1. The rise and fall of the New Mathematics movement in the U.S.

Reflection on beliefs and practices in historical events can shed light on those in our own time. Since the post-war period, the secondary school mathematics curricula in most education systems have undergone significant changes, but arguably the New Mathematics or Modern Mathematics movement in the 1960s has left the deepest trail. As observed by Wong (2001), the New Mathematics movement raised fundamental issues relating to the nature of mathematics and mathematics education. These issues were still under the spotlight in different mathematics education reform movements in subsequent years, indicating that they have not yet been thoroughly resolved. The following somewhat exaggerated classroom episode taken from the work of Kline (1973, pp.1-2) illustrates some well-founded features, but mistakenly interpreted by teachers, of New Mathematics.

“Let us look into a modern mathematics classroom.

The teacher asks, ‘Why is 2 + 3 = 3 + 2?’ Unhesitatingly the students reply, ‘Because both equal 5.’ ‘No,’ reproves the teacher, ‘the correct answer is because the commutative law of addition holds.’

Evidently the class is not doing well and so the teacher tries a simpler question. ‘Is 7 a number?’... [The teacher explains after one student gave an incorrect answer.] ‘Of course not! It is the name of a number. 5 + 2, 6 + 1, and 8 – 1 are names for the same number. The symbol 7 is a numeral for the number.”

The publication of Morris Kline’s Why Johnny Can’t Add and the 1975 Report of the National Advisory Committee on Mathematics Education (NACOME Report, 1975), which pointed out some drawbacks of the New Mathematics reform, brought the reform movement essentially to an end in the U.S. The cry of “back to basics” is still heard from time to time as a reaction to the more recent mathematics curriculum reforms. It would be educationally enlightening to look more closely into the origins and development of this New Mathematics movement with a view to shedding light on the present and future mathematics reforms.

2. The New Mathematics movement in Hong Kong and the United Kingdom

Hong Kong (HK) and the United Kingdom (UK) did not escape from the global trend of the New Mathematics movement, though not without their own distinct background and perspectives. Since HK had been a colony of the UK until 1997, the HK education system has been very much under the influence of that in the UK and in many respects followed the changes taking place in its sovereign state, though often lagging a few years behind in their development and implementation. However, due to the differences in the western and eastern cultures as well as traditions, similar curriculum changes taking place in both places may lead to different results in educational ideology as well as in classroom practices. Therefore, it would be informative to compare and contrast the changes in the secondary school mathematics curriculum in HK and the UK during the New Mathematics era. In making such a comparison, many issues relating to pedagogy/curriculum will be touched upon and issues regarding social justice will also be pointed out wherever appropriate.

II. Curriculum and Curriculum Planning

1. The essential elements and linear model of curriculum planning

As the New Mathematics movement was a curriculum reform, it would be apt to discuss briefly the notions of curriculum and curriculum planning as a basic framework from which to comment on the New Mathematics movement in HK and the UK.

There are many definitions of the term ‘curriculum’. These definitions vary considerably depending on whether the definition focuses on the nature of what we teach, or the planned outcomes of schooling, or students’ experiences and activities in school. According to Cockcroft (1982, p.128),

“We use ‘syllabus’ to denote a list of mathematical topics to be studied but ‘curriculum’ to include the whole mathematical experience of the pupil; in other words, both what is taught and how it is taught. The curriculum therefore includes the syllabus; it is concerned with the way in which the syllabus is presented in the classroom as well as with other matters which are important. For example, problem solving, logical deduction, abstraction, generalization, conjecture and testing should play a part in the work of all pupils.”

Regarding curriculum planning and development, the most influential approach around the period of the 1960s was derived from the four key questions identified by Tyler (1949):

(a) What are the intentions of the development?

(b) What is the content?

(c) What methods are used to deliver it?

(d) How is it assessed?

These key questions are still significant in contemporary thinking about curriculum planning. For instance, according to Morris (1995) the product-based model of curriculum planning involves four stages in a linear model as depicted below.

This linear model matches very well with Tyler’s key questions.

2. The cyclic model of curriculum planning

There are many variations of the above linear model. One example is that of Wheeler (1967) who modifies it to allow evaluation to be used to improve the curriculum. While this modification, as shown below, is cyclic rather than linear, its essential concern or starting point is the identification of intended learning outcomes.

Curriculum planning and development take place at various levels and primarily involves decisions made by governments, schools and teachers. I shall argue that there were fundamental differences in the interaction of these key players in the New Mathematics movement in HK and the UK. The New Mathematics movement in HK was ‘top-down’, initiated and controlled by the government, whereas that in the UK was ‘bottom-up’, initiated by teachers. In fact, the development of the New Mathematics movement in the UK resembled that of a ‘school-based curriculum development’ (SBCD), and more will be said about this later.

III. The Historical Background of the New Mathematics Movement

1. The impact of the launching of the ‘Sputnik’

Many mathematics educators attribute the global New Mathematics movement to the launching of the first manned spacecraft Sputnik by the Russians in October 1957, and the subsequent resolution of the US government to catch up and surpass its superpower rival by improving mathematics and science education in high schools and beyond.

2. Influences beyond the impact of the launching of the ‘Sputnik’

In fact, there was desire for reform even before the Sputnik incident as Wooton, (1965, p.5) has noted:

“In the eyes of many thoughtful members of the mathematical community, the picture of mathematics education in American high schools in 1950 was not a pretty one. In particular, they were dissatisfied both with the content of the course offerings and with the spirit in which the material was presented. They were convinced that the traditional subject matter was inappropriate to the times.”

According to the 32nd Yearbook of the National Council of Teachers of Mathematics in the US (NCTM, 1970), school mathematics curriculum reform in this period was driven by college mathematicians, and the mathematical needs of students entering colleges can be identified as triggering the first major curriculum development at the high school level in this period. Price (1994, p.207) summarizes as follows:

“University pure mathematicians played a leading part in the early American movement during the 1950s, and further impetus for a concerted national effort directed at curriculum reform came from the Soviet launching of Sputnik in 1957.”

Beyond the US, similar dissatisfaction with the traditional mathematics curriculum was not uncommon among mathematicians, mathematics educators and mathematics teachers in the European countries, including the UK (Crawford, 1961).

Several university led high school mathematics curriculum projects were developed in the US in the late 1950s and early 1960s, but as Henderson (1963, p.57) remarked

“The University of Illinois Committee on School Mathematics (UICSM) has been viewed as the progenitor of all current [high school] curriculum projects in mathematics.”

The director of the UICSM, Max Beberman (1958, p.4), describing the philosophy and programme of the UICSM, insisted that

“...the student must understand his mathematics. ... a student will come to understand mathematics when his textbook and teacher use unambiguous language and when he is enabled to discover generalizations by himself.”

These two facets of understanding – precision of language and discovery of generalizations – were central influences in the development of the New Mathematics curriculum in the US as well as in other education systems. The requirement to be precise in the use of mathematical language in the New Mathematics curriculum, taken to the extreme, could be the reason behind the rigour sought by the teacher in Kline’s classroom episode described above, where the teacher emphasized excessively the difference between a number and its representing numeral.

IV. The New Mathematics Movement in the UK

1. The background behind the New Mathematics movement

Watson (1976) accounts for the factors that brought about dissatisfaction with school mathematics in the UK in the 1950s. While school mathematics in 1955 was little different from that of the 1930s, new developments of the subject were occurring rapidly, and whole new areas of mathematics were being explored in universities and industry. It was believed that students’ work in schools should include some material reflecting the changes which had taken place in the advanced study of mathematics. There were other educational reasons for reform, such as the growth of knowledge in developmental psychology, notably the work of Piaget, and its implications for the teaching of mathematics. Developments in the use of mathematics in industry, e.g. operational research, linear programming and computing, also brought concern about the mathematical knowledge of school leavers. It was felt that the school mathematics curriculum and the ways the subject was taught had failed to keep in step with these new developments.

Another rather different source of dissatisfaction dated back as early as the 1940s. Despite the recommendations of the Jeffery Committee (1944) that mathematics should be taught and examined as a unified whole and not as a number of disjoint components – Arithmetic, Algebra and Geometry, the majority of candidates took these three papers in the General Certificate of Education (GCE) Ordinary Level (O-Level) examinations until 1962 (Howson, 1982). The notion of teaching mathematics as a unified course was again brought up in the Blackpool Conference (1958). Members of the Discussion Group on this issue were unanimous that it was desirable to teach mathematics as a unified subject. The Group felt that there was a need for a suitable textbook on connecting the topics in different branches of mathematics. Making connections between various mathematical topics and integrative use of mathematical knowledge in problem solving are indeed the very essence of mathematics learning in any contemporary mathematics curriculum.

To conclude, Birtwistle (1961, pp.3-4), the Editor of the journal Mathematics Teaching, succinctly summarized that

“It is no longer necessary to argue that there is a crisis in the teaching of mathematics; the fact is now accepted. ... Probably no single factor has contributed to the present situation ... But understanding how the situation has arisen merely shows us the problem ...; it does not, of itself, give us a solution.”

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

(i) The School Mathematics Project (SMP) and other similar projects

Set against the background described in the previous section, the New Mathematics movement progressively developed in the UK. However, Howson (1982) has observed that whilst a date could not be definitely defined on which the New Mathematics reform first started, the most influential project representing the New Mathematics reform – the School Mathematics Project (SMP) – was formally established in 1961. SMP was masterminded by the Senior Mathematics Masters (Heads of Mathematics Departments) of four independent grammar schools and a university professor of mathematics, and started as a semi-private experimental venture in eight grammar schools. From the point of view of Marsh et al. (1990), SMP is really a SBCD project, an essentially teacher-initiated grass roots phenomenon. Besides SMP, there were more than ten other similar mathematics projects developed in this period, such as the Contemporary School Mathematics (CSM) Project, which had its roots in St. Dunstan’s College; the Midland Mathematics Experiment (MME), which started with a group of eight lower-status secondary schools: two grammar, four technical and two modern comprehensive ; and the Mathematics in Education and Industry (MEI) Project, which received some funding from industry (Mathematical Association, 1976; Howson, 1978). There were also many small-scale teacher-designed experiments to try out different ideas of New Mathematics. For instance, Brissenden (1962) experimented with set theory in teaching geometry topics in his grammar school mathematics syllabus. Therefore, it can be seen that the New Mathematics movement in the UK was remarkably teacher-initiated without any government intervention or support.

Among all the New Mathematics curriculum projects, SMP rapidly became the most successful one in England and Wales. According to Price (1994), it had been estimated that of around 3,500 schools involved in New Mathematics programmes, probably about 3,000 were connected in some way with SMP. For this reason, in the following discussion, I will take SMP as a representative of the New Mathematics movement in the UK.

(ii) The characteristics of the New Mathematics movement as exemplified by the SMP

The director of SMP, Bryan Thwaites (1972, p.6) explained the rationale behind the project as follows:

“Of over-riding importance for us, however, is that syllabuses and the associated methods [i.e. curriculum in Cockcroft’s perspective] should be developed as a practical outcome of classroom experience, rather than as a result of theoretical discussions round committee tables. ... if any claim needs to be made for the SMP’s work it will rest primarily on the experimental teaching and the experience of it gained in a group of schools.” (emphasis added)

Hence it can be seen that the proponents of the New Mathematics reform in the UK regarded the school mathematics curriculum as something to be determined by school teachers and not to be greatly influenced by theories of curriculum development or the mathematical preferences of pure mathematicians, as was the case in the US. As a result, the new materials produced were usually considered as teachable to students of the type to be found in the classrooms of the project schools. Curricula were usually constructed without reference to theoretical arguments. As reported by Thwaites (1972), ‘practical experience’ rather than curriculum theory had been the guide for the development of SMP. Curricular experimentation at the school level that targeted at the needs and interests of students is in the spirit of SBCD as described by Marsh et al. (1990). Furthermore, Thwaites explained that the teaching materials and teaching approaches of SMP were tried out in classrooms and subsequently evaluated for necessary modifications and refinement. This evaluation as a feedback in the developmental process fits well into Wheeler’s cyclic framework mentioned earlier.

According to Thwaites (1972, p.3), the ultimate objectives of SMP were:

• the evolution of a syllabus, for the whole grammar school range of 11+ to 18, which would adequately reflect the modern trends and usages of mathematics; and

• the production of a complete set of associated textbooks and teachers’ guides.

It is clear from Thwaites’ Report that the developers of SMP had other intentions, such as

• the contents chosen were to bridge the gulf which at that period separated university from school mathematics – both in content and in outlook – and impart a knowledge of the nature of mathematics and its uses in the modern world; and

• to make school mathematics more exciting and enjoyable so as to encourage more students to pursue the study of mathematics further.

“Bridging the gulf separating university from school mathematics’ is still an issue being addressed in the report Making Mathematics Count – The Report of Professor Adrian Smith's Inquiry into Post-14 Mathematics Education published in February 2004. Smith (2004, p.55) remarks that

“The traditional programme of full-time study was increasingly seen as a less than adequate preparation for work or for ... Higher Education, which required a broader range of knowledge and skills ... In addition, considerable concern was expressed ... about the lack of mathematical fluency of those entering Higher Education courses requiring more specialist mathematics skills.”

(iii) The pros and cons of the New Mathematics movement in the UK

SMP’s curriculum objectives and process of development as a whole seem to be pedagogically sound. However, Flemming’s critique (1980) commented that no attempt appeared to have been made to set out the course objectives as detailed learning outcomes and select subject content and teaching methodologies in accordance with any curriculum development model such as the linear or cyclic one described earlier. Furthermore, though the teaching materials were tried out in schools, no summative evaluation against the declared objectives had been carried out. Hence, the design of SMP could not be considered as ‘standard’ with respect to conventional curriculum development theories.

Regarding the choice of teaching materials, the syllabuses for the SMP GCE examinations in various examination boards reflected many new ideas. One of the main changes lied in the increased emphasis on algebraic structures. Sets, relations, matrices, vectors and groups were included together with concepts such as commutativity, associativity and distributivity. Logical symbols like andwere introduced to help students understood what mathematicians meant by ‘if then’, ‘if and only if’ and converse statements. In geometry, the training of deductive reasoning offered by the formal Euclidean geometry of traditional mathematics was largely replaced by a study of geometrical transformations and three-dimensional geometry, which was intended to give students better appreciation of spatial sense and relationships. Coordinate geometry was introduced earlier to emphasize the interplay between algebra and geometry. In particular, matrix form was used to represent geometrical transformations leading to simple examples of groups. Time for studying new topics was made by reducing the complexity of manipulations of traditional topics like algebraic fractions, indices, solving equations, etc. The emphasis had shifted from complicated applications of these topics to understanding of the processes involved. There were substantial in-service teacher training programmes conducted by SMP to better equip teachers in dealing with the new content as well as new approaches of teaching.

Though it has been said that the New Mathematics movement in the UK had not followed the US approach of attuning to the thinking of pure mathematicians, the new content areas listed above reflected to some extent the importance of precise language and symbolism as well as the discovery of generalizations, the two facets of understanding underpinning the US New Mathematics movement as highlighted above by Beberman. Furthermore, the choice and organization of the curriculum content of New Mathematics reflected to a certain extent Bruner’s (1975, p.454) emphasis on the structure of a subject that

“... the curriculum of a subject should be determined by the most fundamental understanding that can be achieved of the underlying principles that give structure to the subject. Teaching specific topics or skills without making clear their context in the broader fundamental structure of a field of knowledge is uneconomical in several deep senses.”

However, this radical change of content in New Mathematics was not totally appreciated without objection. In his critical review of the Synopses for Modern Secondary School Mathematics, Goodstein (1962, p.72) concluded that “Proposals as extreme and eccentric as those under review can I fear only serve to damage the case for reform.” Fletcher (1962, p.178) responded to Goodstein’s critique by pointing out that

“The case for expanding school instruction in the direction of modern algebra ... rests on recent psychological work by Piaget and on pedagogical research in the classroom.”

Hence it can be seen that, though the development of the New Mathematics curriculum was not grounded on any theory of curriculum development, there had been serious debate based on pedagogical considerations.

Thwaites (1972) claimed that the changes in the subject content and teaching methodologies of New Mathematics, as represented by SMP, did bring about positive effects on the learning of mathematics by students. He reported some genuine progress has been made in making mathematics a more attractive subject in school as reflected by the increased excitement and enthusiasm in classroom work. Teachers were also surprised at the changed classroom atmosphere as a result of the concentration on discovery of ideas rather than the acquisition of technical skills. Even today, mathematics teachers are still striving hard for an enlivened learning atmosphere and enhanced enthusiasm for discovering ideas.

(iv) ‘Happy’ ending of the New Mathematics movement in the UK

A challenge to the New Mathematics movement was posed by the restructuring of the UK secondary school system in the mid 1960s, in which the ‘tripartite’ system was largely replaced by a ‘comprehensive’ one (Howson, 1982). If the New Mathematics movement, which started with grammar schools as its target sector, was to prosper, it has to address the problems of catering also for comprehensive school students with widely different attainments, background knowledge and aspiration. Hence, another series of SMP ‘letter books’ were developed to cater for the needs of the comprehensive school students. The debate between curriculum entitlement of differentiation of students and their curricula raised the social justice issue. The questions whether setting was socially divisive and whether an alternative, compatible with prevailing social ethos, that could be translated into effective classroom practice are still being debated till now. At the same time, judging from the syllabuses of the GCE O-Level examinations, traditional mathematics began to evolve to incorporate some elements of New Mathematics. From mid 1960s to early 1970s, the two originally quite polarized curricula continuously transformed and approached each other in terms of content and teaching approaches, and finally merged to become a unified course of study.

Much of the information given in the above account is confirmed by some recollections of one of the originators of SMP, Mr. Douglas A. Quadling, whom I have interviewed personally at his home. He emphasized that SMP, like other mathematics projects in that period, was an experimental attempt by a group of teachers to look for alternative content and teaching approaches in secondary school mathematics education. There was no intention to totally displace traditional mathematics, but to find ways to give the teaching of secondary school mathematics a more ‘modern’ perspective. The ultimate merging of New Mathematics and traditional mathematics into one single unified course of study was considered by Mr. Quadling a ‘happy’ ending of the New Mathematics movement in the UK. When asked whether SMP had a theoretical curriculum framework underpinning, Mr. Quadling remarked that the language of curriculum development was not commonplace in the 1960s. This lack of theoretical underpinning behind curriculum development in England may still be true even now. In response to an invitation to describe the principles underpinning the recent arithmetic curriculum, leading British mathematics educator Margaret Brown (2001, p.35) writes “To admit to having any principles is a most un-English thing to do.”

Here marks the end of Part 1 of this article. In Part 2, which will be published in the next issue of this publication, I will continue to describe the New Mathematics movement in HK and compare the two curriculum initiatives in HK and in the UK.

Notes:

After several revisions and as at 1958, the UICSM materials were divided into 11 units, covering essentially all the topics in the usual four-year high school programmes. The 11 units are (1) arithmetic of the real numbers, (2) generalizations and algebraic manipulation, (3) equations and inequalities, (4) ordered pairs and graphs, (5) relations and functions, (6) geometry, (7) mathematical induction, (8) sequences, (9) elementary functions, (10) circular functions and trigonometry and (11) polynomial functions and complex numbers. The unit format and the refusal to use the traditional designations, such as algebra and geometry, were a significant break from the past and a partial realization of a recurring recommendation in mathematics education.

The Jeffery Committee proposed an ‘Alternative Syllabus For Elementary Mathematics’ (called 1944: The Jeffery Proposals) consisting of seven branches of mathematical content: Numbers; Mensuration; Formulae & Equations; Graphs, Variation & Functionality; Two-dimensional Figures; Three-dimensional Figures; Practical Applications. There would be three papers in the corresponding examination, each paper may contain questions on any part of the syllabus and the solution of any question may require knowledge of more than one branch of the syllabus.

Following the 1944 Education Act in the UK to extend compulsory education beyond primary schooling, secondary education in most parts of the UK basically adopted a ‘tripartite’ system of grammar, technical and secondary modern schools. However, around mid 1960s, the ‘tripartite’ system was largely replaced by a ‘comprehensive’ one. New comprehensive schools began to become the mainstream with classes comprising students who differed widely in attainments, knowledge and aspiration.

The five persons initiating SMP were Dr. H. M. Cundy of Sherborne School, Mr. T. A. Jones of Winchester College, Mr. T. D. Morris of Charterhouse, Mr. D. A. Quadling of Marlborough College and Prof. B. Thwaites of Southampton University.

When the term SBCD was first introduced, it usually referred to curriculum development projects initiated within one school. However, the term is now also widely used to denote curriculum development projects initiated by a group of schools. Such kind of group projects is getting more and more common among school communities.

The Organization for European Economic Co-operation (OEEC) conducted a seminar at Royaumont, France in the autumn of 1959 to facilitate cross-fertilization of ideas on New Mathematics between America and Europe. Then a major collaboration resulted in the publication of an international consensus by the OEEC in 1961 and the publication was called Synopses for Modern Secondary School Mathematics.