I. Prologue

In Part 1 of this article, I described the historical background, development and characteristics of the New Mathematics movement in the United Kingdom (UK). Now I shall continue to describe the New Mathematics movement in Hong Kong (HK) and compare the two curriculum initiatives in the two places.

A few mathematics educators in HK have written extensively on various issues relating to the New Mathematics movement in HK, e.g. Wong (2000 & 2001), Leung (1974, 1977 & 1980), etc. I will try to recapitulate in Section II below the main features behind the historical background, development and characteristics of this curriculum movement.

II. The New Mathematics Movement in HK

1. The background behind the New Mathematics movement

Till early 1970s, most of the secondary school mathematics textbooks used in HK were imported from the UK. As a result, there was similar dissatisfaction as in the UK (see Section IV.1 of Part 1) with the secondary school mathematics curriculum among HK mathematics educators and teachers. In particular, there was dissatisfaction with the tedious calculations involved in the use of imperial units such as miles, furlongs, yards, feet and inches; gallons, quarts and pints. However, such dissatisfaction was less extensive as tertiary education (there was only one university) and industry were relatively under-developed in HK in the 1950s and 1960s.

Elements of New Mathematics were first introduced into university mathematics courses in 1959. In the Foreword of the book Elementary Set Theory – Part 1 by Leung & Kwok (1964), Prof. Y. C. Wong, the Head of the Mathematics Department of The University of Hong Kong (HKU) in that period, explicated that set theory was introduced into the 1st Year Mathematics Course in 1959, followed by inclusion of modern algebra in 1960. Set theory and symbolic logic were then introduced into the 1964 Advanced Level Pure Mathematics Examination Syllabus. Concerning the introduction of New Mathematics into the secondary school curriculum, Prof. Y. C. Wong attended the seminar on New Mathematics at Southampton University in 1961 and brought back to HK ideas of this new movement. Subsequently HKU conducted a series of seminars in the summer of 1962 on New Mathematics and these seminars were attended by a considerable number of secondary school mathematics teachers. Some teaching materials based on SMP were distributed during the seminars. Around the same period, R. F. Simpson, a senior lecturer in the Faculty of Education, HKU, independently introduced the ideas of New Mathematics in public talks given to mathematics teachers of the Hong Kong Teachers’ Association. Simpson’s speech (1962) was published in the official publications of the Association, in which the opportunity offered to students by New Mathematics in self-discovery and creative thinking was emphasized. This new pedagogy, in contrast to the expository approach of teaching and emphasis on rote learning by students prevailing in the traditional mathematics curriculum, appealed to the more enthusiastic teachers. In fact, this new pedagogy is compatible with the contemporary theories of learning. Furthermore, there were articles (e.g. Kwok, 1962) in the Mathematics Bulletin, a periodical edited by the Mathematics Section, Advisory Inspectorate of the former HK Education Department (ED), discussing the New Mathematics movement happening outside HK. These series of events set the scene for the launching of the New Mathematics movement in HK. It is significant to note that mathematicians and mathematics educators in HKU were particularly keen on promoting and supporting the New Mathematics movement in this preliminary stage. As remarked by Leung (1980), the modernization of university mathematics education in HK was one of the driving forces behind the reform in secondary school mathematics.

2. The development of the New Mathematics movement and its characteristics

Many professionals in the HK mathematics community were very excited by the series of events described above. However, since the school mathematics curriculum was centrally controlled under the jurisdiction of the Mathematics Section of ED, teachers themselves could not initiate New Mathematics teaching in secondary schools, in contrast to what had happened in the UK. In the 1962/63 school year, the HK Secondary School Mathematics Project Committee was established by the Mathematics Section to explore ways to experiment with New Mathematics in secondary schools. Based substantially on the content of SMP in the UK, the Committee devised a draft of a set of guidelines on the content of New Mathematics and tried it out in the 1964/65 school year in Queen Elizabeth’s School, a government secondary school. From a retrospective point of view, the implementation of the New Mathematics movement in HK tended to follow, at least in the initial stage, Fullan’s notion (1991) of “Think big, start small” in curriculum development. The experimentation was soon extended from one to ten secondary schools, though mostly prestigious ones of the grammar school type. The number of secondary schools experimenting with New Mathematics in the 1967/68 school year increased to over 24 (Poon, 1978), while the percentages of secondary schools adopting the New Mathematics curriculum in the 1969/70 and the 1972/73 school years rose to about 49 and 61 respectively. These figures show that the New Mathematics movement was rapidly spreading among the HK secondary schools and, to a certain extent, fulfilling the aspiration of Y. C. Wong made in 1968 that “concerning the New Mathematics movement, HK would not be satisfied to be just a spectator” (Wong, 2001, p. 31). It is interesting to note that in spite of its key role in the development of the New Mathematics movement, ED had been adopting quite a low profile in publicizing the new development. Among all the Annual Summary and Triennial Survey issued by ED within the period from 1960 to 1975, only the 1964/65 Annual Summary compiled by Gregg (1965, p. 10) contained the following brief indication:

“In connexion with a project in Modern Mathematics, which was begun in a government Anglo-Chinese secondary school, a temporary outline syllabus for Forms 1 to 5 and a detailed teaching syllabus for Form l in Modern Mathematics were prepared.”

It might be speculated that ED was cautious when first launching the New Mathematics project as an experiment and soon became impatient and aggressive in pushing the movement forward as a mainstream policy by hinting that the traditional mathematics syllabus would be phased out very soon in public examinations (Wong, 2001). This could indirectly lead to the rapid increase in the number of schools adopting New Mathematics. The original “start small” concept was soon forgotten and rapidly gave way to a tendency for full-scale implementation. The government’s strong intervention in the New Mathematics movement could also be discerned in the following incident. There were only two series of locally developed textbooks on New Mathematics and the most widely used one was Modern Mathematics of the Mathematics Study Monoid published in 1965. The Mathematics Study Monoid (a textbook writing group) consisted mainly of civil servants working in government secondary schools and teacher training colleges. This fact might have been interpreted by the public to mean that the government was very much behind the promotion of the New Mathematics movement, since civil servants would not be allowed to write commercially published textbooks without the permission of the government.

However, New Mathematics had not totally replaced traditional mathematics (see statistical data given in a later section) as in the UK. Some teachers/schools were skeptical about the benefits of New Mathematics, particularly on the de-emphasis of deductive plane geometry and more complex manipulative skills in algebra. Papers on traditional mathematics were offered side by side with New Mathematics in public examinations for secondary school leavers throughout the New Mathematics era.

According to Leung (1974), the following patterns of thought among the proponents of New Mathematics formed the theoretical framework underpinning its development in HK schools:

• The structure of mathematics and the rigour of this structure were considered as the foundation of New Mathematics. Since the structure of mathematics was developed by logical deduction expressed through the language of set theory, therefore set language and symbolic logic were very much emphasized.

• Mathematics was regarded as a theoretical system with common properties or characteristics. Taking this view, one representative example found in the HK New Mathematics curriculum was the treatment of the number system. Starting from the set of natural numbers, the set of whole numbers was constructed, then the set of rational numbers, the set of irrational numbers, the set of real numbers and finally the set of complex numbers. The commutative, associative and distributive properties together with the existence of the identity and inverse elements relating to the operations of these kinds of numbers were formally discussed.

• The concise and precise use of mathematical symbolism and language was essential in mathematics learning. Therefore, simple algebraic equation like 2x – 4 = 0 was regarded as an open statement with a certain truth set. Solving the equation was considered as finding the elements of the relevant truth set and the solution x = 2 had to be presented as “{2} is the solution set of the open statement”.

In contrast to the UK scenario where the development of New Mathematics was essentially based on pragmatic classroom experiences, the development in HK claimed to be very much based on a principled view of the nature of the subject. With respect to this view, the New Mathematics movement in HK could be regarded as more pedagogically meaningful than that in the UK, i.e. the curriculum was based on what is important or worth learning in mathematics rather than what students are capable of learning in mathematics. However, the development in HK still lacked a sound underpinning curriculum development framework.

The aims of the New Mathematics movement in HK were to emphasize the structure and concepts of mathematics expressed through precise mathematical language, to reduce complexity in calculations, eliminate the more difficult parts of plane geometry and replace geometric proofs by algebraic deduction. The first public examination of the New Mathematics syllabus took place in the year 1969. The above stated aims were reflected to a certain extent in the aim of the examination as announced by the HK Certificate of Education (English) Board (1968):

“The aim of the examination is to test ability to understand and to apply mathematical concepts rather than to test ability to perform lengthy manipulations. Candidates will be expected to do some deductive thinking and to do some simple proofs. Credit will be given to a clear and systematic presentation of an argument. Symbolic expressions are often helpful in making statements concise and precise, and candidates will be expected to be familiar with the use of approved symbols which are listed in the syllabus.”

It is interesting to note that throughout the New Mathematics era in HK, much more emphasis was placed on the subject content rather than the teaching approaches, except the remark of R. F. Simpson made in his public talks during the preliminary stage of development.

Though the proclaimed rationale and aims of the New Mathematics movement in HK were rather different from those in the UK, the content of the New Mathematics curriculum in HK was surprisingly similar. In fact, as mentioned earlier, the guidelines on the content of New Mathematics set down by the HK Secondary School Mathematics Project Committee was essentially based on SMP. In contrast to the opponents of the New Mathematics curriculum in the UK, who criticized the insufficient training on deductive reasoning because of the removal of formal Euclidean geometry from the syllabus, it is interesting to note that the proponents of New Mathematics in HK argued that the training of logical reasoning could be better taught through algebraic deduction, including set and symbolic logic, than through proofs in traditional plane geometry (Tsiang, 1961). This argument is quite contrary to the observation of Davis & Hersh’s (1981, p.7) that “... as late as the 1950s one heard statements from secondary school teachers, reeling under the impact of the ‘new math’, to the effect that they had always thought geometry had ‘proof’ while arithmetic and algebra did not”. Basically much of the traditional plane geometry and more tedious mechanical manipulations like finding cube roots of given numbers were deleted to give way for new topics like concepts of modern algebra, statistics, probability, coordinate and transformational geometry, etc. Similar to the situation in the UK, many in-service teacher training programmes were organized by the government to familiarize teachers with the principles and practices of the New Mathematics movement.

In the late 1960s and early 1970s, there had been a rather rapid expansion of secondary schools in HK, which led to an increased number of students with diversified learning capabilities receiving compulsory secondary school education. It is interesting to note that the social justice issue concerning curriculum entitlement against differentiation of students and their curricula did not arise in HK as was the case in the UK. The same curriculum was offered to all schools adopting New Mathematics and setting was also uncommon in HK schools during that period.

When New Mathematics was first introduced into HK, many members of the mathematical community were enthusiastic and hopeful that it would resolve some of the unsatisfactory elements in the teaching and learning of the traditional mathematics curriculum. However, in the early 1970s, heated debates on the advantages and disadvantages of New Mathematics began to surface. Some schools started abandoning New Mathematics and reverted back to the traditional mathematics curriculum. As summarized by Leung (1977), the problems of the implementation of New Mathematics in HK were formality replacing substantiality; presentation format replacing mathematical content; emphasizing trivial concepts/properties but not important skills; putting immaterial concepts and theories before practice and applications of mathematics; and as a result,「只見樹木，不見森林」. Leung’s criticism resonated with Goodstein’s view of New Mathematics as “extreme and eccentric” as mentioned earlier. Kline’s classroom episode where the teacher emphasized the supposedly important yet intuitively trivial commutative property of addition of numbers could also have happened in HK classrooms. As a compromise between the two camps of New Mathematics and traditional mathematics, and following the good Chinese tradition of ‘not going to the extreme’ and adopting a ‘middle road’, a third curriculum called Amalgamated Mathematics was developed by integrating the ‘good’ elements of the two extremes. This amalgamated curriculum was first introduced to secondary schools in the school year 1975/76, where there were three mathematics curricula - new, traditional and amalgamated - for schools to choose from. Similar to what had happened in the UK, this amalgamated curriculum then became the only one to be offered in schools in the school year 1981/82.

III. Comparison and Contrast of the New Mathematics Movement in Hong Kong and the UK

The following table sets out the key features of the New Mathematics movement in HK and in the UK as a comparison.

	UK	HK
Origins	The ideas of New Mathematics originated from members of the local mathematical community, though with some influence from what happened in the US and Europe.	The ideas of New Mathematics were mainly ‘imported’ from the UK.
Reasons of development	The new developments of subject content for university mathematics, the increased mathematical use in industry, the new developments in teaching theories and desirability of learning mathematics as a unified subject were the main reasons behind the dissatisfaction with the traditional mathematics curriculum among members of the mathematical community.	There was similar dissatisfaction but to a lesser extent, probably due to the state of development in tertiary education and in industry.
Ownership	New Mathematics was developed solely as SBCD, initiated mainly by school teachers without government intervention and support.	New Mathematics was initiated and developed by the government, with support from tertiary mathematicians and mathematics educators.
Mode of development	New Mathematics was introduced and remained as a teaching experiment and there were various projects developing in the same period for schools to choose from.	New Mathematics was first introduced as a teaching experiment, and soon the government tried to push the initiative as a mainstream policy to be implemented in all schools. There were also no alternative projects for schools to choose from.
Design of development	The development of New Mathematics was not based on any curriculum development model, but developed on a pragmatic approach of finding alternative content teachable to students of the schools taking part in the project.	Although also not based on any curriculum development model, it was claimed that New Mathematics was developed to reflect the nature of the subject.
Objectives and choice of content	Although there were no explicit curriculum objectives, content of New Mathematics was developed and field-tested in classrooms with subsequent modifications and refinement.	Although there were explicit curriculum objectives based on the rationale that New Mathematics was to reflect the nature of the subject, the teaching content was essentially modeled on SMP.
Emphasis	New Mathematics emphasized both on new content (structure and language of the subject) and new teaching approaches (discovery of generalizations by students).	New Mathematics seemed to emphasize only on new content (structure and language of the subject).
In-service training	SMP organized substantial in-service teacher training programmes to better equip teachers.	There were also substantial in-service teacher training programmes, but organized by the government.
Advantages	New Mathematics seemed to improve learning atmosphere and enhance enthusiasm for discovering ideas.	There seemed to be no particular report on the good effects of New Mathematics.
Main criticism	New Mathematics was criticized as extreme and eccentric.	New Mathematics was criticized as「只見樹木，不見森林」in mathematics learning.
Social justice issue	The replacement of the ‘tripartite’ system with a ‘comprehensive’ one posed the problems of curriculum entitlement against the differentiation of students and their curricula.	Such social justice issue had not been noted in spite of the obvious widening of the range of attainments and needs of secondary school students.
Relationship with tradition mathematics	New Mathematics had not totally replaced the traditional mathematics curriculum. The number of O-Level candidates taking New Mathematics constituted only about 20% of the national entry at its peak in 1977.	Similarly, New Mathematics had not totally replaced the traditional mathematics curriculum. The number of candidates taking New Mathematics constituted about 67% of the territory-wide entry at its peak in 1979.
Evaluation	There was no official evaluation on the effectiveness of New Mathematics. However, its content was continuously adjusted to cater for the needs of students in both grammar and comprehensive schools.	Similarly there was no official evaluation and the content remained quite stable though the range of students’ attainments and needs widened as secondary school education was provided to more and more primary school leavers.
Final outcome	New Mathematics and traditional mathematics gradually merged into a unified course of study in mathematics.	There had been a compromise between New Mathematics and traditional mathematics by creating a third curriculum called Amalgamated Mathematics as a transitional arrangement until the Amalgamated Mathematics became the only mathematics curriculum to be offered in all schools.

IV. Concluding Remarks

Three interesting points emerge regarding the New Mathematics movement in HK and the UK. Firstly, unlike the situation in the US where the New Mathematics movement was essentially brought to an end when the NACOME Report (1975) indirectly announced it as a failure, New Mathematics had never been officially evaluated in the UK and HK to the extent of passing a final judgement on its success or failure. Most members of the mathematical community in both places regarded New Mathematics as a worthwhile experiment to tackle the problematic issues of traditional mathematics. In the end, New Mathematics and traditional mathematics interacted with each other and had gradually evolved and transformed into a more unified and well-structured course of study.

Secondly, the New Mathematics movement in the UK could be considered as a well-intentioned initiative tried out by groups of enthusiasts in mathematics education, mainly consisting of secondary school teachers. Their intention was to tackle the unsatisfactory situation prevailing in school mathematics in the 1950s by a series of teaching experiments. In many respects, mathematics projects like SMP were pioneers of SBCD. However, the proponents of the New Mathematics movement in the UK had not adopted a systematic and comprehensive approach to work out a solution in response to Birtwistle’s appeal (1961). As Flemming (1980) observes, before curriculum changes could be effected, teachers had to be convinced of its desirability. Innovations tended to come about slowly by a piecemeal process through the influence of textbook writers, education researchers and agencies like the Mathematical Association. By contrast, the New Mathematics movement in HK in the 1960s was basically the government’s decision to keep up with the global trend of mathematics education, particularly following the footstep of its sovereign state.

Lastly, the two facets of understanding underpinning the New Mathematics movement - precision of language and discovery of generalizations – still leave their deep trail in recent mathematics curriculum initiatives in the UK as well as in HK. For instance, the importance of precise and concise use of mathematical language is still strongly recognized by the Cockcroft’s Report (1982, p. 1) that “We believe that all these perceptions of the usefulness of mathematics arise from the fact that mathematics provides a means of communication which is powerful, concise and unambiguous” and the Smith’s Report (2004, p. 11) that “Mathematics provides a powerful universal language and intellectual toolkit for abstraction, generalization and synthesis”. Furthermore, the National Numeracy Strategy in the UK also emphasizes the importance of exact and accurate use of mathematical vocabulary by publishing a booklet for teachers’/students’ use (DfEE, 1999). Concerning the discovery of generalizations, Cockcroft recommends that investigation work should be an integral part of classroom practice for students of all ages and “is fundamental both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to extend knowledge and to solve problems in very many field” (Cockcroft, 1982, p.73). In HK, the recent Curriculum Reform stresses the importance of developing students’ communication skills and inquiry skills in mathematics learning at all school levels (Curriculum Development Council, 2000). Furthermore, considerable amount of subject content introduced into New Mathematics in the 1960s, such as statistics, probability, transformational geometry, etc., are still regarded as important and worth learning in the current mathematics curriculum in HK and the UK.

Notes:

The Hong Kong Education Department (ED) was re-structured as the Education & Manpower Bureau (EMB) in January 2003 as part of an overhaul of the government administrative structure in Hong Kong. EMB was further re-structured to become the current Education Bureau (EDB).

Before the year 1977, the public examination HK Certificate of Education Examination was under the jurisdiction of the Examination Section of ED. In 1977, the Hong Kong Examinations Authority (HKEA) was established as an independent statutory body to run all public examinations in HK. HKEA was further re-structured to become the current Hong Kong Examinations & Assessment Authority (HKEAA).

According to Flemming (1980), the number of candidates taking O-Level New Mathematics rose from 919 in 1964 to 62,691 in 1977 which constituted about 20% of the national entry. However, it was remarked that the influence of SMP had almost certainly been greater than these figures has suggested. More than half of the schools in the UK were said to be making some use of SMP materials.

According to the Annual Reports of the HKEA, the number of candidates taking New Mathematics in the HK Certificate of Education Examination rose from 330 in 1969 to 48,590 in 1979 which constituted about 67% of the territory-wide entry.