These Guidelines are intended to provide a resource for teachers of Junior Certificate mathematics. In particular they are designed to support the revised Junior Certificate syllabuses (introduced in September 2000) during the first cycle of implementation: that is, until the end of the academic year 2002/2003.
The current draft version of the Guidelines can be updated after 2003 to reflect the experience and wisdom of teachers who have taught the new syllabuses during their introductory period.
The document aims to address a variety of questions that may be asked about the syllabuses. Questions can be grouped under the headings "why", "what" and "how". The opening sections of the Guidelines focus especially on providing answers to questions of "why" type. Section 1 describes the background to and reasons for the revision of the mathematics syllabuses; Section 2, dealing with the specification of aims and objectives, emphasises the rationale and principles underlying their design; and Section 3, on syllabus content, outlines reasons for including specific topics. These sections also address issues of "what" type. For example, Section 3 highlights the changes that were made from the preceding syllabuses and the differences that may be expected in student knowledge and attitude at entry to the junior cycle following the introduction of the revised Primary School Curriculum. In Section 4, "how" questions are considered: how the work might be organised in the school or classroom, and how specific topics might be taught. This section describes some of the insights and practice of experienced teachers who have "made it work" in Irish classrooms. Against this exposition of where the learning of mathematics should be going and how it might get there, Section 5, on assessment, indicates ways in which information can be obtained on whether or not the students have arrived at the intended goals. A number of appendices provide reference material.
Thus, the different parts of the Guidelines are likely to be used in different ways. The initial parts offer a general orientation and invite reflection on the purposes of mathematics education in the junior cycle. The central sections provide a resource for the day-to-day work of mathematics teaching: a compendium of ideas, to which teachers may turn when they wish to improve the quality or vary the style of learning in their classrooms. The section dealing with assessment is likely to be of special interest to those preparing students for state examinations, but its brief is wider, with the focus being on formative as well as summative assessment. Altogether, it is hoped that the emphasis on meaningful and enjoyable learning will enrich the quality and enhance the effectiveness of our students' mathematics education.
The Guidelines are designed to support the teaching of mathematics in the junior cycle in such a way as to meet a wide range of learning needs. However, students with a mild general learning disability often face particular challenges in the area of mathematics. Additional guidelines for teachers of students with a mild general learning disability have been drawn up, and provide a further resource for all teachers of mathematics.
Many people too many to mention individually have contributed to the Guidelines. A special tribute must be paid to the teachers who provided the "lesson ideas" which form a key part of the document and also to those who made inputs to other sections. The engagement of so many people from outside the course committee, as well as of committee members, is a welcome feature of the process leading to the production of the draft Guidelines document. It represents a significant advance in the sharing of mathematics teaching methodologies at national level. As indicated above, the intention is that even more members of the mathematics education community will contribute their ideas over the period of operation of the draft Guidelines. The final version can then provide a resource that reflects as much good practice as possible in mathematics teaching in Ireland.
Context of the Changes
1.1 HISTORY OF THE SYLLABUSES
DEVELOPMENT OF THE JUNIOR CYCLE SYLLABUSES
The development of the current syllabuses can be traced back to the sixties: that period of great change in mathematics courses world-wide. The changes were driven by a philosophy of mathematics that had transformed the subject at university level and was then starting to penetrate school systems. It was characterised by emphasis on structure and rigour. Starting from sets, the whole edifice of mathematics could be built up logically, via relations and in particular functions; the structure laws (such as those we know as the commutative, associative and distributive properties) were golden threads tying the different parts of mathematics together. As a summary of the mathematics of the day, it was splendidly conceived and realised by the famous Bourbaki group in France; but it was devised as a state-of-the art summary of the discipline of mathematics, not as an introduction to the subject for young learners. Nonetheless, many countries embraced the philosophy or at least its outcome, revised subject-matter and rigorous presentation with great enthusiasm. In Ireland, teachers attended seminars given by mathematicians; the early years of the Irish Mathematics Teachers' Association were enlivened by discussions of the new material.
The first changes in the mathematics syllabuses in this period took place in the senior cycle; new Leaving Certificate syllabuses were introduced in 1964, for first examination in 1966. Meanwhile, thoroughly "modern" syllabuses were being prepared for the junior cycle. They were introduced in 1966, for first examination in 1969, and were provided at two levels: "Higher" and "Lower". (Prior to 1966, two levels had also been offered, but the less demanding of the two had been available only to girls.) These revisions brought in some topics which we now take for granted, such as sets and statistics, as well as a few, such as number bases, which have not stood the test of time. The new syllabuses also addressed a problem in their predecessors: students were finding difficulty with the (comparatively) traditional presentation of formal geometry difficulty, it can be said, shared by students in many other countries; so Papy's system, based on couples and transformations of the plane, was introduced alongside the existing version. For the examinations, papers clearly delineated by topic arithmetic, algebra and geometry were replaced by two papers in which a more integrated approach was taken, in keeping with the "modern" vision of mathematics.
The syllabuses introduced in 1966 ran for seven years. In 1973, revised versions were implemented in order to deal with some aspects that were causing difficulty. Two main alterations were made. First, the hybrid system of geometry was replaced by one entirely in the style of Papy. Secondly, the examination papers were redesigned so that the first section on each consisted of several compulsory multiple-choice items, effectively spanning the entire content relevant to the paper in question. The revised syllabuses were even more strongly Bourbakiste in character than their predecessors, with the ideas of set and function being intended to unify very many aspects of the syllabuses. However, it was hoped that emphasis would be given to arithmetic calculation and algebraic manipulation, which were felt to have suffered some neglect in the first rush of enthusiasm for the modern topics.
The size of the cohort taking the Intermediate Certificate increased in the 1970s. The rather abstract and formal nature of "modern" mathematics was not suitable for all the cohort, and further revisions were needed. In the early eighties, therefore, it was decided to introduce a third syllabus, geared to the needs of the less able. Also, some amendments were made to the former Higher and Lower level syllabuses. The package of three syllabuses, then called Syllabuses A, B and C, was introduced in 1987, for first examination in 1990. In the examination papers for Syllabuses A and B, question 1 took on the role of examining all topics in the relevant half of the syllabus, but the multiple-choice format was dropped in favour of the short-answer one that was already in use at Leaving Certificate level. A very limited choice was offered for the remaining questions (of traditional "long answer" format). For Syllabus C, a single paper with twenty short-answer questions was introduced.
The unified Junior Certificate programme was introduced in 1989. As the Intermediate Certificate mathematics syllabuses had been revised so recently, they were adopted as Junior Certificate syllabuses without further consideration (except that Syllabuses A, B and C were duly renamed the Higher, Ordinary and Foundation level syllabuses). Consequently, there was no opportunity to review the syllabuses thoroughly or to give due consideration to an appropriate philosophy and style for junior cycle mathematics in the 1990s and the new millennium.
The focus so far has been on syllabus content, and has indicated that the revolution in mathematics education in the 1960s has been followed by gradual evolution. Accompanying methodology received comparatively little attention in the ongoing debates. Choice of teaching method is not prescribed at national level; it is the domain of the teacher, the professional in the classroom. However, pointers could be given as to what was deemed appropriate, and the Preambles to the syllabuses introduced in 1973 and 1987 referred to the importance of understanding, the need for practical experience and the use of appropriate contexts. Now, with the increase in student retention and in view of the challenges posed by the information age, greater emphasis on methodology has become a matter of priority.
RECENT CHANGES AT OTHER LEVELS IN THE SYSTEM
To set the scene properly for the current revision of the Junior Certificate syllabuses, it is necessary to look also at what precedes and follows them in the students' education: the Primary School Curriculum and Leaving Certificate syllabuses.
The Leaving Certificate syllabuses were revised in the early 1990s. The revised syllabuses had to follow suitably from the then current Junior Certificate syllabuses, to fit into the existing senior cycle framework, and also to meet the needs of the world beyond school. This imposed some limitations on the scope of the revision. However, content was thoroughly critiqued for current relevance and suitability, some topics being discarded and a limited number of new ones being introduced. For the new Foundation level syllabus brought in as the Ordinary Alternative syllabus in 1990, and amended slightly and designated as being at Foundation level in 1995 some recommendations were made with respect to methodology. They emphasised the particular need for concrete approaches to concepts and for careful sequencing of techniques so that students could find meaning and experience success in their work. Much of the work was built round the use of calculators, which were treated as learning tools rather than just computational aids.
The introduction of calculators is also a feature of the revised Primary School Curriculum which was published in 1999. The revision was the first to be undertaken since the introduction of the radically restructured Primary School Curriculum in 1971. The revised curriculum is being implemented on a phased basis, with the mathematics element scheduled for introduction in 2002. As is the case for the second level syllabuses, the revolution of the earlier period has been followed by evolution. The 1971 curriculum emphasised discovery learning, and this has led to considerable use of concrete materials and activity methods in junior classes. The revised Primary School Curriculum focuses to a greater extent on problem-solving and on the need for students to encounter concepts in contexts to which they can relate. Students emerging from the revised curriculum should be more likely than their predecessors to look for meaning in their mathematics and less likely to see the subject almost totally in terms of the rapid performance of techniques. They may be more used to active learning, in which they have to construct meaning and understanding for themselves, rather than passively receiving information from their teachers. The detailed changes in content and emphasis likely to affect second level mathematics are outlined in Section 3.4.
1.2 DEVELOPMENT OF THE REVISED SYLLABUSES
EVALUATIONS OF THE 1987 SYLLABUSES
Under the jurisdiction of the NCCA, the mathematics (junior cycle) course committee was first convened in November 1990, and was asked to analyse the impact of the new junior cycle mathematics syllabus. A somewhat similar brief was given in 1992, coinciding with a wider review: that of the first examination of syllabuses introduced at the inception of the Junior Certificate in 1989. The committee produced a report in response to each request. Among the difficulties identified in one or both of these reports were
- the length of the Higher level syllabus (which had actually been shortened in 1987, but the restricted choice in the examination meant that greater coverage was required than before)
- aspects of the geometry syllabus, especially at Higher level
- the proscription of calculators in the examinations, and consequently their restricted use as learning tools and computational aids in the classroom
- design of the Higher and Ordinary level examination papers (the restricted choice being endorsed, but the absence of aims and objectives giving problems in specifying criteria for question design and for formulation of marking schemes).
Both reports included favourable comments on the appropriateness of the Foundation level syllabus newly introduced at that stage for most of the students who were taking it.
BRIEF FOR THE REVISIONS
In Autumn 1994, the course committee was asked to critique the Junior Certificate mathematics syllabuses, this time with a view to introducing some amendments if required. Because of the amount of change that had taken, and was taking, place in the junior cycle in other subject areas, it was specified that the outcomes of the reviewwould build on current syllabus provision and examinationapproaches rather than leading to a root and branchchange of either. Thus, once more, the syllabuses were to berevised rather than fundamentally redesigned. The review was to take into account
- the work being done by the NCCA with respect to the curriculum for the upper end of the primary school
- the earlier reviews carried out by the committee
- changing patterns of examination papers over recent years
- analysis of examination results since 1990.
The following tasks were set out for the committee:
- To identify the major issues of concern regarding the existing syllabuses in their design, implementation and assessment;
- To address the issues surrounding Foundation level mathematics;
- To draft an appropriate statement of aims and objectives for each of the three syllabuses in line with Junior Certificate practice;
- To prepare Guidelines to assist in improving the teaching of mathematics.
EXECUTION OF THE TASKS
The committee had already identified the main issues, as described above. Consultation with the constituencies, for example at meetings of the Irish Mathematics Teachers' Association, tended to confirm that these were indeed the areas of chief concern to mathematics teachers. Chief Examiners' reports and international studies involving Ireland provided further insights. Altogether, the information pointed to strengths of Irish mathematics education such as its sense of purpose and focus and the very good performance of the best students but also to weaknesses, for example with respect to students' basic skills and understanding in some key areas of the curriculum, their communication skills and their ability to apply knowledge in realistic contexts. Consequent changes eventually made to the syllabuses and proposed for the examinations are outlined in Sections 3.3 and 5.4 of this document.
Further consideration of the Foundation level syllabus led to endorsement of its main thrust and its appropriateness for many students at the weaker end of the mathematical ability spectrum. Against this background of general approval, however, the committee recognised some difficulties, both with the syllabus content and with the format of the Junior Certificate examination. Changes were needed to enrich the content and improve the standing of the syllabus and to give students more opportunity to show what they had learnt. The problem of students following the Foundation level syllabus, or taking the examination, when they are capable of working at the Ordinary level, also needed to be addressed. Again, consequent changes are described elsewhere in these Guidelines (see Sections 3.3 and 5.4).
FROM INTENTION TO IMPLEMENTATION
The course committee duly presented the final draft of the syllabus to the NCCA Council, and this was approved by Council in May 1998. Later in the same year, the Minister for Education and Science announced his decision to implement the syllabus. It was introduced into schools in Autumn 2000 for first examination in 2003.
Introduction of the syllabus is being accompanied by incareer development for teachers of mathematics. This has been planned so as to focus, not only on the changes in syllabus content, but also on the types of teachingmethodology that might facilitate achievement of the aims and objectives of the revised syllabus. These Guidelines are intended to complement the in-career development sessions. They can also support further study involving teachers or groups of teachers throughout the country.
Aims, Objectives and Principles of Syllabus Design
One of the tasks given to the Course Committee was to write suitable aims and objectives for Junior Certificate mathematics. Accordingly, aims and generalobjectives are specified in the Introduction to the syllabus document. They apply to all three syllabuses. To augment these broad aims and general objectives appropriately, each syllabus is introduced by a further specification of its purpose, by means of
- a rationale, describing the target group of students, the general scope and style of the syllabus, and aspects deserving particular emphasis in order to tailor the syllabus to the students' needs
- a statement of level-specific aims, highlighting aspects of the aims that are of particular relevance for the target group
- a listing of assessment objectives: a subset of the general objectives, to be interpreted in the light of the levelspecific aims and hence suitably for the ability levels, developmental stages and learning styles of the different groups of students.
Sections 2.2 to 2.4 of the Guidelines discuss these features and set them in context. Against this background, Section 2.5 outlines certain principles that guided and constrained design of the syllabuses to meet the aims.
2.2 AIMS FOR JUNIOR CERTIFICATE MATHEMATICS
The aims formulated for Junior Certificate mathematics are derived from those specified for the current Leaving Certificate syllabuses, with appropriate alterations to suit the junior rather than the senior cycle of second level education. The Leaving Certificate aims were based on those specified in the booklet Mathematics Education:Primary and Junior Cycle Post-Primary produced by the Curriculum and Examinations Board in 1985. In the absence of a specific formulation for the Junior Certificate, these were taken as the best approximation to contemporary thinking about mathematics education in Ireland.
The syllabus document presents a common set of aims for the three syllabuses (Higher, Ordinary and Foundation level). They can be summarised and explained as follows.
It is intended that mathematics education should:
- A) Contribute to the personal development of the students.
This aim is chiefly concerned with the students' feelings of worth as a result of finding meaning and interest, as well as achieving success, in mathematics.
- B) Help to provide them with the mathematical knowledge, skills and understanding needed for continuing their education, and eventually for life and work.
This aim focuses on what the students will be able to do with their mathematics in the future: hence, on their ability to recognise the power of mathematics and to apply it appropriately.
Section 3.5 of this document describes one vision (not the only possible one) of how these aims might be addressed in the different content areas.
2.3 GENERAL OBJECTIVES FOR JUNIOR CERTIFICATE MATHEMATICS
The aspirational aims need to be translated into more specific objectives which, typically, specify what students should be able to do at the end of the junior cycle. As with the aims, the general objectives are modelled on those for the current Leaving Certificate syllabuses, notably in this case the most recently formulated set produced for Foundation level.
The objectives listed in the syllabus document can be summarised and explained as follows.
A. Students should be able to recall basic facts.
That is, they should have fundamental information readily available for use. Such information is not necessarily an end in itself; rather, it can support (and enhance) understanding and aid application.
B. They should be equipped with the competencies needed for mathematical activities.
Hence, they should be able to perform the basic skills and carry out the routine algorithms that are involved in "doing sums" (or other exercises), and be able to use appropriate equipment (such as calculators and geometrical instruments) and they should also know when to do so. This kind of "knowing how" is called instrumental understanding: understanding that leads to getting something done.
C. They should have an overall picture of mathematics as a system that makes sense.
This involves understanding individual concepts and conceptual structures, and also seeing the subject as a logical discipline and an integrated whole. In general, this objective is concerned with "knowing why", or so-called relational understanding.
D. They should be able to apply their knowledge.
Thus, they should be able to use mathematics (and perhaps also to recognise uses beyond their own scope to employ) hence seeing that it is a powerful tool with many areas of applicability.
E. The students should be able to analyse information, including information presented in unfamiliar contexts.
In particular, this provides the basis for exploring and solving extended or non-standard problems.
F. They should be able to create mathematics for themselves.
Naturally, we do not expect the students to discover or invent significant new results; but they may make informed guesses and then critique and debate these guesses. This may help them to feel personally involved in, and even to attain a measure of ownership of, some of the mathematics they encounter.
G. They should have developed the necessary psychomotor skills to attain the above objectives.
Thus, for example, the students should be enabled to present their mathematics in an orderly way, including constructions and other diagrams where relevant, and to operate a calculator or calculator software.
H. They should be able to communicate mathematics, both verbally and in writing.
Thus, they should be able to describe their mathematical procedures and insights and explain their arguments in their own words; and they should be able to present their working and reasoning in written form.
I. They should appreciate mathematics.
For some students, appreciation may come first only from carrying out familiar procedures efficiently and "getting things right". However, this can provide the confidence that leads to enjoyable recognition of mathematics in the environment and to its successful application to areas of common or everyday experience. The challenge of problems, puzzles and games provides another source of enjoyment. Aesthetic appreciation may arise, for instance, from the study of mathematical patterns (those occurring in nature as well as those produced by human endeavour), and the best students may be helped towards identifying the more abstract beauty of form and structure.
J. They should be aware of the history of mathematics.
The history of mathematics can provide a human face for the subject, as regards both the personalities involved and the models provided for seeing mathematics as a lively and evolving subject.
It is important to note that the objectives, like the aims, are common to all the Junior Certificate syllabuses (Higher, Ordinary, and Foundation level). However, they are intended to be interpreted appropriately at the different levels, and indeed for different students, bearing in mind their abilities, stages of development, and learning styles. This is particularly relevant for assessment purposes, as discussed in Section 5. It leads to the formulation of level-specific aims and their application to assessment objectives, discussed in Section 2.4.
2.4 RATIONALE, LEVEL-SPECIFIC AIMS AND ASSESSMENT OBJECTIVES
The aims and objectives together provide a framework for all the syllabuses. However, the fact that there are three different syllabuses reflects the varying provision that has to be made for those with different needs. Each syllabus has its own rationale, spelled out in the syllabus document. Key phrases in the three rationales are juxtaposed in the following table in order to highlight the intended thrust of each syllabus.
|RATIONALE FOR THE HIGHER LEVEL||RATIONALE FOR THE ORDINARY LEVEL||RATIONALE FOR THE FOUNDATION LEVEL|
|[This] is geared to the needs of students of above average mathematical ability.... However, not all students ... are ... future users of academic mathematics.||[This] is geared to the needs of students of average mathematical ability.||[This] is geared to the needs of students who are unready for or unsuited by the mathematics of the Ordinary [level syllabus].|
|For the target group, particular emphasis can be placed on the development of powers of abstraction and generalisation and on an introduction to the idea of proof.||For the target group, particular emphasis can be placed on the development of mathematics as a body of knowledge and skills that makes sense and that can be used in many different ways hence, as an efficient system for the solution of problems and provision of answers.||For the target group, particular emphasis can be placed on promoting students' confidence in themselves (confidence that they can do mathematics) and in the subject (confidence that mathematics makes sense).|
|A balance must be struck, therefore, between challenging the most able students and encouraging those who are developing a little more slowly.||[It] ... must start where these students are, offering mathematics that is meaningful and accessible to them at their present stage of development. It should also provide for the gradual introduction of more abstract ideas.||[It] must therefore help the students to construct a clearer knowledge of, and to develop improved skills in, basic mathematics, and to develop an awareness of its usefulness.|
In the light of these rationales, level-specific aims emphasise the various skills in ways that, hopefully, are appropriate to the levels of development of the target groups. The table opposite presents the aims for the three levels, as set out in the syllabus document.
|SPECIFIC AIMS FOR HIGHER LEVEL||SPECIFIC AIMS FOR ORDINARY LEVEL||SPECIFIC AIMS FOR FOUNDATION LEVEL|
[This] is intended to provide students with
- a firm understanding of mathematical concepts and relationships
- confidence and competence in basic skills
- the ability to formulate and solve problems
- an introduction to the idea of proof and to the role of logical argument in building up a mathematical system
- a developing appreciation of the power and beauty of mathematics and of the manner in which it provides a useful and efficient system for the formulation and solution of problems.
[This] is intended to provide students with
- an understanding of mathematical concepts and of their relationships
- confidence and competence in basic skills
- the ability to solve problems
- an introduction to the idea of logical argument
- appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems.
[This] is intended to provide students with
- an understanding of basic mathematical concepts and relationships
- confidence and competence in basic skills
- the ability to solve simple problems
- experience of following clear arguments and of citing evidence to support their own ideas
- appreciation of mathematics both as an enjoyable activity through which they experience success and as a useful body of knowledge and skills.
Against this background, objectives can be specified for assessment leading to certification as part of the Junior Certificate. Mathematics to date in the Junior Certificate has been assessed solely by a terminal examination; consideration of alternative forms of assessment was outside the scope of the current revision. This has a considerable effect on what can be assessed for certification purposes a point taken up more fully in Section 5 of this document.
The assessment objectives are objectives A, B, C, D(dealing with knowledge, understanding andapplication), G (dealing with psychomotor skills) and H(dealing with communication). These objectives, while the same for all three syllabuses, are to be interpreted in the context of the level-specific aims as described above. Further illustration is provided in Section 5.
2.5 PRINCIPLES OF SYLLABUS DESIGN
The aims and objectives are important determinants of syllabus content; but so also is the context in which the syllabuses are implemented. The most attractive syllabuses are doomed to failure if they cannot be translated into action in the classroom. With this in mind, several principles were specified in order to guide the syllabus design. They are particularly important for understanding the inclusion, exclusion, form of presentation, or intended sequencing of certain topics. The principles are displayed (in "boxes") overleaf.
A. The mathematics syllabuses should provide continuation from and development of the primary school curriculum, and should lead to appropriate syllabuses in the senior cycle.
Hence, the syllabuses should take account of the varied backgrounds, likely learning styles, potential for development, and future needs of the students entering second level education. Moreover, for the cohort of students proceeding from each junior cycle syllabus into the senior cycle, there should be clear avenues of progression.
B. The syllabuses should be implementable in the present circumstances and flexible as regards future development.
They should therefore be teachable, learnable and adaptable.
The points regarding teachability, learnability and adaptability can be considered in turn.
(a) The syllabuses should be teachable, in that it should be possible to implement them with the resources available.
- The syllabuses should be teachable in the time normally allocated to a subject in the Junior Certificate programme.
- Requirements as regards equipment should not go beyond that normally found in, or easily acquired by, Irish schools.
- The aims and style of the syllabuses should be ones that teachers support and can address with confidence, and the material should in general be familiar.
(b) The syllabuses should be learnable, by virtue of being appropriate to the different cohorts of students for whom they are designed.
- Each syllabus should start where the students in its target group are at the time, should move appropriately from the concrete to or towards the abstract, and should proceed to suitable levels of difficulty.
- The approaches used should accommodate the widest possible range of abilities and learning styles.
- They should cater for the interests and needs of all groups in the population.
- The materials and methods should be such that students are motivated to learn.
(c) The syllabuses should be adaptable designed so that they can serve present needs and also can evolve in future.
C. The mathematics they contain should be sound, important and interesting.
In order to cater for the differing interests of students and teachers, a broad range of appropriate aspects of mathematics should be included. Where possible the mathematics should be applicable, and the applications should be such that they can be made clear to the students (now, rather than in some undefined future) and can be addressed, at least to some extent, within the course.