The Chief Examiner's Report

The Chief Examiner's Report in Mathematics for Junior Cert  2015 was published this week on examinations.ie. The full report is linked here it includes a number of recommendations for teachers and students.

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

In advance of the examination
Many of the points below are good habits that should be developed over the course of the students’ studies in mathematics. It 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 papers.
•   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 syllabus.
•   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 learning.
•   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 developed.
•   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 operations.
•   Students should understand the difference between an expression and an equation, and what operations can be validly done to each.
•   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 practice.
•   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 are.
•   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 appropriately.
•   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 examination.
•   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 quickly.
•   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 them.
•   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 way.
•   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 familiar.