, Reprinted in Archambault, R. D. (Ed.) John
Dewey on Education, Selected Writings, Chicago, US: University of Chicago Press
Education & Manpower Bureau (EMB) (2005) Circular Memorandum 33/2005 – Specialized
Teaching in Aided Primary Schools, Hong Kong: Education & Manpower Bureau
Education Commission (EC) (1996) Education Commission Report No. 6 – Enhancing Language
Proficiency: A Comprehensive Strategy, Hong Kong: Hong Kong Education Department
Education Commission (EC) (2000) Learning for Life, Learning through Life – Reform Proposals for
the Education System in Hong Kong, Hong Kong: Hong Kong Education Department
Floden, R. E. & Buckmann, M. (1989) Philosophical Inquiry in Teacher Education, Michigan, US:
The National Centre for Research on Teacher Education, Michigan State University
Fung, C. I. (1999) Pedagogical Content Knowledge versus Subject Matter Knowledge: An
Illustration in the Primary School Mathematics Context of Hong Kong, Hong Kong: Unpublished
Ph.D. Dissertation, The University of Hong Kong
Goulding, M & Suggate, J. (2001) Opening a can of worms: investigating primary teachers’ subject
knowledge in mathematics, Mathematics Education Review, Volume 13, pp.41-54
Huckstep, P., Rowland, T. & Thwaites, A. (2002) Primary Teachers’ Mathematics Content
Knowledge: What does it look like in the classroom? Paper presented at the Annual Conference of
the British Educational Research Association, University of Exeter, UK in September 2002
Law, N. (Ed.) (1997) Science and Mathematics Achievements at the Mid-Primary Level in Hong
Kong – A Summary Report for Hong Kong in the Third International Mathematics and Science
Study (TIMSS), Hong Kong: TIMSS Hong Kong Study Centre, The University of Hong Kong
Leinhardt, G., Putnam, R., Stein, M. & Baxter, J. (1991) ‘Where subject matter knowledge matters’ in
Brophy, J. (Ed.) Advances in research on teaching Volume 2, pp. 87-113, Greenwich, CT: JAI
Press
Ma, Liping (1999) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of
Fundamental Mathematics in China and the United States, New Jersey, US: Lawrence Erlbaum
Associates, Inc.
Martin, G. & Harel, G. (1989) Proof frames of pre-service elementary teachers, Journal for Research
in Mathematics Education, Volume 20, No. 1, pp. 41-51
McNamara, D. (1991) Subject Knowledge and its Application: problems and possibilities for teacher
educators, Journal of Education for Teaching, Volume 17, No. 2, pp. 113-128
National Council of Teachers of Mathematics (NCTM) (2000) Principles and Standards for School
Mathematics, Virginia, US: NCTM
Organization For Economic Co-operation and Development (OECD) (2004) Learning for
Tomorrow’s World – First Results from PISA 2003, OECD
Quality Assurance Division (QAD) (2000, 2001 & 2002) Quality Assurance Inspection Annual
Report, Hong Kong: Hong Kong Education Department
13
Rosenshine, B. & Stevens, R. S. (1986) ‘Teaching Functions’ in Wittrock, M. C. (Ed.) Handbook of
Research on Teaching 3rd Edition, pp. 376-391, New York, US: Macmillan
Rowland, T., Martyn, S., Barber, P. & Heal, C. (2001) ‘Investigating the mathematics subject matter
knowledge of pre-service elementary school teachers’ in van den Heuvel-Panhuizen, M. (Ed.)
Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics
Education, Volume 4, pp. 121-128, Utrecht, the Netherlands: Freudenthal Institute, Utrecht
University
Schwab, J. J. (1978) Science, Curriculum and Liberal Education, Chicago, US: University of
Chicago Press
Shulman, L. S. (1986) Those Who Understand: Knowledge Growth in Teaching, Educational
Researcher, Volume 15, No. 2, pp. 4-14
Shulman, L. S. (1987) Knowledge and Teaching: Foundations of the New Reform, Harvard
Educational Review, Volume 57, No. 1, pp. 1-22
Statistics Section (2003) Teacher Statistics 2002/03 – A Summary of Key Findings, Hong Kong:
Education & Manpower Bureau
Tsang, F.K.W. (2004) The Subject Matter Knowledge of Hong Kong Primary School Mathematics
Teachers, Cambridge, UK: Unpublished M.Phil. Dissertation, The University of Cambridge
Correspondence: tr202@cam.ac.uk
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