THE SECONDARY-TERTIARY INTERFACE
3. BETWEEN SCHOOL AND TERTIARY
STUDY
3.1 Bridging courses
3.2 Aims of bridging courses
3.3 Success of bridging courses
4. FIRST YEAR OF TERTIARY
STUDY
4.1 Pedagogical
changes at first year university level
4.2 Routine algebraic skills
4.3 Academic support services
5. OTHER ASPECTS OF TRANSITION
5.1
Systemic solutions: senior or junior colleges
5.1.1 Extended degree
programmes
5.2 Orientation
5.2.1
Interaction with teachers in secondary schools
5.2.2 School examinations
So, compared to other discipline areas, mathematics is in a privileged position in the school curriculum. In most countries, the study of mathematics is compulsory at least until the minimum school leaving age. There has been a strong message that mathematics is important for university study in a wide range of disciplines and that not studying mathematics can severely limit one's choice of degree programme. Added to this is the success of female students in secondary mathematics study. In the past 20 years, the numbers of female students completing high levels of secondary mathematics has increased dramatically. Even so, gender differences in advanced mathematics achievement favours males (IEA, 1995-6) in a majority of countries7.
But university mathematicians are complaining. Students are more numerous and more diverse, at least at fírst year level. Pedagogical and curriculum changes at preuniversity level have had an impact on students' skills and ideas of rigour. Technology is changing the way mathematics is done and changing mathematies itse1f. There are ongoing questions about the quality of graduates and the integrity of the discipline. lt is harder to attract good quality students into higher mathematics study at university. Indeed, ¡t is getting more difficult to attract students to higherlevel mathematics study at secondary school. This is in contrast to the pervasiveness of mathematics in other disciplines and in the workplace (Australian Acaderny of Sciences, 1996).
Much can be done to improve the response of university mathematicians to these changes. There are positive outcomes from the changes to secondary education. The diversity of students in mathematics classes can lead to new ways of working with mathematics. It also raises questions such as: "How necessary are algebraic manipulation skills when computing power is so great?" and "How important is the need for rigour for all students when there is a huge diversity of mathematical careers?`
For some students it can be a long time as they take a break from study in general, and mathematics and particular. Others have not even succeeded in finishing secondary school. This is frequently an area of political priority where governments fund special programmes to increase the numbers and the success rate of disadvantaged students in tertiary education.
Therefore, this overview will include bridging programmes for mature-aged students returning to study as well as school leavers who have not studied the appropriate level of mathematics for the degree programme that they wish to undertake.
Changes to the teaching and leaming of mathematics at secondary level have been successful in opening mathematics to more students, especially those from non-academic backgrounds. However the profile of mathematics students has changed making it necessary for tertiary mathematicians also to adapt. Two main transition adaptations are bridging courses between school and university and changes in the first year university mathematics curriculum. Systemic changes, such as the creation of senior high schools and junior colleges are briefly discussed in a later section.
A range of programmes can be called bridging courses because they bridge between the academic needs of school and tertiary study. For ease of comparison we will discuss their aims and target groups, rather than their length, cost or who teaches them. Bridging courses prepare students for the academic concepts, skills and attitudes of their degree programmes as opposed to orientation programmes that prepare students for university life. Orientation programmes will be considered in the next section.
Some of the earliest bridging mathematics programmes were introduced after the Second World War for soldiers returning to study. Since then bridging programmes have reflected societal changes and government priorities. In the 1970s, programmes were established to assist women to enter and succeed in tertiary study. These programmes were often for women only and contained a component of confidence building and career information. There were several interesting names, such as WOW (work opportunities for women) and Mathematics Assertiveness. For women in their thirties, forties and fifties, bridging courses opened doors to universities; equal pay and equal opportunity policies opened the workforce to them. This idea of bridging courses as a way of compensating for discrimination is still the dominant philosophy in many universities. Following the success of programmes for women, other programmes were devised for disadvantaged groups, such as Native Americans, Hispanics and so on. With an increased access to universities and an emphasis on equal opportunity for disadvantaged groups, the number and variety of bridging courses has exploded. In countries, such as the Republic of South Africa, political and social goals require greater access to higher education for disadvantaged groups so there is great interest in devising bridging programmes.
Another type of bridging course was very successful with particular groups in the 1980s. There were waves of refugees from the war areas of South East Asia migrating to North America, Europe, Australia and New Zealand. These students had fractured schooling, poor language skills in their adopted country and had suffered the trauma of war. Lecturers devised innovative teaching methods to teach language, mathematics and study skills in the same programme. Conventional syllabuses and materials were not appropriate and lecturers moved to condense years of mathematics into short courses. These were very successful with motivated students.
Bridging courses allow lecturers to be innovative because they are not part of the mainstream mathematics degree programmes and are therefore not contingent on the same constraints. Frequently young, inexperienced and non-tenured lecturers were used to teach the programmes. The teaching and learning ideas developed in teaching these groups are now being applied to mainstream classes.
So bridging programmes are not new. They have been used for political
and social purposes to prepare particular groups for tertiary studies.
The philosophy and style of the bridging programme depends on the target
group and the political climate. There are also bridging courses for students
who have just finished school. These courses cater for the many students
who have not studied the correct level of mathematics for their tertiary
studies either through bad advice, disadvantaged schooling or through changing
their minds as to career choice.
A second aim of bridging courses is to prepare students for first year mathematics courses in which results are proved. In many secondary schools, proofs have almost completely disappeared. The result is that for many students the confrontation with proofs at tertiary level is difficult. The target groups for these types of programmes are students who have succeeded in secondary studies and wish to move into tertiary mathematics.
A third aim is to develop the attitudes and language to become a successful mathematician or student of mathematics. The target groups for these programmes are students who come from non-academic backgrounds, students who are new to the education system such as international students and students who are studying in a different language to their mother tongue. These students may have excellent mathematical skills but may not be able to read textbooks quickly enough or understand what is required for assessment. Many university mathematicians argue that all students should be explicitly taught the discourse of advanced mathematics.
Other aims, not so directly related to mathematics, include the development of computing skills in cases where students will be required to use technology in their mathematics study at university and the development of study skills. Students may be required to partake in more than one bridging programme.
Examples of the first aim are one year or one semester bridging programmes for adults and young students who did not study enough mathematics at secondary school, such as those offered in Linkoping University, Sweden and Harvard University, USA. Other universities have short summer programmes to bring students up to the level expected, for example, Vrije Universiteit Brussel, Belgium, University of Cape Town, Republic of South Africa and University of Sydney, Australia.
Some universities have a comprehensive suite of bridging courses aimed at different groups of students, some have diagnostic tests, such as Warwick University, UK, and some have different levels and content depending on which degree programme you will be entering. An example is the Vrije Universiteit, Brussels, which has been offering bridging courses in mathematics for more than 20 years. The bridging courses are offered at different levels of difficulty and with different content, aimed at students entering different majors. They are offered during the month of September, before the start of the academic year, for a period varying from 2 to 4 weeks and they are taken on a voluntary basis (Grandsard, 1996). Part of the bridging programme includes a computer package on methods of proof and introductory problem solving (Grandsard, 1988).
London University (UK) has a bridging course that fulfils the second
aim. Students studying mathematics there have high entry grades but it
was felt that
students were finding difficulty with the transition to formal ideas.
For two weeks in September prior to entry to the university, students are
offered a voluntary 'top-up' course with the aim of introducing formal
ideas informally. Lecturers use a narrative style with informal explanations
rather than rigorous proofs. There are lectures in the mornings and problem
sessions in the afternoons.
Another trend in bridging courses is that bridging courses are offered for credit towards a degree. Bridging courses have become mainstrearned as more students require the skills and as lecturers realise that careful development of skills and attitudes in first year can lead to more effective leaming in later years: indeed it can lead to more students taking mathematics in later years. With the recognition that not all students enter tertiary study with the same background, new courses have emerged to cater for this clientele. Tall (1992) and others have documented the difficulties of the transition to tertiary mathematics. Students who have studied low levels of mathematics at secondary level may be able to study high school mathematics for credit at university but it may slow their progress through a degree or add more years to the length of a normal degree. There is some opposition to this trend, as students receive credit for what is essentially school level mathematics.
This is similar to trends in other subject areas such as physics, which few students now study to a high level at secondary school. What may have been considered school level physics in the past is now taught at the tertiary level. Indeed many disciplines such as economics, psychology and sociology are rarely offered at secondary school and must start from scratch at university. Mathematics needs to adapt to these changing circumstances and make all students who want to study mathematics at tertiary level welcome. Lecturers in statistics have made great strides in making statistics accessible to all students: mathematicians should do the same.
There are examples of subjects that aim to introduce students to advanced mathematical thinking. These subjects can be considered as bridging subjects even though they are for credit as part of a degree. Often there is opposition from mathematics faculty for the introduction of a subject of this type as these 'ideas' subjects take time away from process or content subjects. Mathematicians often find it easier to teach mathematics itself rather than take a step back and help students examine the ideas of mathematics. Faculty at Concordia University in Montreal, Canada have developed a course to assist students to refine their mathematical thinking. This course is a credit course that develops ideas about proof, proof techniques, and the use of precise language and notation (Hillel, 1999). Whilst this could not be considered a secondary level course it takes the role of bridging students to advanced mathematical thinking.
Another example of a subject that aims to bridge students to mathematical
thinking and mathematical reading and writing is on offer at the University
of Technology, Sydney, Australia. This subject was established to teach
students the skills of reading, writing, listening and speaking mathematics.
Again this is not a secondary level subject but it does bridge students'
academic language for university study. This subject is for all mathematics
majors regardless of their background and has been used to orient students
to the study of mathematics at university (Wood and Perrett, 1997). There
is evidence that students' reading skills in mathematics have improved
significantly.
Other questions concern bridging courses themselves. Are they successful in meeting their objectives? Do their graduates succeed in tertiary mathematics studies? In many cases formal evaluations are not done, though there is ample anecdotal evidence of success. All lecturers would have examples of students from disadvantaged backgrounds who have succeeded at tertiary study after completing a bridging course. As in much educational research, evaluation is difficult. Students are often choosing to do the bridging courses. Would these students have been successful anyway? Are other factors more important, such as peer groups and changes in assessment in first year studies? The situation is rarely static enough to allow easy comparisons between treatments.
Reactions by students on course experience questionnaires for bridging programmes have always been very positive. The numbers taking the courses are increasing which can be taken as a positive outcome. However, there are doubts about the long-term effects of remedial programmes. As mastery is required in the long term, it is important that students keep reviewing prerequisites regularly. And finally, the programme does not work for very weak students: no one can make up in one month for deficiencies that may have accumulated over six years.
There are groups of academics with a common interest in bridging programmes. They hold regular sessions at ICME congresses and these often reflect world-wide priorities, such as women returning to study or adults returning to mathematics. Another group that meets regularly is the Bridging Mathematics Network in Australasia. This is a group of university mathematicians involved in Mathematics Centres and bridging programmes. They have been holding biyearly conferences since 1992. The conferences cover many aspects: administration of bridging and support programmes, content, teaching methods and resource development. Teaching methods and resource development for disadvantaged groups are particularly emphasised.
There are major adaptations to be made by new tertiary students of mathematics.
Classes are often large and impersonal. Students have trouble coping with
large amounts of new material in a short time. Academic staff seem unapproachable
and there may be little support for students with difficulties. Students
are expected to do much of the work by themselves. There are many new computer
programs to learn. Tertiary educators are taking note of student difficulties
and are making changes but the question remains, are we expecting too much
of first year tertiary students?
Individual assistance. Some students may require individual assistance. Students with a record of failure or students with low confidence may improve significantly with individual assistance. Students with a disability may also require assistance. The assistance can take the form of counselling by psychologists in student learning or may be individual tuition by a member of the mathematics faculty or a combination of both. Though this is expensive in terms of time and personnel, many students will only require a few sessions to gain enough confidence and skills to succeed.
Enabling tutorials. Where a group with similar problems can be identified, a regular weekly tutorial is provided. This tutorial may be taught by a member of the mathematics faculty or by a professor who may be appointed to work particularly with students who are having difficulties with mathematics and statistics.
Workshops. Short workshops (3-6 hours) are provided for groups of students with particular problems, including examination review. These are generally held on Saturdays or in the evenings. Many universities have found these popular especially with students undertaking mathematics as a service subject in their engineering, business or science degrees.
Review mathematics. Students who clearly will not succeed with their mathematics subjects are encouraged to defer their mathematics study until after completion of some basic mathematics review.
Advice for staff. Because mathematics faculty is frequently dealing with students who have difficulty with mathematics and statistics, academic staff in all faculties may enquire about areas in their curriculum that may cause difficulty for their students.
Self-study materials. Review mathematics materials may be available in print, video or on the Web for students who do not wish to or are unable to attend class.
Academic support has been a very successful way to change attitudes to mathematics and to assist students to succeed. Peer tutoring (the use of later year students as tutors or mentors) in Academic Support Centres has also been successful.
The aim of orientation programmes has moved from information, such as finding the library and classrooms, to making the transition to university a more personal 'whole person' approach that includes how to find friends, join clubs and enjoy studying. There is significant evidence that affective domain variables have an important effect on the transition to university study and the decision to stay as shown in Montague and Hepperlin's (1997) study comparing the experiences of first year undergraduates who leave with those who stay.
In response to this study, the University of Technology, Sydney (Australia)
has introduced a Semester Zero, a coordinated orientation for new
first year students. Students are offered a range of activities in the
month between enrolment and commencing classes. Counsellors talk to students
about practical issues, such as housing and money, and provide a general
introduction to study skills. Campus clubs and the Student's Union provide
social and sporting activities and each faculty provides a more academic
programme. The University-wide activities start first and narrow down to
the student's own faculty just before classes begin. At enrolment, students
are given a detailed flow chart and a self-evaluation questionnaire to
check whether they have the prerequisite knowledge for their courses. Students
are referred to appropriate bridging courses (mathematics, chemistry, English)
where necessary.
The dialogue between secondary and university faculty has for too long been in one direction. As secondary school curricula change, reflecting increased use of technology as well as alternate pedagogical approaches, such as group activities, we (at the university level) need to listen to what the high school teachers have to say. This communication needs to be two- This section reports on educational systems where successful interaction has occurred.
In France there have been recent initiatives to increase the cooperation between secondary and tertiary mathematics teachers. The meetings included representatives of the academic mathematicians at universities and Grandes Ecoles and the mathematics societies (GRIAM) France also has for many years an Institut du Recherche sur L'Enseignment des Mathematiques in each Academy. These institutes are situated in universities. Secondary and tertiary teachers work together to produce innovations, make proposals for syllabuses, provide teacher in-service and are closely connected with didactic research. The institutes work both ways; university teaching is informed by the presence of secondary teachers.
In Colombia many tertiary mathematicians are now engaged in teacher pre- and in-service training for secondary schools. This was a government decision which came with financial incentives for universities to organise and supervise the initial teacher training and the continuing education courses required of teachers to increase their ranking and salary in their profession. Secondary school teachers influence and challenge the university professors with questions derived from their experience and the inventiveness of their pupils, with problems of philosophical, epistemological, historical or didactical import. This cooperation was achieved in a few years by the combined push of economic pressures on the public universities to self-finance part of their budget, and of economic incentives to engage in designing and orchestrating pre- and in- teacher education. One added advantage to this is that many university and secondary school mathematicians have built an atmosphere of solidarity for the cause of mathematics.
Another cooperative model is at the University Eduardo Mondlane in Mozambique
where staff is working in partnership with secondary schools, technical
institutes and teacher training colleges to improve education and leaming.
In other school systems, Belgium (Flanders) for example, university
mathematicians have little influence on secondary school outcomes, however,
recently an entrance examination for medical school has been installed,
and this will have some influence on secondary school teaching. In Australia
(South Australia), other disciplines on the secondary board governing upper
secondary curriculum have worked together to reduce the number of hours
for mathematics. This political aspect of mathematics education and control
of upper secondary curriculum and examinations has an important influence
on tertiary mathematics.
It is clear that the pre-eminent position of mathematics has been challenged, particularly in Western countries. Its position has been taken over by computing and computer-based technology. Fewer students are studying mathematics at higher levels in secondary schools, due to competition for time in the secondary curriculum. Fewer students are studying mathematics as a major at tertiary level. However the numbers studying some mathematics as part of a degree continues to increase, especially statistics and mathematical computing.
Universities are reacting to this in several ways. Some continue to define mathematics as a discipline associated with proof and analysis, some have moved to more applied areas of statistics, operations research and financial mathematics, some teach mathematics as a service subject to other disciplines. In all of these areas, university mathematicians find their students have changed. Reactions to changes include: bridging courses, more interaction with secondary teachers, changes in curriculum at first year tertiary level, and support for students with difficulties, Perhaps it is time to stop reacting and being more proactive in the transition to mathematics curriculum at tertiary level.
Crawford, K., Gordon, S., Nicholas, J. and Prosser, M. (1998). Qualitatively different experiences of learning mathematics at university. Learning and Instruction, 8, 5, 455-468.
de Guzmán, M., Hodgson, B.R. Robert, A. and Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. Documenta Mathematica. Extra Volume ICM, 747-762.
Grandsard, F. (1996). Bridging Courses to smoothen the transition from high school to university. Paper presented at ICMI 8, Sevilla.
Grandsard, F. (1988). Computer-assisted courses on Problem Solving and Methods of Proof. In H. Burkhardt, S. Groves, A.H. Schoenfeld and K.C. Stacey (Eds.), Problem Solving - A World View, pp. 274-278. Nottingham: Shell Centre.
Hillel, J. (1999). A Bridging Course on Mathematical Thinking. In W. Spunde, R. Hubbard and P. Cretchley (Eds.), The Challenge of Diversity, pp. 109-113. Toowoomba: Delta '99.
IEA (1995-96). Third International Mathematics and Science Study (TIMSS).
McInnis, C., James, R. and McNaught, C. (1995). First Year on Campus. Canberra: Australian Government Printing Service.
Tall, D.O. (1992). The transition to advanced mathematical thinking: functions, limits, infinity and proof. In D.A. Grouws (Ed.) Handbook of Research on Mathematics Teaching and Learning. pp. 495-511. New York: Macmillan.
Wood, L.N and Perrett, G. (1997). Advanced Mathematical Discourse. Sydney: UTS.
Leigh Wood, University of Technology, Sydney, Australia
leigh@maths.uts.edu.au