Assessment

5.1 INTRODUCTION

Assessment is an integral part of teaching and learning. It is intended to "support the learning of important mathematics, and furnish useful information to both teachers and students" (NCTM, 2000). Assessment is traditionally categorised as being either formative or summative: formative assessment focuses on providing feedback which helps students to learn, while summative assessment describes the levels they have reached at the end of some section of their education. Obviously the two roles are not entirely distinct, but the former is of daily concern, whereas the latter is associated chiefly with national certification and hence (for the junior cycle) the Junior Certificate examinations. Formative assessment is discussed in Section 5.2. The remaining sections consider assessment for the Junior Certificate: choice of mode, design and marking of examination papers, and development of grade criteria.

5.2 FORMATIVE ASSESSMENT

Formative assessment is chiefly the business of the classroom teacher. Throughout the junior cycle, students need feedback to help them monitor their progress and develop their concepts and skills. They receive this feedback in a multitude of ways ­ perhaps most obviously in being told whether their answers to questions are right or wrong, but more importantly in being helped to understand why. In this way they not only correct errors and improve procedural skills, but also develop their understanding of concepts and their ability to apply them to the solution of problems. Moreover, they can be affirmed and encouraged, enabling them not only to enhance their mathematical expertise but also to grow as people.

There are many ways in which formative assessment can be carried out. International studies of mathematics education indicate that we have a strong tradition in this country with regard to some of these ways. In particular, we often give students exercises to do on their own in class, and we set frequent short homework tasks; these typically provide reinforcement of the students' procedural skills. Class tests and school examinations are other tools familiar in an Irish setting ­ though examinations, in particular, lose much of their formative role if teachers do not provide students with appropriately detailed feedback on their performance. Tests and examinations tend to focus chiefly on procedural skills. Our strong tradition as regards assessment of these skills may be complemented by putting more emphasis on encouraging reflection, discussion and exploration in the mathematics classroom. These are activities which help students to build concepts, identify and clarify their misconceptions, and (in however small a way) create mathematics for themselves. In terms of the syllabusobjectives, therefore, perhaps our formative assessment has tended to emphasise the lower-order ones (objectives A and B), reflected in classes in which the students engage busily in doing routine exercises; but perhaps it has paid less heed to those of higher order (objectives C, D, E, F and H), concerned with the students communicating mathematically, connecting and extending ideas, exploring consequences and becoming actively involved in their own learning.

Students accustomed to "busywork" classrooms find it difficult to take control of their own learning. They find the transition to a reflective and analytical learning style quite hard to achieve. However, they can eventually learn more meaningfully and with greater enjoyment, and may achieve the kind of independence to which we aspire on their behalf. The activities suggested in Section 4 of thisdocument provide opportunities for students to developthese higher-order behaviours and for teachers to assesstheir development. Many of the objectives can be assessed by observing the students as they work, by listening to their discussions among themselves, and by talking to them individually, encouraging them to communicate their insights and difficulties. These processes are timeconsuming and demanding for teachers. However, they can provide crucial information, complementing that obtained from the routine correction of classwork, homework, tests and examinations. In particular, the assessment of relational understanding ­ the aspect of learning to which so much of these Guidelines is devoted ­ is greatly enhanced by the use of such processes.

It is appropriate here to mention diagnostic assessment: assessment which aims to identify specific areas of difficulty (and strength) for a given student. Teachers carry this out informally when they use their professional expertise to identify areas in which individual students are having problems with their mathematics. More formal approaches to diagnostic assessment are outside the scope of these Guidelines. However, it would be inappropriate to ignore the contribution that such assessment can make to identifying and catering for the needs of individual students.

5.3 ASSESSMENT FOR CERTIFICATION: SCOPE AND CONSTRAINTS

The point was made in Section 2.4 that summative assessment in Mathematics ­ assessment for the Junior Certificate ­ at present takes place solely by means of a terminal examination. Forms of summative assessment other than terminal examinations are unfamiliar in mathematics education in Ireland.

Unfortunately, not all skills are easily assessed by terminal examinations. The syllabus points to the consequences of this.

Written examination at the end of the Junior Cycle can test the following objectives (see section 1.3 [of the syllabus document]): objectives A to D, G and H, dealing respectively with recall, instrumental understanding, relational understanding and application, together with the appropriate psychomotor (physical) and communication skills.

As pointed out in Section 2.4 of these Guidelines, this limits the assessment objectives to those specified in the syllabus (A, B, C, D, G and H). In the context of a general review of assessment in the junior cycle, the list of assessment objectives might be extended. Such a movemight necessitate a substantial revision of the rest of thissection of the Guidelines.

The Junior Certificate mathematics syllabus highlights the following points with regard to summative assessment.

Assessment...is based on the following general principles:

• candidates should be able to demonstrate what they know rather than what they do not know;
• examinations should build candidates' confidence in their ability to do mathematics;
• full coverage of both knowledge and skills should be encouraged.

If examinations are to satisfy the first two of these principles, they must allow students to show the conceptual knowledge and procedural mastery that they have developed. This can be done by asking students to answer questions many of which should be well within their scope. The third principle points to the importance of spanning all major content areas and as many types of skill as can validly be assessed under examination conditions (bearing in mind the age of the candidates and their inexperience in dealing with major examinations).

NOTE

Given the exclusion of some objectives from the summative assessment process, it is all the more important to ensure that these objectives are addressed during the students' mathematical education. The revised Primary School Curriculum gives a higher priority to these objectives than does its predecessor. Ongoing development of Junior Certificate mathematics might address ways in which they can receive some coverage even in formal examinations; there are models in other countries' assessment systems and in recent international studies (for example, the OECD Programme for International Student Assessment (PISA)). The limited brief given for the current revision prevented their inclusion at this stage. It may be noted here that Transition Year provides opportunities for students to broaden their range of learning styles in mathematics, and to bring multiple intelligences to bear on the subject. This can provide them with a stronger platform for moving on to the Leaving Certificate, as well as developing abilities greatly valued in the workplace: applying their work in real-life contexts, solving nonstandard problems, and communicating their findings clearly and succinctly when required.

5.4 SPECIFICATIONS FOR THE DESIGN OF THE JUNIOR CERTIFICATE EXAMINATIONS

The formal specifications for the Junior Certificate examinations involve describing the number of examination papers and the time allocated to each, the format of each paper, the distribution of questions per topic on each paper and the marks allocated to individual questions, and finally the structure of each question. The specifications proposed by the NCCA Course Committee for Junior Certificate mathematics were reflected in the proposed sample assessment materials circulated in Autumn 2000. However, in the light of developments in assessment at junior cycle, some of these proposals may not be implemented. The sample examination papers, issued by the Department of Education and Science, will indicate the number of papers at each level, the time allocation per paper, and a typical distribution of questions for each topic. Comments can be made here about the format of the papers and the design of individual questions.

PROPOSED FORMAT OF THE PAPERS

It is proposed that the examination papers for at least some levels will be in booklet form. Instead of reading the questions from an examination paper and doing their work in a separate booklet, candidates will write their answers in the booklet that contains the questions. The booklet will be handed in at the end of the examination.

Where a booklet is used, each part of each question will be followed by a solution box designed to accommodate candidates' answers, along with supporting work where relevant. A special symbol () will appear in solution boxes where work to support answers must be shown in order to earn full marks. When this symbol is absent, supporting work can be included at the discretion of the student. (Examples are given in Section 5.6 below.)

The change in format is intended to serve a number of ends. First, it will give a new, and hopefully more student-friendly, look to the examination papers. This is in keeping with the intended emphasis on making the subject more enjoyable for the student. Secondly, students may find it easier to complete questions when they do not have to transfer attention from examination paper to script and back again at frequent intervals. They can also be helped to work through individual question parts, where appropriate, by the inclusion of "prompts". Less able students, in particular, may find it easier to keep working in the more structured setting that the booklets will provide. Thirdly, the requirement that

students show their working in at least some cases can support the drive for learning with understanding and can test communication skills. It should be noted that the introduction of booklet-style examination papers in other subject areas in the Junior Certificate has been deemed very worthwhile from the student's point of view.

DESIGN OF INDIVIDUAL QUESTIONS

Questions will typically have the three-part structure ­ "part (a)", "part (b)", and "part (c)" ­ currently used for the Leaving Certificate. Again as for the Leaving Certificate, questions will display a "gradient of difficulty", leading students through easier work to more difficult tasks.

With regard to the objectives, typically:

• part (a) of the question will test recall (objective A), very simple manipulation (objective B) or basic relational understanding (objective C);
• part (b) will test the choice and execution of routine procedures or constructions (objective B), or various aspects of relational understanding (objective C);
• part (c) will test application (objective D).

The characteristics of the various "parts" are considered in more detail below, in the context of indicating grade criteria at each level of the examination. It should be noted that this proposed format is a change for the Foundation level.

NOTE

For the remainder of this section of the Guidelines it is taken as a working assumption that the Ordinary level and Foundation level papers will be in booklet form, but that the Higher level papers will be of traditional format (with the students presenting their work on examination scripts as heretofore).

5.5 THE MARKING SCHEME

The marking scheme is crucial in implementing the examination design and in ensuring that the assessment objectives are appropriately tested. In particular, to reflect the aims and objectives of the revised syllabus, objectives C and H ­ those concerned with relational understanding and communication ­ need to be given greater emphasis than in the past. In order to make such demands reasonable, the marking scheme must be appropriately transparent and familiar to teachers (and indeed students), and the terminology used on the examination papers must be well understood.

In this section, therefore, relevant features of the marking scheme are presented and the meanings to be ascribed to frequently-used terms and expressions are set out. The following section relates the marking scheme and the assessment objectives to the development of grade criteria.

AWARD OF MARKS

Typically, but not inevitably, the marks for the three parts of the question will be in the ratio 1:2:2. Thus, for a fifty-mark question, there will be ten marks for a part (a), twenty for a part (b), and twenty for a part (c).

Marking schemes are designed so as to give maximum reward to candidates whose answering meets the requirements of the questions asked. This includes adhering to instructions that are embedded in questions. For example, the use of a protractor may not be allowed, answers may be required in a certain form or an earlier result may have to be used to perform a task. For the foreseeable future, it is envisaged that candidates' work will be marked in the traditional way: that is, with marks usually being deducted for "slips" (minor errors) and "blunders" (major errors) but awarded for "attempts" where relevant, though some questions are scored on a "hit or miss" basis. Procedural mistakes tend to be categorised as slips and conceptual ones as blunders, but no general rule can be given, as the distinction is heavily dependent on the context. Details are specified in published marking schemes for individual Junior Certificate papers; schemes from past examinations are available on the Department of Education and Science website (http://www.irlgov.ie/educ).

Where a booklet format is used, the proposed provision of solution boxes will allow the examination paper to offer candidates considerable guidance as to what type of answer is required. In

particular, the presence of the symbol is a clear indication that supporting work is essential and that a correct answer on its own will not be awarded full marks. Similarly, the absence of the symbol in a solution box will reassure candidates that they will not be penalised for presenting a correct answer without work. In general, however, incorrect answers standing alone are worthless. It is therefore in candidates' interest to show their working where possible; this enables examiners to identify slips and blunders, deduct only such marks as should be lost, and award the remainder. Thus, work leading to answers should be clearly presented wherever it is practical to do so, even in the absence of the symbol.

Also in general, all correct mathematical solutions are accepted for full marks. For example, an Ordinary level candidate may solve a quadratic equation using the formula even though this is not in the Ordinary level syllabus. However, there is one exception to this rule. With regard to the synthetic geometry at Higher level, it should be noted that what constitutes a correct mathematical proofis dependent upon the context within which it ispresented, in this case the particular system of geometryspecified in the syllabus. It must be emphasised again (see Section 3.3) that proofs using transformations are not acceptable. In proving any theorem, candidates must build their arguments upon the theorems and facts that precede the one concerned, as set out in the syllabus. Further details are supplied in Appendix 2.

TERMINOLOGY

There are some terms and expressions that occur frequently in examination questions and that carry particular meanings and expectations during the marking process. The principal ones are briefly explained with a view to helping students in their answering and to facilitating understanding of published marking schemes.

• "Construct ..."
  "Construct" means to draw according to specific requirements, usually with instruments such as ruler, compass and protractor. Accurate measurements are required and construction lines such as arcs should be shown clearly. Free-hand drawings are not acceptable. The marking of constructions involves measuring of the candidates' work by examiners. For each measurement a small tolerance is allowed without penalty.
• "Draw the graph ... / Graph ..."
  Graphs are likely to be required in questions on coordinate geometry, statistics and functions. Where relevant, they should be presented in solution boxes, in the spaces in which gridlines are provided ­ or, if necessary, on separate sheets of graph paper. They should be distinctly drawn and sufficiently large to ensure clarity. Axes should be perpendicular and clearly labelled. Appropriate scales should be chosen and indicated. Graduations should be marked clearly on both axes. Plotted points should be accurately positioned and identified. Where the points are to be joined this should be done in the appropriate manner; for example, a smooth curve is necessary for a quadratic function whereas line segments are required for a trend graph.
  
  It should be noted that an ogive (cumulative frequency curve) is a smooth curve. In examination answers it should always start on the X-axis, since its initial point will represent the value below which it is known there are no data.
  
• "Estimate .../Show how to calculate an approximatevalue of ..."
  Questions involving estimation or approximation require candidates to use their own judgement regarding the level of rounding that is appropriate in order to lead to a value close enough to the exact calculation to be useful. In a solution box, the layout of solutions may be prompted by the provision of appropriate blank spaces, and candidates should be alert to this source of guidance. Examiners will focus on the depth of understanding of the approximation process displayed by candidates as well as on their ability to perform the mechanical steps. This means that flexibility will be exercised in the marking of the numerical results presented and that usually a specific estimate will not be required for full marks.
  
• "Give your answer in the form ..."
  For full credit, candidates must adhere closely to instructions of this type. For example, an answer of 1.87 will be penalised if the question requests a result to one place of decimals. In the same way, 25 will not suffice if candidates are told to give the answer in the form 5n.
  
• "Hence..."
  The word "hence" is generally used to connect two tasks which the candidate is expected to perform, one after the other, with the outcome of the first helping the second. It points candidates to the method or approach which examiners expect.
  It is important to note that when "hence" is used is this way, candidates may be penalised if the first result is not used in order to perform the second task.
  More commonly, the phrase "hence, or otherwise" is used. This indicates that any approach of the candidate's choosing can be taken to the second task. However, a helpful lead-in is always provided by the first part, and candidates usually fare better if they follow this rather than make a fresh start at the second part.
• "Prove ..."
  In accordance with the syllabus, the idea of proof will be addressed only in Higher level questions. Candidates may be required to prove the theorems marked with asterisks in the syllabus as well as "cuts" arising from the results (theorems and "facts").
  All the steps in proofs must be written down in logical order. Each assertion in the proof should be accompanied by a reason. Further details are given in Appendix 2.
  Proofs should be accompanied by diagrams wherever they serve to clarify the argument being presented. However, it should be noted that information marked on diagrams will not be accepted as a substitute for written steps.
  
• "Show ..."
  Normally, when students are asked to show a result, any correct mathematical method is awarded full marks assuming that it is properly applied. One exception is that measurement from diagrams using a ruler, protractor, or other instrument is not accepted unless this approach is specifically requested. In cases where a particular method is required, the question will give clear directions ­ for example, "Show, by calculation, that " ­ and these directions must be followed.
  
• "Sketch ..."
  When a sketch is required, diagrams are not expected to conform to specific measurements. Examiners will be assessing candidates' intuitive feel for the task at hand. For example, in sketching images of shapes under transformations, examiners will be looking for evidence that the shape has the correct orientation and is in roughly the correct location.
  
• "Use your graph to show that ..."
  To earn full marks, candidates must display evidence that they have extracted their answers from their graphs. It is not acceptable to use other methods of arriving at the required results even if the alternative methods are mathematically correct and accurately applied. For example, candidates who successfully solve the equation using the quadratic formula cannot be awarded marks if the instruction given is "Use your graph to estimate the roots of the equation ".
  
• "Verify that ..."
  Verifying a solution of an equation in algebra involves substituting the value into the given equation and showing that the result is a true statement. It is important to note that solving the equation is not acceptable if verification is sought.

5.6 QUESTIONS, OBJECTIVES AND STANDARDS: THE ROUTE TO GRADE CRITERIA

Knowledge and skills displayed by the students can be related to standards of achievement, as reflected in the different grades awarded for the Junior Certificate examinations. The three-part design of questions (with the typical relationship to objectives described at the end of Section 5.4 above), taken together with the marking scheme which allocates marks approximately in the ratio 1:2:2 to the three parts, is intended to operationalise the gradecriteria. Thus:

• recall alone (accounting for about twenty per cent of the marks) should not be enough to enable a candidate to achieve a D grade;
• recall together with some instrumental and relational understanding ­ execution of familiar and well-learnt techniques, and grasp of concepts ­ together with the ability to communicate the results, should be necessary for a D grade;
• recall together with good instrumental and relational understanding ­ diligent and accurate execution of familiar and well-learnt techniques, and grasp of concepts ­ together with the ability to communicate the results (accounting jointly for some sixty per cent of the marks), should be required for a C grade;
• the ability to apply, or to execute more difficult examples of familiar exercises, or to demonstrate understanding at the upper level of the syllabus, is needed for a higher grade;
• evidence of abstraction and/or better application, with good communication, is needed for a top grade.

It remains to give fuller descriptions of the characteristics of the three question parts, to provide examples of questions suitable for each part at each level, and to indicate key aspects of the solutions required to obtain full marks. It is important to note that the solutions given hereare sample solutions, provided in order to illustrate thetype of detail and level of communication expected. Theyare not the only (or necessarily the best!) possibilities. In view of the assumption that the Higher level papers will be of traditional format, as indicated in Section 5.4, it should be noted that the "boxes" shown for the Higher level examples are not solution boxes; they represent portions of the student's examination script. It should be stressed thatstudents are encouraged to show detailed working in orderto obtain maximum marks.

The examples also serve to indicate the style and depth of coverage expected in some areas of the syllabus. Moreover, they indicate ways in which the candidates' relational understanding and communication skills can be demonstrated and rewarded.

CHARACTERISTICS OF A "PART (a)"

A"part (a)" is intended to have the following characteristics.

• It should allow candidates to demonstrate what they know, and also permit them to "limber up" for the remainder of the question. As indicated above, questions typically test facts, very straightforward skills, or basic understanding.
• It should be presented as straightforwardly as possible, so that the required mathematics is tested directly. For example, candidates with poor reading skills should not be handicapped.
• Ideally, all (credible) candidates should get "part (a)" right ­ except for the slips that can strike even the best candidates in examinations.

EXAMPLES OF "PART (a)" QUESTIONS

"a" 1: Foundation level Find the value of 3(x+y) when x = 2 and y = 1.

"a" 2: Ordinary level Find the image of the point (3, -2) under the translation (1, -3) (2, 4).

"a" 3: Ordinary level Find the values of x for which 5+2x=13,

"a" 4: Ordinary level VAT at 21% is added to a bill of 130. Calculate the total bill.

"a" 5: Ordinary level Divide 1506 by 0.6 and express your answer in the form a×10n ,where 1=a< 10

"a" 6: Ordinary level Given that tanA = 3.3544 find the value of A to the nearest degree, where

"a" 7: Higher level Given y=ax+a3 and x=3-2a2

1. express y in terms of a and simplify the result
2. evaluate y when a = 2.

"a" 8: Higher level Simplify (2+ 7)(3- 7).

CHARACTERISTICS OF A "PART (b)"

A"part (b)" is intended to have the following characteristics.

• It should allow the candidates to demonstrate their ability to execute diligently practised procedures, or to display non-trivial understanding (for example, interpreting a mathematical statement, reading and using information from graphs, recognising the solution of an equation, or ­ for Higher level candidates ­ writing out a proof).
• In some cases the "part (b)" may be divided into subparts, containing (say) slightly easier and slightly harder "sums" in the required area.
• The candidate who just deserves a D grade on a particular question should be able to get around half of the marks available for this part of the question. For example, in the case of a "part (b)" with two sub-parts, the candidate may get the first sub-part correct except for slips, and may earn an attempt mark on the second sub-part.
• Candidates deserving a C grade for a particular question should be able to get the "part (b)" right (except perhaps for slips).

It should be noted that, although "part (b)" questions are not intended to test the higher-order skills associated with problem-solving, they may be formulated as simple "word problems". Thus, in accordance with the thrust of the revised syllabus, information may be presented in a context. The skills of identifying and interpreting the relevant information, and then processing it, should involve relational and instrumental understanding rather than genuine problem-solving. A hallmark of the latter is that the method of solution should not be immediately apparent. For "part (b)" questions, however, candidates ­ having duly understood the principles involved in such work and practised many examples ­ should be able to see at once how they should tackle the question.

EXAMPLES OF "PART (b)" QUESTIONS

"b" 1: Foundation level 60 students were asked to choose a colour to paint the school hall. 15 said yellow, 20 said blue and the rest said white.

1. How many said white?
2. Draw a pie chart to show this information.

"b" 2: Ordinary level Given that cosA = 0.5, find the value of sinA and the value of tanA. Give your answers correct to two places of decimals.

"b" 3: Ordinary level Síle is six years older than Seán. The sum of their ages is 30 years.

1. Letting x = Seán's age write down an equation inx to represent this information.
2. Hence, find Síle's age.

"b" 4: Ordinary level

1. Sketch an equilateral triangle with sides of length 2 units.
2. Calculate the perpendicular height in surd form.
3. Hence, find sin60°.

"b" 5: Higher level It takes 4 hours and 20 minutes to travel a journey at an average speed of 120 km/hr. How many hours and minutes will it take to travel the same journey if the average speed is reduced to 100 km/hr?

"b" 6: Higher level By putting the smallest number first, place the following numbers in order:

"b" 7: Higher level

1. Write the following as a single fraction:
2. Evaluate your answer when x = 2.

CHARACTERISTICS OF A "PART (c)"

A"part (c)" is intended to have the following characteristics.

• It should provide a challenge, so that good candidates (relative to the level they are taking) can demonstrate their ability to apply their mathematics, solve problems, and so forth, and to cope with the subject-matter that is at the highest conceptual level on the relevant syllabus.
• To obtain a safe C grade on a given question, candidates should be able to attempt the "part (c)" (or at least the first sub-part of the "part (c)", if it is divided into sub-parts) for most of the questions on the paper. However, full marks on a "part (c)" may be gained in general only by very competent candidates.

Obviously, "word problems" presented in a "part (c)" will be more complex than those appearing as a "part (b)". Typically they involve application (objective D). Thus, information presented verbally may have to be translated (non-trivially) into mathematical form, and suitable approaches and techniques chosen. However, in examination conditions, it would not be fair to present information in very unfamiliar guise, or in highly complex fashion. Moreover, the solution must be obtainable at the end of a few minutes' work. The type of problem-solving involved is therefore rather limited.

EXAMPLES OF "PART (c)" QUESTIONS

"c" 1: Foundation level

A rectangular space on a bathroom wall measures 1 m by 2 m. It is to be covered with square tiles, each of which measures 10 cm by 10 cm. How many tiles will be needed?

• Lower edge measures 2 m or 200 cm
  Number of tiles needed along lower edge is 200/10=20
  Left edge of space is 1 m or 100 cm
  Number of tiles needed along left edge is 100/10=10
  Total number needed to fill rectangular area is 20 ×10 tiles
  Answer: 200 tiles

"c" 2: Ordinary level A solid sphere made of lead has radius 6 cm.

1. Calculate its volume in terms of .
   This sphere is melted down and all of the lead is used to make smaller solid spheres each of radius 3 cm.
2. How many of these spheres are made?

"c" 3: Higher level

1. State the theorem of Pythagoras
2. In the triangle xyz, xwyz
   Prove that xy2 + wz2 = yw2 + xz2

NOTE

Use in certificate examinations of the three-part questions, with the different parts aiming to test different objectives and with the question as a whole displaying an appropriate gradient of difficulty, has been a positive development in mathematics education in Ireland. If students are sitting for a paper at a level appropriate to them, they can be confident that they will be able to tackle the earlier parts of the questions and so earn their reward for good understanding and diligent work.

However, there is a danger that some students may focus unduly on the first two parts of the questions, targeting a middle grade in the examinations. While this may sometimes be appropriate, or inevitable, in examination conditions, it would be unfortunate if the skills associated with a "part (c)" were to be regarded as a "bolt-on extra" in the teaching and learning of mathematics. Just as for the currently non-examinable objectives (see Section 5.3), objectives such as application and appropriate problem-solving should be addressed by all students in their mathematical education.