**Recommendations to Teachers and Students**

** **The following advice is offered to teachers and students preparing for Junior Certificate Mathematics examinations.

## 4.1 In advance of the examination

Many of the points below are good habits that should be developed over the course of the sh1dents’ studies in mathematics. lt is unlikely that candidates will be successful at checking over work effectively, or at performing algebraic manipulations accurately, on the day of the examination if these skills and habits have not been developed over a period of time before the examination.

- Teachers and students should cover the full syllabus. This is of particular importance as there is no choice on any of the examination
- Teachers should use the support material produced by the Project Maths Development Team and the National Council for Curriculum and Assessment. It has been developed specifically to support the kind of learning envisaged in the current mathematics
- Close to the time of examination, questions from past and sample examination papers provided by the State Examinations Commission should be used for practice. However, examination papers should not be relied on excessively during the main period of learning, as this might unnecessarily restrict the range of student
- Students should get into the habit of showing supporting work at all times. This will help them tackle more difficult problems, and will allow them to check back for mistakes in their work.
- Students should develop strategies for checking their answers. One of these is to have an estimate of the answer in advance. In real-life problems, check if the answer makes sense. For example, it is unlikely that someone’s net income will be greater than their gross income, or only a tiny fraction of it. In addition to techniques for identifying that an error has been made, techniques for finding those errors quickly and calmly, including getting to know one’s own weaknesses, should also be
- Teachers should provide frequent opportunities for students to gain competence and accuracy in basic skills of computation and algebraic manipulation. Students should be particularly careful with signs, powers, and the order of
- Students should understand the difference between an expression and an equation, and what operations can be validly done to
- Students should always round their answers to the required level of accuracy, and include the appropriate unit where relevant. These are skills that are not conceptually challenging, and they should be developed to a high standard through regular
- Students should get used to describing, explaining, justifying, giving examples, etc. These are skills that are worth practising, as they will improve understanding, as well as being skills that may be assessed in the examination. Students do not need to be able to produce word-perfect statements of results or definitions, but they do need to be able to state or explain reasonably clearly what these
- Students should make sure that they have geometric instruments and should practise using them accurately. This applies particularly to students at Ordinary and Foundation level, where there was evidence that many candidates either did not have the requisite geometric instruments with them in the examination or were unable to use them
- Teachers should provide opportunities for students to apply the skills and knowledge from one strand to material from another strand. Mathematics is not a list of discrete rules and definitions to be learned but rather a series of interconnected principles that can be understood and then applied in a wide variety of contexts. While compartmentalising knowledge may help keep it organised, it will restrict the ability to cope with unfamiliar questions, particularly those requiring the synthesis of knowledge and skills from several strands.
- Students should practise different ways of solving problems – building up their arsenal of techniques on familiar problems will help them to tackle unfamiliar ones. Students at Higher level should pay particular attention to algebraic methods of solving problems, as there may well be questions that require such methods in the
- When using trial and improvement, students should develop methods for systematically improving the answer. For example, does an increase in the input lead to an increase in the output? If so, this may allow the problem to be solved more
- Teachers should provide students with opportunities to practise solving problems involving real-life applications of mathematics and to get used to dealing with messy data in such problems. Students should also be encouraged to construct algebraic expressions or equations to model these situations, and/or to draw diagrams to represent
- Teachers should provide students with opportunities to solve unfamiliar problems and to develop strategies to deal with questions for which a productive approach is not immediately apparent. Students should be encouraged to persevere with these types of question – if the initial attempt does not work, they should be prepared to try the question a different
- Students at Higher level should learn the five examinable formal geometric proofs. They should understand the logic of each proof and the meaning of each result. They should also pay attention to the method of proof: statements built up one after the other, with a reason given for each one based on previous statements or previous results. Many candidates struggle when asked to apply the results of axioms and theorems to prove a result with which they are not

# 4.2 During the examination

- It is intended that Junior Certificate Mathematics examination papers will have the easier questions near the start, and the more difficult questions near the end. However, there may be some areas in which a particular candidate is more confident than others. It is generally a good idea to try to start with one’s better questions. However, candidates need to remember to answer all of the questions the examination paper.

- Candidates should read each question carefully, paying particular attention to key words. For example,
*calculate*and*measure*are asking for different things, as are*solve*and Giving an answer*correct to the nearest metre*is not the same as giving it*correct to the nearest centimetre.*Candidates should make sure that they answer the exact question asked, and give the answer in the required form with the appropriate unit – many candidates fail to do this. - The question often contains clues regarding the nature of the answer. For instance, if it asks to
*find the value of x**,*there should only be one answer, while if it asks to*find the values of**x,*one might expect more than one. Similarly, attention should be paid to whether the question indicates that the answer is a natural number, integer, real number, etc. - Candidates should concentrate when answering, as careless en-ors will result in marks being needlessly lost. Errors can further disadvantage candidates as they can make the work that follows more
- Candidates should present their work as neatly and tidily as possible. This will help when checking back over it, and will help the examiner to find relevant work for which marks can be
- Candidates should show all of their work. In some questions, full marks will not be awarded unless candidates show supporting work. Furthermore, marks are generally not awarded for an incmTect answer without any supporting work. However, if candidates show what they are thinking, then they may get credit for
- lf possible, candidates should have an estimate of the answer in advance. They should check that the answer makes sense, including the appropriate unit – if it does not, they should review their
- When candidates believe that they have made a mistake, it is often most beneficial to start again. They should draw a single line through the incorrect work. They should not use corrector fluid or otherwise make the work illegible – if there is valid work presented and still visible, it may be awarded some
- Candidates should be prepared for the unexpected. The syllabus states that they should be able to solve problems in familiar and unfamiliar contexts, so the examination paper wouldnot be fit for purpose if the candidates recognised everything on it. Candidates should expect that there will be questions on the examination paper that, at first glance, they will not know how to complete. They should stay calm, and use the problem-solving skills that they have developed throughout their studies. The less familiar the territory, the more credit is likely to be awarded for attempts at applying appropriate strategies. It is always advisable to make an attempt at these questions.

- Candidates should use all of the time in the examination. If they finish the examination paper early, they should review their work, checking as many answers as