I. Introduction
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 postwar
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.12) illustrates some wellfounded 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 productbased 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
‘topdown’, initiated and controlled by the government, whereas
that in the UK was ‘bottomup’, initiated by teachers. In fact,
the development of the New Mathematics movement in the UK resembled
that of a ‘schoolbased 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 (OLevel)
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.34), 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 semiprivate experimental venture in eight grammar
schools. From the point of view of Marsh et al. (1990), SMP is really
a SBCD project, an essentially teacherinitiated 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 lowerstatus 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 smallscale teacherdesigned
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
teacherinitiated 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 overriding 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 Post14 Mathematics Education published in February
2004. Smith (2004, p.55) remarks that
“The traditional programme of fulltime 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 threedimensional 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 inservice 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 OLevel 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 unEnglish 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 fouryear 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; Twodimensional
Figures; Threedimensional 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 Cooperation (OEEC) conducted
a seminar at Royaumont, France in the autumn of 1959 to facilitate
crossfertilization 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.
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