Syllabus Structure and Content

3.1 INTRODUCTION

The way in which the mathematical content in the syllabuses is organised and presented in the syllabus document is described in sections 3.2 and 3.3. Section 3.3 also discusses the main alterations, both in content and in emphasis, with respect to the preceding versions. The forthcoming changes in the primary curriculum, which will have "knock-on" effects at second level, are outlined in Section 3.4. Finally, in Section 3.5, the content is related to the aims of the syllabuses.

3.2 STRUCTURE

For the Higher and Ordinary level syllabuses, the mathematical material forming the content is divided into eight sections, as follows:

• Sets
• Number systems
• Applied arithmetic and measure
• Algebra
• Statistics
• Geometry
• Trigonometry
• Functions and graphs

The corresponding material for the Foundation level syllabus is divided into seven sections; there are minor differences in the sequence and headings, resulting in the following list:

• Sets
• Number systems
• Applied arithmetic and measure
• Statistics and data handling
• Algebra
• Relations, functions and graphs
• Geometry

The listing by content area is intended to give mathematical coherence to the syllabuses, and to help teachers locate specific topics (or check that topics are not listed). The content areas are reasonably distinct, indicating topics with different historical roots and different main areas of application. However, they are inter-related and interdependent, and it is not intended that topics would be dealt with in total isolation from each other. Also, while the seven or eight areas, and the contents within each area, are presented in a logical sequence ­ combining, as far as possible, a sensible mathematical order with a developmental one for learners ­ it is envisaged that many content areas listed later in the syllabus would be introduced before or alongside those listed earlier. (For example, geometry appears near the end of the list, but the course committee specifically recommends that introductory geometrical work is started in First Year, allowing plenty of time for the ideas to be developed in a concrete way, and thoroughly understood, before the more abstract elements are introduced.) However, the different order of listing for the Foundation level syllabus does reflect a suggestion that the introduction of some topics (notably formal algebra) might be delayed. Some of these points are taken up in Section 4.

Appropriate pacing of the syllabus content over the three years of the junior cycle is a challenge. Decisions have to be made at class or school level. Some of the factors affecting the decisions are addressed in these Guidelines in Section 4, under the heading of planning and organisation.

3.3 SYLLABUS CONTENT

The contents of the Higher, Ordinary and Foundation level syllabuses are set out in the corresponding sections of the syllabus document. In each case, the content is presented in the two-column format used for the Leaving Certificate syllabuses introduced in the 1990s, with the lefthand column listing the topics and the right-hand column adding notes (for instance, providing illustrative examples, or highlighting specific aspects of the topics which are included or excluded). Further illustration of the depth of treatment of topics is given in Section 5 (in dealing with assessment) and in the proposed sample assessmentmaterials (available separately).

CHANGES IN CONTENT

As indicated in Section 1, the revisions deal only with specific problems in the previous syllabuses, and do not reflect a root-and-branch review of the mathematics education appropriate for students in the junior cycle. The main changes in content, addressing the problems identified in Section 1, are described below. A summary ofall the changes is provided in Appendix 1.

Calculators and calculator-related techniques

As pointed out in the introduction to each syllabus, calculators are assumed to be readily available for appropriate use, both as teaching/learning tools and as computational aids; they will also be allowed in examinations.

The concept of "appropriate" use is crucial here. Calculators are part of the modern world, and students need to be able to use them efficiently where and when required. Equally, students need to retain and develop their feel for number, while the execution of mental calculations, for instance to make estimates, becomes even more important than it was heretofore. Estimation, which was not mentioned in the 1987 syllabus (though it was covered in part by the phrase "the practice of approximating before evaluating"), now appears explicitly and will be tested in examinations.

The importance of the changes in this area is reflected in two developments. First, a set of guidelines on calculators is being produced. It addresses issues such as the purchase of suitable machines as well as the rationale for their use. Secondly, in 1999 the Department of Education and Science commissioned a research project to monitor numeracy-related skills (with and without calculators) over the period of introduction of the revised syllabuses. If basic numeracy and mental arithmetic skills are found to disimprove, remedial action may have to be taken. It is worth noting that research has not so far isolated any consistent association between calculator use in an education system and performance by students from that system in international tests of achievement.

Mathematical tables are not mentioned in the content sections of the syllabus, except for a brief reference indicating that they are assumed to be available, likewise for appropriate use. Teachers and students can still avail of them as learning tools and for reference if they so wish. Tables will continue to be available in examinations, but questions will not specifically require students to use them.

Geometry

The approach to synthetic geometry was one of the major areas which had to be confronted in revising the syllabuses. Evidence from examination scripts suggested that in many cases the presentation in the 1987 syllabus was not being followed in the classroom. In particular, in the Higher level syllabus, the sequence of proofs and intended proof methods were being adapted. Teachers were responding to students' difficulties in coping with the approach that attempted to integrate transformational concepts with those more traditionally associated with synthetic geometry, as described in Section 1.1 of these Guidelines.

For years, and all over the world, there have been difficulties in deciding how ­ indeed, whether ­ to present synthetic geometry and concepts of logical proof to students of junior cycle standing. Their historical importance, and their role as guardians of one of the defining aspects of mathematics as a discipline, have led to a wish to retain them in the Irish mathematics syllabuses; but the demands made on students who have not yet reached the Piagetian stage of formal operations are immense. "Too much, too soon" not only contravenes the principle of learnability (section 2.5), but leads to rote learning and hence failure to attain the objectives which the geometry sections of the syllabuses are meant to address. The constraints of a minor revision precluded the question of "whether" from being asked on this occasion. The question of "how" raises issues to do with the principles of soundness versus learnability. The resulting formulation set out in the syllabus does not claim to be a full description of a geometrical system. Rather, it is intended to provide a defensible teaching sequence that will allow students to learn geometry meaningfully and to come to realise the power of proof. Some of the issues that this raises are discussed in Appendix 2.

The revised version can be summarised as follows.

• The approach omits the transformational elements, returning to a more traditional approach based on congruency.
• In the interests of consistency and transfer between levels, the underlying ideas are basically the same across all three syllabuses, though naturally they are developed to very different levels in the different syllabuses.
• The system has been carefully formulated to display the power of logical argument at a level which ­ hopefully ­ students can follow and appreciate. It is therefore strongly recommended that, in the classroom, material is introduced in the sequence in which it is listed in the syllabus document. For theHigher level syllabus, the concepts of logicalargument and rigorous proof are particularlyimportant. Thus, in examinations, attempted proofsthat presuppose "later" material in order to establish"earlier" results will be penalised. Moreover, proofsusing transformations will not be accepted.
• To shorten the Higher level syllabus, only some of the theorems have been designated as ones for which students may be asked to supply proofs in the examinations. The other theorems should still beproved as part of the learning process;students should be able to follow the logical development, and see models of far more proofs than they are expected to reproduce at speed under examination conditions. The required saving of time is expected to occur because students do not have to put in the extra effort needed to develop fluency in writing out particular proofs.
• Students taking the Ordinary and (a fortiori)Foundation level syllabuses are not required to prove theorems, but ­ in accordance with the level-specific aims (Section 2.4) ­ should experience the logical reasoning involved in ways in which they can understand it. The general thrust of the synthetic geometry section of the syllabuses for these students is not changed from the 1987 versions.
• It may be noted that the formulation of the Foundation level syllabus in 1987 emphasised the learning process rather than the product or outcomes. In the current version, the teaching/learning suggestions are presented in theseGuidelines (chiefly in Section 4), not in the syllabus document. It is important to emphasise that the changed formulation in the syllabus is not meant to point to a more formal presentation than previously suggested for Foundation level students.

Section 4.9 of this document contains a variety of suggestions as to how the teaching of synthetic geometry to junior cycle students might be addressed.

Transformation geometry still figures in the syllabuses, but is treated separately from the formal development of synthetic geometry. The approach is intended to be intuitive, helping students to develop their visual and spatial ability. There are opportunities here to build on the work on symmetry in the primary curriculum and to develop aesthetic appreciation of mathematical patterns.

Other changes to the Higher level syllabus

• Logarithms are removed. Their practical role as aids to calculation is outdated; the theory of logarithms is sufficiently abstract to belong more comfortably to the senior cycle.
• Many topics are "pruned" in order to shorten the syllabus.

Other changes to the Ordinary level syllabus

• The more conceptually difficult areas of algebra and coordinate geometry are simplified.
• A number of other topics are "pruned".

Other changes to the Foundation level syllabus

• There is less emphasis on fractions but rather more on decimals. (The change was introduced partly because of the availability of calculators ­ though, increasingly, calculators have buttons and routines which allow fractions to be handled in a comparatively easy way.)
• The coverage of statistics and data handling is increased. These topics can easily be related to students' everyday lives, and so can help students to recognise the relevance of mathematics. They lend themselves also to active learning methods (such as those presented in Section 4) and the use of spatial as well as computational abilities. Altogether, therefore, the topics provide great scope for enhancing students' enjoyment and appreciation of mathematics. They also give opportunities for developing suitably concrete approaches to some of the more abstract material, notably algebra and functions (see Section 4.8).
• The algebra section is slightly expanded. The formal algebraic content of the 1987 syllabus was so slight that students may not have had scope to develop their understanding; alternatively, teachers may have chosen to omit the topic. The rationale for the present adjustment might be described as "use it or lose it". The hope is that the students will be able to use it, and that ­ suitably addressed ­ it can help them in making some small steps towards the more abstract mathematics which they may need to encounter later in the course of their education.

Overall, therefore, it is hoped that the balance between the syllabuses is improved. In particular, the Ordinary level syllabus may be better positioned between a more accessible Higher level and a slightly expanded Foundation level.

CHANGES IN EMPHASIS

The brief for revision of the syllabuses, as described in Section 1.2, precluded a root-and-branch reconsideration of their style and content. However, it did allow for some changes in emphasis: or rather, in certain cases, for some of the intended emphases to be made more explicit and more clearly related to rationale, content, assessment, and ­ via the Guidelines ­ methodology. The changes in, or clarification of, emphasis refer in particular to the following areas.

Understanding

General objectives B and C of the syllabus refer respectively to instrumental understanding (knowing "what" to do or "how" to do it, and hence being able to execute procedures) and relational understanding (knowing "why", understanding the concepts of mathematics and the way in which they connect with each other to form so-called "conceptual structures"). When people talk of teaching mathematics for ­ or learning it with ­ understanding, they usually mean relational understanding. The language used in the Irish syllabuses to categorise understanding is that of Skemp; the objectives could equally well have been formulated in terms of "procedures" and "concepts".

Research points to the importance of both kinds of understanding, together with knowledge of facts (general objective A), as components of mathematical proficiency, with relational understanding being crucial for retaining and applying knowledge. The Third International Mathematics and Science Study, TIMSS, indicated that Irish teachers regard knowledge of facts and procedures as particularly important ­ unusually so in international terms; but it would appear that less heed is paid to conceptual/relational understanding. This is therefore given special emphasis in the revised syllabuses. Such understanding can be fostered by active learning, as described and illustrated in Section 4. Ways in which relational understanding can be assessed are considered in Section 5.

Communication

General objective H of the syllabus indicates that students should be able to communicate mathematics, both verbally and in written form, by describing and explaining the mathematical procedures they undertake and by explaining their findings and justifying their conclusions. This highlights the importance of students expressing mathematics in their own words. It is one way of promoting understanding; it may also help students to take ownership of the findings they defend, and so to be more interested in their mathematics and more motivated to learn.

The importance of discussion as a tool for ongoing assessment of students' understanding is highlighted in Section 5.2. In the context of examinations, the ability to show different stages in a procedure, explain results, give reasons for conclusions, and so forth, can be tested; some examples are given in Section 5.6.

Appreciation and enjoyment

General objective I of the syllabus refers to appreciating mathematics. As pointed out earlier, appreciation may develop for a number of reasons, from being able to do the work successfully to responding to the abstract beauty of the subject. It is more likely to develop, however, when the mathematics lessons themselves are pleasant occasions.

In drawing up the revised syllabus and preparing the Guidelines, care has been taken to include opportunities for making the teaching and learning of mathematics more enjoyable. Enjoyment is good in its own right; also, it can develop students' motivation and hence enhance learning. For many students in the junior cycle, enjoyment (as well as understanding) can be promoted by the active learning referred to above and by placing the work in appropriate meaningful contexts. Section 4 contains many examples of enjoyable classroom activities which promote both learning and appreciation of mathematics. Teachers are likely to have their own battery of such activities which work for them and their classes. It is hoped that these can be shared amongst their colleagues and perhaps submitted for inclusion in the final version of the Guidelines.

Of course, different people enjoy different kinds of mathematical activity. Appreciation and enjoyment do not come solely from "games"; more traditional classrooms also can be lively places in which teachers and students collaborate in the teaching and learning of mathematics and develop their appreciation of the subject. Teachers will choose approaches with whichthey themselves feel comfortable and which meet thelearning needs of the students whom they teach.

The changed or clarified emphasis in the syllabuses will be supported, where possible, by corresponding adjustments to the formulation and marking of Junior Certificate examination questions. While the wording ofquestions may be the same, the expected solutions may bedifferent. Examples are given in Section 5.

3.4 CHANGES IN THE PRIMARY CURRICULUM

The changes in content and emphasis within the revised Junior Certificate mathematics syllabuses are intended, inter alia, to follow on from and build on the changes in the primary curriculum. The forthcoming alterations (scheduled to be introduced in 2002, but perhaps starting earlier in some classrooms, as teachers may anticipate the formal introduction of the changes) will affect the knowledge and attitudes that students bring to their second level education. Second level teachers need to be prepared for this. A summary of the chief alterations is given below; teachers are referred to the revised Primary School Curriculum for further details.

CHANGES IN EMPHASIS

In the revised curriculum, the main changes of emphasis are as follows.

There is more emphasis on

• setting the work in real-life contexts
• learning through hands-on activities (using concrete materials/manipulatives, and so forth)
• understanding (in particular, gaining appropriaterelational understanding as well as instrumentalunderstanding)
• appropriate use of mathematical language
• recording
• problem-solving.

There is less emphasis on

• learning routine procedures with no context provided
• doing complicated calculations.

CHANGES IN CONTENT

The changes in emphasis are reflected in changes to the content, the main ones being as follows.

New areas include

• introduction of the calculator from Fourth Class (augmenting, not replacing, paper-and-pencil techniques)
• (hence) extended treatment of estimation;
• increased coverage of data handling
• introduction of basic probability ("chance").

New terminology includes

• the use of the "positive" and "negative" signs for denoting a number (as in +3 [positive three], -6 [negative six] as well as the "addition" and "subtraction" signs for denoting an operation (as in 7 + 3, 24 ­ 9)
• explicit use of the multiplication sign in formulae (as in 2 ×r , l ×w).

The treatment of subtraction emphasises the "renaming" or "decomposition" method (as opposed to the "equal additions" method ­ the one which uses the terminology "paying back") even more strongly than does the 1971 curriculum. Use of the word "borrowing" is discouraged.

The following topics are among those excluded from the revised curriculum:

• unrestricted calculations (thus, division is restricted to at most four-digit numbers being divided by at most two-digit numbers, and ­ for fractions ­ to division of whole numbers by unit fractions)
• subtraction of negative integers
• formal treatment of LCM and HCF
• use of formulae which the children have not developed
• two-step equations and "rules" for manipulating equations (that is, emphasis is on intuitive solution)
• sets (except their intuitive use in developing the concept of number)
• Pi and advanced properties of circles.

(Some of these topics were not formally included in the 1971 curriculum, but appeared in textbooks and were taught in many classrooms.)

NOTE

The reductions in content have removed some areas of overlap between the 1971 Primary School Curriculum and the Junior Certificate syllabuses. Some overlap remains, however. This is natural; students entering second level schooling need to revise the concepts and techniques that they have learnt at primary level, and also need to situate these in the context of their work in the junior cycle.

3.5 LINKING CONTENT AREAS WITH AIMS

Finally, in this section, the content of the syllabuses is related to the aims and objectives. In fact most aims and objectives can be addressed in most areas of the syllabuses. However, some topics are more suited to the attainment of certain goals or the development of certain skills than are others. The discussion below highlights some of the main possibilities, and points to the goals that might appropriately be emphasised when various topics are taught and learnt. Phrases italicised are quoted or paraphrased from the aims as set out in the syllabus document. Section 5 of these Guidelines indicates a variety of ways in which achievement of the relevant objectives might be encouraged, tested or demonstrated.

SETS

Sets provide a conceptual foundation for mathematics and a language by means of which mathematical ideas can be discussed. While this is perhaps the main reason for which set theory was introduced into school mathematics, its importance at junior cycle level can be described rather differently.

• Set problems, obviously, call for skills of problem-solving; in particular, they provide occasions for logical argument. By using data gathered from the class, they even offer opportunities for simple introduction to mathematical modelling in contexts to which the students can relate.
• Moreover, set theory emphasises aspects of mathematics that are not purely computational. Sets are about classification, hence about tidiness and organisation. This can lead toappreciation of mathematics on aesthetic grounds and can help to provide a basis forfurther education in the subject.
• An additional point is that this topic is not part of the Primary School Curriculum, and so represents a new start, untainted by previous failure. For some students, therefore, there are particularly important opportunities for personal development.

NUMBER SYSTEMS

While mathematics is not entirely quantitative, numeracy is one of its most important aspects. Students have been building up their concepts of numbers from a very early stage in their lives. However, moving from familiarity with natural numbers (and simple operations on them) to genuine understanding of the various forms in which numbers are presented and of the uses to which they are put in the world is a considerable challenge.

• Weakness in this area destroys students' confidence andcompetence by depriving them of theknowledge, skillsand understanding needed for continuing theireducation and for life and work. It therefore handicaps their personal fulfilment and hencepersonaldevelopment.

The aspect of "understanding" is particularly important ­ or, perhaps, has had its importance highlighted ­ with advances in technology.

• Students need to become familiar with the intelligent and appropriate use of calculators, while avoiding dependence on the calculators for simple calculations.
• Complementing this, they need to develop skills in estimation and approximation, so that numbers can be used meaningfully.

APPLIED ARITHMETIC AND MEASURE

This topic is perhaps one of the easiest to justify in terms of providing mathematics needed for life, workand leisure.

• Students are likely to use the skills developed here in "everyday" applications, for example in looking after their personal finances and in structuring the immediate environment in which they will live. For many, therefore, this may be a key section in enabling studentsto develop a positive attitude towards mathematics as avaluable subject of study.
• There are many opportunities for problem-solving, hopefully in contexts that the students recognise as relevant.
• The availability of calculators may remove some of the drudgery that can be associated with realistic problems, helping the students to focus on the concepts and applications that bring the topics to life.

ALGEBRA

Algebra was developed because it was needed ­ because arguments in natural language were too clumsy or imprecise. It has become one of the most fundamental tools for mathematics.

• As with number, therefore, confidence andcompetence are very important. Lack of these underminethepersonal development of the students by depriving them of the knowledge, skills and understanding needed forcontinuing their education and for life and work.
• Without skills in algebra, students lack the technical preparation for study of other subjects in school, and in particular their foundation for appropriate studies lateron ­ including further education in mathematics itself.

It is thus particularly important that students develop appropriate understanding of the basics of algebra so that algebraic techniques are carried out meaningfully and not just as an exercise in symbol-pushing.

• Especially for weaker students, this can be very challenging because algebra involves abstractions andgeneralisations.
• However, these characteristics are among the strengths and beauties of the topic. Appropriately used, algebra can enhance the students' powers of communication, facilitate simplemodelling and problem-solving, and hence illustrate the power of mathematics as a valuablesubject of study.

STATISTICS

One of the ways in which the world is interpreted for us mathematically is by the use of statistics. Their prevalence, in particular on television and in the newspapers, makes them part of the environment in which children grow up, and provides students with opportunities for recognition and enjoyment of mathematics in the worldaround them.

• Many of the examples refer to the students' typical areas of interest; examples include sporting averages and trends in purchases of (say) CDs.
• Students can provide data for further examples from their own backgrounds and experiences.
• Presenting these data graphically can extend students'powers of communication and their ability to shareideas with other people, and may also provide anaesthetic element.
• The fact that statistics can help to develop a positiveattitude towards mathematics as an interesting andvaluable subject of study ­ even for weaker students who find it hard to appreciate the more abstract aspects of the subject ­ explains the extra prominence given to aspects of data handling in the Foundation level syllabus, as mentioned earlier. They may be particularly important in promotingconfidence and competence in both numerical and spatial domains.

GEOMETRY

The study of geometry builds on the primary school study of shape and space, and hence relates to mathematics in the world around us. In the junior cycle, different approaches to geometry address different educational goals.

Synthetic geometry is traditionally intended to promote students' ability to recognise and present logicalarguments.

• More able students address one of the greatest of mathematical concepts, that of proof, and hopefully come to appreciate theabstractions andgeneralisations involved.
• Other students may not consider formal proof, but should be able to draw appropriate conclusions from given geometrical data.

MATHEMATICS

• Explaining and defending their findings, in either case, should help students to further their powersof communication.
• Tackling "cuts" and other exercises based on the geometrical system presented in the syllabus allows students to develop their problem-solving skills.
• Moreover, in studying synthetic geometry, students are encountering one of the great monuments to intellectual endeavour: a very special part of Western culture.

Transformation geometry builds on the study of symmetry at primary level. As the approach to transformation geometry in the revised Junior Certificate syllabus is intuitive, it is included in particular for its aesthetic value.

• With the possibility of using transformations in artistic designs, it allows students to encounter creative aspectsof mathematics and to develop or exercise their own creative skills.
• It can also develop their spatial ability, hopefully promotingconfidence and competence in this area.
• Instances of various types of symmetry in the natural and constructed environment give scope for students' recognition and enjoyment of mathematics in the worldaround them.

Coordinate geometry links geometrical and algebraic ideas. On the one hand, algebraic techniques can be used to establish geometric results; on the other, algebraic entities are given pictorial representations.

• Its connections with functions and trigonometry, as well as algebra and geometry, make it a powerful tool for the integration of mathematics into a unified structure.
• It illustrates the power of mathematics, and so helps to establish it with students as a valuable subject of study.
• It provides an important foundation for appropriatestudies later on.
• The graphical aspect can add a visually aestheticdimension to algebra.

TRIGONOMETRY

Trigonometry is a subject that has its roots in antiquity but is still of great practical use to-day. While its basic concepts are abstract, they can be addressed through practical activities.

• Situations to which it can be applied ­ for example, house construction, navigation, and various ball games ­ include many that are relevant to the students' life, work and leisure.
• It can therefore promote the students' recognition andenjoyment of mathematics in the world around them.
• With the availability of calculators, students may more easily develop competence and confidence through their work in this area.

FUNCTIONS AND GRAPHS

The concept of a function is crucial in mathematics, and students need a good grasp of it in order to prepare a firm foundation for appropriate studies lateron and in particular, a basis for further education inmathematics itself.

• The representation of functions by graphs adds a pictorial element that students may find aesthetic as well as enhancing their understanding and their abilityto handle generalisations.
• This topic pulls together much of the groundwork done elsewhere, using the tools introduced and skills developed in earlier sections and providing opportunities forproblem-solving and simple modelling.

For Foundation students alone, simple work on the set-theoretic treatment of relations has been retained. In contexts that can be addressed by those whose numerical skills are poor, it provides exercises in simple logical thinking.

NOTE

The foregoing argument presents just one vision of the rationale for including the various topics in the syllabus and for the ways in which the aims of the mathematics syllabus can be achieved. All teachers will have their own ideas about what can inspire and inform different topic areas. Their own personal visions of mathematics, and their particular areas of interest and expertise, may lead them to implement the aims very differently from the way that is suggested here. Visions can profitably be debated at teachers' meetings, with new insights being given and received as a result.

The tentative answers given here with regard to whycertain topics are included in the syllabus are, of course, offered to teachers rather than junior cycle students. In some cases, students also may find the arguments relevant. In other cases, however, the formulation is too abstract or the benefit too distant to be of interest. This, naturally, can cause problems. Clearly it would not be appropriate to reduce the syllabus to material that has immediately obvious applications in the students' everyday lives. This would leave them unprepared for further study, and would deprive them of sharing parts of our culture; in any case, not all students are motivated by supposedly everyday topics.

Teachers are therefore faced with a challenging task in helping students find interest and meaning in all parts of the work. Many suggestions with proven track records in Irish schools are offered in Section 4. As indicated earlier, it is hoped that teachers will offer more ideas for an updated version of the Guidelines.