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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

The 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 despite 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.

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 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 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.

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

Mastery of relevant SMK

General scoring principles

0

Insecure

Not attempted, no progress towards a solution

1

Insecure

Partial and incorrect solution

2

Insecure

Correct in parts, incorrect in parts

3

Secure

Correct solution with small errors, explanations acceptable but not completely convincing

4

Secure

Full solution with convincing and rigorous explanations (not necessarily 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.

1

2

3

4

5

6

7

8

9

10

% Secure

48.5

36.6

87.3

61.9

17.2

51.5

32.1

28.4

54.5

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. 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 a = 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.

Pearson’s correlation coefficient between the mean of scores and the mean of ‘Self Audit’ = 0.539 (significant at a = 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 a = 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 a = 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.

Pearson’s correlation coefficient between the teachers’ total scores and their number of years of teaching experience = - 0.256 (significant at a = 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 a = 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 a = 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 a = 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.

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

  63´37=63´(40-3)=63´40-63´3=2520-189=2331

´

 

3

7

 

1

8

9

   
 

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

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.

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.

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.

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.

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Correspondence: tr202@cam.ac.uk

This document was added to the Education-Line database on 11 November 2005