Assessment: an integral part of teaching and learning
Assessment is a continuous, dynamic and often informal process. It is a continuum, ranging from classroom observation to standardised tests. Equally important are questioning and dialogue, homework, and structured tests developed by teachers. Assessment provides information that can be used in decision-making about how the teacher can realistically answer the needs of the child. It must be an integral part of the educational process and should not become an end in itself. A balance must be struck between time spent on assessment and the time spent on teaching and learning.
The constructivist approach to mathematics encourages the use of guided discoverylearning and dialogue. Teaching and the gathering and analysis of assessment information should run concurrently, with the results of assessment feeding back into the teaching and learning process. Assessment should be a positive experience for the child, as this makes his/her learning more effective. Teacher-designed tasks and tests that are linked to the actual teaching provide a wealth of information. The role of the teacher is paramount in helping the child to develop his/her own learning.
Roles of assessment: why assess?
- Assessment is particularly helpful in mathematics because of the highly structured nature of the subject. It is also important in the identification of the kinds of difficulty experienced by children in developing mathematical concepts and skills.
- Assessment has a formative role. It enhances the child's learning by providing accurate feedback for both the child and the teacher. It informs the teacher of the childs' strengths and weaknesses and indicates the child's readiness to proceed to a new topic. Assessment assists the teacher in his/her planning and in the pacing of mathematics lessons and activities. Learning a new concept in mathematics is dependent on the child having a firm grasp of all aspects of previous concepts: for example, it is impossible for the child to progress in the area of decimals if he/she has an incomplete concept of place value. Assessment also has an evaluative role in planning. The results of assessment encourage the teacher to examine the suitability of the curriculum content for his/her particular class or group and also the methodologies and approaches being used.
- Diagnostic assessment helps the teacher to identify children who may have difficulties in specific areas of mathematics. It helps the teacher to identify children with special needs, for example the mathematically more able child or the child with learning difficulties in mathematics, so that that child's needs may be more closely met.
- Summative assessment is the recording of a child's mathematical development in a systematic way at the end of a specified time, for example a week, a term, or a year. This information is essential when reporting to parents and providing information for other teachers.
Assessment in the mathematics curriculum: what should be assessed?
The emphasis in assessment should be on finding out what pupils know, what they can do, and how they do it, rather than focusing on what they cannot do. Assessment should look at the whole child and consider both the processes of the child's learning and the products of that learning. The cognitive and affective areas that should be assessed to provide this information include the following:
- conceptual knowledge and understanding is assessed in the application ofmathematical concepts, for example the conservation of length. This must be done in a variety of contexts, including observation of the child performing a task or noting the child's application of a concept in a real-life situation.
- problem-solving ability is assessed by evaluating the approaches, strategies and processes the child uses in dealing with mathematical tasks and theconnections he/she makes within mathematics itself and within other subjects computational proficiency includes assessing the use of number, the appropriate application of the four number operations, and the ability to compute numbers efficiently, both mentally and in written situations.
- recall skills are assessed in the recall of number facts, terminology, definitions and formulae and in their efficient use within a given situation. This is a particularly important skill in the area of estimation.
- mastery of specific content areas (for example number, algebra, measures, shape and space, data) is assessed through the application of these areas in practical, everyday contexts.
- the ability to communicate and express mathematical ideas and processes and the correct use of mathematical language in oral and written form can be assessed by observation while the children are engaged in a mathematical task. Discussion of their own work can reveal gaps in their knowledge and skills. Incomplete understanding of mathematical terminology or processes can also be identified. When recording, children can communicate pictorially, orally or in written form using words and/or symbols.
- attitudes towards mathematics, including confidence, interest, willingness to take risks, and perception of the usefulness of mathematics, are assessed by observing the enthusiasm with which the child approaches a task. Attitudes also encompass the interest the child shows in completing tasks and in using mathematics confidently in other curricular areas and in real-life situations. Teachers' observations of such attitudes contribute to an overall picture of the child's mathematical development and are continuing and informal.
Assessment tools: how to assess
Although proficiency in computation is essential, assessment should encompass examination of the child's understanding of mathematical concepts and skills and his/her ability to verbalise that understanding.
Assessment tools must also consider the child's use of mathematical language and symbols.
A broad range of assessment tools is available in mathematics. It is suggested that teachers use a variety of tools in assessing mathematics, for example a portfolio that includes samples of a child's work, observation records, mastery check-list results, and the results of both teacher-designed tests and standardised tests.
Teachers assess children every day as they observe them at work, correct homework or class work, and engage them in discussion. Many of these observations are done informally but indicate to the teacher how the child is responding to a particular topic as it is being taught.
This type of continuing assessment includes observation of the child's activity, written work, discussion and questioning during class or group work. It is useful to have a notebook to hand in which to note the strengths or difficulties a particular child may have during an activity, for example a child who frequently chooses an inappropriate measuring tool or a child who constantly approaches addition tasks by adding the tens first. These short observations help teachers in planning the next step of a lesson or in assessing the child's readiness for a new topic and in the building up of a pupil profile.
Discussing a child's work with him/her can be very revealing, particularly when he/she is asked to explain how an assignment was completed, either individually or in groups. The responses will often indicate gaps in knowledge and skills, and appropriate action can then be taken.
This type of observational assessment also includes analysis of the child's written work to identify types and patterns of error and is a useful way of establishing how he/she is performing in relation to his/her peers.
Teacher-designed tasks and tests
Teacher-designed tasks and tests, used regularly, provide information useful in planning for children of differing ability and in matching the programme and methodology to the needs of those children. They also enable teachers to determine the level of progress of each child and provide information for reappraisal and modification of the mathematics programme. They are directly linked to the instructional objectives of a particular class and can be used to provide formative, diagnostic and summative data on children's progress.
By providing a variety of formats in the presentation of teacherdesigned tasks and tests the teacher can help the child become comfortable with assessment. A broad range of presentations helps children who have different learning styles.
Some examples of such presentations would be:
- oral tests of recall skills (tables, continuation of number patterns)
- written tests of numerical competence
- problem-solving exercises that use a variety of mathematical skills
- projects that require compilation of data, construction of a model or
- drawing a diagram.
In examining and recording the results of these tests and tasks the teacher can also note the processes used by the child in performing the task, for example using a separate sheet for rough estimates or choosing the correct tool for the task (long ruler, protractor, number line).
Work samples, portfolios and projects
These are systematic collections of children's work kept in a folder or file, and they provide a tangible record of development over a term or a year. They provide a basis for discussion with both the child and the parent and can be passed on to the next teacher. Models of portfolio assessment include representative sampling of progress through written work or subject-based portfolios that contain all work done in that area. Manageability is an issue in the compilation of a portfolio, and consideration must be given to the quantity and value of the work that is kept. The child can take an active part in the compilation of his/her own portfolio by sometimes choosing a piece of work for inclusion.
Curriculum profiles allow the teacher to make an overall judgement about the achievement of an individual child. They allow for the interpretation of a wide span of learning outcomes. This requires the teacher to look at the child's ability to select materials and processes for particular mathematical tasks, to select and use appropriate strategies for completing a task, or to identify the solution to a simple problem. The teacher then decides whether the child in question has developed these skills or whether they are still in the developmental stage.
Diagnostic tests identify learning difficulties in particular areas of mathematics, and the results can then be used in the remediation of a problem. Commercial diagnostic testing kits often provide schemes of work that are specifically aimed at the skill or skills that the child needs to improve. This type of assessment is often undertaken by a remedial teacher. However, analysis of a child's work can also fulfil a diagnostic function, and tests can be designed by the teacher. Persistent errors in a child's work can be analysed to identify areas of difficulty. The use of early screening tests at infant level means that children who are experiencing problems in mathematics can be identified at an early stage and appropriate remediation provided at this point. This type of analysis also indicates the child's strengths, and the results can be used by the teacher in providing extension work.
Standardised tests comprise norm-referenced tests and criterion-referenced tests. Norm-referenced tests compare pupils with other pupils or with national standards. They consist of highly structured tasks that have associated with them a set of scoring rules. Standardisation refers to the uniformity of procedures in administering a test. All children take the same test under the same time limits and instructions. These rules must be adhered to rigidly in order to produce a standard score and maintain the validity of the test Administering the same test to all children under the same conditions means that achievement can be judged independently of external factors.
Criterion-referenced tests provide information on the child's functional performance level, but, unlike norm-referenced tests, this is not made in relation to the performance of others. They allow a teacher to estimate the amount of specified content an individual pupil has learned and are based on sets of instructional objectives or on course content.
Mastery records and check-lists are one type of criterion-referenced test and are used to keep track of mastery in certain elements of the curriculum in a structured manner. This form of assessment can be based on teacher-made tests or may be part of a mathematics textbook or scheme. Unlike more formal tests, these are not administered in a strictly standardised manner, and the child's scores cannot be interpreted with reference to class or agelevel norms. They are, however, extremely useful in providing diagnostic information on a pupil's achievement.
Standardised tests should be used judiciously. They can be diagnostic if errors are analysed and are used as a means of identifying children's strengths and their readiness for further learning.
A balanced approach to assessment
Tests must be evaluated with regard to their aims and suitability for the children for whom they are intended. Teacher-made tests, purchased tests and check-ups in textbooks all have different purposes and applications. It is important to consider variety in the types of test given to children, for example a dictated test that requires short written answers, tests where the child has to show how they worked out the answer, and multiple-choice tests.
The language used in a test must also be considered, as it can militate against the performance of a child with a reading difficulty.
Manageability of tests
The manageability of tests is an important issue. Tests that can be administered to a whole class are useful for screening but are not usually diagnostic. Where an area of weakness has been identified, a more detailed test will need to be given to a smaller group or an individual child. Tests must be easy to administer, as many teachers operate in a shared or multi-class situation.
Recording and communicating
Reporting the results of assessment
The results of assessment must be meaningful. At school level it can be decided to have a common format for reporting to ensure that accurate information is carried from class to class. Assessment results for parents should also cover more than just numerical proficiency. The use of a portfolio-type system that includes areas such as perseverance, presentation of work and ability to work in a group gives an informative and rounded view of the child's mathematical ability. This provides an opportunity for parental feedback. The analysis of results on a school or class level can show areas of weakness or strength, which can then be developed.
Pupil profile cards
Pupil profile cards allow the teacher to systematically record the progress of the children and include some examples of observations that the teacher has noted throughout the year. These profiles provide an overall description of the child's progress in mathematics and are completed over the course of the school year. They contain information derived from various forms of assessment, for example standardised tests, teacher-designed tests and tasks, and teacher observation. They are then used to provide accurate information for parents and other relevant parties. The recording system should complement sound instructional practice and reflect the breadth of learning outcomes implicit in the curriculum. Each school should develop a co-ordinated policy on record-keeping, which sets out the types of information to be gathered, the frequency of the data-gathering, and the uses to which it will be put.