Junior Certificate Mathematics
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AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA MATHEMATICS Junior Certificate GUIDELINES FOR TEACHERS THESE GUIDELINES SYLLABUS STRUCTURE & CONTEXT INTRODUCTION • changes in the primary curriculum • teaching mathematics • linking content areas with aims CONTEXT OF THE APPROACHES TO PLANNING CHANGES & METHODOLOGY • history of syllabuses • lesson ideas • development of the • examples revised syllabuses ASSESSMENT AIMS, OBJECTIVES & • specifications PRINCIPLES OF • marking scheme SYLLABUS DESIGN PLUS helpful hints, changes and clarifications, and much more... MATHEMATICS CONTENTS INTRODUCTION TO DRAFT GUIDELINES 1 1 CONTEXT OF THE CHANGES 3 2 AIMS, OBJECTIVES AND PRINCIPLES OF SYLLABUS DESIGN 7 3 SYLLABUS STRUCTURE AND CONTENT 13 4 APPROACHES TO PLANNING AND METHODOLOGY 23 5 ASSESSMENT 89 6 APPENDICES 101 1. Corrections, clarifications and changes 102 2. Notes on geometry 106 3. Resources 111 4. Tables for planning and recording 114 MATHEMATICS Introduction to the Draft Guidelines π 1 MATHEMATICS INTRODUCTION 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. he current draft version of the Guidelines can be mathematics teaching: a compendium of ideas, to which Tupdated after 2003 to reflect the experience and teachers may turn when they wish to improve the quality or wisdom of teachers who have taught the new syllabuses vary the style of learning in their classrooms. The section during their introductory period. dealing with assessment is likely to be of special interest to those preparing students for state examinations, but its brief he document aims to address a variety of questions is wider, with the focus being on formative as well as that may be asked about the syllabuses. Questions T summative assessment. Altogether, it is hoped that the can be grouped under the headings "why", "what" and emphasis on meaningful and enjoyable learning will enrich "how". The opening sections of the Guidelines focus the quality and enhance the effectiveness of our students’ especially on providing answers to questions of "why" mathematics education. type. Section 1 describes the background to and reasons for the revision of the mathematics syllabuses; Section 2, he Guidelines are designed to support the teaching of dealing with the specification of aims and objectives, Tmathematics in the junior cycle in such a way as to emphasises the rationale and principles underlying their meet a wide range of learning needs. However, students design; and Section 3, on syllabus content, outlines reasons with a mild general learning disability often face particular for including specific topics. These sections also address challenges in the area of mathematics. Additional issues of "what" type. For example, Section 3 highlights the guidelines for teachers of students with a mild general changes that were made from the preceding syllabuses and learning disability have been drawn up, and provide a the differences that may be expected in student knowledge further resource for all teachers of mathematics. and attitude at entry to the junior cycle following the any people – too many to mention individually – introduction of the revised Primary School Curriculum. have contributed to the Guidelines. A special tribute In Section 4, "how" questions are considered: how the work M must be paid to the teachers who provided the "lesson might be organised in the school or classroom, and how ideas" which form a key part of the document and also to specific topics might be taught. This section describes some those who made inputs to other sections. The engagement of the insights and practice of experienced teachers who of so many people from outside the course committee, as have "made it work" in Irish classrooms. Against this well as of committee members, is a welcome feature of the exposition of where the learning of mathematics should be process leading to the production of the draft Guidelines going and how it might get there, Section 5, on assessment, document. It represents a significant advance in the sharing indicates ways in which information can be obtained on of mathematics teaching methodologies at national level. whether or not the students have arrived at the intended As indicated above, the intention is that even more goals. A number of appendices provide reference material. members of the mathematics education community will hus, the different parts of the Guidelines are likely to contribute their ideas over the period of operation of the Tbe used in different ways. The initial parts offer a draft Guidelines. The final version can then provide a general orientation and invite reflection on the purposes of resource that reflects as much good practice as possible in mathematics education in the junior cycle. The central mathematics teaching in Ireland. sections provide a resource for the day-to-day work of 2 MATHEMATICS Context of the changes π 3 MATHEMATICS 1.1 HISTORY OF THE SYLLABUSES DEVELOPMENT OF THE he syllabuses introduced in 1966 ran for seven years. JUNIOR CYCLE SYLLABUSES TIn 1973, revised versions were implemented in order he development of the current syllabuses can be traced to deal with some aspects that were causing difficulty. Tback to the sixties: that period of great change in Two main alterations were made. First, the hybrid system mathematics courses world-wide. The changes were driven of geometry was replaced by one entirely in the style of by a philosophy of mathematics that had transformed the Papy. Secondly, the examination papers were redesigned so subject at university level and was then starting to penetrate that the first section on each consisted of several school systems. It was characterised by emphasis on compulsory multiple-choice items, effectively spanning the structure and rigour. Starting from sets, the whole edifice entire content relevant to the paper in question. The revised of mathematics could be built up logically, via relations and syllabuses were even more strongly Bourbakiste in in particular functions; the structure laws (such as those we character than their predecessors, with the ideas of set and know as the commutative, associative and distributive function being intended to unify very many aspects of the properties) were golden threads tying the different parts of syllabuses. However, it was hoped that emphasis would be mathematics together. As a summary of the mathematics of given to arithmetic calculation and algebraic manipulation, the day, it was splendidly conceived and realised by the which were felt to have suffered some neglect in the first famous Bourbaki group in France; but it was devised as a rush of enthusiasm for the modern topics. state-of-the art summary of the discipline of mathematics, he size of the cohort taking the Intermediate Certificate not as an introduction to the subject for young learners. increased in the 1970s. The rather abstract and formal Nonetheless, many countries embraced the philosophy – or T nature of "modern" mathematics was not suitable for all the at least its outcome, revised subject-matter and rigorous cohort, and further revisions were needed. In the early presentation – with great enthusiasm. In Ireland, teachers eighties, therefore, it was decided to introduce a third attended seminars given by mathematicians; the early years syllabus, geared to the needs of the less able. Also, some of the Irish Mathematics Teachers’ Association were amendments were made to the former Higher and Lower enlivened by discussions of the new material. level syllabuses. The package of three syllabuses, then he first changes in the mathematics syllabuses in this called Syllabuses A, B and C, was introduced in 1987, for Tperiod took place in the senior cycle; new Leaving first examination in 1990. In the examination papers for Certificate syllabuses were introduced in 1964, for first Syllabuses A and B, question 1 took on the role of examination in 1966. Meanwhile, thoroughly "modern" examining all topics in the relevant half of the syllabus, syllabuses were being prepared for the junior cycle. but the multiple-choice format was dropped in favour of They were introduced in 1966, for first examination in the short-answer one that was already in use at Leaving 1969, and were provided at two levels: "Higher" and Certificate level. A very limited choice was offered for the "Lower". (Prior to 1966, two levels had also been offered, remaining questions (of traditional "long answer" format). but the less demanding of the two had been available only For Syllabus C, a single paper with twenty short-answer to girls.) These revisions brought in some topics which we questions was introduced. now take for granted, such as sets and statistics, as well as he unified Junior Certificate programme was a few, such as number bases, which have not stood the test introduced in 1989. As the Intermediate Certificate of time. The new syllabuses also addressed a problem in T mathematics syllabuses had been revised so recently, they their predecessors: students were finding difficulty with the were adopted as Junior Certificate syllabuses without (comparatively) traditional presentation of formal geometry further consideration (except that Syllabuses A, B and C – difficulty, it can be said, shared by students in many other were duly renamed the Higher, Ordinary